An Introduction to Dynamic Light Scattering by Macromolecules

An Introduction to Dynamic Light Scattering

by Macromolecules

Kenneth S. Schmitz

Department of Chemistry

University of Missouri-Kansas City

Kansas City, Missouri

A C A D E M I C P R E S S , I N C .

Harcourt Brace Jovanovich, Publishers

B o s t o n San D i e g o N e w Y o r k L o n d o n Sydney T o k y o T o r o n t o

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Copyright © 1990 by Academic Press, Inc.

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Library of Congress Cataloging-in-Publication Da t a

Schmitz, Kenneth S. An introduction to dynamic light scattering by macromolecules/

Kenneth S. Schmitz, p. cm.

Includes bibliographical references. ISBN 0-12-627260-3 (alk. paper) 1. Macromolecules—Optical properties. 2. Polymers—Analysis .

3. Light—Scatter ing. I. Title. QD381.9.066S36 1990

547.7Ό45414—dc20 89-17670

CI Ρ

Printed in the United States of America

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This book is dedicated to Candace, wherever she may be,

and to the memory of DAX, may his principles survive.

Preface

"The gentleman devotes his efforts to the roots, for once the roots are established, the Way will grow therefrom/'

From Lun yu fThe Analects, 1.2) by Confucius (55J-479 B.C.)

Dynamic light scattering (DLS) methods monitor the temporal behavior of the intensity of scattered light, from which an "apparent diffusion coefficient," Z) a p p, is computed. Depending on the experimental circumstances, D a pp pro-vides information about relaxation processes that range from the dynamics of "isolated" macroparticles to the dynamics of a collection of macroparticles as found in congested solutions, melts, and gels.

As a result of the widespread use of DLS methods to characterize the physical properties of macromolecular systems, instruments of varying degrees of sophistication are now commercially available. In the spirit of the above passage from The Analects by Confucius, the present textbook is aimed at providing a strong foundation in both the theory and the application of DLS techniques that may be of value to newcomers in the field, and therefore in the future development of the field.

There exist several excellent books and review articles on the applications of DLS techniques. The focus of each of these books lies in one of three areas: (1) instrumentation, (2) theory, or (3) meeting proceedings. The books that focus on instrumentation generally present a cursory introduction to the mathematical expressions used to interpret DLS data. The books dedicated to the theory of DLS generally emphasize the mathematical rigor of the ex-pressions used to describe the molecular motion, but they present a minimum

XV

XVI P R E F A C E

of experimental systems that illustrate the theoretical results. Meeting pro-ceedings, being the creation of several authors whose contributions may be edited, are usually structured with the assumption that the reader is familiar with the fundamental concepts of DLS, and in most cases different notation may be used in the various chapters. By their very nature, review articles give a brief survey of the current status in a particular area of application of the DLS techniques.

There is a need for a textbook that focuses on the basic concepts of DLS, with a critical assessment of specific examples used to illustrate these concepts and to assist a newcomer in the field in the interpretation of DLS data: These are the aims of this book. To achieve these goals, it is convenient for the reader to have in one text a self-contained presentation of the appropriate equations used to interpret the DLS data. Selections from the literature are chosen to illustrate how the mathematical expressions are applied to actual data. In many cases a brief discussion of the chemical and/or physical nature of the system is given in order to provide some insight as to the type of specific information that can be gained about the system using DLS techniques.

It is emphasized that DLS techniques alone may not suffice to provide unambiguous information about the systems of interest to the experimenter. The format of this textbook also includes reviews and discussions of results from complementary studies on the specific systems in order to assist in the interpretation of the DLS data. There is no attempt to provide a detailed discussion of these complementary methods, since to do so would require a prohibitive number of additional pages in the text and would detract from its major focus.

This textbook is directed to (1) advanced undergraduate students and graduate students in the chemical, physical, and biological sciences; (2) scientists who might wish to apply DLS methods to systems of interest to them but who have no formal training in the field of DLS; and (3) those who are simply curious as to the type of information that might be obtained from DLS techniques. In this respect, I have developed this textbook with no as-sumptions regarding the reader's prior knowledge of the DLS methods or the physical and chemical properties of the systems used to illustrate the analysis of the DLS results. I do assume, however, that the reader has a working knowledge of algebra and elementary calculus.

In an introductory textbook one cannot represent all areas in which DLS methods are applied. I have chosen to focus on the solution properties of macromolecules, since these applications represent a major fraction of the papers in the literature, and hence comprise the most likely areas of interest of the reader. Omitted, for example, are detailed discussions on pure liquids and critical phenomena.

Since the main focus of this book is on the interpretation of dynamic light scattering data, I have omitted detailed discussions on the experimental design

P R E F A C E XVll

for light scattering techniques. Interested readers are referred to the detailed discussions on instrumentation found in the book by Ben Chu entitled Laser Light Scattering (1974, Academic Press), which is scheduled for revision in 1990.

The material in this text is developed in accordance with increasing complexity of the system, ranging from dilute solutions of noninteracting particles to the more complex multicomponent systems of strongly interacting particles. The relevant concepts presented in each section are illustrated by a critical analysis of selected systems reported in the literature. Alternative interpretations are sometimes presented if the data are consistent with a different point of view.

Perhaps to the dismay of the more established practitioners in the field of DLS, I have found it necessary at times to employ a notation not generally found in the literature in an attempt to distinguish between two or more concepts. For example, I have used an extensive superscript/subscript system to identify D a pp with particular molecular processes under examination as determined by the experimental conditions.

Because this book is aimed at a wide range of readers, the material in this text is partitioned into three major divisions. The topics covered in the first five chapters address the more fundamental questions and applications regarding the technique of quasi-elastic light scattering. The material in Chapters 6 through 9 is directed to basic concepts and methods used in the study of the more complex systems of interacting particles and applied external fields. I have therefore included relevant background material for the type of system under examination in order to provide a foundation for the interpretation of the DLS results. In Chapter 6, for example, I have derived the relevant expressions for the hydrodynamics of a sphere with the intent to provide a theoretical basis for the series expansion describing multisphere hydro-dynamic interactions. The material in the tenth chapter focuses on four specific observations for which there is as yet no consensus as to the proper interpretation of the data, thus representing current research problems in the field.

It is my hope that since the material is presented in this manner, the reader will become better aquainted with the DLS technique, its advantages and shortcomings, and the wealth of information that can be gained about a system when complementary methods are correlated with the DLS results.

Acknowledgements

I wish to thank the many scientists who have submitted material for consideration for inclusion in this textbook. Their cooperation and willingness to share information has been a very refreshing experience for me. Special recognition is given to Bruce Ackerson, Sow-Hsin Chen, Maurice Drifford, Ot to Glatter, Norio Ise, Alex Jamieson, Charles S. Johnson, Jr., Rudolf Klein, Albert Philipse, Peter Pusey, Manfred Schmidt, Ikuo Sogami, and David A. Weitz for discussions and correspondence regarding their work, and to Luc Belloni for a copy of his doctoral dissertation. I am particularly grateful to Magdaleno Medina-Noyola for a critical assessment of my summary of his theoretical work on the tracer friction factor for multicomponent systems that appears in this volume, and to John Hayer for comments concerning the sections on the Ornste in-Zernike relationship and methods of solving these integral equations. My special thanks to Mickey Schurr, who stimulated my interest in the solution properties of macromolecules.

I am extremely grateful to Shuu-Jane Yang for her encouragement during the presentation of the manuscript.

I give many thanks to the personnel at Linda Hall Library of Science and Technology in Kansas City for assistance in the search of the literature.

Finally, I wish to acknowledge the National Science Foundat ion for providing time to complete this manuscript.

xix

About the Cover

"BUTTERFLIES"

(c) The Escher image "Butterflies," reproduced with permission by M. C. Escher Heirs c/o Cordon

A r t - B a a r n - H o l l a n d .

The Escher image "Butterflies" reflects the intended scope of this book in regard to the information that can be gained about the solution properties of macromolecules through dynamic light scattering methods. The lower portion of the image exemplifies the dilute solution regime, where the detailed shape of the isolated butterfly (translational diffusion of the isolated molecule) and internal pattern of the wing (internal motion of the isolated macro-molecule) are readily discernible. As one proceeds to the top of the image, the information about isolated butterflies (individual macroparticles) becomes more obscure, eventually reaching a point where the butterflies become a collective unit (collective and cooperative motions under congested polymer solution conditions).

xx

C H A P T E R 1

Introduction

"Consider this small dust, here in the glass by atoms moved: Could you believe that this the body was of one that loved:

And in his mistress' flame playing like a fly, Was turned to cinders by her eye:

Yes; and in death, as life unblessed, To have it expressed,

Even ashes of lovers find no rest."

The Hour Glass by Ben Jonson (1573-1637)

1.0. Brownian Motion

In a privately printed pamphlet from 1828 entitled "A Brief Account of Microscopical Observations Made in the Months of June, July, and August, 1827, on the Particles Contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies", Robert Brown described his observations on the motion of suspended pollen grains of Clarkia pulchella. In honor of Robert Brown, this random motion of solute particles is called Brownian motion.

The introductory poem, The Hour Glass by Ben Jonson, aptly illustrates the ubiquitous nature of Brownian motion, in which the interaction with atoms resulted in the restless movement of the much larger ash particles. What is perhaps remarkable about this poem is that it was written almost two centuries before Robert Brown's observations, and almost three centuries before

1

2 1. I N T R O D U C T I O N

Einstein's landmark 1905 paper that related Brownian motion to the random thermal motions of solvent molecules colliding with the suspended particles. The mystique of spontaneous motion of suspended particles has thus stimulated the imagination of both poets and scientists. It has only been within the last three decades, with the invention of the laser, that the spontaneous motion of submicroscopic particles in solution could be monitored by techniques generically referred to in this text as dynamic light scattering (DLS) methods. DLS methods monitor the time-dependence of the intensity of light scattered by the medium. I(t). The rate at which I(t) fluctuates about its average value in turn depends upon the rate at which the scattering elements move in solution, the latter being characterized by an apparent diffusion coefficient, D a p p.

The primary advantage of using DLS methods is that one can rapidly and accurately obtain a value for D a p p. This value is obtained from either the autocorrelation function of the phototube current or the linewidth of the spectral density profile of the scattered light intensity. It is the interpretation of D a pp for macromolecular systems that is the main thrust of this book.

1.1. Brief History of Dynamic Light Scattering

It has been known since the turn of the century through the works of Smoluchowski (1908) and Einstein (1910) that fluctuations in the density of condensed media result in local inhomogeneities that give rise to light scattered at angles other than the forward direction. These authors did not, however, calculate the spectral profile of the scattered light. Brillouin (1914, 1922) showed that fluctuations that propagated with a velocity ν gave rise to "doublets" that were frequency-shifted by an amount proportional to ± ν from the frequency of the incident light. Gross ( 1930,1932) experimentally observed this doublet and, in addition, a central peak of unshifted frequency. Landau and Placzek (1934) correctly interpreted the central, or Rayleigh, peak as being due to nonpropagating entropie fluctuations. They showed that the ratio of frequency integrated intensities of the central peak « / > c ) to the shifted peak « / > s ) was < / > c / < / > s = (C P — C v ) / C v , where C P is the heat capacity at con-stant pressure and C v is the heat capacity at constant volume. Even though there was a wealth of information to be obtained from the spectral profiles of scattered light, the intrinsic linewidth of the incident radiation was too broad to allow meaningful information contained in the relatively small frequency shifts to be gained except under the most unusual circumstances. This situation was changed in the 1960s with the invention of the laser.

In his doctoral thesis in chemistry under the direction of Professor Bersohn in 1962, Pecora (1964) showed that the frequency profile of the scattered electric field was broadened by the diffusion processes of the macromolecules. The half-width at half-height of the central peak was a direct measure of the

1.2. Time Scales 3

translational diffusion coefficient (Dp), viz, Δ ω 1 / 2 = D®K2, where the scatter-

ing vector Κ is related to the index of refraction of the solvent (rc0), wave-length of incident light (λ0\ and scattering angle (0) by the expression Κ = (4TOoMo)sin(0/2).

The first experimental report using lasers as the source of incident radiation for the study of macromolecular solutions [poly(styrene)] was by Cummins et al. (1964), and that for pure fluids near the critical point was by Ford and Benedek (1965). The ensuing ten years focused on testing the theoretical predictions for simplified systems, such as theories for translational and internal relaxation modes of spheres and of rodlike and coillike particles.

The first major variation in the light scattering technique occurred in 1971 when Ware and Flygare (1971, 1972) reported DLS studies on bovine serum albumin (BSA) in the presence of a static electric field, E°. The effect of E° is to superimpose a constant drift velocity, proportional to the electrophoretic mobility of the species, on the random Brownian motion of the charged particles. The resulting spectral density is thus composed of peaks that are Doppler-shifted from the central position. Ware and Flygare were able to resolve BSA monomers and dimers. This technique is referred to as Doppler shift spectroscopy (DSS) or electrophoretic light scattering (ELS).

Early applications of DLS methods focused on the determination of molecular weights and shapes. It was not an accident that the development of the hydrodynamic theories for complex, irregular-shaped particles paralleled the development of DLS techniques. There was almost a two-decade gap between the pioneering works of Kirkwood and Riseman (1948), Kirk wood ( 1949,1954) and Zimm ( 1956) on the beaded string subunit model for polymers before progress was made in this field by Bloomfield et al. (1967), Rotne and Prager (1969), Yamakawa (1970), Yamakawa and Tanaka (1972), Yamakawa and Yamaki (1972, 1973), McCammon et al. (1975), McCammon and Deutch (1976), and Garcia de la Torre and Bloomfield (1977).

Largely because of the successful determinations of molecular weights and shapes, DLS methods were accepted by the scientific community as a whole in the mid-1970s. As more groups began to use these methods, new information about specific systems began to emerge. Technological advances in the instrumentation led to more precise and accurate data. Minor discrepancies between theory and experiment soon became apparent. These discrepancies led to more sophisticated theories and methods of analysis, and to the development of new techniques for the preparation and handling of samples.

1.2. Time Scales

The relaxation of fluctuations in the polarizability of the medium occurs over a wide range of times associated with the very small solvent particles as well as with the largest macromolecules in the solution. It is very important, both in


theory and practice, to take into consideration the relative values of the data collection interval Δί and the characteristic time τ associated with a relaxation process. Rudolf Klein, of the University of Konstanz, illustrates the im-portance of these differences to his students in the following manner (personal communication). At time intervals such that Δί < 10 13 s, the momenta and positions of all the particles in a solution must be taken into consideration because the time interval between collisions involving solvent particles occurs on this time scale. The dynamics of the system is then described by the Liouville equation,

dp(pm,qm,Po*qo,0 ^ , Λ n~ u — = ^ . p ( p m , q m , p 0 , q 0 , r ) , (1.2.1) at

where p(pm, q m, p 0 , q 0 , r) is a density function that contains information about the momentum (p) and position (q) of the solvent particles (p 0,qo) a n <^ macroparticles ( p m, q m) , and Sf is a Liouville operator that depends upon the 6 χ (m + 0) coordinates. For Δ t » 1 0 " 1 3 s, the solvent dynamics have relaxed, and the dynamics of the macroparticles are no longer coupled to the solvent. This condition is manifested by a new distribution function P ( p m , q m , t\

P(Pm ,q m,0 P(Pm > Qm, Po > Qo> 0 dPo dq0, (1.2.2)

which obeys the Fokker-Planck equation of the form,

d P ( p m, q m, 0 ôt = n.P(pm,qm,0, (1.2.3)

where Ω is an operator involving the 6 χ m coordinates of the macroparticles and the average interactions of the solvent particles with the macroparticles. The Fokker -P lanck equation describes the evolution of the macroparticle system over the time range 10~ 1 3 s « Δί < 10~ 8 s. For A i » 1 0 ~ 8 s , the momenta of the macroparticles have relaxed, and the system is described by a new distribution function,

G ( q m , 0 = P ( p m , q m , i ) d p m , (1.2.4)

which obeys the Smoluchowski equation,

3 G ( q m, t )

dt = Q . G ( q m, i ) , (1.2.5)

where 0 is an operator that contains 3 χ m spatial coordinates of the macroparticles. The effect of a wide separation of relaxation times is that the equation of motion and the subsequent interpretation of the data are simplified.

1.3. Organizat ion of the Textbook 5

1.3. Organization of the Textbook

The material in this textbook is presented in order of increasing complexity of the systems under examination, ranging from dilute solutions of noninteract-ing particles to concentrated multicomponent solutions of strongly interact-ing particles and gels. Literature examples are interspersed in the step-by-step mathematical descriptions to illustrate the concepts being developed. Prob-lems are presented at the end of each chapter to emphasize these concepts. Since a major emphasis of this textbook is the interpretation of DLS data obtained by polarized light scattering studies on macromolecular solutions, the results of complementary experimental techniques are also presented in order to gain insight into the dynamics of these systems. The reader should not, therefore, assume that the interpretations presented in this text are accepted by all researchers in these fields. This is especially true of the material presented in Chapter 10 on the more complex systems.

Chapter 2 is partitioned into three parts. In Part I, the basic principles of DLS are presented in detail. It is shown, for example, that fluctuations in the polarizability of the medium give rise to light scattered at angles other than the forward direction. Part II focuses on static, or total intensity, light scattering. Intraparticle and interparticle structure factors are presented. These concepts are then extended in Part III to dynamic processes. It is shown that DLS methods provide reliable values for the mutual diffusion coefficient (Dm), even for molecules that absorb light.

The development then proceeds to systems of noninteracting particles whose dimensions are smaller than the wavelength of the incident light. The information obtained by using DLS techniques on these systems is the rotationally averaged diffusion coefficient. This information then can be used to compute molecular weights and gain insight as to the shape of the macromolecule, which is the major emphasis of Chapter 3. The differences between the mutual and tracer friction factors are highlighted.

In general, the correlation function reflects more than one decay process, whether these modes arise from molecular-weight inhomogeneity of the preparation, size of the particles, and /or interactions between particles. In Chapter 4, the effect of polydispersity on the intensity of scattered light is discussed. Simulated correlation functions are then analyzed by a variety of techniques, and the results are compared with the known parameters associated with the simulated function. The polydispersity analysis methods include expansion techniques (viz, cumulant analysis) and Laplace transform methods. The various distribution functions (amplitude, amplitude-weighted decay rates, and number) are compared.

The next level of complexity arises from particles of size comparable to the wavelength of incident light. Chapter 5 focuses on the internal relaxation processes of rigid and semiflexible molecules. The correlation functions for


very large particles are discussed for flexible coils and rigid rods. Theories based on the first cumulant (initial rate of decay) are emphasized. As the concentration of the solution is increased, interactions between particles due to overlap of molecular volumes begin to influence the dynamics of the particles. The scaling laws for the semidilute and gel regimes are discussed. Particular emphasis is given to reptation and crossover models. Probe diffusion of large particles through solutions of smaller particles or isore-fractive particles and gel matrices is discussed.

Discussed in detail in Chapter 6 are the hydrodynamic interactions between spherical particles that may also be coupled through a weak interaction potential (no long-range order in the solution). Hydrodynamic interactions are treated by both the method of reflections and the method of induced forces. Of particular relevance to these systems is the concept of a "time-dependent" Z) a pp that results from the relaxation of the "background matrix" and is manifested as a "memory function". The interpretation of Z) a pp is shown to be dependent upon the value of the product Κ AR, where AR is the average distance between centers of neighboring spheres. If KAR « 1, then D a pp is identified with the mutual-diffusion coefficient, whereas if Κ AR » 1, then D a pp

is identified with the self-diffusion coefficient.

Attention is focused in Chapter 7 on solutions of polyelectrolytes in which the role of small ions may be described as either passive or active. The small ion-polyion coupled modes are examined in detail, where the X-dependence of D a pp is developed for the first time. The Medina-Noyola formalism (1987) is used to examine the effect of small ions on the tracer friction factor of polyions. Condensation of small ions onto the surfaces of highly charged polyelectro-lytes is discussed. The effect of ionic strength on the persistence length of a very flexible polyelectrolyte is examined, with the result that the polymer remains quite flexible even under "zero added salt" conditions.

Chapter 8 describes colloidal systems in which direct polyion-polyion interactions play a major role in determining the dynamics of the polyions. The D L V O (Der jaguin-Landau-Verwey-Overbeek) theory (Verwey and Overbeek, 1948) is presented for the case of spherical particles with a "thick" ion cloud. The cross-correlation data of Clark and Ackerson (1980) indicate that "local ordering" occurs in the "liquid regime" of colloidal solutions, as well as "long-range ordering" in the crystalline regime. The solution structure factor for highly interacting particles is developed in detail. The interparticle interactions are described in the context of the Sogami model (Sogami, 1988), which takes into consideration the distribution of the small ions. Fractal objects and the rate of their formations are also discussed.

The response of macromolecular solutions to external perturbations is discussed in Chapter 9. In regard to electric fields, the response of charged polymers to static, pulsed square-wave and sinusoidal-wave forms is ex-amined. Application of a shear field is shown to disrupt the crystalline-

1.3. Organizat ion of the Textbook 7

like structure of colloidal systems. Mechanical excitation of gels provides information about the elastic modulii of the crosslinked systems. The effect of high pressure on the glass transition and bimolecular association kinetics is also examined.

Chapter 10 focuses on current areas of research, and therefore the in-terpretation of the data may be somewhat controversial. Admittedly, one cannot provide an adequate introduction to all areas of research currently in progress, so the four systems examined in this chapter were selectively chosen to function as extensions of material presented in earlier chapters.

-Δω ς + Δ ω δ Η

Fig. 1.1. Schematic of the spectral density profile for the incident and scattered light. The

spectral density profile for the incident laser light is very narrow, as shown in the upper right hand corner. The random motions of the molecules broadens the peak centered at ω 0 (entropie fluctuations). Superimposed directional mot ion results in peaks of shifted by an amount + Δ ω (propagat ion fluctuations). Information about the long-wavelength molecular motions (viz, center-of-mass diffusion) is contained in the linewidth at half height ( Δ ω 1 / 2) and peak position (Δωδ). Raman lines, which provide information about the vibrational states of local groups within the molecule, are presented for comparison. Note that the amplitudes of the propagat ion fluctuation peaks at ± Δ ω 5 are the same, whereas the Stokes peak is of larger ampli tude than the anti-Stokes peak.


Whenever possible, I have retained the notation used in the original literature cited in this textbook. In many cases, however, the same symbol has been used in different sources to denote different parameters. It was deemed necessary in these cases to introduce a new notation in order to maintain a degree of self-consistency within this textbook. A glossary is therefore provided to identify the more frequently used symbols in these chapters.

There is a wealth of information about the dynamics of macromolecular systems contained in the frequency distribution of the scattered light intensity. This information is manifested in the location and linewidth of the peaks that are present in the spectral density profile. Low frequency motions such as center-of-mass and long-wavelength internal motions are within the domain of DLS methods. High-frequency internal vibrational modes are studied by Raman scattering techniques. A schematic of the spectral density of incident and scattered light is shown in Fig. 1.1.

1.4. Nomenclature

Today there are many variations of the original light scattering technique that was introduced in 1964. For the purpose of clarity in this text, the following terms are used to describe the various techniques as distinguished by either the method of detection or the process being examined. The reader is cautioned that the proposed terminology does not necessarily correspond to the useage in the literature.

Dynamic light scattering (DLS) is a generic term encompassing all of the light scattering methods that provide information about the molecular dynamics. Electrophoretic light scattering (ELS) or Doppler shift spectroscopy (DSS) refers to the DLS experiment in which a square-wave electric field of alternating polarity separated by "rest periods" is applied across the sample, and data are collected only during the period of time when the applied electric field has reached a constant value. Laser Doppler velocimetry (LDV) refers to the DLS experiment in the presence of a laminar-flow field. Photon correlation spectroscopy (PCS) refers to the experimental technique of photon counting in the computation of the autocorrelation function, as opposed to using an analog signal as employed in the earlier correlators. Quasielastic light scattering (QELS) is used to describe experiments that are performed in the absence of electric or hydrodynamic flow fields. Quasielastic light scattering I periodic pulsed electric field ( Q E L S - P P E F ) refers to the DLS experiment in which a square-wave electric field of alternating polarity is applied across the sample with no rest period between pulses, accompanied by continuous acquisition of data. Quasielastic light scattering j sinusoidal electric yfc/i/(QELS-SEF) refers to the

1.4. Nomenclature 9

DLS experiment in the presence of a single-frequency sinusoidal electric field, accompanied by continuous acquisition of data.

The DLS methods only provide information about the "long-wavelength" properties of the system. Complementary techniques such as gamma ray, X-ray, and neutron scattering methods provide information on a much shorter wavelength scale. Typical ranges for the wave vectors accessible by these various scattering methods are:

Method Abbreviation K-range (cm

l)

(approximate)

Dynamic light scattering D L S 5 χ 104

- 4 χ 105

Small-angle X-ray scattering SAXS 2 χ 106

- 4 χ 107

Small-angle neutron scattering SANS 7 χ 105

- 9 χ 106

Wide-angle neutron scattering W A N S 1 χ 108

- 5 χ 108

Quasielastic gamma ray scattering Q E G S 1 χ 107

- 1 χ 109

To illustrate how these scattering techniques provide complementary information about a system, the combined light scattering and SAXS studies of Schaefer et al. (1984) on aggregated silica particles are given in Fig. 1.2. The

2.1

SAXS

O.OOOI

r~ n —

o . o o i o . o i

Κ ( A "1 )

'—1 1

I

0.1

Fig. 1.2. Combined light scattering and small-angle X-ray scattering from colloidal silica. The

power law — 2.1 indicates that the aggregates are fractal objects, and the power law — 4.0 indicates that the monomeric units that make up the aggregate are intact. Neu t ron scattering data would also appear in the region of the SAXS results (cf. E8.6 for a more detailed discussion of this work). [Reproduced with permission from Schaefer et al. (1984). Phys. Rev. Lett. 52, 2371-2374. Copyright 1984 by the American Physical Society.]


power law —2.1 for the scattered light intensity indicates that the aggregates are fractal objects. At the short wavelength regime accessible to SAXS, the power law —4.0 indicates that the monomer units that make up the aggregate remain intact.

A complete understanding of a macromolecular system may also require complementary techniques other than scattering methods. Additional hydro-dynamic methods include sedimentation velocity and viscometry, which provide information about the gross structural properties of the system. Local structural changes can be monitored by "point probes", using such methods as circular dichroism and fluorescence techniques. The examples provided in this text will also discuss these additional complementary techniques if it is necessary to clarify the "physics" of the system.

A new unit of diffusivity, which is patterned after the unit in sedimentation velocity, is also used in this textbook. Molecular weights can be computed from the ratio of the sedimentation coefficient (sT) and the diffusion coefficient [D(T)]. The unit of sedimentation is the Svedberg, defined as 10~

1 3 s. By

analogy through application of the Svedberg equation, a unit of diffusivity, 1 fick = 1 0 ~

7c m

2/ s , has appeared in the literature. Although this unit is

convenient for the diffusivities of a wide range of polymers, or when computing molecular weights from the Svedberg equation [D(T) / s x = 10

6] ,

this unit is not consistent with the SI system of units, as has been pointed out by Walter Stockmayer (personal communication). In honor of Adolf Eugen Fick (1829-1901), the unit of diffusivity is defined in this text as 1 fick ( F ) = 1 m

2/ s .

C H A P T E R 2

Basic Concepts of Light Scattering

"The White Rabbit put on his spectacles. 'Where shall I begin, please your Majesty?' he asked.

'Begin at the beginning/ the King said, very gravely, 'and go on till you come to the end: then stop. '

There was dead silence in the court, whilst the White Rabbit read out these verses: - "

From Alice's Adventures in Wonderland by Lewis Carroll (1832-1898)

2.0. Introduction

As suggested by the above passage, one should begin any story at the beginning in order to lay the proper foundations for the material that follows. In this chapter the basic concepts of light scattering are introduced, starting with the wave description of light and how light interacts with matter (Part I), and then proceeding to static (Part II) and dynamic (Part III) light scattering by a system of macromolecules.

11

12 2. BASIC C O N C E P T S O F L I G H T S C A T T E R I N G

PART I I N T E R A C T I O N O F L I G H T W I T H MATTER

2.1. The Nature of Light

Light is a nonperturbative probe that can be used to obtain information about the structure and dynamics of molecules. Maxwell's equations form the basis of the description of all electromagnetic phenomena. These equations identify light as a transverse electromagnetic wave that oscillates in both space and time, i.e., the direction of oscillation is perpendicular to its direction of propagation. The electric field associated with the light at location r, measured from the origin of a laboratory reference frame, and time t is given by the expression

E 0(r ,r) = E 0exp( iK?-r ) exp( - i a> 0t ) , (2.1.1)

where E0(r, t) is written as a vector quantity to denote the spatial orientation of the oscillation (polarization) for a field strength of magnitude E0. The distance between adjacent maxima defines the wavelength (λ0) of light, which in turn defines the magnitude of its wave vector |K 0 | = 2πη0/λ0, where n0 is the index of refraction of the medium. The angular frequency ω0 is related to the speed of light (c) by ω 0 = |K 0 | c /w 0.

2.2. The Electrical Nature of Matter

The mode of interaction of light with matter depends upon the electronic structure of the material as determined by its quantum mechanical properties. If the energy of the incident photon (hœ0/2n, where h is Planck's constant) is equal to the difference in energy between two states in the system, then the photon might be absorbed by the system. As an oscillating electric field, light also distorts the distribution of the charges in the system that, as accelerated charges, emit radiation in the form of scattered light. If there is no exchange of energy between the photon and the system, then the frequency of the scattered light, ω ν, is equal to ω 0 , and the process is referred to as elastic light scattering. If energy is exchanged between the photon and the system, the cos differs from ω 0 , and the process is referred to as inelastic light scattering.

The response of a system to an external electric field is called the polarization of the system. The magnitude of the polarization is dependent upon the amplitude of the applied electric field and the ability of the charge distribution to be "deformed" by the external stimulus. The capacity of the system to be distorted is referred to as the polarizability of the system and is represented by the symbol a. In general, the direction of the response of the system to light is not constrained to the direction of polarization of the light. The polarizability of the system is then represented as a tensor quantity (i.e., an Ν χ Ν matrix) denoted by a(r, t). The polarization of the system at the location r and time t

2.3. The Scattered Electric Field 13

is then given by,

P(r,r) = a(r , t) .E 0(r , f ) , (2.2.1)

where the individual components are given as

px ^xy y-xz Ex

= ^yx Otyy *yz • Ey

Pz *zx *zz Ez

and the subscripts refer to directions in a laboratory fixed coordinate system, and the explicit statement regarding the spatial and time dependence of these quantities has been omitted to simplify the notation.

2.3. The Scattered Electric Field

The polarization of a volume element d3r , whose center is at a point r, arises

from the distortion of the local electron distribution in the material. The response of the local polarization to the light is described by the equation of motion for a damped harmonic oscillator wth a sinusoidal driving force. To illustrate, it is assumed that the time dependence is described by cos(coi). The solution to this problem has two components, one that varies in time as cos(coi) and another that varies in time as sin(coi) (cf. Prob. 2.1). The amplitude of the sin(cot) function is important only if ω 0 is in the vicinity of the natural frequency ω 0 8 of the material oscillator. This component of the response function corresponds to absorption of the light. The amplitude of the cos(a;r) function is positive when ω 0 < ω 0 8 (in phase) and negative when ω 0 > ω 0 8

(180° out of phase). The discussion is limited to the region ω 0 < ω 0 8, in which light is not absorbed by the system and the amplitude of the response is independent of ω 0 .

The electric field radiated by a "patch" of material located at r with volume element d

3r is given by

dEs(R,t) = ^d

- ^ ß d \ (2.3.1)

where R is the location of the detector in the laboratory reference frame, R = |R|, and t is the time of detection of the light radiated at the earlier time t\

t> = t -| Κ

-Φ

~ ° . (2.3.2) c

The quantity |R — r | n 0/ c is the time it takes light to travel from the scattering center at r to the detector. This lag time is the basis of dynamic light scattering methods.


If a(r, t') is a slowly varying function of time in comparison to ω 0 , then substitution of Eqs. (2.1.1) and (2.2.1) into Eq. (2.3.1) results in the expression

d E s ( R , i ) = - ^ r « ( r , i , ) - E 0 ( r , i ' ) d 3 r . (2.3.3)

The total electric field that falls upon the detector from all the patches is obtained by integration of Eq. (2.3.3) over the scattering volume Vs:

E s(R ,t) = k 5

Rc a ( M ' ) - E s ( r , i ' ) d 3 r . (2.3.4)

We now simplify Eq. (2.3.4) by introducing a specific laboratory frame of reference. The ζ direction is defined as the direction of propagation of the light beam, and χ is perpendicular to the plane described by the propagation direction of the incident beam, the sample cell, and the detector. The polarization of the laser beam is along the χ axis. The scattering geometry is illustrated in Fig. 2.1.

For simplicity, it is assumed that the polarizability of the scattering medium is isotropic, i.e., (α)0 = 0 if i φ j and (<x)n = a, where i, j = x, y, or z. The polarization of the scattered light is therefore parallel to that of the incident light; hence the pre-exponential factors reduce to the scalar product a £ 0 . The wave vector for the scattered electric field, K s, is defined by the vector

incident light

scattered light

LABORATORY FRAME Fig. 2.1. The scattering geometry. The scattering angle θ is in the zy plane of the laboratory

reference frame.

2.4. Fluctuations in the Polarizability 15

product

( R - r ) r . K s JR

-f K

" ° . (2.3.5) c

Substituting Eqs. (2.1.1), (2.3.2), and (2.3.5) into Eq. (2.3.4) and defining Κ = K 0 - K s gives for £ S(R, t),

Es(Rj) = - ω

°η°

Ε° cxp(/K s

T - R ) e x p ( - f o 0i ) a ( K , i ' ) , (2.3.6)

c R

where the spatial Fourier transform of the polarizability from real space (r) to momentum space (K) is given as

α(Κ,ί') = a ( M ' ) e x p ( i Kr. r ) d

3r (2.3.7)

vs

2.4. Fluctuations in the Polarizability

In general, the polarizability of a system is not homogeneous throughout. One can express the polarizability as an average part « α » and a contribution which represents local inhomogeneity, both in space and in time [<5oc(r, f')]:

α(Κ,ί') = <<χ> e x p ( i Kr. r ) d

3r + δα(τ, i ' ) exp( iK

r · r)d

3r. (2.4.1)

Jvs Jvs

The first integral has the property

Since Κ = 0 results when K s = K 0 , this term contributes only in the forward scattering direction. It is concluded, therefore, that the scattering of light at any angle other than the forward direction is a result of fluctuations in the polarizability of the medium.

For elastic scattering one has |KS| = |K 0| ; hence, from the conservation of momentum diagram,

|K| = 2 | K 0 | s i n | = ^ s i n ^ (2.4.3)

where θ is the scattering angle as measured from the direction of propagation of incident light. By varying the scattering angle Θ, one monitors fluctuations with different momentum vectors K. Light scattering techniques may be referred to as spectroscopic techniques in momentum space.


PARTII

TOTAL INTENSITY L I G H T SCATTERING

2.5. Total (Integrated) Intensity of Scattered Light

The photomultiplier tube does not detect the electric field directly at any instant in time i, but rather the intensity of the scattered light, /(R, i), as defined by

/(R,0-Ô eE s*(R,O r-E s(R,O, (2.5.1)

where Qe represents the quantum efficiency of the photomultiplier tube and £S*(R, t') is the complex conjugate of the electric field scattered at the earlier time t'. The time-average integrated intensity is equated with the ensemble average </(R)>,

</(R)> = Ôe<Es*(R, t'Y . ES(R, t')>. (2.5.2)

The quantities α and r have thus far been associated with a region in the medium, which contains contributions from both the solvent and the macromolecules, viz,

</(R)> = </(R)> s o lv + </(R)>macro (2.5.3)

The macromolecules can be thought of as a collection of scattering units held together by the chemical bonds. The location of the pth scattering unit of the P th macromolecule at time t' [denoted as R(i')Fn] can then be expressed as the vector sum of two terms, the location of the center-of-mass [R(i')p] a n

d the relative location of the pth segment within the macromolecule [r ( f ' ) Ρ ρ] . It is assumed that the presence of a scattering center affects neither the amplitude nor the phase of the incident light. The intensity of the scattered light then depends only on the spatial arrangement of the scattering centers. This type of scattering is referred to as Rayleigh-Gans scattering, or Rayleigh-Debye scattering. If the relative magnitude and phase of the electric field are altered by a scattering center, then the characteristics of the electric field are dependent upon its location within the particle. This type of scattering is referred to as Mie scattering. We limit our development in this text to Ray le igh-Gans -Debye scattering.

If the segment polarizability a s is the same for all of the segments, then the polarizability can be expressed as

a(r, t') = as X £ δ{τ - [R(i')P + r(t')Pp]}, (2.5.4)

F = 1 Ρp = 1 where N p is the number of polymers in the solution, ns is the number of scattering subunits associated with each polymer, and ô(x) is the Kronecker delta function, which is equal to unity if the pth segment of the P th particle is

2.5. Total (Integrated) Intensity of Scattered Light 17

/ Np Np rts ns

χ ( Σ Σ Σ Σ \Ρ= 1 Q = l Ρρ=ί Qq=l

χ expi-iK7"- [ R ( î ' ) p - R ( i ' ) c + r{t')pP - Γ ( ί ' ) ο β] } ^ ( 2 . 5 . 5 )

The total intensity can be partitioned into the sum of two parts: Npn^ terms that contribute as a "molecular self par t" </(R)> s e lf (Ρ = β) , and Np(Np — l)n s

2

terms associated with a "molecular pair part" </(R)> p a ir (Ρ Φ Q). We shall define "interference" or "structure factors" that are normalized

to the number of terms in the sum. Upon defining the incident intensity / 0 = | £ 0 |

2ß e , the generalized molecular self term is

< / ( R ) > . „ f = J V i ,2 — - £ a ,

2

1 * P

Σ Σ Σ Npns p=i Pp=i Pq

xexp{ïKT.[r(t')Pp-T(t')Pq]})9 ( 2 . 5 . 6 )

which reduces for the case of identical particles to

</(R)> s e lf = Np ( - ° - £ *2

snlP{K Ar), (2.5.7) ωο

c j R2

where the particle structure factor P(K Ar) is defined by

P(K Ar) = (-Υ/ χ χ exp{/KT · [r(t')Pp - r(t')Pq]} ) . ( 2 . 5 . 8 )

\nj \Pp=l Pq=l

The molecular pair contribution to </(R)>maCro is

</(R)> p a ir = ( - ) ^ α2[ Λ Τ ρ ( Ν ρ - 1 ) η

2]

1 Np Np ns ns

Σ Σ Σ Σ ΝΡ(ΝΡ - l)n s

2 ^ ^ , pfciatii

exp{ - iKT - [R(f % - R ( i ' ) e + r(i')p, - r ( t ' ) e, ]} ) . (2.5.9)

in the solution "patch" at the time t' (x — 0) and is zero otherwise. </(R)>macro due to the Np macromolecules in the scattering volume Vs is given by the expression

</(R)>macro = f ^ Ï Ï ^ Y « s 2 e e


We digress at this point to clarify some instances in the literature where it has been erroneously argued that the terms in the brackets < > vanish because of "spherical averaging". The bracketed terms represent "interference effects" that result when the relative distances between scattering centers cannot span all available space in the integration. Chemical bonding prevents sampling all phase space for the internal structure factor. In the case of the "pair term", the distances between scattering centers located on two different polymers is determined by a pairwise interaction potential l/(r), where r is the distance between the two centers. In the most general case, therefore, the intensity of light scattered by a single scattering center in a solution of macromolecules is modified by internal interference with other scattering centers on the same macromolecule, and also by interference effects by scattering centers on all of the other macromolecules. The relationships between the center-of-mass and internal coordinates are shown in Fig. 2.2.

</(R)> — </(R) s l o v, and hence a s, is directly related to the excess high frequency dielectric constant increment, Δε = ni — ni, where ns is the index of refraction of the solution. Approximating nl — η$ ~ 2n0(ns — n 0) , then the polarizability a s becomes (Flory, 1953)

a = . (2.5.10) 2n{NvnJVs)

We further use the approximation (ns — n0)/cp ~ (dns/dcp)Ttfl>9 where (dns/dcp)j μ> is the index of refraction increment at constant temperature Τ and

Fig. 2.2. Molecule-fixed reference frame. The center-of-mass of molecule J is located a distance R, from the origin of the laboratory reference frame. The relative location of scattering units within the molecule (r,j) are measured from the origin of the molecule-fixed frame of reference.

2.5. Total (Integrated) Intensity of Scattered Light 19

constant chemical potential of the remaining species (μ') and c p (in g /cm 3

Hence,

< / ( R ) - / ( R ) S O J I> = /eÇ4

Ρ/Τ,μ'

\ 2nR

MpVscpP(KAr) + c

2

pV2S'{KAR' (2.5.11)

where cp = MpNp/NAVs, Mp is the molecular weight of the polymer, NA is Avogadro's number, Np/Vs is the number density of the particles, P(K Ar) is given by Eq. (2.5.8), and S'(K AR) is the intermolecular structure factor where AR' is the separation distance between scattering units of the different polymers (Ρ φ Q),

S'{KAR') 1

(%"s)2

Np Np ns ns

Σ Σ Σ Σ ρ = ι ö=i Ρρ=ί Qq= ι

χ e x p { - / K r . [R(Op - R(î')q + Γ(ί')Ρρ - r(0QJ}.) (2.5.12)

Eq. (2.5.11) can be rearranged more conveniently so that all of the instrument parameters can be on one side of the equation with all of the molecular parameters on the other side of the equation. The Rayleigh ratio Re is defined as

Re = m <J(R)Xn,

KI0

•R2 = F(0)Im,R

2, (2.5.13)

where / t i ls is the normalized dimensionless scattered intensity per unit volume at the scattering angle θ implicit in the vector distance R , and F(0) is a geometric correction term. The parameter Η is an instrumentation parameter, viz,

Ί2

Η = Ρ/Τ,μ'.

ΝΑλΪ

The total intensity data can thus be represented by

1

Re MpP(K Ar) + VSNAS'(K AR)cp

{2.5.14)

(2.5.15)

If light is polarized in the direction perpendicular to the plane defined by the source, the scattering cell, and the detector, then the beam is said to have vertical polarization, and F(6) = 1 in Eq. (2.5.13). If the polarization of both

2. BASIC C O N C E P T S O F L I G H T S C A T T E R I N G

the incident and the scattered light is in the vertical direction, then the experiment is referred to as polarized light scattering, denoted by 7 V V. Depolarized light scattering, when the horizontal component of the scattered light is monitored, is denoted by 7 V H.

Unpolarized light can be thought of as a combination of several polarized beams having random orientation about the axis of propagation. If unpolar-ized light is used in the scattering experiment, then F(6) = 2 / [ l + cos

2(#)]

in Eq. (2.5.13). This "correction term" takes into consideration the geometric relationship between the orientation of the polarization components of the unpolarized light and their projection onto the phototube surface. An analogy would be the viewing of a coin, where the "zero angle" is defined by viewing the coin face-on so that the coin appears perfectly circular. As the coin is rotated relative to this vantage point, the coin first appears elliptical and, at 90°, a vertical line. It is noted that the correction term reduces to unity when θ 0 or 180°. The theories and experiments discussed in this text are limited to 7 V V

A comment should be made at this time regarding the use of the index of refraction increment at constant chemical potential [_(8nJôcp)T μ \, which arises naturally in the grand canonical ensemble description of dielectric constant fluctuations (Shogenji, 1953; Ooi, 1958; Vrij and Overbeek, 1962). Most studies use (dns/dcp)T c>, where c' denotes constant concentration of all other species (Kirkwood and Goldberg, 1950; Stockmayer, 1950). The two expressions are equivalent for a two-component system.

2.6. Light Scattering by Small, Noninteracting Particles

If the particles are small enough that Kr · [r{t')Pp — v(t')Pq] « 1, then

P(K Ar) ~ 1. Likewise, if the particles are nonintereacting, then S'(K AR) ~ 0. Physically this means that the scattering centers are not confined to be at "preferred distances" from each other as dictated by intermolecular potentials. Hence,

Hence, / t i ls is a quantity that is weighted in proportion to the molecular weight of the polymer. For preparations that are heterogeneous in the molecular weight,

studies.

(homogeneous preparation). (2.6.1)

(2.6.2)

{2.6.3) < M > W =

2.7. Light Scattering by Large, Noninteract ing Particles 21

Example 2.L Dilute Insulin Solutions The first protein whose amino acid sequence was determined by Sanger and coworkers in 1956 was insulin. The monomer unit has a molecular weight of 5773 Daltons and is composed of two polypeptide chains. In the crystalline form, six monomers from a quaternary structure that resembles an equilateral triangle about two central zinc atoms. The function of this hormone is to control carbohydrate metabolism and transmembrane transport of glucose into muscle and adipose tissue. Absence of insulin leads to a hyperglycemic state in the blood, and the body shifts to alternative metabolic pathways to increase intracellular glucose. Temporary relief from this condition is effected by injection of insulin or, for more prolonged relief, protamine zinc insulin. Bohidar and Geissler (1984) examined the aggregation properties of insulin using static and dynamic light scattering methods. These authors used the following parameters in their analysis: (dns/dcp)TtC> = 0.192 c m

3/ g ; and

n0 = 1.336. Compute the aggregation number after 10 days on the basis of the following hypothetical parameters: λ0 = 488 nm; R = 1 meter; and Aiis = 6 χ 1 0

- 9 c m

- 3 for cp = 0.00140 g / cm

3. [Assume dilute solution con-

ditions such that Eq. (2.6.3) is valid.] Solution: From Eq. (2.5.14),

[2π(1.336)(0.192)]2

(6.02 χ 102 3

)(488 χ 1 0 ~7)

4

= 7.608 χ 10~7 mole g~

2 c m

2 m o l e c

- 1.

The Rayleigh ratio is computed from Eq. (2.5.13) to be Re = (6 χ 10"

9) (100

2) = 6 χ 10~

5 c m "

1, and from Eq. (2.6.2),

6 χ 1 0 "5

< M >» = (7.608 χ 1 0 - ^ X 0 . 0 0 1 4 0 )

= 5 6'

3 32 8 / M° '

E-

Since the monomer weight is 5773, the average number of units in the aggregate after 10 days is 56,332/5773 = 9.75.

2.7. Light Scattering by Large, Noninteracting Particles

If the dimensions of the macromolecule are comparable to the "probe length" 1 /X, then the instantaneous amplitude of the incident light beam will differ for various regions within a single molecule. These differences in phase lead to interference effects in 7 t i ls as the range of Κ is spanned. If the solution is sufficiently dilute such that the second term in Eq. (2.5.15) is negligible, which can be achieved by taking data at several concentrations and extrapolating to c p 0, then for homogeneous preparations,

He. 1


In the limit Κ Ar « 1,

Ρ ( Κ Δ γ ) „ 1 _ £ l < £ g [ r ( t ' ) p p _ r ( i 2 > + . . .

P(KAr)~\- — (R2

G}, X

2

T (2.7.2)

where the linear term averages to zero and < # g > is defined as the mean-square radius of gyration of the macromolecule. The operating equation for data interpretation is now

Hcn 1

Re M N

« K R G > « 1). (2.7.3)

Table 2.1

Intramolecular Structure Factors

Solid sphere (X = KRS, Rs = radius of the sphere)

P(X) = y^{sin(X)-Xcos(X)}

Hollow sphere (X = KRS, y = K , /R s, = inner radius)

3 P(X)- - j - 3-{sin(X) - sin(yX) - Xcos(X) + yXcos(yX)} Χ

Λ(\ — y

J)

Thin Rod (X = KL, L = length of the rod)

P(X) 2

: sin(Z)

dZ 2 X — sin — X 2

Disk, infinitely thin (X = KR, R = radius of the disk, JX{2X) = Bessel function of the first order)

P(X)-2 1 - JX{2X)

Circular cylinder (L = length of the cylinder, Rc = radius of the cylinder, X = KLcos(ß), Y = KRcsin(ß))

Γη/2

P(KL) = 2J(X/2)Jl(Y)

sin(ß)dß

Gaussian Coil {Χ = KRG, RG = radius of gyration)

2 P(X) = jïlexp(-X

2) + X

2 - 1]

2.7. Light Scattering by Large, Noninteract ing Particles 23

Fig. 2 .3. Internal structure factors. Κ = (47rno/Ao)sin(0/2) is the scattering vector and dc

represents the characteristic dimension of the molecule. The inside/outside ratio of radii for the

hollow sphere was 1/10. The solid line shows the structure factors for a rigid rod; the dashed line,

for a solid sphere; the dot-dashed line, for a hollow sphere; and the dotted line, for a Gaussian coil.

The initial effect of internal interference is to decrease the intensity of scattered light. Note that is independent of the shape of the particle and easily can be computed from the slope of Hcp/Re versus K

2 [or s in

2(0/2)] . In order to

extract more detail about the shape of the particle, data must be taken over a wider range of angles, and the full expression given by Eq. (2.7.1) must be employed. Since the double summations over the internal indices of the particle are independent, one can write

P{KdC) = \y

where Vp is the volume of the particle and dc is a "characteristic dimension" of

the particle. In the case of a rod, for example, the parameter dc represents the

length L of the rod. The quantity in the { } brackets defines the particle form factor. Expressions for P(Kd

c) for simple geometries are given in Table 2.1 and

plotted in Fig. 2.3.

exp[-iKT.r(t')-]dVp} , (2.7.4)

Example 2.2. Internal Structure Factor for a Hard Sphere The volume element for a sphere is dVp = 4nr

2 dr over the integration limits

a = 0 to b = Rs, where Rs is the radius of the sphere. Changing variables to


χ = Kr, one has

P(KRS) = 3 / 4 π

4 π Ρ | \K* x

2 exp( — ix) dx

3 [s in(KP s) - K P s c o s ( K P s ) ] (2.7.5) P(KRS) =

K3Ri

Example 2.3. Radius of Gyration for Τ 7 DNA Deoxyribonucleic acid (DNA) is a double-stranded molecule whose primary structure (sequence of the purine and pyrimidine bases) determines the genetic code for living systems. The diameter of the double helix is ~ 2 7 Â, and the contour length varies according to source, exceeding one meter in length in some cases. The secondary stucture can be in one of several forms, as inferred from X-ray scattering data and circular dichroism spectra, where the most common form is the "B" form. There is some indication that other secondary structures may be important in gene expression since the flexibility of DNA changes with its secondary structure. The flexibility of the DNA is reflected in its radius of gyration; hence it is important that the data be in the "linear region" before Eq. (2.7.3) can be applied in the analysis. Two of the problems with large molecules, however, are the presence of dust and the reflection from the scattering cell walls when one attempts to go to very low scattering angles. It is for these reasons that earlier studies were limited to angles above 30°. By using a very elongated scattering cell, Harpst (1980) was able to obtain data for θ < 10° for T7 DNA. Representative low-angle scattering data of (Hcp/Re)Cp 0 for T7 DNA in BPES buffer (0.006 M N a 2 H P 0 4 , 0.002 M N a H 2P C > 4 , 0.001 M N a 2 - EDTA, and 0.179 M NaCl, pH 6.8) are shown in Fig. 2.4. Using the values (dns/dcp)TtC> = 0.166 mL/g, n0 = 1.329, and / 0 = 546 nm, calculate the radius of gyration for T7 DNA in BPES buffer.

Solution: According to Eq. (2.7.3), the slope of a plot of Hcp/Re versus sin

2(#/2) is given by \6π

2η\βΜρλ%. Correcting for the multiplicative

factor for the ordinate scale, the estimated values of the slope and intercept of the plot in Fig. 2.4 are 54 χ 10~

7 and 0.34 χ 10"

7,

respectively. The mean-square radius of gyration is

è> 3(slope) ΜΡλ

2

\6π2η

2

2 _(3)(54 χ 10"7)(29 χ 10

6)(546 χ 1 0 "

7)

2

< G> ~ (16)(3.14)

2(1.329)

2

 = 5.02 χ 1 0 "9c m

2,

or a root-mean-square radius of gyration of 7096 Â.

2.8. Light Scattering by Small, Interacting Particles 25

100 -40°

i 80-

I0

7

60-X

cr 40- 20° J

\ Q.

ο X

20-10° !

C 1 1 1

) 0.05 0.10 0.15

s i n2( 0 / 2 )

Fig. 2.4. Zimm plot for T7 DNA. Hcp/Re for the cp = 0 are shown above for the angle range 0° < θ < 50° for T7 DNA. [Reproduced with permission from Harpst (1980). Biophysical Chem-

istry. 11, 295-302 . Copyright 1980 by Elsevier Scientific Publishers.]

2.8. Light Scattering by Small, Interacting Particles

The functional form of Hcp/Re for small [_P{Kdc) = 1] interacting particles is

1

Re {l + [ W ( X A K ) c p / M p ] (2.8.1)

Hcp 1 VsNAS(KAR)cO

where AR is the center-of-mass separation distance and

S ' ( K A R ) ~ - W £ Σ e x p { i Kr. [ R ( a - R ( O e ] } \ .

N*\VÂ=l I (2-8.3) Further simplification is achieved if pairwise interactions are assumed to be the dominant interaction. The difference R(f')p ~ R(î')q i

s expressed as

R ( i ' ) ! — r(f ') , where R ( i,) i locates the center of mass of one of the particles

and r(f ') represents the location of particle 2 from a coordinate system whose origin is at particle 1. Integration of R ( i ' ) i

o v er a^ space gives the volume

Vs. The second integration involves the pair distribution function g{2)

2{r).

VSS'(KAR) = lg?\(r) - l ] e x p ( / Kr . τ)dr. (2.8.4)


Relative to noninteracting particles, repulsive interactions [g(i\(r) < 1] lead

to a decrease in scattered light intensity, whereas attractive interactions lg

(i\(r) > 1] lead to an increase in the scattered light intensity. B 2 for a system

of identical interacting particles is given by

B 2 = 1

[0 (iHW-l]dr. (2.8.5) 0

2

Hence Eq. (2.8.2) becomes, upon defining A 2 = iVAB2/Mp,

H ç E ^ _ L + ^ c E = _L + 2 A ( 2. 8. 6 )

Re M p Ml M p

Example 2.4. Excluded Volume Contribution to B 2

The interaction potential for a system of identical hard spheres is defined for two regions: U(r) = o o for r < 2 P S , hence g

(2)

2(r) = 0; and U(r) = 0 for r > 2 P S , hence g

(i\(r) = 1. Thus B 2 = + 4F S, where Vs is the hard sphere

volume for one particle. Example 2.5. B 2 for Weak, Long-Range Interactions It is assumed that U(r) = (a/r) exp( — br), which is the form of the potential for many types of screened interactions, such as the screened Coulomb interaction and screened hydrodynamic interactions. If the interaction is weak, so that 9

{U2 ~

1 - LU(r)/kTl then

1 m 00

a B 2 = - -J t-aexp(-br)-]4nrdr = ^ . (2.8.7)

Because short-range and long-range interaction potentials dominate over different regions in space, the respective values of the second virial coefficients are additive, giving B 2 = (a/2b

2) + 4VS. If the long-range term is attactive,

then it is possible to have B 2 = 0, which is referred to as the theta condition.

2.9. Light Scattering by Large, Interacting Particles: One-Contact Approximation

The full expression of Eq. (2.5.15) must be used to describe completely the system of large interacting particles. It is not possible, however, to calculate exactly the interactions between all of the segments of different chains at arbirary separation distances. Zimm (1948) considered the special case in which only one segment from each chain is in physical contact, where the distance between the interacting segments is AR(t')Pp Qq = R(f ' ) P +

r(

t')pP ~

R(i ' ) Q - r(t')Qq (cf. Fig. 2.5). By associating one of the internal coordinates with the center-of-mass of the

particle, the sum over the remaining internal coordinates has in the

2.9. Light Scattering by Large, Interacting Particles 27

Fig. 2.5. Single-contact model for flexible coil molecules. In the Zimm theory for interacting

coil molecules, only one subunit of each molecule is assumed to be in physical contact.

exponential argument the difference expression Ar'(t')pq = r(t')Pq — r\t')PO. The resulting expression is similar to the form given for P(Kd

c) [cf. Eq. (2.5.8)].

In a similar manner, one defines r'(t')Qq to recover a second factor of P(Kdc)

for molecule Q. The term < e x p [ / Kr · AR(t')Pq Qq]} is associated with B 2 .

Hence,

Re MpP(Kdc) 1 - l2B2NAP(Kd

c)cp/Mp-]

(2.9.1)

In the dual limits of cp -> 0 and 0 - • 0, one has

He,

Mt

ι 1 6 π2κ

2< Κ

2) . 2

Θ ,

2^ a B 2

+ 3AgM p

Sm 2

+ Ml

(2.9.2)

A plot of Hcp/Re versus /csin2(0/2) + k'cp, where k and k' are an arbitrary

constants, is referred to as a Zimm plot.

Example 2.6. B 2 and R s for Poly(y — benzyl Glutamate) Poly (y-benzyl glutamate) (PBG) is a polypeptide that undergoes a sharp transition from coil to α-helix with a change in solvent or temperature conditions. Schmidt (1984) attempted to determine the structure of PBG in dimethylformate (DMF) from Zimm plots for several molecular weight preparations of PBG (cf. Fig. 2.6). The slope computed from these data is (7.8 - 4.8) χ \0~

6/2 = 1.5 χ 10~

6 mole c m

2/ g with an intercept given by

1/<M>W = 4.8 χ 10"6, or < M > W = 2.1 χ 10

5. From the slope one has

A 2 = (slope)Zc' χ 1000/2 = 2.5 χ 10~4 c m

3 m o l / g

2. We can estimate the hard


0 5 10

K2x l ö ' ° + k * C p / c m "

2

Fig. 2.6. Zimm plot for poly(y-benzyl glutamate). The data are for poly(y-benzyl glutamate) of molecular weight M w = 210,000, with a polydispersity ratio of 1.6 < M w / M n < 2. The value of k' is 0.34 cm/g. [Reproduced with permission from Schmidt (1984). Macromolecules. 17, 553-560. Copyright 1984 by the American Chemical Society.]

sphere radius Rs from B 2 = 2.5 χ 1 ( Γ4 χ (2.1 χ 10

5)

2/ 6 x Ι Ο

2 3 = 1.83 χ

1 0 "1 7 cm

3/part icle . Hence R s~[(3/4π)(1.83 χ 1 ( Γ

1 7) / 4 ]

1 /3 = 103 χ 10"

8 cm.

As a point of reference, Schmidt (1984) reported RH = 175 χ 10~8 cm as

computed from D a p p.

It is emphasized that the "single-contact approximation" is not valid for particles that interact through a long-range potential where "noncontact" multiple interchain interactions are important.

2.10. Solution Structure Factor

The previous discussions separated the "self" and "pair" particle contributions to the total intensity of scattered light, where B 2 and A 2 are directly related to the interference terms of particles in "preferred separation distances" due to l/ 1 > 2(r). In practice, however, the self and pair contributions to the total intensity cannot be separated. The solution structure factor S(K AR) is thus defined by the generalized expression

j ίω

ο 'tils

dns

dc~t Ρ/Τ,μ' M c P(K Ar)S(K AR). (2.10.1) J

A R

Comparison with Eqs. (2.5.6), (2.5.11), and (2.8.3)-(2.8.5), leads to the identity

S(KAR) = 1 + </W>pair

</(R)>self

2.10. Solution Structure Factor 29

where S'(KAR') is defined by Eq. (2.5.12). The ratio lVsS'(KAR')/P(Kdc)]

cannot be further factored since the center-of-mass and internal contributions to S'(K AR) cannot be separated.

Special Case: Single-Contact Flexible Polymer

As shown in Sec. 2.9, the single-contact model for flexible polymers gives S'(KAR') ~ B2P

2(Kd

c). The solution structure factor for this particular

system is therefore

S(KAR) = 1 - 2^B2P(Kdc). (2.10.3)

' s

S(KAR) for multiple-contact systems has been studied for homopolymer mixtures (Benoit and Benmouna, 1984) and copolymers (Benmouna and Benoit, 1983).

Special Case: Spherical Particles

In the case of spherical particles, the double sum involving the differences rPp(t') — rQq(t') can be written as the square of the integral over one index [cf. Eq. (2.7.4) and Example 2.2]. In this case the double sum is expressed as a product of two terms, each of which is independent of the identity and location of the macromolecule. S'(KAR) is now proportional to P(Kd

c) instead of the

square of this quantity. Hence,

S(KAR)= 1 - 2 ^ B 2 (2.10.4)

S(KAR)= 1 + 4 π ^ lg[2Mr)- iy

2 dr (2.10.5)

Example 2.7. S(K AR) for Lithium Dodecyl Sulfate Neutron and light scattering are complementary techniques. The long wavelengths of visible light enable one to probe large distances, whereas the short thermal wavelengths of neutrons (on the order of 5 Â) enable one to probe much shorter distances. The neutrons are scattered by the atomic nuclei instead of the electronic structure of the particles. Since the atomic nuclei differ in their scattering efficiencies, one can tailor the composition of the molecule to the needs of the experimenter. For example, the scattering amplitude of hydrogen is negative ( — 0.3742) whereas that of deuterium is positive ( + 0.6671) (Zaccai and Jacrot, 1983). By varying the H 2 0 / D 2 0 ratio in the solvent, one can enhance the scattering efficiency of specific regions of a molecule.


Bendedouch et al. (1982) used neutron scattering techniques to determine both P(Kd

c) and total intensity ( / t i n s) for spherical particles, viz,

/tins = I(K) ~ P(Kdc)S(K AR), (2.10.6)

where

P(Kdc) = ν χ ( Ρι - p2)P(KR1) + V2(p2 - Po)P(KR2) (2.10.7)

and Vj is the spherical volume of component j having a scattering length density P j and radius R,. [The difference ( P i — P j) is the neutron scattering analogue to the index of refraction increment in light scattering.] In this study, the ionic detergent lithium dodecyl sulfate (LDS) was examined for 0.2 M added LiCl and zero added salt conditions at 37°C. P(Kd

c) was determined

under high salt conditions in D 2 0 , where the interparticle electrostatic interactions were assumed to be screened by the salt. P(Kd

c) was assumed not

to change with a decrease in the salt concentration. Neutron scattering measurements are shown in Fig. 2.7.

P(Kdc) for LDS resembles that for the spherical objects shown in Fig. 2.3.

These authors reported that the two-shell model with the parameters Ri = 14.5 Â, R2 = 23 Â, P l = - 0 . 6 χ 1 0 "

6 Â~

2, p2 = 5.0 χ 10~

6 Â "

2, and p0 =

6.34 χ 10~6 Â

- 2 gave a good fit to the experimental curve. The value of R2

was about 2 Â smaller than the hydrodynamic radius, thus indicating a shell of bound solvent. S(KAR) exhibits a maximum at Κ ~ 0.078 Â

- 1, corre-

sponding to an interparticle distance of AR = 2π/Κ ~ 80 Â

3 0 Q

ω 2 0 0 >

σ ω 100

S(KAR)

1.0

ι ι ι ι ι ι ι ι I ι ι ι ι ι ι ι 0.10

ι ι ι ι ι ι ι ι ι I I I I I I I I

0 0.10

Α Κ ( Α )

Β Fig. 2.7. Neutron scattering studies on lithium dodecylsulfate: (A) the particle structure factor

P(Kdc) (open circles) and total intensity Itins(K) (solid circles) determined from neutron scattering

measurements; (B) the calculated solution structure factor S(K AR) for lithium dodecyl sulfate (LDS) in D20 at 37°C. The concentration of the LDS was 4 % (g/dL). [Reproduced with permission from Bendedouch et al. (1982). J. Chem. Phys. 76, 5022-5026. Copyright 1982 by the American Institute of Physics.]

2.12. Electric Field and Intensity Correlat ion Functions 3 1

PARTIII D Y N A M I C L I G H T SCATTERING

2.11. Time-Dependent Total Intensity

The intensity of scattered light is dependent upon the spatial arrangement of the scattering centers at any instant in time. The macromolecules, however, are undergoing constant motion because of collisions with the solvent molecules. The instantaneous value of 7 t i ls therefore fluctuates in time about the average intensity, where the rate at which these spontaneous fluctuations decay to the equilibrium value is directly dependent upon the dynamics of the molecules. Information regarding the time dependence for a set of values {x(t)} is contained in the autocorrelation function C(t), defined as

%T/2

x(t')x(t' + t)dt' = <x(0)x(i)>. ( 2 . 1 1 . 1 ) -T/2

C(i) = lim 1 Γ - ο ο

1

2.12. Electric Field and Intensity Correlation Functions

The photomultiplier tube (PMT) is a square-wave detector that generates a current J(t) whenever light falls on its photocathode surface, where the illuminated surface has an area A. The correlation function <J(0)J(i)> is

<J(0)J(i)> - < J2> + <(SJ(0)<5J(i)>, ( 2 . 1 2 . 1 )

where ôJ(t) is the fluctuation about the average value <J> in the phototube current at time t. The instantaneous value of J(t) relayed to the autocorrelator is the sum of all of the events that occur on the photocathode surface. The quality of the function < J(0) J(i)> is therefore dependent upon the number of "coherence areas" on the photocathode surface. A measure of the coherence area Acoher is given by the expression v 4 c o h er = λ^/Ω, where Ω is the solid angle subtended by the source and the detector. The number of coherence areas is NCoher ~ A/Acoher. If the current generated in region Ax is not correlated with the current generated in region A2, then the pairwise product for these two events does not contribute to (ôJ(0)ôJ(t)}. According to Eqs. ( 2 . 3 . 6 ) , ( 2 . 3 . 7 ) ,

( 2 . 4 . 1 ) , ( 2 . 4 . 2 ) , and ( 2 . 1 1 . 1 ) , the electric field correlation function is

<E s*(R ,0)r - E s(R,i)> = Β<α*(Κ ,0)

τ · α(Κ,r)>exp(- i<M), ( 2 . 1 2 . 2 )

where Β is a constant. The normalized (to unity at t = 0) electric field correlation function is denoted by g

{l)(t).

Since the average value of the intensity is not zero, the intensity correlation function has two terms (the scattering by the solvent is assumed to be negligible):

< / ( 0 ) / ( t ) > = </(0)2> + β

2< | Ε * ( 0 )

Γ . Ε( ί ) | |Ε*( ί )

Γ · E ( 0 ) | > , ( 2 . 1 2 . 3 )


C(t)

04 , * Tt

Fig. 2.8. Effect of the coherence area on the correlation function: ( ) one coherence area; ( ) several coherence areas.

hence the general form

C(t) = </(0)/(i)>/</(0)>2 = 1 + f(Ncoher)g^(t). (2.12.4)

The parameter f(Ncoher) < 1 is a constant that depends upon the number of coherence areas that generate the signal. For a Gaussian distribution of the intensity profile of the scattered light, one has the relationship

9{2\t) = \g

{1)(t)\

2. (2.12.5)

The effect of coherence areas is shown in Fig. 2.8.

2.13. Center-of-Mass Diffusion

It is assumed that ACp(r, i) obeys the macroscopic diffusion equation (Fick's second law of diffusion),

dàCp(r,t)_n „ 2 et

= DmW2 A C P ( M ) , (2.13.1)

where C p is in moles per liter, Dm is the mutual diffusion coefficient, and V2 is

the Laplacian operator. According to Eq. (2.3.7), the concentration that is needed is not in r space but rather in Κ space. Substituting

/ 1 \ 3 / 2 C+oo

ACp(r,i) = i — J I ACÔexpi- iK^.r)^3^ (2.13.2)

into Eq. (2.13.1) and solving the rate expression gives

ACp(K, t) = AC p(K, 0) exp( -DmK 2

t). (2.13.3)

2.14. Effect of Ratio C p / K3 on Dm 33

The molecular correlation function [G^K, r)] for center-of-mass motion is therefore a first-order decay process,

^ ( Κ , ί ) = <AC p(K,0)*AC p(K,0)>exp(-Z) mX2i ) . (2.13.4)

Hence g(1)(t) = g

(l)(K,t) = e x p ( - D m K

2i ) .

In general, Dm computed from DLS data is dependent upon Κ because of interparticle interactions. In the limit Κ - • 0, Dm is related to the mutual friction factor fm and the osmotic susceptibility (dn/dcp)T ' = (1000/M p) (dn/dCp)Tß>, where μ' denotes constant chemical potential for all species except p, by the generalized Stokes-Einstein relationship,

In the limit of infinite dilution, (dn/dCp)Tifl> NAkT = RT.

2.14. Effect of Ratio C p / K3 on D m

One of the early concerns of dynamic light scattering was the identification of the "scattering volume". If the scattering volume Vs is defined as 1 /K

3, then the

magnitudes of the concentration fluctuations are relatively large. Questions may then be raised about the mathematical foundations of dynamic light scattering, such as the validity of Fourier analysis of the concentration fluctuations and linear tansport theory.

Oh and Johnson (1981) examined the Κ dependence of Dm for bovine serum albumin (BSA) over a series of concentrations, i.e., 2 < Cp/K

3 < 27,000. BSA

has a molecular weight of 69,000 and an equivalent hydrodynamic shape of a prolate ellipsoid with an axial ration of ~ 3 / l . Because of its variety of ionizable groups (carboxyl, imidazole, ε-amino, and phenolic), the charge on BSA can vary from + 5 to —55 over the pH range 4.3 to 10.5. Commercial preparations of BSA are contaminated with dimers and trimers, however. The BSA samples used by Oh and Johnson were first purified by gel filtration on a Sephacryl S-200 Superfine column to obtain a monodisperse sample of monomers. The sulfhydryl groups were blocked to prevent self-aggregation. The pH was adjusted to the isoelectric point (pH 4.7) by the addition of N a O H . Two lasers were used to expand the range of X-values, where λ0 = 632.8 nm ( H e - N e laser) and λ0 = 457.9 nm (Ar ion laser). Data were collected at 20°C in a buffered solution of ionic strength 0.12 M. Plots of Dm as a function of Κ are shown in Fig. 2.9.

It can be concluded from these data that Dm is independent of the scattering vector Κ over the range 2 < Cp/K

3 < 27,000. These authors also reported that

Dm dependence upon the concentration is

(2.13.5)

Dm(20°C) = D p (1 - 0.0166c'p), (2.14.1)


CVJ

Ο

ο Ε

5.5H

5.0H { I n * { 4

Ε 4.5

0 5 10 15 2 0 25 30 35 4 0

KxlO"4 (cm"')

Fig. 2.9. The wave vector dependence of Dm for bovine serum albumin. The experimental conditions for the data shown are: c'p = 1.00 g /dL at λ0 = 457.9 nm ( • ) ; c'p = 1.02 g /dL at λ0 = 632.8 nm (O); and cj, = 6.74 g /dL at Λ.0 = 632.8 nm (Δ ) . [Reproduced with permission from Oh and Johnson (1980). J. Chem. Phys. 74, 2717-2720. Copyright 1980 by the American Institute of Physics.]

where Dp = 5.92 χ 10"7 c m

2/ s is the infinite dilution value of Dm and

1.0 < c p < 10.7 is in g/dL. Another conclusion that can be drawn from these studies is that the scattering volume is defined by the illuminated region of the solution and not by 1/X

3. It is also concluded that Dm for a system of

noninteracting particles is independent of K.

2.15. Osmotic Susceptibility Correction

The tracer diffusion coefficient, D T r, is defined in the absence of a concen-tration gradient and is related to Dm through the osmotic susceptibility by (DJDjr) = (l/NAkT)(dn/dCp)Ttfl. (cf. Sec. 6.5). Hall and Johnson (1980) examined this relationship for hemoglobin (Hb) and met-Hb. Photon correlation spectroscopy (PCS) was used to obtain Dm for Hb, and the osmotic susceptibility was computed from

where y is the activity coefficient. Eq. (2.15.1) was obtained from viscosity and sedimentation data by Minton and Ross (1978), where the partial specific volume is vp = 0.0092 dL/g. The values of Dm and DTr as a function of c'p are shown in Fig. 2.10 along with D X r data of Keller et al. (1971).

d\n(c'p)

1 + lvpc'p + 22(vpc'p)2 + 43.45(z7pCp)

3

+ 67 .74 (^Cp)4 + 95.57(i7pCp)

5 + 158.52(r pc p)

6 (2.15.1)

2.16. Dynamic Light Scattering by Absorbing Molecules 35

8

H 'δ 8-

0 4 8 12 16 20 24 28 32 36 Cp (g/dL)

Fig. 2.10. Mutual and tracer diffusion coefficients for hemoglobin: ( • ) obtained from PCS

data; (O) obtained from the d iaphragm method in the absence of a concentration gradient; ( χ )

obtained from the d iaphragm method in the presence of a concentration gradient; (solid line)

computed by Eqs. (2.13.5) and (2.15.1) using the PCS data. [Reproduced with permission from

Hall and Johnson (1980). J. Chem. Phys. 72, 4251-4253. Copyright 1980 by the American

Institute of Physics.]

It can be concluded from these data that Dm φ D T r, and the data also appear to support Eq. (2.13.5). Hence PCS is a reliable technique for determining diffusion coefficients and thus for studying the solution properties of macromolecules.

2.16. Dynamic Light Scattering by Absorbing Molecules

The effects of light absorption and multiple scattering must be considered if dynamic light scattering is to be of general applicability in studies of macromolecular systems.

Absorption of light by the scatterer would lead to local heating of the scattering volume; hence Dm would be affected through the ratio Τ/η0. Following Hall et al. (1980), we assume cylindrical symmetry along the direction of propagation (cf. Fig. 2.1), whereby the incident intensity of light is of the form

where w is the beam waist (\/e2 radius) and μ 3 is the absorption coefficient for

the medium. To briefly outline the development, the rise in temperature (AT) is first estimated from the expression V

2T = — μΆΙ(Γ,ζ)/κτ, where κτ is the

Γ-2Ι-Η ε χ ρ ( - μ 3 ζ ) , (2.16.1) I(r, z) = I0 exp


thermal conductivity. The rise in temperature AT at radial distance a is then

where P0 = (πνν 2/2)/ 0 is the total power and β = 1.781. Using the viscosity data for pure water to obtain Dm(T) as a linear function of AT (from 293°K) and then assuming an intensity average diffusion coefficient, the following expression was obtained:

where Β = 0.029° AT1, f(a/w) = \n{a/w) + 0.29168, zx and z 2 are the limits of the scattering volume along the beam, and

In the limit of small values of μ 3ζ, one can use the average distance <z> = (zj + z 2) /2 , for which Q = exp( — μ 3 < ζ » .

The experiments were performed with a H e - N e laser and a dye laser. Both single-beam (dye laser only) and dual-beam experiments were carried out in the studies on hemoglobin. The dual-beam experiment was performed on systems with high absorbances ( > 2 ) and low concentrations (<1 .5 g/dL), conditions that resulted in low signal levels. In the dual-beam experiment, the H e - N e laser was used for the probe beam, and the detection was controlled by having vertical polarization for the H e - N e beam and horizontal polarization for the dye laser. The solutions were made in a phosphate buffer (0.044 M phosphate, pH 6.7). The index of refraction for the hemoglobin systems was found to be n(Ub system) = 1.334 + 0.00197c p, where c'p is given in g/dL.

According to Eq. (2.16.3), Dm is a linear function of the power of incident light. This prediction is verified by the C O - H b data in Fig. 2.11. The "best fit" value of κτ for the 0.044 M phosphate buffer was 7 m W/(cm-°K).

Hall et al. (1980) also considered multiple scattering contributions to the correlation function. Multiple scattering has the effect of introducing faster components to the decay process. After a detailed mathematical analysis, they concluded that the angle dependence of the apparent diffusion constant was a sensitive test for multiple scattering.

From their studies on oxyhemoglobin, CO-hemoglobin, and unliganded hemoglobin, Hall et al. concluded that: (1) convective motion and thermal lensing effects were not significant, whereas temperature increases were significant; (2) accurate values of Dm can be obtained for absorbing molecules by linear extrapolation to F 0 = 0; and (3) multiple scattering is negligible for these systems (up to 15 g/dL). (Multiple scattering is again addressed in Chapter 4.)

(2.16.2)

(2.16.3)

ß(*l,*2) = e x p ( - ^ a z 1 ) + e x p ( ^ a z 2

2 (2.16.4)

2.17. Evaluation of <AC P(K,0)*AC P(K,0)> 37

3 0 -

2 9 -ω

\ ΓΟ

Ο 28-

L- 27-

26-

I 1 1 1 —

0 10 20 30 P 0( m w )

Fig. 2.11. The decay constant as a function of laser power for the CO-Hb system. [Reproduced

with permission from Hall et al. (1980). J. Phys. Chem. 84, 756-767 . Copyright 1980 by the

American Chemical Society.]

2.17. Evaluation of <ACp(K, 0)*ACp(K, 0 »

It is assumed that the relative probability, P R, of observing a concentration fluctuation ACp(r,0) away from the uniform concentration < C p> u = Np/VSNA, which is also assumed to be the equilibrium concentration, is given by

P[AC p( r ,0)] •AG[ACp(r,0)]

kT (2.17.1)

where AG[AC p(r ,0)] is the free energy associated with the fluctuation. If the particles are noninteracting (AH = 0), then AG[AC p(r ,0)] = — TAS[AC p( r ,0 ) ] . The solution is divided into unit cells of volume ΔΚ, with cell concentrations

Cp(r,0) ( = <n,)u + δηι

AVNA, (2.17.2)

where <«,>„ is the uniform (equilibrium) number of particles in the volume and <5n, is the number fluctuation. The relative probability P R is then

n1!n2!n3!. <ni> u!<n2>u!<«3>u! .

(2.17.3)

Expressing Stirling's approximation for ln(PR) as a series expansion to the first two terms gives the Gaussian form,

Pr = exp -ΣΟΚ)

i

2<n(>u

-l(àCi)2AV

= exp 2C n

(2.17.4)

with <C P> U = C p . If one now substitutes an integral for the summation and (5Cp(r,0) for <5C;, and then expresses <5Cp(r,0) as the Fourier transform of

38 2. BASIC CONCEPTS O F LIGHT SCATTERING

(5CP(K,0), one obtains

PR = A exp" 2VCP

£ [ < 5 C P( K , 0 ) ]2] . (2.17.5)

Since the Κ modes are independent, the amplitude is simply

<<5Cp(K,0)*<SCp(K,0)> = VSCP. (2.17.6)

Summary

Light is scattered at angles other than the forward directions by a medium because of fluctuations in the polarizability of the medium. In the case of a solution of macromolecules, the information that one can obtain from the total intensity, or integrated intensity, is the (weight average) molecular weight, the radius of gyration of the molecule and internal spatial arrange-ment of the scattering centers, and the virial coefficients. Since the molecules are undergoing Brownian motion, the instantaneous intensity of scattered light varies with time. From the temporal behavior of the total intensity, as characterized by the autocorrelation function, one can obtain information about the mutual diffusion coefficient, and hence the equivalent hydro-dynamic radius. Studies directed to the effect of concentration, heating by absorbing molecules, and multiple scattering verified the reliability of this technique as a rapid means of studying the dynamics of macromolecules.

Problems

2.1. If E(t) = E0cos((Dt), solve the equation of motion for an electron of mass rae and charge e,

d2Y dY

me ^ 2 - + 7^T + /cY= -eE(t)9

where γ and k are damping and restoring force constants.

2.2. The Langevin equation for a particle of mass m p in the presence of a random fluctuating force F(t) is

dvjt)

where i;p(r) is the instantaneous velocity of the particle and ξρ is the Newtonian friction coefficient. Find an expression for vp(t + τ). [Hint: first multiply the equation by exp(i^ p/m p) . ]

2.3. Using the result in Prob. 2.2, find an expression for j " ' 0 <ι;ρ(0)ΐ;ρ(τ)> dr. What conclusions can be drawn about the motion of the particle in the time limits t —• 0 and t —• oo?

Problems

2.4. Deoxyribonucleic acid (DNA) can exist in both linear and circular forms. Assume that you have at your disposal a total intensity light scattering facility [ H e - N e laser (λ0 = 632.8 nm) with 10° < θ < 170°]. You have a dilute solution of circular D N A having a molecular weight of 132,000 Daltons, where the solvent conditions are identical to those used by Harpst (cf. Example 2.3). At what value of θ would you expect to see a 5% change in 7 t i ls for the circle rod transition? Assume 660 Daltons per base pair and 3.4 Â between base pairs for each conformation. [Hint: use Eq. (2.7.3).]

2.5. Assume conditions similar to those in Prob. 2.4, except that you have a tunable laser with the wavelengths λ0 — 752.5, 647.1, and 520.8 nm, and θ = 90° is fixed. What are the percentage changes in intensity at each λ0Ί

2.6. The nucleosome is the repeat unit structure of chromatin, in which 200 base pairs of DNA are wrapped about a histone aggregate composed of two copies each of the histones H2A, H 2 B , H3, and H4. The hydrodynamic radius of the nucleosome is ~ 55 Â. It is assumed that the nucleosome unfolds from the compact spherical structure to a rodlike structure of length equivalent to that of the stretched-out In-form D N A contained within the nucleosome. Assuming that you have an argon ion laser (λ0 = 488 nm), what is the percentage change in the light scattering intensity expected at θ = 90°? The weight ratio his tone/DNA is —1.07 (Olins et al., 1976). The index of refraction increment is (dnjdcp) = 0.184 m L / g (Harrington, 1980), and the index of refraction for the solution is assumed to be 1.333. (Hint: to compute M p , use the information given in Prob. 2.4 for the D N A component.)

2.7. At what concentration of hemoglobin will Dm differ by 10% of Om(Cp - • 0) = Dp [Cf. Eqs. (2.13.5) and (2.15.1).]

2.8. Compute the contribution to D m of hemoglobin (cp = 20 g/dL) in 0.044 M phosphate due to heating using the results of Hall, Oh, and Johnson (1980) [cf. Eq. (2.16.3)]. Assume P0 = 1 0 0 m W a t A 0 = 540 nm with a beam waist w = 36 μιη, a = 0.5 cm, and the average path length ζ = 1 cm. The extinction coefficient [ l o g 1 0( / / 7 0) 5 4o ] is 0.66 L/g.

2.9. Estimate the coherence area ^ l c o h er (cf. Sec. 2.12) for a light scattering facility in which a pinhole of diameter 0.5 mm is placed 20 cm from the sample with R = 100 cm.

2.10. Bantle et al. (1982) studied polystyrene latex spheres in toluene (fi0 = 1.496) using both static and dynamic light scattering methods (A0 = 647.1 nm). From \/P{Kd

c) vs. K

2 in Fig. 2.12, calculate 

1 /2

and the hard sphere radius Rs.


P(Kdc

i.QL

π r 2 3

κ2 χ i o - ' ° U2;

Fig. 2.12. Κ2 dependence of the inverse particle form factor for poly (sty rene) latex spheres.

[Reproduced with permission from Bantle et al. (1982). Macromolecules. 15, 1604-1609.

Copyright 1982 by the American Chemical Society.]

2.11. Hall, Oh, and Johnson (1980) obtained the value /)£ = 6.75 χ 1 0 "

7 c m

2/ s at 20°C. What is RH of hemoglobin?

2.12. Hall, Oh, and Johnson (1980) obtained the relationship £>m(20°C) = - 0.0064cp), where c p is in g /dL and Dp = 6.80 χ 10~

7 c m

2/ s ,

for oxygenated Hb, and Dm = - 0.00568c p) for Hb, where D° is given in Prb. 2.11. Compute c p values that given identical values for Dm

for these two forms of Hb.

2.13. / t i ls measurements were made on biopolymer A using an argon ion laser. The index of refraction increment at 488 nm was measured to be (dnJdcp)Tß> = 0.192 c m

3/ g in the aqueous medium (n0 = 1.333) The

following table of Rayleigh ratios (Re χ 108 c m

- 1) was determined:

Values of Re for Hypothetical Biopolymer A

c'P(g/dL)

θ 0.5 1.0 1.5 2.0 2.5

10 6.65 7.04 7.21 7.30 7.35

25 3.91 5.16 5.77 6.14 6.38

40 2.27 3.49 4.26 4.78 5.16

55 1.44 2.42 3.13 3.67 4.09

70 1.00 1.77 2.37 2.86 3.27

Calculate the molecular weight, the radius of gyration, and the second virial coefficients A 2 and B 2 for biopolymer A. Give possible interpre-tations as to the origin of A 2 and B 2 .

2.14. Patterson and Jamieson (1985) employed a H e - N e laser (Λ0 = 632.8 nm) to examine the static properties of poly(acrylamide) (PAAm)

Problems 41

0 I 2

S i n2( 0 / 2 ) + 2 0 0 0 C p ( m g / m l _ )

Fig. 2.13. Zimm plot of poly(acrylamide). [Reproduced with permission from Patterson and

Jamieson (1985). Macromolecules, (1985), 18 ,266-272 . Copyright 1985 by the American Chemical

Society.]

at 20°C in 0.1 M NaCl. The index of refraction increment at constant chemical potential was measured to be (dnJdcp)T μ. = 0.170 m L / g at λ = 546 nm. A representative Zimm plot is given in Fig. 2.13. Assuming n0 = 1.333, calculate M p , (Rq}

1/2, and B 2 for this preparation.

2.15. Patterson and Jamieson (1985) also used QELS methods in their study of the PAAm system (cf. Prob. 2.14), with the result D{J(20°C) = 2.50 picoficks (1 fick = 1 m

2/ s ) . Calculate the apparent hydrodynamic

radius (RH). From the results in Prob. 2.14, calculate the ratio RH/RG. How does this value compare to that expected for a sphere? What conclusions can be drawn from this comparison?

2.16. Cohen et al. (1976) reported DLS studies on beef liver glutamate dehydrogenase (GDH) at 25°C in 0.2 M sodium phosphate at pH 7.0. D a pp was found to be a strong nonlinear function of the G D H con-centration, indicating extensive aggregation. The infinite dilution value was reported to be D£ D H(25°C) = 4.4 χ 1 0 "

7 c m

2/ s . Calculate RH for

this particle.

2.17. Pullulan is an extracellular polysaccharide. Kato et al. (1984) examined the M p dependence of D a p p, < # g >

1 / 2, and [rç] of dilute aqueous

solutions of pullulan at 25°C. Sample P800 was reported to have the following characteristics: DJJ(25°C) = 9.20 χ 1 0 "

8 c m

2/ s , and

< # g >1 / 2

= 37.5 nm. Calculate the hydrodynamic radius and the ratio RH/(Rl}

1/2. Why might this ratio differ from unity?

2.18. Photon correlation spectrocopy counts the number of photons n(t) arriving at the photocathode surface during a predetermined time


interval. The correlation function is

<n(i)n(0)> = <1δη(ή + <π>][<5π(0) + <«>]> - <δη(ήδη(0)> + <n>2,

where (5n(£) is the number fluctuation about the average <n> at time t. Assuming a Poisson distribution for n [P n = «n>"/w!)exp( —<n»] , show that <w(f)w(0)>/</?>

2 decays from a value of 2 at ί = 0 to 1 at

t = oo.

Additional Reading

Berne, B. J., and Pecora, R. (1976). Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics. Wiley, New York.

Chu, Β. (1974). Laser Light Scattering. Academic Press, New York. (Soon to be revised.) Dahneke, Β. E. (1983). Measurement of Suspended Particles by Quasi-elastic Light Scattering. John

Wiley, New York.

Huglin, M. B. (1972). Light Scattering from Polymer Solutions. Academic Press, New York.

Pecora, R. (1985). Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy. Plenum, New York.

CHAPTER 3

Translational Diffusion—Hydrodynamic

Dissipation

"So quiet little Martin William Seagull, startled to get caught under his instructor's fire, surprised himself and became a wizard of low speeds. In

the lightest breeze he could curve his feathers to lift himself without a single flap of wing from sand to cloud and down again."

From Jonathan Livingston Seagull by Richard Bach (1936- )

3.0. Introduction

Martin William Seagull was able to change his motion through the air by simply altering the position of his feathers relative to the air movement in a light breeze. In a similar manner, the shape of a molecule influences the manner in which the molecule moves through the solution.

3.1. Macroscopic Description of Mass Transport

One way to describe any transport process is by the force (X) and flux (J) relationship,

J = L X , (3.1.1)

where Jß = Cß\ß and the elements of L are the coupling coefficients. The energy dissipitation function, To, (units of energy/volume-second) is given by

Ta = XT.J= Σ Σ

χΜ · ί

3·

1·

2)

4 3

44 3. T R A N S L A T I O N A L D I F F U S I O N — H Y D R O D Y N A M I C D I S S I P A T I O N

The Ν components are not independent, however, but are coupled by the constraint of no net flow,

Σ ÇXf = 0, (3.1.3)

i - 1 and the G i b b s - D u h e m expression,

£ Ci(\pi)T,P = 0. (3.1.4)

Ί = Ι Solving Eq. (3.1.3) for J J · X 0 = C0VJ · X 0 , To becomes,

Τσ = Y C,.(V,. - V0)R · Χ,· = Y(J?Y • Xf (3.1.5)

i = 1 i = 1 where J [

e l is the flux of component i relative to the solvent, and X[

e l is the

conjugate force in the solvent-fixed reference frame. An important result of Eq. (3.1.5) is that energy dissipation only occurs when the solute particles move relative to those of the solvent. Comparison of Eqs. (3.1.3), (3.1.4), and (3.1.5) leads to the identity

ΧΓ' = -(V^r.P. (3.1.6)

Observations are made, however, in the laboratory-fixed reference frame. We follow the arguments of Phillies (1974) for the transformation from the solvent-fixed to the laboratory-fixed reference frames in an incompressible fluid. The Galilean transform for the polymer and solvent fluxes, J p and J 0 , respectively, are J p

a b = J

r

p

el + CPV

REL and 3x$h = C0V

REL, where VREL is the

relative velocity of the solvent. Multiplying these expressions by φρ and φ0, respectively, and then adding the two expressions, one obtains the equality VREL = -φρ3ρ

]. Hence,

J p

a b = Jf(l - φρ). (3.1.7)

Since the energy dissipated is independent of the reference frame, To satisfies the relationships

Τσ = (Jr

p

eY - X p

e l = ( J p

a b)

r - X p

a b. (3.1.8)

It necessarily follows from Eqs. (3.1.7) and (3.1.8) that

ylab _ X

pe l _ ( Ρ)Τ,Ρ ] Q\

which is precisely the thermodynamic driving force employed by Batchelor (1976). We can also arrive at Eq. (3.1.9) using the chemical potential gradient,

3.2. Dm and the Osmotic Susceptibility 45

— ( V / i ) r p, as the driving force for both the solute and the solvent. Thus,

To = ( J p

a b) r . (-ΝΜΡ)Γ,Ρ + (J™)T · (-ΝΜ0)τ,Ρ· (3.1.10)

From Eq. (3.1.4), ( V ^ 0 ) T P = — (Cp/C0)(\ßp)TP and the incompressible fluid

condition J p

a b^ p + Jo bô = 0>

o n e finds Jo b = _(^P/ô)Jp

ab> where i p and

y 0 are the partial specific volumes of the particle and solvent, respectively.

Noting that 1 + Cpvp/C0v0 = (C0v0 + Cpvp)/C0V0 = 1/φ0 = 1/(1 - φρ\ one

obtains from Eq. (3.1.10)

Τ Σ = (3ρ*Ύ'(-ΝΜΡ)τ,Ρ ( 31 n )

1 - Φρ The conjugate force in Eq. (3.1.11) is therefore identical to the conjugate force defined by Eq. (3.1.9). As pointed out by Batchelor (1976), there is no thermodynamic driving force on the solvent particles if the driving force acting of the solute particles is defined by Eq. (3.1.9).

Following Batchelor (1976), X p

a b is adopted in this text as the correct

thermodynamic driving force, viz,

= fmVp (3.1.12)

where fm is the mutual friction factor. The coupling coefficient L p p is determined from the relationships

JjTb = C pv p = L p pXjT

b = L p p/ m v p , (3.1.13)

hence L p p = Cp/fm. J p

b for an incompressible fluid is

J , a b = ^ ( V ^ P fm 1 - Φρ

3.2. D m and the Osmotic Susceptibility

The gradient in the chemical potential can be expressed in terms of the osmotic susceptibility, (dn/dCp)TP:

Noting that for an incompressible fluid (dn/dCp)TiP/(l — φρ) = (dn/dCp)T >, where Μ' indicates constant chemical potential for all other species, one can express the flux J p

b as


which, compared with Fick's first law of diffusion,

J p

a b= - D m ( V C p ) r , P , (3.2.3)

yields an expressions for D m for an incompressible fluid:

Dm = -J— (^Λ (laboratory frame). (3.2.4)

For a compressible fluid with isothermal compressibility κτ, Dm is (Allison et al., 1979; Schurr, 1982):

1 f du

ο . . W./-YYR„. ( 3 2 5 )

3.3. Effect of External Field—Sedimentation

In calculations involving external fields, it is convenient to express the chemical potential as

μ = μ° + με - Mex> (3.3.1)

where pc = kT ln(C) is the concentration component of the total chemical potential, and μ ε χ represents the alteration of the chemical potential due to the external field. In a rotating frame with angular velocity ω for a particle of mass m p, μ ε χ at the radial distance r is given by,

where the density increment is (dps/dcp)Tμ, and ps is the density of the solution. The flux J p is now a scalar quantity along the radial direction of rotation:

p f J m

— kT dCO (DPS

C p dr P

V ^ p / r , M '

(3.3.3)

where fm is the same as that in Eqs. (3.2.4) and (3.2.5). Assuming dCp/dr ~ 0 and writing v r a d i al = dr/dt, one has

^ = °τ = ψ(Ρ) (3-3.4) ω ν fm \dcpJTtli,

In honor of Theo Svedberg, the inventor of the analytical ultracentrifuge, the

svedberg is defined as 1 χ 1 ( Γ1 3 s = S.

3.4. Friction Factors Associated with D Tr and Dm 47

3.4. Friction Factors Associated with D X r and Dm

The Langevin equation for the isolated particle i is

^ + ^ v , . ( 0 = fs(r), (3.4.1)

dt m,

where f s(i) represents a fluctuating force exerted on the particle by random

collisions with the solvent molecules, vf(f) is the particle's instantaneous velocity, and / is a friction coefficient whose identity is to be determined.

The tracer friction factor is defined in terms of a "self" velocity correlation, where Eq. (3.4.1) is multiplied on the left by v t(0)

T:

v , (0)r · ^ + - v , (0)

r · ν,(ί) = ν,·(0)

Γ · P(t), (3.4.2)

at τ

and τ = mjf is the relaxation time for the process. If fs(t) is independent of

v,.(0), then <v,(0)T · f

s(t)> = 0 and

<ν,·(0)Γ · ν,.(ί)> = <ν ((0)

Γ · v((0)> e x p ( ^ ) . (3.4.3)

We now integrate over time and, using the equipartition theorem

<v,(0)T · v,(0)> = 3kT/mh we obtain the expression

. , 3/CTT <Vi(0)T>v,(t)>dt =

m, 1 — exp(

The tracer friction factor, / T r, is defined for t - • oo, hence

3kT / T .

(3.4.4)

(3.4.5)

(vtiOf.yMydt

The mutual friction factor fm reflects the influence of the other particles on the motion (velocity) of the "probe" particle, denoted by the subscript i. If the effect is assumed to be pairwise, then Eq. (3.4.1) is first multiplied by the sum of the velocities of all the other particles, viz, ]Γ ν,·. Since the averages for all pairs of velocities, (yf · v ,* , ) , are identical, fm is defined by

Ai reo / 3 4 ß\ [ < V ! ( 0 )

r . V l( i )> + ( % - L ) < V l( 0 )

r - v 2 ( I ) > ] dt. K ' ' }

Since friction interaction with the solvent occurs only on the exposed macromolecular surface, its magnitude depends on the shape of the molecule. Molecular shapes considered in this chapter can be divided into three


RANDOM COIL

Fig. 3.1. Hydrodynamic modeling shapes for macroparticles. Regular solids (sphere, cylinder,

prolate and oblate ellipsoids of revolution). Irregular solids (subunit spherical bead model for

solenoid, open helix, and random coil). Flexible polymers (linear, circular, and branched coils;

supercoil circles; star and trifunctional branched coils).

categories: (1) regular solids: (2) irregular rigid structures; and (3) flexible polymers (cf. Fig. 3.1).

Example 3.1. fm and fTr for Poly (ethylene oxide) Brown et al. (1983) employed quasi-elastic and static light scattering, sedimentation velocity, and Fourier t ransform-pulsed field gradient nuclear spin-echo techniques to study aqueous solutions of poly(ethylene oxide) (PEO) at 25°C. The QELS and sedimentation data provided independent measurements of fm. The spin-echo technique provided a value for the tracer, or self, friction factor fTr. The molecular weight range of P E O used in these studies was 73,000 < M w < 661,000. The ratio M w / M n was less than 1.05 for the lower-molecular-weight preparations, but was 1.10 for the highest-molecular-weight sample. Values of fm were computed from s 2 5, w using 1

— VpPo = (dps/dcp)Tß>, where ϋρ = 0.833 mL/g for P E O was determined

in their laboratory. QELS data were used to compute fm from Eq. (3.2.4), where the osmotic susceptibility was determined from static light scattering measurements. The data are presented in Fig. 3.2 as f°/f.

3.5. Determinat ion of the Molecular Weight 49

1.0

F 0 . 5

\

V +

0 5 10 15

C p ( k g / m3

)

Fig. 3.2. Mutual and tracer friction factors for poly (ethylene oxide): based on ( · ) N M R spin

echo, (O) sedimentation velocity, and ( + ) photon correlation spectroscopy. [Reproduced with

permission from Brown et al. (1983). J. Poly. Sei. 21, 1029-1039. Copyright 1983 by John Wiley

and Sons.]

These data clearly show that fm obtained from sT is identical to fm obtained from QELS data. There is a clear discrepancy between fm and fTr at finite concentrations of PEO. Altenberger and Tirrell (1984) suggested that the cross terms in Eq. (3.4.6) are the origin of these differences in values for fTr and fm.

3.5. Determination of the Molecular Weight

It is recognized that sT and Dm must be determined under the same solvent conditions. If this were not the case, then the Svedberg expression [Eq. (3.3.4)] cannot be expected to provide reliable values for M p .

Example 3.2. Hexameric Phosphofructokinase Phosphofructokinase (PFK) is an allosteric enzyme that plays an important role in the glycolytic pathway. In the presence of M g

2 +, this enzyme transfers

a phosphate group from adenosine triphosphate (ATP) to D-fructose-6-phosphate to form D-fructose-l,6-diphosphate. The enzyme is activated for low values of the ratio [ A T P ] / [ A M P ] and inhibited by high levels of citrate. The enzyme therefore is most active when the cell requires energy or building blocks for synthesis. Depending on the source, the enzyme in the active form may be composed of four, six, or eight identical subunits with one catalytic site per subunit. The molecular weight of the subunit is 80,000 Daltons.


Paradies and Vettermann (1979) used sedimentation, QELS, and viscome-try methods to study the hydrodynamic properties of P F K obtained from Dunaliella salina, a marine green alga. By crosslinking with dimethyl suber-imidate, these authors were able to purify a hexameric form of PFK. The concentration dependence of s 2 0, w

a nd Dm(20°C) was determined and for

CP -> 0, s 5 0 tW = 14.7S and D°p (20°C) - 2.88 χ 1 0 "7 c m

2/ s (or 28.8 picoficks).

Given vp = 0.74 mL/g, calculate M p and verify that it is a hexamer.

Solution: The density increment is (dpjdcv) = 1 — vpp0 = 0.26, where p0 ~ 1. Substitution of these values and fm = kT/Dp into Eq. (3.3.4) yields M p = 4.78 χ 10

5. Given that the subunit molecular weight is

80,000, the degree of polymerization is 478,000/80,000 - 6.

3.6. Determination of the Equivalent Hydrodynamic Shape of Regular Solids

The discussion in this section is limited to the translational motion of regular solids, viz, the sphere, cylinder, and ellipsoids of revolution. The values of DjT

and D m are the rotational averaged values, i.e., <D> =^(Dl + D2 + D 3) , where the subscripts denote the three principal axes of the molecule. The hydro-dynamic radius is

Rh _ \dcJT.„ _ kT NA 6πη0Ωη 6πη0ΟΎΐ'

A measure of the molecular asymmetry is the ratio fTr/fs = RH/RS, where # s is

Table 3.1

Theoretical Expressions for Ru/Rs for Regular Solids

(Rs = spherical radius, a = major axis, b = minor axis) Prolate ellipsoid (p = b/a)

RH

RK 1 + d

Oblate ellipsoid

RH (Ρ2 - I)

1

Rs p2/3tan-

l(p

2 - 1)

Long thin cylinder (rod) (p = L/2RC, L = length, Rc = radius)

Rs (3/2)1 / 3

21n(2p)

3.6. Determinat ion of the Equivalent Hydrodynamic Shape of Regular Solids 51

4 J

3 H

7 13 19 25 31

MAJOR AXIS/MINOR AXIS Fig. 3.3. K H/ / ? S vs. axial ratio for regular solids: ( ) oblate ellipsoid; (- - -) prolate ellipsoid;

( ) cylinder with end effects; ( ) cylinder without end effects.

the equivalent spherical radius, i.e.,

The regular solid shapes are illustrated in Fig. 3.1, and the theoretical expressions for RH/RS are given in Table 3.1. The ratio RH/RS is plotted as a function of the axial ratio for these structures in Fig. 3.3. The divergence of RH/RS for L/2RC -» 1 for the cylinder without end effects indicates that caution should be used when applying this formula to rods or cylinders of finite length.

The ratio RH/RS may be greater than unity for spherical particles that are solvated, i.e.,

where δν0 accounts for the volume of the solvation shell.

Example 3.3. Hydration Shell of Phosphofructokinase Assume that phosphofructokinase in the hexameric form is a solid sphere. Using the data in Example 3.2, estimate the thickness and the weight of the shell of bound water from the ratio RH/RS - Do these values seem reasonable?

Solution: The apparent radius computed from D£(20°C) and Eq. (3.6.1) is RH = (1.38 χ 1(Γ

1 6)(293)/(6π)(0.01)(2.88 χ 1(Γ

7) = 74.4 χ K T

8 cm.

R\i = Γνρ + δν0 (hydrated sphere), (3.6.3)


The "dry radius" computed from vp and M p is Rs — [(3/4π)(0.74 χ 4.78 χ 1076.02 χ 1 0

2 3) ]

1 / 3 = 52 χ ΙΟ"

8 cm. The increment δν0

can now be computed from Eq. (3.6.3): δν0 = [_(RH/Rs)3 —

1 ] ^ P = [(74.4/52)

3 - 1] χ 0.74 = 1.42 mL/g. The increment δϋ0 = 1.42 mL/g

is greater than that for pure water (v0 — 1 mL/g), hence the value RH/RS = 1-43 cannot be attributed to bound water alone. It can be concluded that shape asymmetry contributes to the ratio.

3.7. Determination of the Equivalent Hydrodynamic Shape of Irregular Rigid Structures

The first attempt to describe the architecture of irregular shaped particles was the bead model of Kirkwood (1948, 1949, 1954), in which a collection of ns

identical spherical beads act as point sources of friction interaction with the solvent. The relative velocity about bead i (vt) is given by (cf. Chapter 6)

vf = U - F'T^F,., (3.7.1) 7=1

where U is the relative solvent flow in the absence of the other beads, the prime on the summation indicates that i = j term is omitted, and Tu is a tensor that describes the interaction between beads i and j . We define ξί = 6π^ 0σ, as the friction factor for bead i of radius at. The total force on the molecule is the sum of the forces on each bead:

F = Σ F

i = Σ £<vi = Ξ · U (3.7.2)

i= 1 i=l where Ξ is the total friction tensor. This expression is quite general. We proceed with the development due to Garcia de la Torre and Bloomfield (1977), where G7 is defined by,

F, = ^ G r U (3.7.3)

Substitution of Eq. (3.7.3) into Eq. (3.7.1) and comparison with Eq. (3.7.2) leads to the identities

S = Σ É/Çi (3-7.4) t = 1

and

J=l

where I is the identity matrix. The physical interpretation of G, is that of a shielding coefficient, where G t = I indicates total exposure to the solvent whereas G; = 0 indicates complete shielding from the solvent by the other

3.7. Determinat ion of the Equivalent Hydrodynamic Shape of Irregular Rigid Structures 53

beads in the structure. One proceeds to make successive approximations to obtain a self-consistent set of shielding coefficients, {G}, using the G a u s s -Seidel method,

G (*) I - Σ IJTY · - Σ ijlu'Qr" j=l j=i+l

(3.7.6)

where (k) represents the kth iteration. The matrix elements of the interaction tensor Τ 0· of Garcia de la Torre and Bloomfield (1977) are given as

(TYW = (TtiU + (TS G-B\

where

(RiMRij)

is the Oseen interaction tensor and

( Τ Γ % = (8 π

Λ )

(3.7.7)

(3.7.8)

(3.7.9)

is the Garcia de la Torre-Bloomfield ( G - B ) correction term for beads of infinite volume and of variable size. The other parameters are the following: α, β = χ, y, ζ; δαβ is the Kronecker delta function; Ru is the scalar distance between the centers of bead i and bead j ; and (# ί 7) α is the vector difference in the component between the beads i and j . The G - B term reduces to the Ro tne -Prage r (1969)/Yamakawa (1970) correction term when o-x = σ 7.

It may not be possible to use the full expression for the interaction tensor for large irregular structures because of the large number of matrix elements that must be computed. Garcia de la Torre and Bloomfield (1977) introduced a "diagonal approximation", in which all of the off-diagonal elements of T f j are set equal to zero.

It is desirable to have an analytical expression for the molecular friction factor, i.e., / T r. To compute / T r, the friction tensor is inverted to obtain the diffusion tensor D since / = /cT/<£>> where <D> = iTr (D) , where D = kTE'

1. One proceeds by substituting Eqs. (3.7.7)-(3.7.9) into D and then

inverting by a Taylor series expansion.

Bloomfield- D alt on- Van Holde Approximation

Bloomfield, Dalton, and van Holde (1967) considered the finite size of the beads by assuming point friction sources at their surfaces. The resulting analytic expression permitted beads of different sizes in / T r, viz,


Kirk wood Approximation The original Kirkwood expression (1954) is obtained from Eq. (3.7.10) for a collection of ns identical subunits of radius σ and friction contribution £ s:

σ ns ns 1

ns i=i j = l Ku

(3.7.11)

According to the computations of Garcia de la Torre and Bloomfield (1977), the Kirkwood expression may underestimate the friction factor by as much as 30%. These authors further note that it is more important to maintain the volume of the irregular shape than the outside dimensions when constructing the object from spherical subunits.

Example 3.4. Shielding Coefficients for a Rod and Solenoid An average shielding coefficient for bead i has been defined as (Schmitz, 1977) <G>i = iTr(Gj), where Tr denotes the trace of the matrix. The full interac-tion tensor given by Eqs. (3.7.7)—(3.7.9) was used in these calculations, where a self-consistent set of shielding coefficients was first obtained by the iterative procedure outlined by Eq. (3.7.5). The beads were identical with a diameter of 110 Â. The results are given in Fig. 3.4.

<S>,

0.6_

0.5_

0.4_

0.3;

0.2_

0.1

*·-χ· ·χ··• χ χ-/

Ί—Γ 4 5

~ι—ι—ι—ι—I I Γ

6 7 Θ 9 10 II 12 bead number

Fig. 3.4. Garcia de la Torre-Bloomfield hydrodynamic shielding coefficients for a beaded rod: ( · ) rod; ( • ) solenoid, 4 beads / turn ; ( χ ) solenoid, 6.4 beads/ turn . [Reproduced with permission

from Schmitz, Κ. S. (1977). Biopolymers 16, 2635-2640. Copyright 1977 by John Wiley and Sons.]

Example 3.5 Kirkwood Shape for Phycocyanin Phycocyanin, a protein obtained from blue-green algae, constitutes part of the photosynthesis apparatus. Phycocyanin is composed of two chains, α and β, and exhibits varying degrees of aggregation.

3.8. Anisotropie Translat ional Diffusion of Cylinders 55

Kato et al. (1974) isolated the I IS species from Phormidium luridum using a series of chromatographic techniques. It was stated that this preparation did not exhibit reversible association-dissociation characteristics. Ζλ ηη was

APP

obtained at only one concentration for two reasons: (1) phycocyanin ab-sorbs at 488.0 nm, hence higher concentrations could not be used because of heating and the possibility of absorbing the scattered light (recall Sec-tion 2.16); and (2) lower concentrations could not be used because of the low levels of scattered light. The reported parameters were: D(20°C) = 4.73 χ 10~

7 cm

2/ s , S2o,W = 10.2 S, and vp = 0.75 mL/g (taken from the literature).

Assuming the subunit has a molecular weight of 35,000, estimate the number of subunits in phycocyanin. If these subunits are arranged in a ring, estimate the radius of the subunit using the Kirkwood equation for fTr.

Solution: From Eq. (3.3.4) one has M p = 209,100 Daltons. The number of subunits is therefore 209,100/35,000 - 6.

For hexameric phycocyanin, the distance from the center of the structure to the center of any subunit bead is twice the radius of the bead, hence /Ύτ/6πη0 = 2Λ22σ. The equivalent hydrodynamic radius is Rapp = kT/(6^0Dm) =45 χ 10~

8 cm. The radius of the subunit bead

is therefore σ = 45 χ 10_ 8

/2 .122 = 21.2 χ 10~8 cm. One might now

compare this value of σ with the "dry sphere" radius computed from Eq. (3.6.2), Κ 8=[(3/4π)(0.75)(35,000)/6.02χ 1 0

2 3) ]

1 / 3- 2 2 . 2 χ 1 0 "

8 cm.

The agreement is quite good.

3.8. Anisotropic Translational Diffusion of Cylinders

The general expressions for the diffusion coefficient perpendicular and parallel to the symmetry axis of a cylinder of length L and radius Rc are

kT[ln(p) + YJ D = —

aj (3.8.1)

Απη0Ε

and

2 ^ L ' ( 1 8

·2 )

where yL and Y(| are "end effects" and ρ = L/2RC. Broersma (1960a, 1960b) first examined the "end effects" on the translational diffussion of cylinders. Tirado and Garcia de la Torre (1979) examined the end effects for both a "capped" cylinder and an "open-ended" cylinder, where they used spherical subunits to mimic the cylinder surface (the "shell model"). Their calculations involved a series of bead radii and an extrapolation to σ = 0, which would be the classical limit to a smooth surface. Because of symmetry,


With end plates Without end plates

y± I'LL y± y\\

sc

= 12

5 1.028 1.216 0.604 1.219 0.607

10 2.056 1.049 0.290 1.051 0.291

20 4.112 0.963 0.086 0.964 0.087

43 8.840 0.927 - 0 . 0 2 7 0.929 - 0 . 0 2 7

sc

= 25

10 1.114 1.134 0.531 1.143 0.537

20 2.227 0.973 0.217 0.980 0.218

40 4.455 0.893 0.013 0.899 0.015

70 7.796 0.867 - 0 . 0 7 1 0.870 - 0 . 0 7 5 99 11.026 0.861 - 0 . 1 1 0 0.865 - 0 . 1 0 9

a Number of 12- or 25-sided polygons composing the cylinder. b ρ = L/2RC, L = length of the cylinder, Rc = radius of the cylinder.

c Model A, where the centers of the beads for consecutive rings are

placed above each other.

[Reproduced with permission from Tirado and Garcia de la Torre

(1979). Copyright 1979 by the American Institute of Physics.]

/ T r= 3 / [ 2 / ( ~ ) x x + 1/(Ξ)„], where (Ξ)« = 4 ^ 0 L / [ l n ( p ) + y j and (Ξ)„ = 27"70L/[ln(p) + y H ] ; thus,

/ ? = Γ Τ Τ ^ . (3-8-3) \n(p) + y

where y = (y± + y\\)/2. Knowing the value of ρ and computing the matrix elements of the friction tensor, Tirado and Garcia de la Torre obtained values for y L and y Y as a function of ρ for cylinders that are capped and open-ended, as given in Table 3.2. The values for y for the hollow cylinder for σ ^ 0 can be fitted to the functional form

η Λ™ ° ·4 7 38

° ·4 1 67

° ·3 3 94

. η . , γ(σ -•()) = 0.32 + + = =—. (3.8.4) Ρ Ρ Ρ

A comparison of the various hydrodynamic models is given in Table 3.3 for a rigid cylinder.

Table 3.2

End Effects for Rigid Cylinder

3.9. Diffusion of Random-Flight , Linear Molecules 57

Table 3.3

Compar ison of Cylinder Models"

*app (A)

Long thin cylinder [Eq. (3.8.3) with y = 0] 265.6 Kirkwood beads [Eq. (3.7.11)] 212.7 Garcia de la Torre/Bloomfield [Eqs. (3.7.3)-(3.7.9)] 218.9 Tirado/Garc ia de la Torre [Eqs. (3.8.3) and (3.8.4)] 231.8

a L/2RC = 12; L = 1320 Â; Rc = 55 Â

Example 3.6. Cylinder Dimensions of Core Particle DNA Fulmer et al. (1981) extracted the duplex D N A from chicken erythrocyte core particles. (A nucleosome unit described in Problem 2.6 has a substructure comprised of the DNA/his tone core particle and "linker D N A " that con-nects core particles.) For added salt in excess of 0.1 M, Z)£(20°C) = 3.0 χ 10~

7 c m

2/ s (30 picoficks) was reported for the DNA. Assuming a base pair

spacing of 3.4 Â and a diameter of 27 Â (Elias and Eden, 1981), verify that free DNA of 150 base pairs is hydrodynamically equivalent to a rigid cylinder.

Solution: The length of the 150 base pairs of DNA is 3.4 χ 150 = 510 Â, hence an axial ratio ρ = 510/27 ~ 19. The end effect correction is γ ~ 0.35 according to the data in Table 3.2 and Eq. (3.8.4). The expected value for D£(20°C) is fcT[ln(p) + y ] / 3 ^ 0 L = 2.8 χ 10~

7 c m

2/ s . The

experimental value was reported to be 3.0 χ 10~7 c m

2/ s . The small

discrepancy may be due to the fact that the theoretical calculations were for a hollow cylinder, whereas D N A is better represented as a solid cylinder; or it may be a slight effect of flexibility of the DNA, since the length 510 Â is comparable to the persistence length of 500 Â.

3.9. Diffusion of Random-Flight, Linear Molecules

It is assumed that the linear structure can be divided into ns rodlike units that are free to rotate about universal joints. The length LK of each unit is referred to as a Kuhn length (Kuhn, 1936, 1939), which represents the average "step size" in a random-walk process. We fix our position at one end of the chain. The distance ζ located the center-of-mass of the chain from our vantage point, r, is the distance from the reference end to the ith link, and is defined as s f = rf — ζ (cf. Fig. 3.5).

The mean-square radius of gyration for a polymer composed of ns identical

subunits is defined as


By definition of the center of mass, ]Ts t = 0. Hence £ r f = nsz. Substitution of this result for z

2 in Eq. (3.9.1) leads to a double sum involving rf · r,- =

(rf + r) — rfj)/2, where r l7 = rf — r,-. Hence,

<^>=ΛΣ Σ 4 (3-9.2) zns ,· = ι j= ι

The double sum is now expressed as

X X . = 2(LK)2Z Σ ( y - i ) = ^ J L . (3.9.3)

ί = 1 j = 1 i = 1 j = i + 1 J

<#g> f °r a

random-flight chain is therefore

(R2

G> = ί φ < (random flight), (3.9.4) 6

where L = n s L K is the contour length. The end-to-end mean-square distance <r

2> is given by the double sum over the average projection of Tj on the Kuhn

length, (rf · r,). Since all orientations are possible, (rf · γ,·) = 0 for i Φ j and,

(r2} = ns(LK)

2 = LLK. (3.9.5)

3.10. Diffusion of Linear Polymers under Theta Conditions

Interactions between segments of the chain—either through attraction or repulsion or through excluded volume effects—and constraints placed on the allowed bond angles and polymer flexibility will alter the chain dimensions in comparison with those of the random-flight polymer. The intersegmental interaction effects can be eliminated by using a theta solvent (cf. Example 2.5). It is convenient, therefore, to characterize the inherent deviations from random-flight statistics by the parameter C(0),

/ 2\ 02

\ c(

0)

= i7 T = T r (3-10-1)

3.10. Diffusion of Linear Polymers Under Theta Condit ions 59

where the argument "0" denotes measurements performed under theta conditions. The deviations from random-flight behavior are contained in the projections (rf · r7>, which in turn reflect the chain statistics.

Let us consider a chain having ns identical repeat units that are free to rotate about a bond at a fixed angle 0 with respect to the adjacent repeat unit. The projection of bond i + 1 along bond i is — L Kcos(0) ; hence,

<Γι

Γ·Γ;> = L

2

K(ns - /c)[ -cos(0)]f e (k = 0 , 1 , . . . n s - 1). (3.10.2)

It follows that (rf · Γ , · > < 0 for sufficiently large values of k, which means that the two segments i and j are no longer correlated in regard to their relative orientation. One can rewrite Eq. (3.10.1) as

nx - x

n

<r2y

<r2>e

Θ = j 2(x 2x 1 - x

1 - ;

1 - cos(0)

L L K ~ 1 + cos(0)

dx

(ns » 1), (3.10.3)

where χ = — cos(0). Bulky side-chain constituents will severely hinder free rotation about the bond. If the potential to hinder rotation is symmetric, i.e., the same value for ± φ, then for small values of φ,

< r2> 0 1 - c o s ( 0 ) 1 +cos(<£)

LLK 1 + cos(0) 1 — cos(^) (3.10.4)

Kratky and Porod (1949) introduced the concept of a wormlike coil model, in which the chain is assumed to be described by a continuous function s[r(L)]. If u, and u, are unit vectors tangent to the curve s[r(L)] at the locations i and 7, respectively, then a persistence length Lp can be defined in terms of the angle 0fj-between the chain directions at the positions i and j :

cos(0o-) = <uj · Uj} = exp (3.10.5)

In this formulation, the persistence length Lp is defined by a bend through an angle 0 = c o s

_ 1[ e x p ( - 1)] = 68.4° (cf. Fig. 3.6).

The dependence of <r2> on Lp for the wormlike chain is

(r2y = 2LLp 1 1 — exp (3.10.6)

In the random coil limit, <r2> = 2 L L p ; hence the relationship between the

persistence length and the Kuhn length is L K = 2 L p . The characteristic ratio is therefore

C ( 0 ) = ^ = 1 1 — exp (3.10.7)


L p / L » I L p / L ~ I L p / L « I

rodlike coillike excluded volume

Fig. 3.6. Statistical regions for linear polymers with a finite persistence length.

Kirkwood and Riseman ( K - R ) (1948) and Zimm (1956) examined the relationship between the translational diffusion coefficient and the mean-square displacement for a random-flight coil with hydrodynamic interactions. The values of (Rl)

l/2/RH were 1.504 ( K - R ) and 1.479 [Zimm, where the

preaveraged Oseen tensor given by Eq. (3.7.8) was used].

Example 3.7. Diffusion of Poly(styrene) in Cyclohexane Varma et al. (1984) reported the value Dp(0) = 1.24 χ 1 0 "

7 c m

2/ s for a PS

preparation with ( M p ) w = 1.21 χ 106 in cyclohexane at 34.5°C (theta con-

ditions). Given η0 = 0.00735 poise for cyclohexane at 34.5°C, calculate RH and L p . Assume that PS is a freely rotating flexible coil about the C - C bonds. In performing this analysis, ignore hindered rotation of the benzyl group. Using the relationship <#G>Ö/2

= 1.50ßH, compute a value for the fixed bond angle Θ.

Solution: Assuming Dp(0) is the zero concentration limit, RH = ΙίΤ/6πη0ϋρ(θ) = 247 χ 10"

8 cm. Since PS is assumed to rotate freely

about the C - C bond, the number of statistical units ns is approxi-mated to be the number of monomer units. Given that the weight of one repeat unit is 104, the number of monomer units in this chain is 1.21 χ 10

6/104 = 8643. The hydrodynamic radius is then related to the

statistical length L K by RH = ( n s / 6 )1 / 2

L K/ 1 .5 . The statistical length

3.11. Excluded Volume—the Flory Limit 61

is now L K ~ 1.5(6/w e)1 / 2

RH = 1.5(6/8643)1 / 2

(247 χ 1(Γ8) - 3.91 χ l ( T

7c m ;

hence L p - L K / 2 = 1.95 χ 1 0 "7 cm.

Huber et al. (1985b) calculated the ratio (Mp/L = 390 g/mole-nm) and L K = 2 nm. for PS in cyclohexane using the Yamakawa-Fuj i i (1973) theory for the diffusion coefficient of a wormlike coil. The length of the Varma et al. sample is estimated to be 1.21 χ 10

6/39 = 3.1 χ

104 A. Substitution into Eqs. (3.10.1) and (3.10.2) gives C(0) = 21.3, or

0 = c o s_ 1

{ [ l - C(0)]/[1 + C(0)]} = 155°. The benzyl groups appar-ently increase the C - C bond angle from the tetrahedral value of 109°.

3.11. Excluded Volume—the Flory Limit

For real chains with an "intrinsic stiffness", the chain statistics are dictated by the length of the chain relative to its persistence length. The molecule is considered to be "rodlike" if the chain length is less than the persistence length. As the chain length is increased beyond the persistence length, the spatial arrangement of the chain segments begins to obey Gaussian statistics. By the definition of L K , ring closure or end-to-end contact is unlikely to occur unless ns > 360768° ~ 6 [cf. Eq. (3.10.7)]. Gaussian statistics are followed in com-puting the ensemble average as long as the number of possible overlap configurations is small. As the chain length is increased further, however, the number of configurations with intrachain penetration points becomes significant. Elimination of these "nonphysical" configurations necessarily re-sults in larger chain dimensions than predicted by random-flight statistics. These situations are illustrated in Fig. 3.6. This "crossover" region from Gaussian to excluded volume statistics is denoted by the parameter nc. The problem, therefore, lies in computing the double summation term of Eq. (3.9.2). One can express < r

2>

1 / 2 as a power law in ns:

<r2y

/2~nl (3.11.1)

where ν is a parameter that depends upon the chain statistics: ν = 1 for rodlike molecules; ν = 0.5 for Gaussian chains; and ν > 0.5 for long chains with excluded volume effects. Flory (1953, Chapter XII) computed the value for the scaling exponent ν for a chain of dimensionality d.

We follow the line of development given by de Gennes (1979, Chapter 1) for the mean field approximation for intrasegment interactions up to the pairwise term. The repulsive interaction energy between any two segments in the chain is given by the mean field approximation, G r 2 = (fcT/2)(l — 2 # ) a

d< C m>

2,

where χ < \ is an interaction parameter that depends upon the quality of the solvent, a

d = vm is the monomer volume, C m is the monomer concentra-

tion, and (Cm} = < C m>2 results from a mean field approximation. The total

repulsive interaction energy for the chain is found upon integration over


the chain volume Rd, AGr = kT(\ - 2x)a

d(Cm)

2R

d = kT(l - 2x)a

dn

2/R

d,

where <C m> = nJRd. The elastic free energy is of the form G e l as = kTR

2/nsa

2,

in which case the reduced total free energy of the system is

G t ot (1 - 2χ)αάη2 R

2

The Flory radius RF is defined by minimization G t o t, thus

« f = [(1 -2χ)1Ι{ά + 2)ά]ηΙ, (3.11.3)

where ν = 3/(d + 2). The Flory limit for the three-dimensional chain is there-fore ν = 0.60.

3.12. Excluded Volume Effect on Translational Diffusion of Linear Molecules

Sharp and Bloomfield (1968) examined the excluded volume effect using a modified Daniels distribution function for the chain segments. These authors defined the excluded volume parameter ε by the expression for <r

2>,

<r2> = L ^ s

1 + £, (3.12.1)

and derived the following expression for { R Q } :

(ε + 3 ) L K "

(ε + 2)(ε + 3) 1

2ns(s + 1) (3.12.2)

It is emphasized that this expression is valid only in the asymptotic limit of large molecules, in which the dynamic and static properties scale the same. For example,

/I?2\3/2 M

~ M p " " ~, I

"V~

1' ( 3

'1 2

'3 )

Independent measurements of \_η\ and RG suggest that this relationship is not valid in the "crossover region" in going from theta to good solvent conditions. Weill and des Cloizeaux (1979) suggested that the failure of Eq. (3.12.3) in the crossover region was that the static properties of the system depend upon ru in accordance with Eq. (3.9.3), whereas dynamic properties of the system as described by the Oseen tensor [cf. Eq. (3.7.8)] are a function of

3.12. Excluded Volume Effect on Translat ional Diffusion of Linear Molecules 63

where RO is the "dynamic radius". Assuming that ns » 1 and nc » 1, integrals replace the summations:

R 1/2

2 v - 1 2 v - 1 + 2x(v - 1) 3x

2(2v + 2) ' (2v + l)(2v 4- 2)

" 2v - 1 2v - 1

x(v - 1) 3(v - 2 )x2 ' x

v( l - ν

where χ = njnc. The power laws are obtained form

= dln(RG) V° d\n(ns) '

= AIN(ITD) V° d\n(ns)'

Some values of < # è >1 / 2

< l / # D > a re

given in Table 3.4.

Γ Ι Γ v)(2 - ν) J

(3.12.5)

(3.12.6)

(3.12.7)

(3.12.8)

Table 3.4

Selected Values of the Ratio of Static and Dynamic Radii

Source M p χ 106

Solvent

POLY(STYRENE)

Bhatt et al. (1987) (25°C) 3.8-8.5 T H F 1.30

Huber et al. (1985b) (20°C) 0.0031 toluene 1.09

0.004 M t, 1.21

0.0107 " " 1.44

Bantle et al. (1982) (20°C) 0.783 " " 1.43

1.28 Il II 1.44 " " 2.89 Il II 1.23

POLY(ISOPRENE)

Tsunashima et al. (1987) (34.5°C) 0.326 1,4-dioxane 1.33

0.568 Il II 1.31 2.44 Il II 1.27

Asymptotic limits for <^è>1/2

<V^D> Θ good

Akcasu and Han (1979) 1.506 1.862


Weill and des Cloizeaux related R G and RO through a characteristic relaxation time τ ρ for large-scale motions of the polymer, where [r(] ~ τ ρ / Μ ρ . They obtained < l / r i / ) = (47rf/0//cTTp)<r?>, from which results

The static and dynamic theories can be expressed in terms of expansion parameters (Θ denotes theta conditions): = a D

= ΚΌ/ΚΌ,ΘΊ

a nd

Example 3.8. Molecular Weight and Temperature Dependence of \_η]Ι\_η]θ

for Poly(styrene) Han (1979) tabulated data on poly(styrene) in cyclohexane and benzene at various temperatures and molecular weights to test the predictions of both the static and the dynamic theories of the ratio M / M ö = α^. The relationships to be tested are: = a G (static theory); and = a D a G (dynamic theory). The expansion parameters were computed from the theoretical expressions of Farnoux et al. (1976, 1978) for aD(j/) and (xG(y) with \/y = njnc =

Φ I . o -Γ—I Ρ*

L_L 0.8-\ Ι—1

0.6-CD

0.4-Ο 0.4-_J

0.2 _

0 -

Weill - des Cloizeaux

Flory - Mandelkern

Slope = 0 . 3

I 10 I0 2 I0 3 I 0 4

10°

N / N c

Fig. 3.7. Chain statistics for poly(styrene) in θ solvents. Calculations were based on the expansion parameters of Farnoux et al. (1976, 1978):

2(3 - y) - 3

•y2(3-2y) + 6y-

2:-

2z + 3 2z + 4

where ζ = ν — 1 and y = nc/ns. [Reproduced with permission from Han (1979). Polymer 20, 1083-1086. Copyright 1979 by Butterworth Scientific Ltd.]

3.13. Flexible Circular Molecules 65

M P [ ( T — 6)/Ty/M0occnp, where θ is the theta temperature ( 0 b e n z e ne ~ — 50°C and 0 c y ci OH E X A N E ~ 34.5°C), M 0 = 104 is the monomer weight, np is the number of monomer units in a persistence length, and cccnp ~ 4 (Akcasu and Han, 1979). The experimental values of and the theoretical expressions and < X D< X G were plotted as a function of njnc, as illus-trated in Fig. 3.7, where the Flory value ν = | was used in the theoretical expressions.

Han concluded that both theories give the same asymptotic limits for the power dependence of <χη, and that the apparent lag in the static model to attain the asymptotic limit is attributed to the slower crossover of RO from the theta-solvent to the good-solvent conditions.

3.13. Flexible Circular Molecules

The flexible circle can be considered a special case of the flexible linear coil in which the two ends of the chain are located in the same volume element (<r

2> = 0 in Fig. 3.5). Zimm and Stockmayer (1949) modeled the circle as a

Gaussian distribution of segments described by the density function W(ri}) = C'exp( — 3rfj/2L

2

ini\ where nt is the number of segments (steps) separating the ith bead from the last bead in the chain. In the case of a circular molecule one must consider the simultaneous probability that the n a beads in one direc-tion and the nh beads in the other direction from the ith bead will both be at r 0 , viz,

e x p f f ^ W ^ = e x p ( - ^ Y (3.13.1)

where ns = na + nh. Hence, < r2> - L&\j - i\)(n - \ j - i\)/ns, where na =

\j - i\ and nh - ns - \ j - i\. for ns » 1 is

which is half that for a flexible linear Gaussian coil. Casassa (1965) derived the following expression for the form factor, P(Kd

c),

for flexible circles:

<Rh> = 12 (3.13.2)

(3.13.3)

where u2 = K

2< « è > and

ξ{χ) = exp(r2)

2)dt. (3.13.4)

ο


One has for P(Kdc) in the limit M » 1,

(linear), (3.13.5)

2 Γ 2 P(Kd

c)~^ l+-2

2 12 ~2 + - Ï U U

(circle). (3.13.6) u u

Inversion of Eqs. (3.13.5) and (3.13.6) indicates that both the linear and circular flexible molecules have the same slope (^), but with clearly different intercepts ( 4- \ for the linear coil and — 1 for the flexible ring).

3.14. Flexible Branched Molecules and Stars

Branched and star molecules represent molecules having repetitious struc-tures, or branch units. The branch unit is said to have a functionality ( / ' ) , which is the number of segments emanating from the unit. At one extreme is the subclass of linear molecules, which represent branch molecules of functionality 1 or 2. Stars are branched molecules having only one branch unit with / ' > 3. Gels can be considered to be a "molecule" with / ' > 2.

Zimm and Stockmayer (1949) have dissected the problem by assigning an outer end of one chain as the reference segment. The segments in this chain are then labeled in sequence until the branch unit is reached, from which the segments of each of the other ( / ' — 1) branches are denoted by both their branch labels and their positions along that branch. RG for this subunit is computed relative to the reference segment. If there are additional branch units in the structure, then one or more of the chains in the initial branch unit will also be attached to a second branch unit. The procedure is then repeated as many times as necessary for the system at hand. Zimm and Stockmayer presented their results in terms of the dimensionless parameter g = <

( ^ G ) /<

\ ^ G ) O introducted by Kramers (1946), where the subscript 0 denotes a linear molecule of the same molecular weight as the branched molecule. Zimm and Stockmayer reported that

6 / ' (one branch) (3.14.1) g =

( / ' + ! ) ( / ' + 2)

ϋ = 3 [ 5 ( / ' )

2 - 6f + 2]

fW)2 - 1] (two branches) (3.14.2)

y = 2 [ 1 3 ( / ' )

2- 2 0 / ' + 8]

f'19(f)2 - 9f + 2] (three branches) (3.14.3)

Note that g = 1 for / ' = 1 or 2 in these expressions, viz, a linear molecule is a branched molecule with a functionality of 1 or 2. Because of the power

3.14. Flexible Branched Molecules and Stars 67

dependence of / ' , an increase in the functionality of a branch unit accentuates the differences between linear and branched molecules.

P(Kdc) for a star with / ' identical chains exhibiting Gaussian statistics is

(Benoit, 1953; Burchard, 1974, 1977)

P(Kdc) = | K - [1 - e x p ( - Vy\ + ^ y ^ [ l - e x p ( - K ) ]

2J , (3.14.4)

where V = f'u2/(3f

f — 2). P(Kd

c) reduces to that of a Gaussian coil for / ' = 1

and approaches that for a solid sphere as / ' - • oo, as first noted by Stockmayer and Fixman (1953).

Stockmayer and Fixman (1953) examined the diffusion coefficient ratio Atar/Ainear f °

r molecules of the same molecular weight. The primary

assumption was that the friction factor at infinite dilution ( / ° ) was related to R H by a universal constant Ρ ' , /°/η0 = P'RH, where η0 is the solvent viscosity and R H may or may not obey random-flight statistics. The hydrodynamic factor h was defined as

h = ^.branched ^ (3.14.5) ^H, linear

where, for a star molecule,

( 2 - / • ) ' " £ ( / · - • ) • < 1 R 6)

The corresponding Kramers geometric factor g for { R Q } is

(if - 2)

( / ' )

It is noted that diffusion (viz, h) is a less sensitive measure of branching than geometric characterization (i.e., g

1/2). This is a direct result of the hydro-

dynamic shielding effect of neigboring beads since the additional branch beads are in the interior or near the surface of the structure. A highly branched star is thus hydrodynamically insensitive to additional branch units.

Example 3.9. Hydrodynamic Radius of Star Molecules and the Effect of Branching

Estimate RH of star molecules having 4 and 16 branches, where L K = 10 Â. The respective contour lengths for the linear analogues of these star molecules are 1200 and 4800 Â.

Solution: From Eq. (3.9.4) and the Ki rkwood-Riseman ratio ( R Q }1 / 2

/

RH = 1.504: RH = (1200 χ 10)1 / 2

/3.67 = 29.8 Â; and RH = (4800 χ 10 )1 / 2

/ 3.67 = 59.7 Â. Substitution of theses values into Eqs. (3.14.6) and


(3.14.7) gives: RH s t ar = 29.8(41 / 2

) / [(2 - 4) -h 21 / 2

( 4 - 1)] - 26.6 Â; and * H, S T A R = 59.7(16

1 / 2) / [2 - 16) + 2 ^ ( 1 6 - 1)] = 40.9 Â. Clearly, K H, s t ar

is not as sensitive to variation in M p as RH, linear.

Realistically, the chains in the branches do not follow random-coil statistics. Mansfield and Stockmayer (1980) considered stars composed of branches of equal length (Ll = L/f\ where L is the contour length of the linear coil of equal molecular weight). The initial direction of each branch from the branch point was predetermined, and the subsequent generation of the branch configurations was dependent upon the persistence length L p . In general, the calculated value of g was smaller for the case of maximal avoidance of rays emanating from the branch point (vector sum of all the components is zero) than for random orientation of these initial directions. The discrepancy between these two limiting cases decreases as the number of branches increases. Physically, this means that as the density of branches increases, the number of independent choices of initial directions decreases. It is also noted the value of g decreases as the ratio L p / L increases. For the maximum mutual avoidance condition, Mansfield and Stockmayer (1980) obtained

<R2o>=^ (3f -2)j^- + 2 ( 1 - / ' )

+ 2(f'Xl - 2xx + 2 ) ^ + Χ ί( Χ ί - 2 ) { ^2

(3.14.8)

where xx = 1 — exp( — Ll/Lp). Zimm (1984) examined chain stiffness and Ki rkwood-Riseman preaverag-

ing of the interaction tensor for a collection of up to 25 beads using Monte Carlo methods to generate the star structures. The mean-square end-to-end distance for a chain of Nb bonds is given

(Ί - Ρ)2

where ρ = L p/ (1 + L p) is the "persistence factor" that determines the initial direction of the first bond in the chain. This formula was found not to dif-fer significantly in numerical value from that of the wormlike coil if L = Nh/(2p — p

2). The following formula for < # G > was found to be in good

agreement with the Monte Carlo results:

< R g> - 6NS(/R (N b + / ' - D '

( 1 1 4-

1 0 )

where: A0 = -2 + SNb - 12Νξ + 8Νξ - 2Ν%\ Αχ=5>- 18/Vb + 2ΑΝ\ -\\Ν\ + 3Nt; Α2 = UNb - \6Νξ + 6Νξ - 4; and Α3 = -4Nb + 3Νξ + 1.

3.15. Crossover Exponential , Phase Separation, and Chain Dimensionali ty 69

The values of h computed from the preaveraged interaction tensor (hKR) were significantly smaller than the values of h computed for the nonpre-averaged case, viz, hKR = 0.912 and h = 0.953 for L p = 6 and / ' = 4.

Daoud and Cotton (1982) suggested that, because of density variations, the statistics of a chain in a star changes as one proceeds from the branch point. The local swelling parameter oc(r) was defined by the ratio of local length extensions, £(r)/£ 0(r), where the subscript denotes the unperturbed size. The star was then represented in terms of "blobs" of dimensions equal to the "free volume" for that segment of the chain, i.e., 0 b( r ) = ns(r)L^/ξ

3(r), where L K is

the statistical length and ns(r) is the number of statistical units. Three regimes were identified: (1) the core of constant density, which is limited by the packing properties of the chain; (2) the coil region, in which excluded volume effects are negligible; and (3) an excluded volume region within the blob. Rstar

was found to be

R = [ N f • (

^ ') 3 /2

+

(^

) 3 /2 j

3 / 5 V™*

L* (3 14 11)

where vm is the monomer excluded volume parameter, and Ν is the total number of statistical units in the branch. The asymptotic limits for Rstar for long and short chains, respectively, are N

3/5v^

5(f')

1/5LK and N

1 / 2( / ' )

1 / 4L K .

3.15. Crossover Exponential, Phase Separation, and Chain Dimensionality

The statistics that describes a flexible chain is dependent upon the nature of intrasegmental interactions. The intrasegment free energy up to the three body interaction for a chain of dimensionality d is

= £(D + Ei2) + E

( 3)

kT

fiot ^T (Rô ' dR

d ' dR

R2

2vn: wnl (3.15.1)

where the first term is the elastic energy, < # g > 0 is defined for the unperturbed chain, ns is the number of segments, and ν and w are the two-body and three-body interaction parameters, respectively. There also should be a configurational entropy term, viz, — d\n(R)/T. As shown by Deutch and Hentschel (1986), the inclusion of this term does not alter the conclusions if w > 0; hence it is omitted. Minimization of Gtot/kT with respect to R yields

R = <R2G>O\ R

d + + R2d + (3.15.2)


A negative value of ν indicates the dominance of attractive intersegment interactions that cause the chain to collapse ("poor" solvent conditions). A positive value of υ results from the dominance of repulsive interactions that cause the chain to expand ("good" solvent conditions). The "theta condition", when ν = 0, is therefore the pivotal point for the description of solvent quality. The crossover between these solvent regions can be effected by changing the temperature (cf. Example 3.8) or concentration (cf. Example 5.13). The crossover region therefore may be defined from a plot of the reduced tem-perature τ = (Θ — Τ)Iθ and concentration C r . The approach is to obtain expressions for τ and C r above and below some "critical point" on the τ vs. C r

plot. The intersection of these expressions extrapolated through the critical region thus provides a value for the critical parameter. The fate of the chain depends upon the relative values of E

(2\ and £

( 3 ).

We examine two approaches to the phase separation problem for linear polymers of dimensionality d in which < K G) o ~

ns [cf. Eq. (3.13.2)]. Similar

arguments can be used for branched chains where (RG}0 depends upon the functionality of the chain (cf. Section 3.14).

The extrapolation procedure is based on the power laws of three funda-mental expressions. The first is the relationship between the radius and the number of segments,

Κ ~ φ 8 )ν· . (3.15.3)

The second relationship is between the pairwise interaction parameter and the reduced temperature:

-v(T) = a

-^l = vmr, (3.15.4)

where v(T) < 0. The third relationship pertains to the properties at the crossover point. Daoud et al. (1983) assumed that the crossover from Θ to collapsed-state conditions occurs when the segment-segment attractive inter-action becomes \vc\n

2/Rd

e ~ kT. Hence τ = Rd

e/n2ad, and from Eq. (3.15.3)

one finds the relationship τ ~ (rcs)_ < / >

, where φ = 2 — dve. On the θ side of the crossover point, the power law is determined by setting ν = 0 in Eq. (3.15.2), which for linear chains is ν θ = 2/(d + 1). Hence one has for temperatures above the collapse point

τ ~ Κ Γ2 / (

'+ 1 )

( T > T C ) . (3.15.5)

We write for the collapsed state [cf. Eq. (3.15.3)]

Rc ~ a(nsYc - a(ns)

i/d, (3.15.6)

3.15. Crossover Exponential , Phase Separation, and Chain Dimensionality 71

which is valid for both linear and branched chains. Moving from the collapsed state towards the θ state, Daoud et al. (1983) write the modified function for the radius in terms of the function F(x), R C ~ [ o ( n s )

v e] F [ i ( n s )

V e] . It is assumed

that F(x)-> 1 as the θ state is approached and that F [ i ( n s )v e

] ~ [ φ 8 )ν θ

]α

upon approach to the collapsed state. The value of α is determined from the expression

R C ~ α τα( Μ δ)

[ ( 1 + α ) ν θ 1, (3.15.7)

where a is subject to the constraint imposed by Eq. (3.15.6). Hence α = (1/£/νθ) — 1 = (1 — d)/2d. The monomer concentration C m is given in general by the ratio

C m ~ ^ ~ ( n . r * (3.15.8)

where φ = dv — 1. For a linear polymer in region between the theta and collapse points, one has φθ = άνθ — 1 = (d — l)/(d + 1), whereas φ0 = 0 in the condensed state. In accordance with this model, the appropriate variables to plot are τ(η&)

φ versus Cm(n s)^, viz, t ( / î s )

2 /3 versus C m( n s )

1 /3 for d = 2 and τ(η5)

ί/2

versus C m( n s )3 /4

for d = 3. However, Deutch and Hentschel (1986) criticized the criteria E

(2) « 1 to

define the theta region and £( 2 )

~ 1 to define the crossover region \vc\n2/

RQ ~ \. They point out that triplet interactions are important in the theta region, as is easily seen from Eq. (3.15.2). These authors proposed as an alternative criterion for determining the crossover exponent:

G ^ ^ W ' « . , (1 .5 .9 ,

where Eq. (3.15.3) was used to substitute for RD and φ is defined for Eq. (3.15.8).

These authors suggested that φ rather than φ is the correct power law for describing the crossover region. The apparent stimulus that inspired Deutch and Hentschel (1986) to pursue their line of thought was the prediction by Daoud et al. (1983) that τ ~ for d = 2 at high concentrations, which differed from the Flory -Hugg ins prediction that τ ~ C m . The differences stem from the power law used in the calculation of the free energy,

G = (ns){~

Ved)F[C(nsy

ed~\ φ β ) > ] , (3.15.10)

where Daoud et al. used the exponent y = φ and Deutch and Hentschel used the exponent y = φ. It is not clear at this stage which exponent is the correct choice. Note that φ = φ = \ for d = 3. According to the Deutch an Hentschel model, the appropriate plot should be z(nsY versus Cm(n s)*. A schematic of the two types of plots is shown in Fig. 3.8.


Fig. 3.8. Schematic coexistence curve for a linear chain. The critical concentration (Cc) and

reduced temperature (T c) are determined from the intersection of extrapolated expressions valid in

the regimes above and below the critical region. For the theta-poor solvent crossover region, the

exponential parameter y is φ = 2 — dv0 in the formulation of Daoud et al. ( 1983) or φ = dv0 — 1 in

that of Deutch and Hentschel (1986).

Summary

The friction factor obtained from sedimentation velocity and QELS methods has been identified with the mutual friction factor. The friction factor asso-ciated with the determination of molecular shapes, however, is the tracer (or self) friction factor. The mutual and tracer friction factors are not, in general, equivalent.

Current hydrodynamic theories for irregular-shaped molecules as being composed of subunit beads have advanced to the stage of providing very reliable estimates of the equivalent hydrodynamic shapes of macromolecules. The theories correct for the finite size of the bead and allow for beads of various sizes. Calculations for rigid cylinders in the limit of zero bead size indicate that the end effects are very significant.

Strides have also been made in the hydrodynamic modeling of flexible coils. The statistics of these coils is intimately associated with the length of the coil relative to the persistence length. This length dependence is reflected in the power law of the molecular weight for both dynamic properties (viscosity, diffusion, and sedimentation) and static properties (radius of gyration).

Problems

3.1. Bloomfield et al. (1967) have shown that fTr for a sphere composed of spherical subunits decreases by only 6% if 60% of the beads are

Problems

removed. Use the Kirkwood model [cf. Eq. (3.7.11)] to determine how many beads must be removed from a rod of 12 subunits before the friction factor decreases to 50% of its original value. Does the location of the beads in the rod make a difference?

3.2. Calculate the percentage change in the sT and Dp for a dimerization reaction between two identical beads.

3.3. Consider the rod <-> circle transition for DNA. Assume that a short piece of D N A can be modeled as a rod of 12 identical beads with a diameter of 27 Â. Use the Kirkwood expression for a rigid circle to determine the relative change in the friction factor for this transition.

3.4. Calculate Dp for the rod and circle in Problem 3.3. What is the expected values for the relaxation time at 20°C in an aqueous solvent at Θ = 90° and λ0 = 488 nm? Assume that you have an autocorrelator in which the lower limit for the data collection interval is 0.5 ^s. Determine whether you can follow the reaction at the 90° scattering angle.

3.5. The enzyme pyruvate oxidase (pyruvate: ferricytochrome bl oxido-reductase, EC 1.2.2.2.) catalyzes the oxidative decarboxylation of pyruvate to carbon dioxide and acetate. Raj et al. (1977) reported the infinite dilution values for the native enzyme: s W = 1 0 . 1 S and D£(20°C) = 4.05 χ 1 0 "

7 c m

2/ s (40.5 picoficks). Using the value UP =

0.75 c m3/ g , calculate M p for native pyruvate oxidase.

3.6. On the basis of the information in Problem 3.5, calculate the ratio FP/FS, where FS is the hard sphere value as computed from VP. Use this ratio and the Kirkwood expression [cf. Eq. (3.7.11)] to estimate a possible configuration of the subunit beads, assuming the native enzyme is a tetramer.

3.7. Filson and Bloomfield (1968) used sedimentation velocity measure-ments to experimentally obtain the Kirkwood sum

1 1 s n - s i σ

»= ~ Σ Σ

— = '

J I R

IJ S

2 S

L

where the values in the subscripts denote the number of subunits in the molecule. Derive this expression.

3.8. The Kirkwood double sum is defined as ση in Problem 3.7. Derive an expression for ση in describing Dp instead of s T.

3.9. Kam et al. (1981) examined the linear form of Col E : D N A as a


function of [NaCl] using sedimentation velocity and QELS methods, with the results in the accompanying tabulation.

[Nacl ] 0.2 4.0

D(20°C) (picoficks) 1.99 1.68

s20 (Svedbergs) 16.2 9.5

(-) 0.457 0.314 (-)

Calculate M p at these two salt concentrations.

3.10. Dp for BSA at the isoelectric point (pH 4.7) in a solvent of ionic strength 0.12 M was reported by Oh and Johnson, Jr. (1981) to be 5.92 χ 10

7 c m

2/ s at 20°C. Compute / £ / / s using M p = 69,000 and

vp = 0.75. Assuming that this ratio is due to molecular asymmetry, compute the axial ratio for prolate and oblate shapes for BSA.

3.11. Huber et al. (1985b) examined the molecular-weight dependence of short poly(styrene) chains in toluene (a good solvent) and cyclohexane (a theta solvent). From the tabulated data in toluene at 20°C, esti-mate v D.

M w( g / M ) 1200 3100 4000 10,700

D(20°C) χ 1 0+ 7

( c m2/ s ) 40.1 28.5 25.8 15.7

3.12. Tsunashima et al. (1983b) examined various molecular weight fractions of poly(styrene) at 20.4°C in irans-decahydronaphthalene (n0 = 1.4751 and η0 = 0.02113 poise). Some of their results are summarized in the accompanying tabulation.

M p x l 0 ~6( g / M ) 9.70 5.53 2.42 0.775

D(20.4°C) χ 108 (cm

2/ s ) 1.458 1.935 3.035 5.223

[ rç ] (cm3/g) 240 183 123 71

χ 1011 (cm

2) 7.85 4.56 1.90 0.61

From these data determine the power laws v D and v G, and compare Eqs. (3.12.3) and (3.12.9).

Problems 75

3.13. A hypothetical polymer is composed of 13 identical subunits. The hydrodynamic radius of the subunit is σ = 50 Â. Use the Kirkwood expression [Eq. (3.7.11)] to estimate RH for the following con-figurations: (1) rod; (2) circle; and (3) tetrahedron (four-armed star, three subunits per arm, one subunit at the origin). Also calculate the volume equivalent spherical radius (Vs = 1 3 K s u b u n i t) [cf. Eq. (3.6.2)]. If the experimental accuracy is 3 % , which, if any, of these configurations can be distinguished from the others? Use Eq. (3.14.11) to estimate L K

for the four-armed star configuration. Does this value of L K seem reasonable?

3.14. Consider a rod configuration composed of two bead sizes, σ 3 = 50 Â and oh = 100 Â. Calculate fp for a rod of 12 beads in the following configurations: (1) alternating beads, i.e., abababababab; and block copolymer, i.e., aaaaaabbbbbb. Use the Bloomfie ld-Dal ton-van Holde model [cf. Eq. (3.7.10)]. Compare these results with a rod composed of equal-sized beads having: (1) the same length as the copolymer rod; and (2) the same total volume as the copolymer rod. Use the Kirkwood model [cf. Eq. (3.7.11)] for these latter calculations. What conclusions can be drawn?

3.15. RG for a rigid ring is simply the radius of the ring. This type of model may be applicable for a circular molecule that is very stiff. For flexible rings, however, the more appropriate expression for RG is given by Eq. (3.13.2). Compare the predictions of these two models for a hypo-thetical molecule of total length 3000 Â for the three cases: (1) rigid planar ring; (2) L K = 1000 Â; and L K = 20 Â.

3.16. Coviello et al. (1986) used light scattering methods to study the bacte-rial poly(saccharide) xanthan produced by Xanthomonas campes-tris. Three different preparations were examined: native xanthan (NX); xanthan in which the pyruvate groups were removed by heating in 0.1 M NaCl and 0.002 M oxalic acid at 95°C for two hours (PFX); and xanthan in which the 0-acetyl groups were removed by allowing the solution to stand at room temperature for three hours in 0.1 M N a O H solution (AFX). Static and dynamic light scattering measurements were performed at 20°C. The contour length L was estimated from the linear mass density obtained from a Holtzer plot (viz, the plateau region of the plot of KRe/Hcp vs. K(RG}

1/2; Holtzer, 1955) of the

total intensity data. (It is pointed out that the experimental value of M p / L is approximately twice the theoretical value.) The results of these measurements are summarized in the accompanying tabulation.


Sample Mp χ 1 0 "

6

(Dalton) Mp/L« (g/nm)

D χ 108

( cm2/ s )

<R2

Gy/2

(nm)

NX 2.94 1830 1.55 289.5 PFX 1.37 1240 2.60 240.8 AFX 1.77 1623 2.60 210.0

a Experimental value was used for Mp.

Estimate LK for these polymers on the basis of a wormlike coil [Eq. (3.10.6)] and Ki rkwood-Riseman random flight polymer. What are the numbers of Kuhn segments in these polymers? Assuming that the chain statistics are the same for all of these polymers, estimate the power laws vG and v D. [Hint: use Eq. (3.12.7).]

3.17. Bhatt et al. (1987) reported DLS results on poly(styrene) in tetrahydro-furan (THF) at 25°C.The diffusion coefficients reported in their Table 1 were stated to be in agreement with their previously reported expres-sion, <D(25°C)> = 3.4 χ 1 0 -

4( M p ) w

0 56 (cm

2/s) . These authors also

reported the ratio {RG)l/2/RH f °

r the molecular weights M p χ

1 0 "6 = 3.84, 5.48, and 8.46, whose average is <RGy

/2/RH = 1.30 ±

0.04. Use the expressions of Weill and des Cloizeaux (1979) for RG/nl

/2 and RD/nl

/2 as given by Eqs. (3.12.5 ) and (3.12.6), respectively,

to obtain the ratio χ = ns/nc that corresponds to the average ratio (RG)

1/2/RH. (Assume that ν = v D = v G. ) Using the value of χ as a

criterion, are excluded volume effects important in this system?

3.18. Huber et al. (1985b) presented the values in the accompanying tabulation for poly(styrene) in toluene.

Mp χ 1 0 "3 (Daltons) 3.1 4.0 10.7

(RG>1,2/RH 109 1.21 1.44

<D(20°C)> χ 107 c m

2/ s 28.5 25.8 15.1

Determine the empirical parameters Κ and a in the expression <D(20°C)> = X M p . Noting the relationship M p / M c = njnc = x, es-timate the value of M c using Eqs. (3.12.5) and (3.12.6). What conclusions can you draw from these results?

Additional Reading

Burchard, W. (1983). Adv, in Polym. Sei. 48, 82.

Garcia de la Torre, J., and Bloomfields V. A. (1981). Quart. Rev. Biophys. 14, 81.

CHAPTER 4

Multiple Decay Analysis of the Correlation Function

"In this mood of mind I betook myself to the mathematics and the branches of study appertaining to that science as being built upon secure

foundations, and so worthy of my consideration." (Victor Frankenstein to Robert Walton)

From Frankenstein by Mary Shelley (1797-1851)

4.0. Introduction

We present in this chapter some of the mathematical techniques used in the analysis of QELS data. The methods range from expansion techniques, such as the cumulant method, to the more powerful inverse Laplace transform methods. The general form of the correlation function is assumed to be

C(K,t) = + Β' + ε(ί), (4.0.1)

where at(K) is the relative amplitude of the ith decay process with decay rate yt(K\ Β' is a constant baseline, and ε(ή represents random fluctuating superimposed noise with the constraint that <ε(ί)> = 0.

4.1. Effect of Polydispersity on the Scattering Amplitudes

As discussed in Appendix B, the amplitudes and interference factors cannot in general be separated in performing the double summation over the pairs of scattering units in a multicomponent system. Hence, for spherical particles,

77

78 4. M U L T I P L E DECAY ANALYSIS O F T H E C O R R E L A T I O N F U N C T I O N

one may write

^ = CT<M?><P(/AN>z<S(K AR)), (4.1.1)

where C T = NT/NAVS is the total molar concentration of solute particles and the averages in brackets < > are a function of the component ratios [n0{dnJdc^Tß

2l[n0(dnJdcT)T -]2- (This ratio is rigorously equal to unity

only for homogeneous compositions.)

Example 4.1. (S(K AR)) for a system of interacting spheres The effect of polydispersity on (S(K AR) for a system of hard spheres of homogeneous composition was examined by van Beurten and Vrij (1981). The ratio Re/H (our notation) was computed from the expression found in Vrij (1979) based on a closed-form Perçus-Yevick approximation (cf. Appendix C) for any number of components. The Schulz distribution of hard sphere diameters dh Nt = ~

1 exp( — ad,)] Ad, was assumed where the parameters

a and b are fixed by chosing the location of the maximum (d0) and the standard deviation (σ) of the distribution. After first computing Re/H, ( M p ) , and (P(Kd

c))z, (S(K AR)) was then computed from Eq. (4.1.1). Shown in Fig. 4.1

is (S(KAR)) plotted as a function of Kd0 for φρ = 0.3. These calculations indicate that the solution structure decreases as the polydispersity increases. It is also noted that S(K AR) -> 1 at smaller values of Kd0 as the polydispersity is increased.

1.6

1.2

< S ( K A R ) >

0.8

0.4

ι ι ι 1

5 10 15 20

Kd0

Fig. 4.1. Average solution structure factor for a system of hard spheres, σ is the s tandard deviation of the Schulz distribution, and d0 is the diameter of the sphere at the maximum. [Reproduced with permission from van Beurten and Vrij (1981). J. Chem. Phys. 74, 2744-2748. Copyright 1981 by the American Institute of Physics.]

4.3. Cumulant Analysis

4.2. Single /Double Exponential Analysis

The simplest method of analysis of C(K, t) is to represent the function by a single exponential function (sef) with a characteristic relaxation time T C :

C(K, t)~ A exp -It

(4.2.1)

where the factor of 2 indicates a homodyne configuration and Β is a constant baseline. Since the value of τ characterizes the entire correlation function, its value may vary with the data collection interval Δί.

In order to characterize two widely separated relaxation regimes, one can characterize the correlation function as the sum of two exponential functions (tef):

C(K,t) ~ <M xexp -t

+ A2exp -t

+ A2exp τι _

T2 _

+ B, (4.2.2)

where again the homodyne, or self beat, experiment is assumed. Nonlinear least squares methods to determine the five adjustable parameters (Αί9 Α2,τί9

τ2, and Β) may not converge if either the two amplitudes or the two relaxation times are of comparable value. This is because the iterative incremental adjustment to determine the new values of these parameters on successive calculations may be continually interchanged between the two relaxation modes.

4.3. Cumulant Analysis

Koppel (1972) first introduced the cumulant analysis method for obtaining the average decay rate and standard deviation for a polydisperse system. This method of analysis is based on a series expansion of the exponential functions comprising the normalized molecular correlation function,

p ^ = g«\K,t)= £ a , . (K)exp(-y i f) ,

which in the limit of r —* 0 is given as a power series in i:

l n [ 0( 1

> ( K , t ) ] ~ - K l t + ^K2t2

The nth cumulant K„ is defined as

Γ' ,Η1 κ„ =

G,(X,0)

dt"

(4.3.1)

(4.3.2)

(4.3.3)


The first two cumulants for center-of-mass diffusion are

Ki(K) = (D(K)}zK2 (4.3.4)

and

K2(K) = [<D(K)2> Z - < D ( K ) >

2] K

4, (4.3.5)

where the subscript ζ denotes the "z-average", and the possibility that these parameters depend upon the scattering vector Κ is emphasized. In the case of noninteracting identical particles, (D{K))Z is defined as

M

X PiKd^MfD, (D(K))Z = ^± . (4.3.6)

Σ P(Kdf)NiMf

i = 1 Example 4.2. Polydispersity Analysis of BSA Commercially available BSA (bovine serum albumin) may be contaminated with dimers and trimers. This contamination arises from reactions of the sulfhydryl groups if they are not properly blocked. Doherty and Benedek (1974) analyzed their correlation functions for BSA by the cumulant method. The reported value <D(25°C)>Z - 5.62 χ 1 0 "

7 c m

2/ s for their BSA sample

under isoelectric and high salt (1 M NaCl) conditions is significantly lower than the temperature/viscosity-corrected value of <D(25°C)> computed from the data of Oh and Johnson, Jr. (1981) (cf. Section 2.14), which is <D(25°C)> = (298/293)(1.005/0.8904)(5.92 χ 10~

7) = 6.8 χ 10~

7 c m

2/ s . T h e

discrepancy between these two values was assumed to be due to the poly-dispersity σ = K2/Kj of the preparation. To correct for the polydispersity, Doherty and Benedek used the expression derived by Hocker et al. (1973):

<£>>mono~<£>ap 1 + σ + —-4

(4.3.7)

Using σ = 0.137 in Eq. (4.3.7) for this particular set of data gives the value < £ > m 0n o ~ 6 . 4 x 1 0 -

7c m

2/ s .

Example 4.3. Effect of Polydispersity on (D(K)}Z

Schmidt et al. (1978) used QELS methods to examine solutions of poly(styrene) latex spheres of diameters 109 ± 2.7 nm and 481 ± 1.6 nm. Correlation functions for solutions containing a known mixture of these spheres were analyzed by the method of cumulants (up to the quadratic term) and as the sum of two exponentials. The two-exponential fit was carried out either with the known diffusion coefficients (three adjustable parameters) or with unknown diffusion coefficients (five adjustable parameters). The results of these analyses are shown in Fig. 4.2.

4.5. Expansion Methods Applied to Simulated Da ta 81

Ε 0.48 -,

Ο

A 0

ι.οΗ

Β RO

Ο CO d

ο «Ί-ο Ö ό

sin2 (θ/2)

Fig. 4.2. (A) Cumulant and (Β) two-exponential analysis (Ο, unknown DT; · , known D f) of a mixture of poly(styrene) latex spheres. These da ta clearly indicate that the smaller particles have a greater influence on <D(K)>, as Κ is increased. [Reproduced with permission from Schmidt et al. (1978). Macromolecules. 11, 452-454 . Copyright 1978 by the American Chemical Society.]

4.4. Asymptotic Analysis Method

The asymptotic analysis method (Schmitz and Pecora, 1975) is based on the relative extent of decay of the various modes as a function of Δί, the data collection interval. Correlation functions of Nc + 1 points are obtained at different values of Δί, and the apparent decay rates Kapp(Nc At) are obtained. A plot of Kapp(Nc At) vs. Nc At has at Nc At = 0 the amplitude average decay rate

whereas the asymptotic limit Kapp(Nc At - • oo) = Ks]ow can be obtained either by fitting the Kapp(Nc At) vs. Nc At plot with an appropriate functional form or by computing the intercept for a plot of l/Kapp vs. l/NcAt.

4.5. Expansion Methods Applied to Simulated Data

Since the cumulant method is widely used both in the analysis of data and, as will be shown in subsequent chapters, in the development of QELS theories, it is instructive to apply this technique to simulated data in order to assess its possible limitations and advantages.

Simulated correlation functions of 64 delay points were generated with 75 exponential functions in accordance with Eq. (4.0.1), with Β' = 0 and ε(ί) = 0.

M

Kapp(NcAt = 0)= Σ m ;= ι

(4.4.1)


The decay constants were equally spaced over the interval 1000 s 1

< yx < 10,000 s

- 1, and the amplitudes were generated by

- ί> ( τ , · -<τ»2"

= N' exp <τ>

2 (4.5.1)

where N' is a normalization constant, b = 8.96, and <τ> = 550 μ. The theoretical values for Kt = 3,355 s

_ 1 and K2

Kl and K2 of the heterodyne configuration are = 4.047 χ 10

6 s~

2, giving a polydispersity of σ = 0.36.

The correlation functions were generated with 20 delay times which were logarithmic spaced over the range 5 /xsec < Δί < 100 /*sec. These functions were characterized by a nonlinear least-squares fit to the exponential form

C(t) = A exp + ß . (4.5.2)

Although B' = 0 and ε(ή = 0 were used in the generation of the simulated data using Eq. (4.0.1), Β in Eq. (4.5.2) was allowed to "float" so that the general procedures actually encountered in the laboratory were mimicked. The simulated functions were first analyzed as a single exponential function (i.e., K2 = K3 = X 4 = 0), where the initial guesses for A, B, and Kx were obtained from the Guggenheim method. The nonlinear least-square results from the nth-order expression were then used as the initial guess parameters for the (n + l)th analysis. Correlation functions for which the 32nd point was less than 5% of the initial value were not analyzed beyond the 1st cumulant, since

Table 4.1

Cumulant Analysis of a Single Mode Distribution in Decay Rates

Δί Cumulant Kl χ Ι Ο "3

Κ2 χ Ι Ο "5

Κ3 χ ΙΟ"7 κ2ικ\

order 1/s 1/s2

1/s3

5 1 6.79 2 6.49 0.818 — 0.194 3 6.86 0.713 0.245 0.152 4 6.62 0.858 0.213 0.196

13.6 1 6.05 — —

2 6.39 0.519 — 0.127 3 6.71 0.701 0.144 0.156 4 6.61 0.834 0.177 0.191

Theoretical values (corrected to homodyne conditions)

Kx = 6.71 χ 103 s "

1

K2 = 8.1 χ 106 s~

2

K2/Kj - 0.18

4.5. Expansion Methods Applied to Simulated Da ta 83

most of the function was baseline. The polydispersity in the correlation functions was measured by the ratio σ = K2/Kj. Selected results of these analyses for the homodyne are summarized in Table 4.1. These calculations indicate that the value of Kl for a highly polydisperse sample is dependent upon the degree of the polynomial used in the cumulant method of analysis.

2 3 4

NcAt x l O3

5 (s)

Fig. 4.3. Asymptotic analysis of a correlation function composed of 75 exponential decay

functions. Shown in the figure are the single exponential analysis decay rates for the 75-

exponential correlation function, with an ampli tude distribution shown in Fig. 4.4. The set of

values K,dpp (NcAt) was analyzed by Eq. (4.5.2) (cf. text for results).

0 I 2 3 4 5 6 7 8 9 1 0

Hi x l O- 3

( s "1)

Fig. 4.4. Comparison of distribution types. at = relative ampli tude (intensity-weighted),

= amplitude-weighted decay rate, «, = relative number distribution. The correlation func-

tion with the above ampli tude distribution was analysed by the asymptotic analysis method, the

results for which are shown in Fig. 4.3.


As more polynomial terms are added in the analysis, the value of Κγ

"oscillates" about the theoretical (homodyne) value of 2 χ 3355 = 6710 s- 1

. The values of Kx obtained from first-order cumulant analysis at the various

data collection intervals were used in the asymptotic analysis method and are shown in Fig. 4.3.

It is emphasized that expansion methods do not reflect the actual distribution in amplitudes ah but rather the distribution in decay rates, i.e., affi. The relative distributions at (weighted amplitudes), afyf (weighted decay rates), and n{ = aJR^P^R^ (number distribution for hard spheres) are illustrated in Fig. 4.4 as a function of the decay rate y, for the unimodal distribution under discussion. This plot shows the dramatic effect of the M p on the weighted average decay rate obtained when expansion methods are applied to highly polydisperse data. This plot also underlines the importance of eliminating dust if data are to be collected at very low scattering angles.

Example 4.4. Experimental Verification of the Numerical Equivalence of the Cumulant and Asymptotic Analysis Methods Patterson and Jamieson (1985) used the cumulant method to analyze their data on poly(acrylamide) (PAAm). Values for Kx and σ were determined for a single correlation function for various extents of decay, using only the first 10, 15, 20, 25, and 30 points, respectively, for Δί = 100 ps. The reported values of Kx and the ratio K2/2K \ are plotted in Fig. 4.5 as a function of the total time

4 .8H

τ 1 Γ I 2 3

t (mi Hi seconds)

Fig. 4.5. Cumulant analysis of PAAm for selected extents of decay. [Reproduced with permission from Patterson and Jamieson (1985). Macromolecules. 18, 266-272 . Copyright 1985 by the American Chemical Society.]

4.6. Lambda Depression Analysis 85

window, Nc At. [Their Fig. 2 is labeled K2/K2, whereas the data plotted are the

ratios K2/2K\. Their text reference K2/K\ = 0.4 is correct (A. M. Jamieson, personal communication).]

The polydispersity ratio can also be computed from the X a p p v. t curve using the asymptotic analysis approach. Using the 3 ms and intercept values, the slope of this curve is (4.6 - 3.5) χ 100/(3 χ 10~

3) = 36,670 s

- 1 - K2/2. The

polydispersity ratio K2/Kj is then 73,340/(460)2 = 0 . 3 5 , which is in good

agreement with the reported value σ = 0.4.

4.6. Lambda Depression Analysis The correlation functions may exhibit a high noise level as well as a high degree of polydispersity. It is of interest to introduced at this point the lambda depression (Ad) analysis method proposed by Isenberg and Small (1982) for the analysis of fluorescence decay times. In this method C(X, t) is multiplied by the function exp( — λή to define an exponentially depressed function F(A, X, t):

F(À9K,t) = exp(-Àt)C(K,t). (4.6.1)

F(À, Χ, t) differs from C(X, t) only in that the decay rates but not the relative amplitudes are modified. This is in contrast to the Guggenheim method, in which the decay rates are unaltered but the amplitudes are changed (cf. Problem 4.4). The nth moment of F (λ, Χ, ί), Μ„(λ, Χ, Τ), is defined as

Μ„μ ,Χ , Τ) = τ

t"F(À,t)dt (π = 0,1,2, . . . ) , (4.6.2)

where NcAt = Τ is the total time window for the Nc + 1 point correlation function. Since the function F (λ, X, t) is truncated at the value T, one must perform numerical integration in the evaluation of the moments. It is only in the limit at which F (λ, X, t) decays to the baseline (effectively infinite time) that Μ„(λ, Χ, Τ) becomes independent of the choice of time windows. Upon changing variables [y,(X) + λ]ί = x f with y0(K) = 0, the working equation then becomes

A t n m ν aiWYiin) , BY0(n) . . . . .

where

/*x;(max)

Yi(n) = J xni exp( - x f) dxt. (4.6.4)

It is recognized that when Xj(max)-ôo and hence is indepen-dent of the decay rates y f. This limiting expression for Y^n) is valid only when C(X,i) decays to the baseline. For truncated functions F(A,X,i),


the evaluation of Μη(λ,Κ,Τ) and Y^n) must be determined by numer-ical integration to Xj(max), which is operationally defined by x^max) = — ln(F(/l, Χ, Ν At). Hence the numerical value of x,(max) is dependent upon the choice of λ.

The 2-depression analysis method tends to suppress any residual baseline and random-noise contributions to the correlation function without affecting the relative amplitudes [cf. Eq. (4.6.3) for the larger values of λ].

A measure of the polydispersity in C(K, t) is determined from the moments. Assuming that Β = 0, ax(K) = 1 and at(K) = 0 for i > 1, the decay rate from the nth moment is

K = Y(n)

Mn(lK,T) - λ. (4.6.5)

If the decay function is truly monodisperse with respect to decay rates, then kn = k± for all η > 1.

One can in principle obtain all 2M + 1 values in the sets (α,(Χ)} and {y,(X)} and the baseline from Eq. (4.6.3) and the normalization constraint Σ at(K) = 1. The "Λ-test" for the correct choice of variables is "local flatness" in the plot of ki vs. λ. An incorrect choice of a^K) or Àt(K) will cause the computed theoretical moment to change with λ differently than the moment computed from the data.

4.7. Z-Transform and Method of Spike Recovery

Szamosi and Schelly (1984) proposed a Z-transform and spike recovery method for analyzing decay functions composed of a sum of exponential decay processes. As in the case of the lambda depression analysis method, the Z-transformation analyzes an Nc + 1 point, exponentially depressed function F(s, mAt) = zS(mAt), where ζ = exp(sAi), and

M

S(mAt)= Σ ^ e x p i - ^ m A i ) . (4.7.1)

i= 1

Defining qt = exp( —ytAi), the Z-transform of S(mAt) becomes

m ce fn\m M

a ζ

Z[S(mA0] = Σ Σ Μ- = Σ —^· (4·7·2) i=i m = ο V ζ ) i = iz — qt

Operationally, however, the summation in Eq. (4.7.2) does not cover an infinite time range, but rather arrange up to the finite time window Nc At. Szamosi and Schelly (1984) therefore introduced the correction term Δ to be added to Z[S(mAi)] :

4.8. Linear Programming Method

The following computational algorithm was proposed by Szamosi and Schelly: (1) provide initial guesses for the parameters at and yf for the η functions; (2) add a "spike" a sexp( — ysmAt) to S(mAt) to provide a reference function for the purpose of providing a criterion for selection of the correct set of parameters; (3) calculate Z[S(mAt)~] for (2M + 2) different values of z; (4) calculate Δ for each ζ using Eq. (4.7.3) and add to the appropriate Z[S(mAi)] ; (5) calculate the difference Δα between the terms in step (4) and the values of S(mAt) obtained from Eq. (4.7.1); (6) vary the values of a{ and y ι to minimize £ ( Δ α )

2; (7) repeat after increasing the number of steps M

by 1. The criterion for termination at M functions is that the parameters that described the added spike must be recovered.

In contrast to the /-depression analysis procedure, in which the equations are adjusted to describe the finite time window associated with the actual data, the prescription of Szamosi and Schelly is to adjust the experimental data to fit the limits of the theoretical expression. The finite sum can be performed exactly, with the result.

Clearly Eq. (4.7.5) is identical to Eq. (4.7.3) subtracted from Eq. (4.7.2). Optimization of ah qt, and η by direct application of Eq. (4.7.5) in fitting C(K, t) is equivalent to steps 4 - 6 in the algorithm given above.

4.8. Linear Programming Method

Zimmermann et al. (1985) applied linear programming (LP) methods to the analysis of dynamic light scattering data. It is assumed that the distribution of decay times G(t) in Eq. (4.3.1) can be represented by histograms of width Δτ,· about the central relaxation time <Ty>. C(K, t) given by Eq. (4.0.1) is expressed as a sum of first-order exponential decay functions:

where (sign) is the sign of the difference C(X, i) - B' and the amplitudes xt are proportional to G « t 1 » A t 1 . It is emphasized that x f is a coarse-grained representation of the data that is equal to the average value of at in the range Δτ,. A set of discrete functions Yj is defined by

(4.7.5)

(4.8.1)

M Yj = Σ x fexp + B, (4.8.2)


which are to be matched with the corresponding experimentally determined functions y(tj) by minimization of the sum of the absolute value of the difference Δα,· = y(tj) — Yy

ζ = Χ |Δα,|

with the linear constraint

M

y(tj) = Σ * «e x

P I = 1

+ Β + Δα,·.

(4.8.3)

(4.8.4)

We define Β = x M + l - xM + 2 and Δα,· = x M + l +j - xNc + i + M +j to give the reduced form

2 — Σ (X

M + 2+J X

N C + M + 2 + 7)5

with the constraints

M + 2 + 2NC

Σ XI

AJ , I = Y I T J ) Ί = { i , ^ c } ,

where we have defined

^ M - = exp T;

i = { l , M } ; ; = {l,JVc;

^7, Nc + 1 — —

^ j , Nc + 2 ~ 1 »

^ j , M + 2 + I — A

J , N C + 2+ M + I =

^ J I ·

(4.8.5)

(4.8.6)

(4.8.7)

(4.8.8)

(4.8.9)

The problem is now to solve Eqs. (4.8.5) and (4.8.6) for xh B, and Δα, for input values of <τ7>. Zimmermann et al. employed the Simplex algorithm (Dantzig, 1963; Chvâtal, 1983) to obtain values for these parameters.

4.9. Inverse Laplace Transform Methods—General Comments

The intensity correlation function C(X, f) can be of the discrete form given Eq. (4.0.1), or it can be represented by a continuous distribution of decay rates whose amplitudes are determined by the distribution function G(y):

C(K, t) exp(-yt)G(y)dy + 1 + s(t). (4.9.1)

The inversion of these expressions represents an ill-posed problem because of the presence of the term ε(ί). The nature of the ill-conditioned behavior has been lucidly presented by Bott (1983) and is summarized here. G (γ) is of

4.9. Inverse Laplace Transform Methods—Genera l Comments 89

the form

= YXAjCOsdiûji) + B j S i n ^ y ) ] ,

hence the integral (/) in Eq. (4.9.1) has the analytic form

(4.9.2)

' = Σ οή + t 2 + οή + t (4.9.3)

It now becomes clear that the high-frequency components may make a large contribution to G(y) but a negligible contribution to C(K,t). The situation is further complicated by the fact that the functions C(K,t) are usually truncated, hence the baseline may not be reached in the time win-dow examined. Premature truncation may lead to distortion of the low-frequency contribution.

From the above discussion it can be concluded that there exists a set of solutions {5} to Eqs. (4.0.1) and (4.9.1) that lie within the experimental noise level ε(ί). Within {5} there may be a subset of solutions {{n}} that can be eliminated because they have no physical significance—for example, those with negative amplitudes. This subset can be immediately eliminated. The problem, therefore, is to choose from the subset of feasible solutions {{ /}} the "correct" solution to the problem.

The degree of success in obtaining a reasonable representation of the distribution function G(y) is dependent upon the assumed functional form implicit in the treatment of the data. McWhirter and Pike (1978) and Ostrowsky et al. (1981) have truncated the frequency ω to the maximum value a ) m a x, which more or less defines the noise level of the data. The choice of γ values is then given by

The exponential sampling recipe can provide an accurate reconstruction of amplitude distributions that have a "high-frequency tail", such as that shown in Fig. 4.4. Clearly, much smaller step sizes must be used on the low-frequency end of the distribution than on the high-frequency end of the distribution in order to resolve the detail. This type of sampling scheme is provided by exponentially spaced steps. On the other hand, the exponential sampling procedure does not provide information of comparable accuracy about a distribution with a "low-frequency tail". Sampling procedures using equally spaced linear step sizes may be more appropriate for these types of dis-tributions. Another approach to reduce the effect of noise is to employ a smoothing procedure that restricts curvature in the function G(y) (Phillips, 1962). A smoothing procedure was employed by Provincher in the develop-ment of C O N T I N (Provincher, 1982a, 1982b). Such a procedure, however,

π (4.9.4) 7n = 7„ - i exp ω, max


assumes a relationship between adjacent and neighboring points in the distribution function. It may be difficult to resolve bimodal and multimodal distributions in G(y) since the smoothing reduces fluctuations in the curve.

One must be cautioned that when the data are massaged, unwarranted correlations between points may result, thereby placing a bias on the optimal solution to the problem. Truncation of the data at c o m a x, for example, makes the unjustified assumption that the coefficients of the higher-frequency components are zero. The maximum entropy method (MEM) (Gull and Skilling, 1984; Skilling and Bryan, 1984; Livesey et al., 1986; Livesey et a l , 1987) is based on the premise that the less information contained in the data, the better the results, since fewer artifacts are introduced in the analysis. A portion of the "kangaroo" problem (Gull and Skilling, 1984) is used to illustrate this point. Given that | of the kangaroos have blue eyes and | of them are left-handed, what proportion of the kangaroos both have blue eyes and are left-handed? A positive correlation of these data is that all left-handed kangaroos have blue eyes, or that y of the kangaroos have both characteristics. A negative correlation of these data is that no kangaroo has both character-istics. Both of these answers require information not given in the original data set—i.e., left-handedness and blue eyes may be genetically coupled. A "safe" response is that there is no correlation between the color of the eyes and the handedness of the kangaroos; hence (I)(I) = I of the kangaroos have both characteristics.

The objective in the M E M is to maximize some function of the distribution that does not introduce correlations between the data. That is, any corre-lations between data points must be intrinsic in the data points themselves. The Shannon-Jaynes entropy is the only such function (Gull and Skilling, 1984), which Livesey et al. (1986) have expressed in QELS notation as

S= - ΧΧ

Ρ τ1 η ^ , (4.9.5)

where mx is an a priori model for the distribution, which may be taken as 1 /τ with no further information, and

Γ Γ τ + dt

[Ί. G{T)dx " o o ~

G(x)dx 0

Livesey et al. (1986) point out the useful properties of M E M : (1) the ln(x) term automatically makes G ( T ) positive; (2) G ( T ) is smooth; (3 ) features appear only if demanded by the data; (4) the shape of the spectrum is independent of the number of points used to display it; (5) it is robust to noise; and (6) the M E M solution is unique for data that are linear functions of the spectrum.

4.11. C O N T I N and D I S C R E T E 91

G(ar)

mul center 3.674

plier 2 steps 3

ovei lap

i " ι " Ί

A ï x l O (s"') Β

Fig. 4.6. Overlay histogram with exponential sampling analysis. A total of 75 exponential

decay functions comprised both the unimodal (A) and tr imodal (B) distributions, where the

intensity-weighted distributions are shown by the shaded areas in the figure.

4.10 Overlay Histogram Method with Exponential Sampling

Fletcher and Ramsay (1983) proposed an overlay procedure in which a histogram model is assumed, where a single value of the amplitude acts to represent a region in y-space. Their method analyzes directly the heterodyne function g

(1)(K, t) by first subtracting the baseline and then taking the square

root of each point. The variables in this method are: (1) the number of steps in the histogram; (2) the width of each step; and (3) the center value of y-space to be examined. After the mean of the distribution is first determined by the cumulant method, greater resolution is achieved primarily by reducing the size of each step in y-space until undesirable features appear, such as negative values that are indicative of oscillations. Once a suitable histogram is obtained, the number of steps and the step-size are maintained, and additional histograms are generated by shifting the central value of y. These histograms are then overlaid and averaged to obtain the final histogram representation for the intensity as a function of y.

This method was applied to the simulated data described above, where three runs involving three steps are overlaid to obtain the final distribution. Unimodal and trimodal distributions were analyzed by this method, and the results are given in Fig. 4.6.

4.11. CONTIN and DISCRETE

C O N T I N and DISCRETE are packaged programs that are made available upon request (Provincher, 1982a, 1982b). DISCRETE is a program that interprets C(K, t) in terms of discrete decay functions. C O N T I N assumes a continuous distribution of amplitudes as described by G(y).


The C O N T I N program is composed of a fixed core of 53 subprograms and 13 "USER" subprograms. The memory requirement is ~ 2 0 0 Kbytes. Aside from the initial entry of the experimental parameters (scattering angle, viscosity, etc.; the integration limits for the Laplace variable: and the selection of particle structure factors), C O N T I N is not a user-interactive program. The program runs through two cycles, the first an unweighted analysis of the data to select a trial set of parameters, and the second a weighted analysis to select the "best fit" set of parameters. Numerical information and distribution plots are presented.

4.12. Effect of Noise on the Analysis of C(K, t)

A bimodal distribution involving 75 exponential decay functions was gener-ated as described in Section 4.5, where the two distributions of linear spaced decay rates "overlapped" at y = 400 s

1. The "average" decay rate for the

heterodyne correlation function of each mode is, respectively, 2813 s1 and

3399 s1, with an overall average decay rate of 3152 s

1.

Noise was added to each point of the correlation function by means of a random-number generator. Because of the relatively small number of points

16540^8270

5

· - 4

- 3

— ·. 2

=_==1 ( ( [

32 64 delay point

Β

Λ χ ΙΟ" (s"1) A

Fig. 4.7. /-depression analysis of a first-order correlation function with added noise. (A) The effective first-order rate constants as determined from Eq. (4.6.5). The distribution of amplitudes is shown in Fig. 4.8. (B) The solid circles represent the bimodal correlation function that was generated as described in the text, and the solid lines are the Α-depressed functions (the value of λ is given in the figure).

4.12. Effect of Noise on the Analysis of C{K, t) 93

0 1 2 3 4 5 6 7 8 0 2 4 6 8 10

T x l O "3 (S

H) ι—»

Α Β Fig. 4.8. CONTIN and A-depression/histogram analysis of the "noisy" correlation function.

The generation of the bimodal correlation function with noise is described in the text, and the resulting correlation function is indicated by solid circles in Fig. 4.7. (A) The bimodal distribution of amplitudes a, is shown by the shaded area, and selected points for the "chosen solution" by C O N T I N are given by the solid circles. (B) The overlay histogram analysis of the correlation function is shown by the solid lines, while the shaded area represents the combined overlay histogram analysis of the Α-depressed function after correcting for the "Α-shift", γ — (λ/2).

in the correlation function (Nc = 64), it was found necessary to add a constant to ensure that the average value of the added noise was zero. The "lambda depressed" correlation functions are shown in Fig. 4.7, as well as the decay rates computed from the moments of the lambda depressed homodyne correlation function. Since a monodisperse correlation function results in identical decay rates from the various moments, these plots indicate that C(X, t) is polydisperse.

C O N T I N and the overlay histogram methods were used to analyze the "noisy" correlation function. Selected points in the amplitude profile for the chosen solution in the C O N T I N program are compared in Fig. 4.8, with the amplitude distribution used to generate the correlation function. Even though C O N T I N uses a smoothing procedure, the resulting distribution appears to be broader with a wider separation between modes those in the simulated data. It is suggested that the effect of added noise is to distort the distribution to higher frequencies. The histogram results for the "natural" and "exponential depressed" correlation functions are also shown in Fig. 4.8. The "natural" function analysis (solid lines) is also significantly broadened relative to the generation distribution. Note that both the histogram and C O N T I N analysis of these data give similar distributions. The histogram distribution of F(À, Κ, t) (filled area) has been corrected for the ^-displacement for the heterodyne function, i.e., γ = yeffective

—


Example 4.5. Inverse Laplace Transform Study of Mixtures of Cationic Polymers and Anionic Mixed Micelles

Dubin et al. (1985) used static and dynamic light scattering methods to examine the aggregation properties of mixtures of cationic polymers and anionic mixed micelles. The total intensity (turbidity) measurements on the poly(dimethyldiallylammonium chloride) (PDMDAAC) + 20 m M Triton X-100 titrated with sodium dodecyl sulfate (SDS) revealed two "critical" values of the mole fraction of anionic surfactant Y[Y = as/(ns + as), where as = anionic surfactant and ns = nonionic surfactant]. The first critical mole fraction, Tc = 0.22, was attributed to the formation of a soluble aggregate that results from a variation of the surface charge density upon addition of the polymer. At low ionic strength solvent conditions, Yc appeared to depend upon the square root of the ionic strength (7 S

1 / 2). This type of dependence is

indicative of Debye-Hückel screening (cf. Chapter 7). The second critical mole fraction, Yp, was the irreversible process of precipitation.

DLS methods were used to study the polymer/micelle mixtures for Y < Yc

and Yc < Y < Yp. The inverse Laplace transform program was first tested with mixed solutions of poly(styrene) latex spheres of known dimensions, with adequate success. QELS data were then obtained for Yc < Y = 0.32 < Yp

and Y = 0.22 = Yc for the PDMDAAC/Tr i t on X-100/SDS system in 0.4 M NaCl. Data were also obtained for Y = 0.25 without the polymer present. The distributions of the reciprocal diameters are shown in Fig. 4.9. These data show that two populations of sizes exist for Yc < Y < Yp in the presence of the

0.08

l / 2R H (nm 1

)

Fig. 4.9. Inverse Laplace transform results for mixtures of PDMDAAC/Triton X-100/SDS. [Reproduced with permission from Dubin et al. (1985). J. Coll. Inter/. Sei. 105, 509-515 . Copyright 1985 by Academic Press, Inc.]

4.13. Multiple Scattering—Diffusing Wave Spectroscopy

cationic polymer, where the larger size is that of the soluble aggregate (A , solid line). The dual distribution does not exist at the onset of the soluble aggregation state (Y = Yc) for the same system (Δ, dashed line). That the cationic polymer is necessary for the formation of the soluble aggregate is demonstrated by the curve in the absence of the polymer for Y > Yc.

4.13. Multiple Scattering—Diffusing Wave Spectroscopy

It is possible that a single photon may be scattered by more than one particle, or multiple scattering within one particle, before it finds its way to the photodetector. The effect of multiple scattering is to introduce a broader distribution of wave vector components into the view of the photodetector. This is why studies on systems at high volume fractions are carried out with particles that are nearly isorefractive with the solvent (cf. Section 5.19).

Dhont et al. (1985) approached the problem of multiple scattering from the point of view of K-space. The general formulation quickly becomes involved, and iterative procedures were developed only to the level of correcting for double scattering to obtain the first-order results.

Pine et al. (1988) considered the case of multiple scattering from noninter-acting spherical particles as a "diffusion" processs for the photons. Following Maret and Wolf (1987), g

(l\t) was assumed to be of the form

" o o

ι τ s

P(s)exp

JO ~ τ

- τ *

ds, (4.13.1)

where τ is the delay time, 1 / T 0 = M Q with K0 = 2πη0/λ0, P{s) is the probability that the light travels a distance s, and λ* is the transport mean free path. The effective diffusion coefficient for the "diffusive" process of light transport is defined as ϋλ = cÀ*/3. Light, therefore, is envisioned as undergo-ing a random walk of step size s/λ*, in which case the average decay of g

{1)(t) is

e x p ( - 2 i / T 0) .

The light source was first assumed to be a plane source through a slab of thickness L and of infinite extent, and next, to be a point source. g

(1)(t) for the

plane source was found to be . ,_ . F L s i n [ y i r

1 / 2]

γ λ* sinh -(τ, W2 (4.13.2)

where τΓ = 6τ / τ 0 and y = z 0/A*, where z 0 is the initial distance for the diffusion process. (The choice of z 0 is such that y ~ 1.) The expression for the point source is

g{i)(t) =

ξ sinh

sinh(i) (4.13.3)


Ο Ο

detector time

A Β Fig. 4.10. Multiple scattering and its effect on the correlation function. (A) Illustration of the

multiple scattering process in solution. (B) Illustration of the effect of multiple scattering from

point and plane sources, and the results of diffusive wave spectroscopy analysis expressions as

reported by Pine et al. (1988).

where x0 = ( 6 T / T 0)1 / 2

( L / / 1 * ) . The above expressions were tested on experi-mental data on poly(styrene) latex spheres of diameter 0.479 μηι for both plane and point sources. In this analysis the only fitting parameter was 2*, since the results were independent of y under the conditions employed. The values of /I* obtained in the fitting process were 143 μηι (point source) and 144 μηι (plane source). This agreement in λ* for the two geometries indicates the validity of the procedure. A schematic of the multiple scattering process and the resulting correlation functions for the two sources (with the curve fitting results) are presented in Fig. 4.10.

An increase in the polydispersity of the sample results in a decrease in the structure of the solution as indicated by the average structure factor <S(KAD)>. The angle dependence of the z-average diffusion coefficient, <D(K)>_, is also affected by the intraparticle structure factor P(Kd

c\especially

if the distribution of particle sizes ranges from very small (Kdc < 1) to very

large (Kdc > 1).

Polydispersity analysis methods can be separated into two categories: expansion methods that are primarily applicable to paucity disperse data; and Laplace transform methods that are most applicable to highly polydisperse and multimodal data. Application of selected methods to simulated data that are noise- and baseline-free indicates that the methods examined in this chapter can provide very accurate information about the distribution of decay rates, provided that the sampling scheme mimics that of the actual dis-

Summary

Problems 97

tribution. The Laplace transform methods have been shown to resolve a trimodal distribution for these ideal decay functions. Application of these methods to data with simulated noise indicates that the average decay rate and corresponding distribution of decay rates are distorted toward the high-frequency end of the spectrum, as might be expected since the noise frequency-limits the Laplace transform variables. Exponential depression methods somewhat correct for this distortion, but not completely for the circumstances considered in this text.

Problems

4.1. Show that the average diffusion coefficient computed from the cumulant analysis metod is the z-average value.

4.2. In characterization of a polydisperse preparation, one may report the ratio of the weight-average molecular weight « M > w) to the number-average molecular weight « M >„). Assume that the only pieces of equipment you have at your disposal are a dynamic light scattering facility and an ultracentrifuge. What average molecular weight can you compute from the sedimentation coefficient « s > w ) and <D>Z? Can you determine the ratio <M> W/<M>„ from / t i l s, <D>Z, and <s> w?

4.3. Brehm and Bloomfield (1975) examined the first and second cumulant for poly(styrene) latex spheres of diameters 91 nm and 234 nm. For the intraparticle form factor these authors assumed the Guinier form, P(KRS) = exp( — (xRiK

2), where α depends upon the radial density

profile ( = j for a solid sphere). Show that from Kl9 K2, dK1/dK2, and

dK2/dK2, one can obtain the following moments of the distribution:

(/R1), </Τ2>,<Κ>, and (R2y. 4.4. The Guggenheim method of analysis was originally devised to analyze a

single exponential decay function with a baseline. In this method a new function is defined by the difference F(t) = C(t) — C(t + Δ), where Δ is a fixed time interval. Use the expression for C(t) given by Eq. (4.2.1) to obtain the normalized function N(t) = F(t)/F(0). How do the relaxation time and amplitude of the new function differ from those of the original function?

4.5. Use the Guggenheim method to obtain an expression for the new function F(t) (cf. Problem 4.4) using Eq. (4.0.1) as the functional form of C(t) where ε(ί) = 0. Apply the cumulant method of analysis to the new function F(t). What are the relationships between the first and second cumulants of F(t) and C(i)? Discuss how cumulant analysis of F(t) might be instrumental in determining whether or not there is a residual baseline Β in the original function C(t).


Κ 1 2 3 4 5 20 22 24 26 a{ 0.05 0.10 0.15 0.15 0.10 0.05 0.10 0.20 0.10

Sketch the actual distribution of at v. y f, and superimpose on this plot a Gaussian distribution based on the average and variance of the actual distribution. What conclusions can be drawn from this comparison?

4.7. Assume that the correlation function decays to a sufficiently small value that Eq. (4.7.4) adequately describes the Z-transform, and that the number of relaxation functions η is 2. Derive expressions to be used in the least squares method of minimizing the sum

S = z\z[S(mAtn-^ a-^-}\

I z-qY z-q2) 4.8. Many polymer systems tend to self-aggregate. Using the expression of

Hocker et al. (1973),

<ß>MONOMER = WAPP + * + F)> where σ = 2/ ι ^ calculate the polydispersity ratio required for a 10% deviation in < D > a pp from < D ) m o n o m e r. How does this value compare with σ for poly(acrylamide) as reported by Patterson and Jamieson (1985) (cf. Example 4.4)?

Additional Reading

Chvâtal, V. (1983). Linear Programming. Freeman Press, New York.

Dantzig, G. B. (1963). Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey.

Gouesbet, G. and Gréhan, G. (eds.) (1988). Optical Particle Sizing: Theory and Practice. Plenum Press, New York.

Stock, R. S. and Ray, W. H. (1985). J. Polym. Sei. 23, 1393.

4.6. Calculate the first and second cumulant expected for the parameters in the accompanying tabulation.

CHAPTER 5

Dilute to Congested Solutions of Rods and

Flexible Coils

"It writhes!—it writhes.

From The Conqueror Worm by Edgar Allen Poe (1809-1849)

5.0. Introduction

The above quotation could describe flexible polymers, where their internal motion is "wormlike." As the concentration of the solution is increased, the internal motions are profoundly modified by their neighbors and can be envisioned as a "reptation" motion of the chain through the milieu. Under congested solution conditions, "scaling laws" appear to be adequate in describing the chain dynamics.

5.1. The Autocorrelation Function

It is assumed that there are iVp polymers composed of ns identical scattering units, and that internal and center-of-mass coordinates are not correlated. Hence,

<E(R,0) r-E*(R,I)> ~ a s

2ß e J V p < e x p { - i K

r. IR(0) R(0]}>

Χ Σ exp{- /K r . [ r (0) q - r ( I ) P ]} (5.1.1)

99

100 5. D I L U T E T O C O N G E S T E D S O L U T I O N S O F R O D S A N D F L E X I B L E C O I L S

5.2. The Cylindrical Particle—General Development

We follow the development of Pecora (1968) and Cummins et al. (1969) in the analysis of M(K, ή for a cylindrical particle. Let a sphere inscribe the cylindrical particle, where the coordinate axes are fixed in the laboratory reference frame. A point on the surface of the sphere is defined by extension of the major axis of the cylindrical particle. The orientational motion of the cylindrical particle is thus equivalent to the two-dimensional diffusion of a point on a sphere. This system is illustrated in Fig. 5.1.

The diffusion equation for the point on the sphere is

Gi(K,t) = Npexp(-DmK2t)I

where the internal correlation function M(K, t) is

t)M(K, i), (5.1.2)

(5.1.3)

6ΊΡ[Ω(Γ)-Ω(0), t] dt

= A dsin(0)d

30

χ ρ [ Ω ( ί ) - Ω ( 0 ) , ί ] , (5.2.1)

Χ

Y

Ε.

Ζ

χ

Fig. 5.1. Geometry for orientation of rigid rods in laboratory reference frame.

The first < > bracketed term is simply the center-of-mass diffusion term, which decays in accordance with the function exp( — DmK

2t). The second term

depends only on the orientation of the molecule in relation to the laboratory frame of reference. G^K, t) is therefore of the form

5.2. The Cylindrical Par t ic le—Genera l Development 101

where ρ[Ω(ί) — Ω(0), t] is the probability of finding the point within the angle range Ω(ί)-Ω(0) over the time range 0 i, and De is the rotational diffusion coefficient. Substituting K

r · r = Kr cos(6), the average in Eq. (5.1.3) is

M(K, t) = 1

4π

'L/2 dr-

-L/2

0 J 1 L

ρ [ Ω ( ί ) - Ω ( 0 ) , ί] ί/Ω(ί)ί/Ω(0)

'L/2 dr' e x p { - iKr cos [0(0]} exp{iKr' cos [0(0)]}

-L/2

(5.2.2)

Eq. (5.2.1) is the well-known Legendre equation whose solutions are the spherical harmonics, Υ™(Ω):

Υ>»(θ,φ) = Σ c MP7[cos(0)]exp(im0), m=-j

2j M ) !

4π ( j + M ) !

2j + 1 (j - \m\)\

( - l )m ( m > 0 ) ,

(m < 0),

(5.2.3)

(5.2.4)

(5.2.5) m 4π (y + N ) !

and P™[cos(#)] is the associated Legendre polynomial. Hence,

oo j

Ρ[Ω(Ί)-Ω(0),Ί] = Σ Σ ΐ7[Ω(')]ΐ7*[Ω(°)] 7 = 0 m = — j

χ e x p [ - D e ; ( 7 + 1)ί]. (5.2.6)

Since the particle is cylindrically symmetric, integration over the angle φ results in the survival of only the m = 0 terms of the spherical harmonics in Eq. (5.2.3):

M(K,r) = (An) £ (2j + l)Bj(KL)expl-j(j + l )D ei ] , (5.2.7) j = o

where the coefficients Bj(KL) are

Bj(KL) = 1

XL

'XL/2 jj(Kr)d(Kr)

XL/2

The spherical Bessel functions jj(Kr) are defined by

1 7}(*r) =

4π /}[cos(0)] exp[iKrcos(0)] dcos(0).

(5.2.8)

(5.2.9)

Some values of Bj(KL) are given in Table 5.1. Hence Eq. (5.1.2) for a rigid, cylindrical particle becomes

OO G,(K,i) = N „ e x p ( - D m K

2r ) £ (2/ + l)Bj(KL)expl-j(j + l ) D e i ] . (5.2.11)

j = o


Coefficients Bj{KL)a for the rigid rod at selected values of KL

KL\j 0 2 4 6

0.4 0.9956 7.87 χ 1 0 "7

6.27 χ Ι Ο "13

0.7 0.9865 7.33 χ 1 0 "6

3.45 χ 1 0 "14

—

1.0 0.9726 3.02 χ 1 0 "5

1.73 χ 1 0- 10

—

4.0 0.6443 5.50 χ 1 0 "3

8.83 χ 1 0 "6

2.96 χ 1 0 "9

7.0 0.2743 0.0264 4.49 χ 10~4

1.97 χ 10~6

10.0 0.0961 0.0337 0.0033 6.58 χ 1 0 "5

40.0 5.59 χ 1 0 "3

1.83 χ 1 0 "3

7.33 χ 1 0 "4

5.66 χ 1 0 "4

100.0 9.15 χ 1 0 "4

2.83 χ 1 0 "4

1.22 χ 1 0 "4

8.87 χ 1 0 "5

a Calculated from Eqs. (5.2.8) and (5.2.9) where the variable integration step size was in the range

0.01 < AKr < 0.20, depending upon the value of KL.

5.3. Centrosymmetric Particles—the Rigid Rod for KL < 1

A centrosymmetric particle exhibits symmetry such that a rotation of 180° results in an identical configuration of scattering centers. Integration from — L/2 to + L / 2 eliminates the odd values of j in Eq. (5.2.8). As seen from Table 5.1, g

(l\K, t) for KL < 1 can be described by two terms:

_ exp(-DmK2t)lB0(KL) + 5 £ 2 ( K L ) e x p ( - 6 / y ) ]

B0(KL) + 5B2(KL)

Example 5.1. Polarized Light Scattering Study on TMV Tobacco mosaic virus (TMV) is a rodlike virus of length 300 nm with an outside diameter of 18 nm. The virus is assembled as a series of "lockwashers", or distorted two-layered discs, stacked on top of each other until the length of 300 nm is achieved. Each layer of a disc contains 17 identical subunits of protein, or approximately 2130 subunits in the intact assembly. To initiate the assembly process, a single strand of RNA (ribonucleic acid) is folded into a "hairpin structure" with the base-paired stem and the "initiation loop" located approximately 1000 nucleotides from the 3' end of the RNA. The RNA then serves as a "thread" to connect the protein discs.

One of the pioneering studies on TMV using DLS techniques was reported by Cummins et al (1969). A spectrum analyzer was used to obtain the power spectrum of TMV at neutral pH over the range 20° < θ < 120°, where both homodyne and heterodyne techniques were employed. A plot of the half-width as a function of sin

2(#/2) is shown in Fig. 5.2.

Table 5.1

5.3. Centrosymmetric Par t ic les—the Rigid Rod for KL < 1 103

900-j ο

70θΗ homodyne

ο

ο ο „ ,ο •

heterodyne

0.2 0,4 ο. 0.6 0,8 ι.ο

s in2(9/2)

Fig. 5.2. Homodyne and heterodyne linewidths vs. Κ2 for tobacco mosaic virus. [Reproduced

with permission from Cummins et al. (1969). Biophys. J. 9, 518-546 . Copyright 1969 by The

Rockefeller University Press.]

The value of Dm was computed from the half-width values for θ < 60°. Using this value of Dm for θ > 60°, the adjustable parameters are then B0(KL), B2{KL\ and De. The reported values are: Dm = 0.28 ± 0.006 χ 1 0 "

7 c m

2 s "

1;

and De = 320 + 18 s "1 at 25°C. This value of D m is approximately 25%

too low when compared with the presently accepted value of Dm ~ 0.4 χ 10~

7 c m

2 s

_ 1. In contrast, D0 is in very good agreement with the currently

accepted values.

Example 5.2. Rotational and Translational Motion of Supercoiled

Plasmids are circular duplex DNAs that occur naturally in bacteria and used extensively in recombinant D N A research. Because they are closed circles, certain topological constraints are imposed on their supercoiled structure.

Lewis et al. (1985) used DLS and transient electric birefringence (TEB) techniques to study plasmid DNAs for which 1.5 χ 10

6 < M p < 8.4 χ 10

6

Daltons. From the relationship s20 = 7.44 + 2.43 χ 1 0 ~3M

0 , 58 for super-

helical plasmids (Hudson and Vinograd, 1969), D m was estimated for the pur-pose of comparison with the QELS data. TEB studies were carried out, in which the free decay of the birefringence signal relaxes with a decay rate 6De. The inverse Laplace transforms (ILT) using C O N T I N and DISCRETE (cf. Chapter 4) were presented as a function of the apparent radius (Rapp). In all cases it was found that only one peak was present for the low-angle data, whereas two peaks were discernible at θ = 90°. Representative curves are shown in Fig. 5.3 for the 2.3 kb plasmid (1.5 χ 10

6 Daltons).

Plasmid DNA


θ = 2 5 . 7 ° θ = 9 0 °

CD TD

CL

Ε <

~Ί Ί ' ' 1

I 3 0 100 1000 30 100 1000

Reff (angstroms)

Fig. 5.3. CONTIN analysis of plasm id DNA. The peaks indicated by the arrows are attr ibuted to artifacts or dust. [Reproduced with permission from Lewis et al. (1985). Macromolecules. 18, 944-948 . Copyright 1985 by the American Chemical Society.]

The value Rapp ~ 400 Â computed from s 2 0 f °r t n e

2.3 kb plasmid compares well with the peak located at 430 Â for the ILP. This peak is the "pure translational diffusion" term with a relaxation time of 206 /is. The value âp P ~ 160 Â corresponds to a relaxation time of 80 /is, or a rotational decay time of l / [ ( l /80) - (1/206)] ~ 130 /is [cf. Eq. (5.2.11)]. This value was re-ported to be in good agreement with the TEB measurements after viscosity-temperature corrections from 4.3°C to 37°C, i.e., 98 /is. The relationships of Broersma (1960a, 1960b) for the friction factor of a cylinder (cf. Section 3.8) give 207 /is and 130 /is, respectively, for the translational and rotational re-laxation time for L = 200 nm and 2RC = 17 nm. In contrast, the peak loca-tions of the ILT curves at θ = 90° were virtually the same for both molecular weight samples. The peak did broaden as the molecular weight was increased, which was attributed to more "visible" internal motions of the higher-molecular-weight plasmids. Another possible explanation suggested by these authors was that the form factor for the purely translational diffusion term became so small at 90° that the slowest of the two modes might represent the first internal mode of the plasmid.

As pointed out in the work of Langowski et al. ( 1986), care must be exercised in interpreting the fast relaxation mode in terms of a single internal relaxation process. As shown in Example 7.12, their biexponential analysis of Gj(K, i) indicated that a reliable value for the rotational time was obtained only for K

2 < 2 χ 1 0

1 0c m ~

2, or θ < 50°.

5.4. Centrosymmetric Par t ic les—the Rigid Rod for KL » 1 105

5.4 Centrosymmetric Particles—the Rigid Rod for KL » 1

Hallet et al. (1985) have computed values for Bj(KL) for 0 < j < 140 and 1 < KL < 300. These computations indicated that no approximation for jj(KL) can be made in the range 1 < KL < 20. For KL » 1, the spherical Bessel functions become "smoothed", and the sinusoidal behavior is not a distinguishing feature of these functions (cf. Fig. 3 of their paper). If the polarizability of the rod was along its cylindrical axis, then most of the scattered light is from molecules aligned with the polarization vector of the incident light. If the orientation process is very sluggish relative to Δί, then θ(ή ~ 0(0) ~ π/2, hence sin(ö) ~ 1 at all times. The process reduces to a planar diffusion equation,

δρίΩ(ή - Ω(0)]

dt = - a

' d2 δ

2 '

ρ[Ω(ί) - Ω(0)]. (5.4.1)

ρ[Ω(ή — Ω(0),f] is therefore Gaussian in both θ and φ. Since jj(Kr) is essentially independent of j for j < KL/2 when KL > 40, G t(K,f) can be written as

Gi(K,i) = J V p e x p ( - D ± K2i )

W exp( — x

2)dx, (5.4.2)

where W = (KL/2)(Det)1/2. The model is shown in Fig. 5.4.

Garcia de la Torre et al. (1984) obtained the following expression for De with end effects:

De = 3kT[\n(p) + y]

(5.4.3)

where they reported that γ = - 0 . 6 6 2 + 0.917/p - 0.050/p2 with ρ = L/2RC,

and Rc is the radius of the cylinder. In the limit of large p, the end effects can be neglected. Using these limiting expressions for De [Eq. (5.4.3)] and D±

[Eq. (3.8.1)], Hallet et al. (1985) showed that Gl (K,r) becomes

(5.4.4) 0ί(Κ,ή = Νρ-(γ) [ e r f ( 71/

2) ] e x p

where Y = (3K2kT/4^0L)[\n(p)]t and the error function is erf(y) =

( 2 / π1 / 2

) β β χ ρ ( - ί2) Λ .

Wilcoxon and Schurr (1983a) examined the relaxation of long thin rods in the limit t 0. Using the first cumulant approach of Pusey (1975, cf. Chapters 3 and 6), they obtained

A, = Σ Σ p = l q = l

at v ^ v ^ e x p l . K ^ Ç r ^ - r ^ O ) ] }

σ,(Κ,Ο) (5.4.5)


only small variations I, in θ and 0

!< I

II

diffusion in a plane

Fig. 5.4. Translational diffusion of very long rods. Because the polarizability of the rod is along its axis and the length of the rods, the model assumes that translational diffusion perpendicular to the rod axis is the only observed mode. The diffusion equation is then reduced to describing the motion of the projection onto a plane.

where the ns elements of the rod have indices that range from L/2 < nx < — L/2. The velocities vz(ni9t) are expressed in terms of their angular veloci-ties about the center of rotation (bead location 0): vz(nh t) = v z(0,0) + d[Az(n i 5i ) ] /di with Az(t) = bnicos[ß(t)'], where b is the separation distance between scattering centers and ß(t) is a time-dependent Euler angle. The laboratory-fixed coordinates were transformed into molecule-fixed coor-dinates, which introduced another Euler angle y(t) into the expression for the angular velocities ωχ> and <xy. Considering the limit τ -> 0 and the slow relaxation time for the very long rod, Wilcoxon and Schurr approximated y(t) ~ y(0). The final expressions in the asymptotic limits are

App = D°P= i(D„ + 2 D J (KL « 1), (5.4.6)

A P P = ßpiat = D±+^De (KL » 1). (5.4.7)

Note that the K2 dependence of the rotational diffusion rate arises from the

amplitudes of the composite functions in the cumulant approach, whereas the K

2 dependence of the translational decay rate arises from the diffusion

process. To illustrate the relationship between the two approaches, correlation

5.5 Irregular-Shaped Particles with Cylindrical Symmetry 107

Table 5.2

Simulated Analysis of Correlation Functions0

L(/mi) eod KL (10 -2/ s ) ( i o -

g ßplat

' cm2/ s ) Dx J>3

25 0.92 149 0.101 0.774 0.6528 1.181 0.1202 0.1171 50 0.93 298 0.015 0.450 0.373 0.685 0.697 0.678

100 2.23 888 0.0021 0.256 0.2106 0.390 0.3749 0.3867

a The correlation functions were generated by the H a l l e t - N i c k e l - C r a i g model using Eq. (5.4.2),

where the diameter of the rod was 0.5 μτη. eod is the extent of decay, defined as NcAt/xc, where tc is

the characteristic single exponential relaxation time. De was computed from Eq. (5.4.3). D° was

computed from Eq. (5.4.6), where D± and were computed from Eqs. (3.8.1) and (3.8.2),

respectively. D p l at was computed from Eq. (5.4.7). Dl and D 3 are the diffusion coefficients computed

from the first cumulant using first-order and third-order cumulant analysis methods, respectively.

functions generated by the Ha l l e t -Nicke l -Cra ig model are analyzed by the cumulant method. The cumulant results are compared with the predicted values of the Wilcoxon-Schurr model in Table 5.2.

5.5, Irregular-Shaped Particles with Cylindrical Symmetry

If a particle is not symmetric about its minor axis, then ^ ( K L ) φ 0 for odd values of j upon integration of Eq. (5.2.8) over the symmetric limits —L/2 to + L/2. Also, the centers of scattering and friction may not coincide, as in a "lollipop" shape where the center of friction is in its stem while virtually all of the light scattering occurs at the head. Such a model was first examined by Koopmans et al. (1979) and applied to T-even bacteriophage by Wilson and Bloomfield (1979b).

Example 5.3. T-even Bacteriophage The T-even bacteriophage has four distinct geometric regions: head, tail, baseplate, and tail fibers. The head region contains nucleic acid and is approximately spherical, with a diameter of 1100 Â. The tail is represented as a cylinder of length ~ 800 Â and diameter ~ 160 Â. The baseplate is disk-like, with a diameter of ~ 300 Â. Of no consequence in regard to scattering intensity because of their low mass are the six tail fibers that are attached to the baseplate. These tail fibers do, however, play a major role in determining the friction properties of the bacteriophage. In fact, the tail fibers are thought to be responsible for the slow (extended) and the fast (contracted) forms (cf. Fig. 9.26).

Welch and Bloomfield (1978) examined the sedimentation and DLS prop-erties of both forms of T-even bacteriophage. Since s T and Dm are related to the mutual friction factor, one has sf/ss = D{/Ds, where the subscripts


0 4 0 80 120 160

distance (nm)

(center of frictional resistance)

τ 1 1 1 « 1 0 1 2 3 4 5 6

Κ χ 10 (cm )

Β Fig. 5.5. Amplitudes of spherical Bessel functions as a function of the distance from the center

of friction resistance for T-even bacteriophage. (A) The first three spherical Bessel functions evaluated at the distance indicated. (B) The angle dependence of the first three spherical Bessel functions when the center of rotation is 30 nm from the baseplate. [Reproduced with permission from Wilson and Bloomfield (1979b). Biopolymers. 18, 1543-1549. Copyright 1979, John Wiley and Sons.]

head baseplate ' " '

5.6. Semi-Flexible Linear Polymers 109

f and s denote the fast and slow sedimentation forms, respectively. The mea-sured ratios were sf/ss = 1000/700 - 1.43 and D{/Ds = 3.85 χ 10~

8/3.05 χ

10~8 ~ 1.16. Conventional boundary spreading techniques (Cummings and

KozlofT, 1960) yielded D{/Ds = 3.40 χ 10"8/2.35 χ 1 0 "

8 ~ 1.45. The appar-

ent discrepancy between the QELS determination of D{/Ds and that ob-tained from conventional methods was resolved by Wilson and Bloomfield (1979b). In both forms, the majority of the light was scattered by the head component, whereas the center of friction resistance was at different locations along the tail for the two sedimenting forms. Using the Garcia de la Torre -Bloomfield hydrodynamic formulation (1977) (cf. Chapter 2), these authors calculated that the center of friction resistance for the fast sedimentation form lies ~ 4 5 0 Â from the center of the head. The center of friction resis-tance for the slow form, however, was computed to lie in the tail ~ 1130 Â from the center of the head. Rotational motion thus contributes to the QELS data for the slow form.

Wilson and Bloomfield (1979b) computed values of (2j + l)Bj(Kd) as a function of distance from the center of friction resistance. Since the j = 0 and j = 1 terms make significant contributions to the intensity, the rotational contribution strongly depends upon the choice of the center of friction resistance, as shown in Fig. 5.5.

Wilson and Bloomfield (1979a) employed downward corrections of 8% and 34% for D( and DS9 respectively, as determined from simulated data that were analyzed by the cumulant method (cf. Chapter 4) as if they were experimen-tal data. The "corrected" ratio Df/Ds was 3.27 χ 10"

8/2.28 χ 1 0 "

8 - 1.44,

which compares with s f/ s s = 1.43.

5.6. Semi-flexible Linear Polymers

Flexible linear polymers may undergo at least three types of internal motion: (1) longitudinal extension and compression along the central axis of the polymer; (2) flexing and bending perpendicular to the central axis of the polymer; and (3) twisting about the central axis of the polymer. The average <exp(z'K · r)> in Eq. (5.1.3) is computed on the basis of a Gaussian distribution for each of the spatial internal coordinates. Hence M(K, i) becomes

M(K,i) = X X e x p - — W(p,q9t) , Ä Ä Γ Κ

2

(5.6.1)

where W(p9q9t) = [r(0)q - r ( i ) p ]r · [r(0)< r(i) p] , and thus

W(p9q9t) = [ r(0)2 + r(0)

2

p - 2r(0)[ · r(i) p] , (5.6.2)

and translation in time is assumed, viz, r(t)2

p = r(O)2,.


5.7. The Continuum Model for Linear Polymers

The continuum model assumes that both the mass and the source of friction are assumed to be uniformly distributed along the chain; it is referred to as the K r a t k y - P o r o d (1949) wormlike chain model. The characteristics of this model are the contour length, L; the mass density, p\ the friction factor per unit length, ξ'; the elastic constant of bending, y; and the elastic constant for stretching, β.

Harris and Hearst (1966) presented a dynamic theory for stiff chains. A point along the chain is described by a space curve r(s, t) = r and obeys the Langevin equation

<32r y dr <34r nd2r

where y = 3LKkT/4 and β = 3kTL/(r2>, and <r 2> is given by Eq. (3.11.6) with

LK = 2Lp. The position r(s,i) and fluctuating force f s(s, t) are written as the transforms

oo

r(s,f)= X Q ( M - q ( M ) , (5.7.2) n = 0 ~

oo

f(s,t) = Σ Q(«,s) .h(s , i ) . (5.7.3)

The tensor Q(n,s) transforms the real space coordinates of the bead or segment into the normal coordinates of the coil.

Following Fujime and coworkers (Fujime, 1970; Fujime and Maruyama, 1973; Maeda and Fujime, 1984a and 1984b), separation of variables in Eqs. (5.7.2) and (5.7.3) gives

/ ^ + ξ,^Μ + λΜηή = ΗηΛ (574)

d4Q(n,s) e2Q(n,s) y~^- - ß ^ T ~ = ^Q(n,s), (5.7.5)

Q(H, 5) = c1 cos(ans) + c2 sin(a„s) + c3 cosh(bns) + c 4 sinh(5„s), (5.7.6)

where the c, values are constants. One now multiplies Eq. (5.7.4) by q (n ,0) r, averages over all initial states, and notes that q(n, 0) is uncorrec ted with h(rc, i); one then obtains

q(n,0) 7 • ^ -

L) = - I — )<q(n,0)' .q(n,t)X (5.7.7)

5.7. The Continuum Model for Linear Polymers 111

hence

<q(n ,0) r .q (n , r )> = <q(n,0) r · q(n,0)> e x p ( ^ ) , (5.7.8)

where τ„ = ξ'/λ„ is the relaxation time for the nth mode. An expression for λ„ is obtained from Eq. (5.7.5) and the free end boundary conditions, d

2Q(n,s)/ds

2 = 0 = y[d

3Q(n,s)/ds

3~\ - ß[dQ(n,s)/ds], from which one has

λη = ybt + ßb2 (5.7.9)

b"

= T ("=

1'

2' -

f o r^ »

1) ' (

5·

7·

10)

' η = 2 , 3 , . . . for — « 1 . (5.7.11) L V L K

The elastic potential for bending is given by

< F > = ^ i | L r W , Jdry ds), (5.7.12)

which for free ends is

<ν) = \Σλ*<q(«.°)T ·q(».ο» = \ Σ Κ. (5.7.13)

Invoking the equipartition principle [<K>„ = 3fcT/2] gives

I k Τ 3 Â T T T

< q ( n , 0 ) r . q ( n , 0 ) > = ^ = ^ . (5.7.14)

Note that τ, hence <q(/?,0) r · q(«,0)>, is proportional to

1 1

Κ 3LKkT 4 3/cTL 2

(5.7.15)

<r 2>

From Eqs. (3.11.6) and (5.7.15), a crossover mode number, wc, is defined in which bending and stretching contribute equally to λη, i.e.,

8 N K (5.7.16) c π 2 [ > χ ρ ( - 2 Ν κ ) + 2 Ν κ - 1 ] '

where N K = L / L K . Modes for which η < nc are expected to decay primarily through longitudinal relaxation, whereas modes for which η > nc are expected to decay primarily through bending relaxation. In the limit NK » 1, one has


nc ~ 0.636iVK. Values of /„ were generated for L = 1 χ 104 Â and 1 χ 10

5 Â

for variable lengths L K . Plots of Χ/λη vs. n/NK are given in Fig. 5.6, where the relative contributions of bending (X = yb„) and stretching (X = ßb

2)

are compared. Combining Eqs. (5.7.2) and (5.7.8) and using the orthonormal properties of

Q ( n , s), one has

^ ( s , s ' , i ) = X<q (« ,0 )r-q (n ,0)>

η

χ | Q ( M , S ) · Q ( n , s) + Q ( n , s') • Q ( n , s') - 2Q(n, s) · Q ( n , s')exp

(5.7.17)

I ι 1 1 1 1 1 1 1 1 1 0 0.2 0.4 0.6 0.8 1.0

η / N K

Fig. 5.6. Relative contribution of bending and stretching components to the relaxation of internal decay modes of a wormlike coil. The theoretical limit for the reduced mode number (n/NK) for the crossover region, i.e., nc/NK — 0.636 as indicated by the arrow, obtains for L/LK — NK> 20. The values of γ and β were calculated for 293°K using information following Eq. (5.7.1), and < r

2> was calculated from Eq. (3.11.6). The values for RG were computed from

the relationship RG = « r2> / 6 )

1 / 2.

5.7. The Cont inuum Model for Linear Polymers 113

There are three parameters that must be evaluated: the amplitudes <q(n,0)

r · q(n,0)>; the transformation coordinates Q{n,s); and the relaxation

times τ„.

Flexible Coil—Free Draining Limit

The flexible coil limit is usually defined by the relationship <r2> ~ LLK.

However, in view of the crossover region defined by Eq. (5.7.16) and the calculations shown in Fig. 5.6, we invoke the inequality ß/yb

2 « 1 for all

modes. In the free draining limit D°p = fcT/£'L, and thus,

T« = £ = 4 ^ (» = 1.2.3....)· (5.7.18)

The free end boundary conditions defined by Eq. (5.7.5) require that c

2 = c

3 = c

4 = 0· Q(

n>s) is therefore Q(w,s) = ( 2 / L )

1 /2 cos(nns/L). The

amplitude <q(n,0)r · q(w,0)> is computed from Eqs. (5.7.14) and (5.7.18).

Stiff Coil—Free Draining Limit

The stiff coil limit is defined by the inequality ß/yb2 » 1 (rodlike limit). The

procedures for obtaining τ„, <q(n,0)r · q(n,0)>, and Q(n,s) are the same as

above, viz,

4 L3

τ n = j j ^ r (n = 2,3,4, . . . ) , (5.7.19) W0LKn*l η - -

hence <qn ,0)r · q(n,0)> = 4L

4/{LK[(n — 1 ) π ]

4} . The normal coordinates are

(Fujime and Maruyama, 1973)

Q ( U ) = l , 3

Q ( M = , L

12\112

L

1 \1 / 2

L

2 ~S

C+

(5.7.20)

(5.7.21)

where C+ = cos(bns) -f cosh(fens), C = cos(bns) — cosh(bns), S

+ = s'm(bns) +

sinh(fr„s), S~ = sin(bns) — sinh(6„s), and bn is defined by Eq. (5.7.11). Soda (1973) suggested that <F> given by Eq. (5.7.12) did not adequately

separate bending and stretching motions. The vector tangent to the contour length is u = dr/ds, with the unit length e = (l/u)(dr/ês). Soda stated that if one were to stretch the chain along its contour, du/ds = d

2r/ds

2 can have a

nonzero component along u while in the longitudinally deformed state, (du/s)

T -e T ^ O . It is only in the limit L - > 0 that Eq. (5.7.12) is valid. The


potential energy given by Soda is

y [L$d^

2

<K> = Ts) '

e ds) + (u- $2ds ) . (5.7.22)

When there is no bending, du/ds is parallel to e and the first term vanishes. The second term vanishes when there is no stretching (u = 1).

5.8. The Discrete Model for Linear Polymers

The discrete model for polymers assumes that the source of friction along the chain is beads connected by massless and frictionless springs. The Rouse model (1953) has no hydrodynamic interaction between beads, whereas the Rouse-Zimm model (Zimm, 1956) includes hydrodynamic interaction using the preaveraged Oseen tensor. The characteristic parameters of this spring-and-bead model are the number of beads, ns; the mass of each bead, m 0 ; the friction factor for each bead, ξ0\ the spring constant, g; and the root-mean-square spring extension, b.

In the Rouse -Z imm model, the restoring force exerted on the ith bead, Fh is assumed to depend only on the relative displacement of the (i + l)th and (/ — l)th beads, viz,

Fxi = g(xi - Xi-i) + g(*i - = g(-Xi-i + 2x f - x f + 1) . (5.8.1)

Expressing this equation in matrix notation, F x = g A · x, where, for nearest-neighbor interactions only,

1 - 1 0 0 0 ·· 0 0

- 1 2 - 1 0 0 ·· 0 0

0 - 1 2 - 1 0 ·· 0 0

0 0 - 1 2 - 1 ·· • - 1 0

2 - 1

0 0 0 0 0 ·· • - 1 1

Zimm (1956) used the hydrodynamic interaction matrix, H,

( ϊ ? ) - = ^ + ( ^ ) 1 / 2 " ' ( 5 · 8 3 )

where ôpq is the Kronecker delta function, b = (3kT/g)1/2 is the root-mean-

square extension of the spring, and h = £ο^81 / 2

/[(12π3)

1 / 277 0] is the draining

parameter. The preaveraged value <l/R 0-> = (1/ί>)(6/π|ί — j | )1 / 2

was used in the Oseen interaction tensor [cf. Eq. (3.7.8)]. The ns beads are assumed to obey

5.8. The Discrete Model for Linear Polymers 115

the Langevin equation

d*r dx mo + to —t + 0 H . A - r = P(t), (5.8.4)

where r is the vector location of the beads in the chain. One now proceeds with a transformation to the normal coordinates. The quantities of interest are the eigenvalues of H · A, viz, (Η · A · α)„ = ληαη. The lowest eigenvalue (n = 0) of the matrix A, hence Η · A, is zero, which corresponds to the center-of-mass motion of the beaded spring structure. The interia term in Eq. (5.8.4) must be retained for this mode, whereas this term may be omitted for the other eigenvalues since these modes are highly overdamped. The equations to be solved are

m o ^ i » + i o ^ M = h (0 , „ , , 5 * 5 ,

+ f 4 q ( n . t ) - J - h ( n , t ) ( n > 0 ) . ,5.8.6) οι ζο Co

Hence, τη = ξ0/ρλη. Zimm (1956) obtained the following expressions for the viscoelastic internal

relaxation times for the flexible coil in the free draining and non-free draining model (solvent interior to the polymer is carried along with the polymer):

6AVfoM ( f r e e d r a i n i n g )) ( 5. 8 . 7 )

n2RTn

2 ν »* ^

T" 0.586KTA;

where the intrinsic viscosity is

(non-free draining), (5.8.8)

36Μρη0

The first few values of ^ a r e : ^ = 4.04; λ'2 = 12.79; and λ'3 = 24.2 (Zimm et al. 1956).

It is noted that upon substitution of <r 2> = nsb2/6 and Dp = /cT/nsd;0, one

obtains

<r 2> ~ — 0 2 2 (viscoelastic). (5.8.10) 6D^n

2n

Comparison of this expression with Eq. (5.7.18) indicates that the viscoelastic relaxation times are precisely half of the Langevin relaxation times.


5.9. The Correlation Function for Flexible Coils

If all of the molecular coordinates are equivalent, then <q(n,0) T · q(n,0)> z = ^<q(n, 0 ) r · q(n, 0)>. Hence the boldface vector and matrix notation is dropped. M(K, t) is then expressed in terms of the normal coordinates:

Μ ( κ , ο = Σ Σ β χ ρ κ

Σ<<?(«,0)<?(η,0)>[ρ 2(Μ + ß 2 M ' ) ] Ο η

χ exp Σ ° )>Ö(^ s)Q(n, s') e x p f — (5.9.1)

Expansion of the Base Exponential (K2(Rl} < 1)

The second base exponential in Eq. (5.9.1) is written as

~ K2 ( — t

exp — £ 0)q(n, 0)> Q(n, s)Q(n, sf) exp —

L 3 » \ τ η ,

Σ — \ Σ<Φ> 0)>6(n, s)Q(n, s') e x p f — M

. (5.9.2)

Recalling Eqs. (3.10.5), (3.11.6), (5.7.14), and the form of Q(n,s), one has the identities in the coil limit:

(q(n,0)q(n,0)y\_Q2(n,s) + ß 2( n , s ' ) ]

k2<r

2

g>-

cos-ι — i + cos^ L

(πη) 2 (5.9.3)

/ nns\ ( nns' 4cosl — cosl —

— (q(n,0)q(n,0)> Q(n,s)Q(n,s') = K2<.R

2

G} V

.1. 2

V

3 (ηπ)

(5.9.4)

In the limit L-> oo, the sum in Eq. (5.9.1) of terms in Eq. (5.9.3) is found in closed form:

Σ

COS nns \ ~ / η , ( nns'

(πη)2

2 (s + s')LK is2 + ( s ' ) 2 ] L 2

' 3 L + L

2

Eq. (5.9.2) is now written with reversed order of summation:

M(K,t) = (~)2 £ PM(K

2(R

2

G},t),

\LKJ M= o

(5.9.5)

(5.9.6)

5.9. The Correlat ion Funct ion for Flexible Coils 117

pM(K\R2

Gy, t)

exPl • 2 * 3 < * â > ) f e Y z S e x H - g ' < * â > (s + s ' ) L x

( s2 + s '

2) L

2 1

M !

/ '\ e x

P Z c o s ( — j c o s ^ —

4 K2< f l

2> ^ / w w

(5.9.7)

where (L/LK)2 is the normalization factor. Since 5 and s' are independent

variables, the double sum in Eq. (5.9.7) can be expressed as the square of a single sum, which is then evaluated by integration. The pure translational diffusion term is defined by M = 0, hence (1/M!)[ ]

M = 1 and

L LILK exp

0

s L ^ V ^ sLK

L I L ds

= — exp( * H erf χ

1 .1/2 (5.9.8)

where erf(j/) is the error function defined by Eq. (5.4.4) and χ = K2(Rq). Combining Eqs. (5.9.7) and (5.9.8) gives

ρ (x) = ^ e x p ( — ^ )<[erf χ V 6

1 .1/2 (5.9.9)

Evaluation of PM(x, t) for M > 0 involves the integrals

I(n) = LjLK exp (sLK

2_ s L *

L

/ nns\

X cos[—jds, (5.9.10)

where I(ri) = 0 for odd values of n, whereas for even n,

i ( . u . . - ( - i r » ( i ) ' " = p [ ï - ^ ] « « { - f

χ1 / 2

ίηπ

1T + 2x

Ti

J (5.9.11)

where i = (—1)1 /2

and Re denotes the real part of the function. Since the present focus is on particle sizes for which χ < 1, we consider only Pt(x, t) and P2(x,t): / _ , N

'-2x\ C X P(

P^x, t) = -^-exp

8 x2

P2(x,t) = —3- exp - 2 x \

ΣΣ J n m [/(η + m) + I(η - m ) ]

2

(nm)2

η

e x p i _ t ^

(5.9.12)

(5.9.13)


where 2cos(a)cos(ß) = cos(a — ß) + cos(a + ß) was employed. Numerical evaluation of PM(x,0) indicates that the only two terms that are of any significance for χ < 1 are P0(x,t) and P2(x,t)n = m=i (Pecora, 1968), where

P2(x,t)n = Ax

2

exp 2x

exp -It

I(2)2. (5.9.14)

For the Gaussian coil for which χ < 1, GX(K, f) becomes

P 0 ( x ) e x p ( - D mX2i )

+ P 2(x ,0)exp DmK2 + — t (5.9.15)

Example 5.4. Observation of the Longest Internal Relaxation Time for Poly(styrene)

Tsunashima et al. (1983a) used QELS techniques to examine the dynamics of poly(styrene) in frans-decalin at 25°C for which < M > W = 5.50 χ 10

6 Daltons.

The correlation functions were analyzed by the histogram method with linear spaced steps (Gulari et al., 1979). The histograms were found to be unimodal for θ < 30° and bimodal for θ > 30°. The values of i i nt were computed for Tint = 2 / [ ( l / T F A S T) — ( 1 / T s 1 o w) ] . The results of these analyses are summarized in Fig. 5.7.

The observation that 1 / [ T s 1 o ws i n2( # / 2 ) ] appears to be independent of

θ supports the assignment to translational diffusion with Dp = 1.84 χ 10"

8 c m

2/ s . The internal relaxation decay rate was also found to have a

weak concentration dependence, but a clear dependence on θ that was attributed to increased contributions of the higher-order internal decay modes. At the lower scattering angles, l / t i nt was associated with 1/τ ΐ9 where τι = 908 ps. This value was compared with τ1 = Μνη0[ΐ(]ΙA^RT. Taking into consideration the factor of 2 for the viscoelastic times, the theoretical values for Al are Αγ = π

2/12 = 0.822 for the free draining model and Ax =

0.586 χ 4.04/2 = 1.184 for the non-free draining model. Using the values M p = 5.50 χ 10

6, M = 222 c m

3 g "

1, and η0 = 1.941 χ 10"

2 g c r r r V

1,

they computed Ax = 1.06. It was concluded that poly(styrene) exhibited a hydrodynamic behavior more closely related to a non-free draining coil than to the free draining coil. However, Al (measured) may contain contri-butions from other internal modes (Nemoto and Tsunashima, personal communication).

5.9. The Correlat ion Funct ion for Flexible Coils 119

τ 1 0 0 .5 1.0

s in2(e /2 )

Fig. 5.7. Contribution of internal modes to D a pp as a function of scattering angle for polystyrene

in /nz/isdecalin. [Reproduced with permission from Tsunashima et al. (1983a). Macromolecules. 16,

584-589. Copyright 1983 by the American Chemical Society.)

Expansion of the Supraexponential

Lin and Schurr (1978) expanded the function exp(-f /T„) to the linear term in time, i.e., approximately 1 — ί/τ„, which gives

Α#(Κ , ί)= Σ Σ x e x

P p=l q=l

<ί(π,0)ί(π,0)>

Χ lQ(n,p)2 + β (π ,ρ )

2 - 2β(π,ρ)ρ(η,«)]

e xP Σ 2<g(tt,0)g(rc,0)> (5.9.16)

For the purpose of mathematical simplicity, we focus on the free-draining beaded spring model, viz, Η · A = A. Hence.

1 Γ [{lp - \)nn Q

2(n,p) = — <cos'

Q(n,p)Q(n,q) = —<icos (p + q — \)

nn

+ cos

+ lj>, (5.9.17)

(p - q)nn >. (5.9.18)


For η > 0, the normalization factor for Q is ( 2 / n s )1 / 2

. Making the substitution (q(n,0)q(n,0)) = kT/Ag s in

2 {nn/2ns) and noting the closed form for the sum

2nkr\ Λ 1 — cos

1 "v ;1 \ ns J

w, k Σ

sin nk

2r r ~n~

s (5.9.19)

gives for M(K, t) in the free draining limit

* „ v ι ( + K2kTt\ »- f K

2kT

M(K,i) = exp — — - χ £ e x p {

(5.9.20)

The center-of-mass motion is identified with the lowest eigenvalue of

(5.9.21)

(H · A · a)„, from which the diffusion coefficient D°p is defined

Ço"s ρ= 1 9=1 Ço^s

where ν 0 is that of Zimm (1956). Collecting terms gives

D° kT Gi(K,i) = Np exp i—K t>n*

exp K

2kTt 1 « s « s

+ - Σ Σ e xP

ns p=l q=l

pïq

tkT . \P~q\

(5.9.22)

D t = Dp — (fc77£0ws) is the effective translational diffusion coefficient. Since D t = 0 for v 0 = 1, the translational diffusion process does not contribute to the initial decay of G^K, t) for the free draining polymer.

As Κ oo and ns » 1, the double sum in Eq. (5.9.22) does not contribute to Gi(K, t). Substitution of b

2 = 3kT/g into Eq. (5.9.22) gives as the criterion of

large Κ limit the inequality K2b

2/6 » 1. D p l at computed from Kl is

kT D p l at = —- (segmental diffusion, K

2b

2 » 1). (5.9.23)

Example 5.5. Segment Diffusion for Intact and Nicked DNA Thomas et al. (1980) examined intact and nicked linear 029 DNA under varying conditions of pH and ionic strength. 029 D N A has a molecular weight of 11.5 χ 10

6 Daltons and a G + C content of 34%. The disrupted

bacteriophage were treated with proteinase Κ for 4 - 6 hours at 37°C. The DNA was extracted twice with phenol at 0°C in standard saline citrate

5.9. The Correlation Funct ion for Flexible Coils 121

(SSC = 0.15 M sodium chloride, 0.015 M sodium citrate), precipitated with 3 volumes of 95% ethanol, spooled on a glass rod, redissolved in a buffered solvent (0.1 χ SSC, 2 m M EDTA, 0.05% chloroform), and stored at 5°C. The DNA was characterized in both neutral and alkaline dilute (0.3%) agarose gels. The presence or absence of single strand nicks was tested on pH-denatured samples by gel electrophoresis and the measurement of D° under high salt (1 M NaCl) conditions. The number of single strand nicks (21) was estimated from the molecular weights computed from the Svedberg equation for both the native and the denatured samples of the nicked DNA. By using an argon ion laser with UV optics, these authors were able to extend the range of Κ values to ~4 .4 χ 10

5 c m

- 1. Plots of D a pp vs. K

2 for intact and nicked φ29

DNAs are shown in Fig. 5.8. There does not appear to be any difference in the Κ dependence of D a pp for

the intact and nicked D N A at pH 9.5. Presence of single strand nicks does not significantly affect the dynamics of the DNA. This observation strongly argues that inter strand hydrogen bonds and/or base pair stacking interactions are capable of maintaining the duplex character of the DNA for pH values below the denaturation pH. As the pH is raised to 10.25, the segmental diffusion co-efficient decreases by a factor of about 2, whereas D p remains unchanged. To

0 2 9 DNA

10 (m )

A Β

Fig. 5.8. D a pp vs. K2 for ( · ) intact and (O) nicked DNA: (21 single strand breaks/chain). (A)

D a pp appears to be insensitive to single strand breaks in the D N A at pH 9.5. (B) D p l at for the single

strand break preparat ion of D N A is lower than D p l at for intact D N A , even though the intercept

values are the same for the two preparat ions at pH 10.25. These da ta indicate that there are no

large-amplitude "breathing" modes in the D N A structure under physiological pH and ionic

strength conditions. [Reproduced with permission from Thomas et al. (1980). Biopolymers 19, 1451-1474. Copyright 1980 by John Wiley and Sons.]


explain this observation, it was proposed that kT/ξο was a measure of the friction associated with a torsion mode (crankshaft-like motion), whereas Dp

was more sensitive to flexion modes of the DNA. These authors also examined linear λ D N A with M p = 31.5 χ 10

6 Daltons

and a G + C content of 49%. There was no difference in the Dapp vs. Κ 2 plots

for these two DNAs, thus indicating that QELS data over the range 1 χ 105

c m- 1

< Κ < 4.4 χ 105 c m

- 1 is primarily sensitive to internal motions of

these high-molecular-weight DNAs. It was also reported that the Z) a pp vs, K2

plots were insensitive to the ionic strength of the solvent over the range 1.0 M NaCl to 0.01 M NaCl. Collectively these data indicate that the DNA does not exhibit large amplitude "breathing modes".

5.10. Estimation of the Rouse-Zimm Parameters

The Rouse-Z imm model for linear flexible chains contains four parameters: D p , ξ 0, b, and ns. The parameter Dp can be obtained from the plot of D a pp vs. K

2 in the zero Κ limit. By computer simulation, Lin and Schurr (1978)

obtained the empirical relationship b2 = 8 /X^, where Km is the value of Κ

for which Dapp = (Dp + D p l a t) /2 . The value of ns is determined from ns = {R^ye/b2. The plateau value of D a pp is of the general form

ßplat = £>P + kT(ns - 1)

bnlJ

2 9 (5.10.1)

where F depends upon the value of ns and is given in Table IV of Lin and Schurr (1978) for linear chains. An empirical expression of F, assumed to be independent of b, is

F = 4.35 + Κ - 5)

2.93 + 0.229(rcs - 5) (5.10.2)

Hence,

kT(ns

A plat p + bnl

12 (5.10.3)

5.11. Internal Modes for Circular D N A — t h e Soda Model

Soda (1984) extended the Berg model (1979) for circular chains to include hydrodynamic interactions. The Langevin equation of motion for the discrete bead j was given as

™ 0 ^ + £ ο ^ - ν ? + Σ Α , . Λ = Ρ;, (5.11.1)

5.11. Internal Modes for Circular D N A — t h e Soda Model 123

where the symbols have their usual meaning as in Eq. (5.8.4), except that the matrix A is the sum of two matrices, one of which contains nearest- and next-to-nearest-neighbor interactions for bending modes, and the other of which contains nearest neighbor interactions for the longitudinal modes. The parameter \f is the fluid velocity at r7 that is generated by beads other than j , viz,

v? = ξοΣ η A c dt — V (5.11.2)

where T° t is the Oseen tensor [cf. Eq. (3.7.8)]. Because of the closed nature of the cyclic chain, matrices A and H both are diagonalized by the unitary matrix U with elements U,„ = ( l / n s )

1 /2 exp(i2nnj/ns). The diagonalized elements are

( V- 1

· A · U)„m = H A m , and ( L P1 · H · U)„m = Gnônm, with

G„= £ tf(|;_*|)c0sfe^

|j-*l = o \ ns where H(\j - lc\) = (H) j 7 c. The eigenvalues are

K = j) 16y0 sin41 — I + 4ß0 s in

2

(5.11.3)

(5.11.4)

where the "0" subscripts denote the discrete model for neighbor beads. Normal mode analysis leads to the equation

d2q(n,t) Y dq(n,t) Ρ d t2

+ ^ ^ J T

1 + ΚΦ> t) = h(n, t\ dt

(5.11.5)

where ρ = m0/b and ξη = ξ0/φοη). Note that ξη now depends upon the mode number n, which for the Oseen tensor is

b "s I 1 £π = 3πι/0^1 + - Σ ( — )cos

1 j = 2 YJI

2nn(j — \)b L

(5.11.6)

where < l / r / 1> is the preaveraged reciprocal distance between the reference bead (index = 1) and the jth bead. Hence τη = ξη/λη. At high Κ values,

App = ^l.t = ^ + Σ 2kT

(5.11.7)

In the limits b -+ 0 and ns oo with bns = L, γ ~ Lp/cT, and β ~ 3/c T/Lp, Soda obtained for G^K, t)\

ÇL/2

Gi(K,i) = 2 L a 2e x p ( - D p

)K

2i ) ds

χ exp Σ AkTK2

3L/L

. 2nsn\ 1 — cos( —-— expl

L / \ τ, (5.11.8)

124 5. D I L U T E T O C O N G E S T E D S O L U T I O N S O F R O D S A N D FLEXIBLE C O I L S

where α is the polarizability per unit length and J K(s) ds was substituted for the summation in Eq. (5.11.6), where K(s) = <l / |r — r 0|> with s being the distance along the chain between the two points r and r 0. Correlation functions were generated with the parameters L = 6.60 μιη, L p = 66 nm, and Τ = 298°K, and App w a s obtained by analysis GX(K, t) as: (1) a single exponential with baseline (Dal); (2) a single exponential without a baseline (Da0); and (3) an initial rate of decay using the first cumulant (D t). The relative ordering of these diffusion coefficients was found to be Dt > Dal > D a 0, thus clearly establishing the nonexponential nature of G^K, t). It was also noted that D p l at was attained at lower values of K

2 for the smaller values of nc. The bending modes thus

dominate at smaller values of Κ 2 as nc decreases. Soda (1984) suggested that

D p l at for D N A may not be a characteristic of the Rouse-Zimm model but rather of its rodlike nature for large K, in accordance with the Wilcoxon-Schurr theory for rigid rods (1983a, cf. Section 5.4).

Example 5.6. Segmental Diffusion of Circular λ DNA Soda and Wada (1984) examined the concentration, ionic strength, temper-ature, and Κ dependence of D a pp for the circular form of λ DNA. The λ D N A was obtained by a phenol extraction procedure which was repeated five times. The integrity of the closed circular form of the D N A was checked by ultracentrifugation, with s2o,w = 39.2 S. This value compares very well with that of 40.5 S for circular λ D N A reported by Dawson and Harpst (1971). Values of Dal and Da0 were determined from the QELS data a s described in Section 5.11 and shown in Fig. 5.9. The good agreement of D p l at with that for

7-1

6 -

5" ο ιΖ Q. 4-

CL °

4

Κ 2 χ ΙΟ"14 (m"2) Fig. 5.9. D a pp vs. K

2 for λ DNA. Single exponential analysis of the correlation function with

and without a baseline gave, respectively, the apparent diffusion coefficients Da{ and Da0.

[Reproduced with permission from Soda, and Wada (1984). Biophysical Chemistry. 20, 185-200. Copyright 1984 by Elsevier Scientific Publishers.]

5.12. Effect of Hydrodynamic Interaction on S(K,ÎO) 125

linear λ DNA reported by Thomas et al. (1980, cf. Fig. 5.8) indicates that the segmental relaxation modes are independent of the conformation of the

polymer. Fitting the Dal values using the Soda theory for circular chains yielded a persistence length of L p ~ 100 nm.

5.12. Effect of Hydrodynamic Interaction on S(K, ω)

By using the preaveraged Oseen tensor, the cross term in the diffusion tensor (cf. Chapter 6) is expressed as a product of averages, i.e.,

<D w, exp (*Kr - rm f c)> = <Dm f c><exp(*K

T . rw f c)>. (5.12.1)

Perico et al. (1975a) used the series expansion expression for the internal modes as defined by Eq. (5.9.2) to examine the effect of hydrodynamic interaction on the spectral density profile, 5(Κ,ω). The series expansion included the terms P 0(x), Λ(χ,2) , P2(x, 1,1), P 2(x,2,2), and P 4(x, 1,1,1,1), where χ = K

2(RQ} [cf. Eq. (5.9.7)]. The number of subunits used in these

computations was 100. The draining parameters used in these studies was h = 0 (free draining) and h = 2. It was found that the total intensity, i.e., P(x) = Σ ΡΜ(

χ)ι

w as n ot affected by the inclusion of hydrodynamic effects,

whereas the individual terms P M(x , i) were significantly affected by the inclu-sion of hydrodynamic interactions. Using the line broadening ratio Δ ω 1 / 2/ Dp Κ

2 as a measure of the effect of internal motions on the spectral density,

this ratio was computed in the absence and presence of hydrodynamic inter-actions to be 1.23 and 1.29, respectively. Likewise, the line height changed from a value of 1.86 to a value of 2.95 when hydrodynamic interactions were included.

Perico et al. (1975b) pointed out that approximation techniques used to obtain the eigenvalues /„ to (Η · A · a)„ may have a profound influence on the physical validity of the Rouse -Z imm model under certain conditions. They defined an effective hydrodynamic radius, r e ff = b(n/3ns)

1/2h, where h is given

by Eq. (5.8.3). A plot of λη as a function of φ = nn/ns for several values of hjn\

12 revealed: (1) a maximum in λη, where 0 m ax depended upon the value of

h/nl12; and (2) for sufficiently large values of h/nl

12, λ η < 0, which clearly

has no physical significance. A plot of 0 m ax vs. reff/h indicated that 0 m ax was constant over the range 0 < re{{/b < 0.43, and monotonically decreased for re{{/b > 0.43. The Rouse -Z imm model therefore appears to break down for values of the draining parameter h/nl

12 > 0.43. It is emphasized that the

expansion of the interaction tensor is at the lowest level (Oseen tensor) in the Rouse -Z imm model, where the beads are assumed to be point sources of friction. Inclusion of higher-order correction terms to the Oseen tensor, such as accounting for the finite size of the beads, may partially overcome the shortcomings of the Rouse -Z imm model and extend the value of 0 m ax to higher values of h.


Akcasu and Gurol (1976) examined the effect of using the preaveraged Oseen tensor in the computation of the initial rate of decay Ω(Κ). These authors carried out the computation with and without the preaveraged Oseen tensor over a wide range of values for Kb and a limited range of values for ξ0/2ηφ, where 2b is the bead diameter. The results of their study indicate that the preaveraged Oseen tensor underestimates Ω(Κ), where the largest deviation occurs in the vicinity of Kb ~ 1. The two methods of calculation appear to become nearly equivalent for Kb > 10 and Kb < 0.1. It was also reported that the discrepancy is more pronounced when the hydrodynamic interactions are large. However, the values of r e f f/ b in this study were 0.5, 1.0, and 2.0. From the works of Perico et al. (1975b) and of Akcasu and Gurol (1976) it can be concluded that the maximum error in using the preaveraged Oseen tensor is an underestimate of less than 15%.

Fujime and Maeda (1985) included hydrodynamic interactions in the Harr i s -Hears t model (cf. Section 5.7). The eigenfunctions for this problem were expressed as a series expansion in Q(n,s). The first order eigenvalues were then

On = Κ + Gm = A„(l + /„), (5.12.2)

where Gnn is an integral that measures the hydrodynamic interaction between two normal coordinates averaged over the chain length. The function /„ varies with the mode number and the value of L/b. From their Fig. 1 for L/b = 100, we estimate: f0 ~ 3.6, J\ ~ 2.3, f2 ~ 1.8, f3 ~ 1.6, and fl0 ~ 0.5.

5.13. Intermediate Κ Region

The asymptotic limits examined above are defined by the inequalities K2

< K G > « 1

(small Κ region) and K2b

2 » 1 (large Κ region). Gj(K,i) for these

regions was found to have a cumulant ΚΛ proportional to K2. The K

2

dependence in the small Κ region is due to the decay rate for translational diffusion, whereas the K

2 dependence in the high Κ region is due to the

amplitudes associated with the internal decay modes. The intermediate Κ region is defined by the dual inequalities K

2{RQ} » 1 and K

2b

2/6 « 1.

The first to examine the intermediate Κ region within the context of neutron scattering was de Gennes (1967) for the Rouse chain. This work was extended to include hydrodynamic interactions (Rouse-Zimm chain) by Dubois-Violette and de Gennes (1967). The limits of long time t - • oo and small scattering vector Κ 0 were employed in these studies. It was found that the half-width at half-height of S(K. Δω) for coherent scattering was proportional to Κ

4 for the Rouse chain and Κ

3 for the Rouse -Z imm chain.

Akcasu et. al. (1980) introduced a generalized projection operator formal-ism for studying the normalized correlation function GX(K, i)/Gj(K,0) over the entire range of Κ values. They examined the initial slope, denoted by Ω(Κ), of

5.13. Intermediate Κ Region 127

the normalized correlation function for the Rouse and Rouse -Z imm chains. In the limits Κ -*• 0 and n s -» oo, the following results were obtained:

1 kT

Co (Rouse chain),

and 1 kT

Ω(Κ) = — — Κ 3

6π η0

(Rouse-Zimm chain).

(5.13.1)

(5.13.2)

It is interesting to note that Eq. (5.13.2) predicts that Ω(Κ) is independent of any molecular parameters! This region is therefore referred to as the universal K

3

region. Schurr (1983) examined the boundaries of the universal Κ 3 region. Using

Eq. (5.9.16) and the preaveraged Oseen tensor, Schurr obtained for D a pp

ξο 2 c , f 6 V /2

* T 1 + U 6n^b ns

ξο 1 +-sd

nc

(5.13.3)

exp

sn= Σ («s s= 1

« s - l

Sd = Σ («s - s)exp -

(5.13.4)

(5.13.5)

Clearly, 1 « (6/π) 1 / 2(ξ 0/6π^ 0/?)(2/η 8)5„ must hold if D a pp is to be independent of ξ0. The evaluation of Sn is achieved by integration if K

2b

2nJ6 > 4 and

K2b

2 « 1. Under these conditions (Schurr, 1983), Sn ~ (12ξ0 /6πη0 (\/Kb% and Sd = nsy/(\ - y), where y = e x p ( - K

2b

2/ 6 ) . If (ns - l)/ns ~ 1, then

A,

1 + 12 1

6n^bJ\Kb

1 + 2y

\ - y

(5.13.6)

If K2b

2/6 « 1, 1 « \2(ξ0/6πη0ο)(\/Κν, and y ~ 1 - K 2 / ? 2 / 6 , then

Eq. (5.13.6) can further be simplified to

/cT

6π>7ο (5.13.7)

In order to place boundaries on the validity of the X 3 region for Ω(Κ), Schurr invoked the result of Perico et al. (1975b). Using 0.43 as an upper limit to r e f f/ b


and defining the onset of the validity of Eq. (5.13.7) when the second term in Eq. (5.13.3) is seven times that of the first, Schurr arrived at the criterion Kb < 0.74 for the lower boundary for the universal Κ

3 region. At the other

extreme, the terms containing the sums are nearly independent of ns for K\Rl) = K

2nsb

2/6 > 7. The Κ

3 region is thus bounded by

/ 4 2 \1 / 2

0.74 > Kb > y—j {K3 region). (5.13.8)

5.14. Brownian Dynamics Calculations

Brownian dynamics methods have been applied to the internal motions of molecules whose subunits interact through a Lennard-Jones potential (Gotlib et a l , 1980), Oseen-Burgers tensor [cf. Eq. (3.7.8); Zimm, 1980], or Ro tne-Prager potential (1969) [Eq. (3.7.9) with σ, = σ,·, Allison and McCammon, 1984]. Allison and McCammon employed a bead-and spring model that included nearest-neighbor and next-to-nearest-neighbor con-tributions. The diffusion equation for the ith bead from its initial position r° is assumed to be

r,.(Af) = r? + ^ Σ »(/ · F

? + ».-(Δί), (5.14.1)

kT c e kT f R..R..

6πη0σ~υ '

v" "

tJ/6^0RU ' R

2j

2σ2

ι Rij

Rij (5.14.3)

and R,(Ai) is the displacement of the ith bead due to random bombardment by the solvent; the other terms are the same as in Eq. (3.7.7). The Hagerman and Zimm (1981) model for D N A was adopted, where σ = 15.9 Â ensures that the hydrodynamic volume for the model system was equivalent to that for DNA of radius 13 Â. The initial chain configurations were generated with the algo-rithm of Hagerman and Zimm (1981), and < r

2> / L

2 was reported to be in

good agreement with that predicted by Eq. (3.11.6) for a range of lengths L and L p as long as ns > 100. The DLS "experiment" appeared to be insensitive to the internal motions, hence also the time step size. This result does not come as a surprise, since KL « 1 for L = 317.1 Â. The simulation yielded a value of Dp = 3.90 χ 1 0

- 7 c m

2/ s , which compares well with the value Dp = 3.89 χ

10~7 computed for a cylinder with end plates [cf. Eq. (3.8.3) and Table 3.2]

for a cylinder of length 371.1 Â and diameter of 26 Â. The apparent lack of sensitivity of Dp to internal motions in this calculation may be attributed to the restoration force and inclusion of hydrodynamic interactions between

5.15. Semidilute Solution Regime for Flexible Polymers: the Blob Model 129

the subunits. For example, Harvey (1979) examined the translational diffusion of freely hinged "once-broken rods" having no hydrodynamic interaction or restoring forces, with the result D h i n g e/ D r od = 1.34. Garcia de la Torre et al. (1985) reported that for a hinged dumbbell, Dp at the Ki rkwood-Riseman level of approximation was ~ 2 5 % larger if hydrodynamic interactions were included.

5.15. Semidilute Solution Regime for Flexible Polymers: the Blob Model

In describing the dynamics of flexible coils thus far, it was assumed that the polymers were independent. This situation may not be achieved in practice for very large polymers. De Gennes has examined the motions of chains in the presence of fixed objects (de Gennes, 1971) and entanglement with other mobile chains for the Rouse chain (de Gennes, 1976a) and the Rouse -Z imm chain (de Gennes, 1976b). Several "length scales" were introduced for particular dynamic situations. The correlation length, ξ€, is a measure of the average distance between topological contacts between the chains. The overlap concentration, c*, is defined when the topological contacts become significant:

c * M ^ = M( l - 3 v ) ( 5 1 5 1)

p Ri M p

3 v

Thus c* scales M ~4 / 5

for a "good" solvent in the Flory limit and Mp

1 for a Θ

solvent. A "blob" is that part of a polymer between topological contact points. Random flight statistics is assumed to govern the chain of nh blobs:

<*è> = ^ , (5.15.2)

where £ b is a correlation length that defines the statistical unit identified as a blob. Excluded volume statistics is assumed to govern the subunits within a blob:

U = L K n l , (5.15.3)

where nK is the number of Kuhn segments within a blob. The concentration of blobs at the onset of overlap is = (Μρ/η^/ξΙ for nh blobs per polymer chain. Hence

£ b = L K[ ( 5 v - l ) / ( 3 v - l ) ] p v / ( 3 v - l ) ( c* ) - v / ( 3 v - l ) 5 ^ , 5 ^

where pp = Mp/(LKnKnh) is the linear mass density of the polymer. This situation is illustrated in Fig. 5.10.

Since Eq. (5.15.4) was derived solely on the assumption of the onset of volume overlap, any length parameter should have the same concentration


dependence. Hence,

< . - « ο & Γ " ~ " <5·

1 5·

5>

For ch < c*, the coils diffuse as isolated particles with Dp ~ Ma

p (a = — 0.5 for a random coil, a = — 1 for a Rouse coil). For cp > c*, the chains diffuse with a collective diffusion coefficient D c o ll (de Gennes, 1976a, 1979):

kT

Dcon = τ r = Ac-p* (cp > c*)9 (5.15.6) 6πη0ξ(:

5.16. Linear Polymers in Concentrated Solutions, Gels, and Melts 131

ro

£ ο

g ιο"ΐ4 oQ-

__y___vZpl__

4 — 0---0---0-

I ι Ulli π—ι ι 11 mi Η—I I I I I III Η—I I I I III!

10 10 10 10'

(mg /m L )

Fig. 5.11. D f a st as a function of concentration for poly(adenylic acid) in 0.1 M NaCl at 25°C:

( • ) M p = 240,000; (V) Mp = 420,000; (O) M p = 760,000. [Reproduced with permission from

Mathiez et al. (1980). J. de Physique. 41, 519 -523 . Copyright 1980 Les Editions de Physique.]

where £ c has the concentration power law given by Eq. (5.15.5), A is a constant, and β is the scaling parameter.

Example 5.7. Dilute-Semidilute Transition for the Sodium Salt of Poly(Adenylic Acid)

Mathiez et al. (1980) reported QELS studies on the sodium salt of poly(adenylic acid) through the dilute-semidilute solution regimes at 25°C, where M p = 2.4 χ 10

5, 4.2 χ 10

5, and 7.6 χ 10

5 Daltons. The experiments

were carried out in 0.1 M NaCl, pH 7.4, to minimize electrostatic interac-tions. The correlation functions were analyzed as the sum of two exponen-tial functions. Disappearance of the slow mode after a period of time was interpreted as evidence for defects in a pseudolattice. The present focus, however, is the analysis of D f a st for the poly(A) system. The plot of Z) f a st vs. c p is reproduced in Fig. 5.11.

The critical concentration c p * is experimentally determined by the inter-section of the extrapolated linear regions of the two regimes. These data are in agreement with the de Gennes theory in that c p * depends upon M p and Z) f a st is independent of M p for c p > cp'*.

5.16. Linear Polymers in Concentrated Solutions, Gels, and Melts: the Reptation Model

It is envisioned that continued increase in the polymer concentration leads to virtual immobilization of the polymer because of interstrand contacts and


entanglements. Such a situation may occur in concentrated solutions, in the movement of polymers through gels, and in melts of polymers. It was proposed by de Gennes (1971) that the motion of a probe polymer through a dense matrix may be described as "snakelike", or reptation, motion about fixed obstacles.

It must be mentioned at the onset of this discussion that there does not appear to be very strong experimental support for the original model of reptation in polymer solutions, as judged by the predicted and experimental power laws. The model does appear to be more applicable to gels and polymer melts, and it is of pedagogical value for the description of the various solution regimes.

In the reptation model, the "free volume" surrounding the probe particle at any point in time is defined as a "tube" defined by the length of the polymer, L, and the distance to the neighboring polymers, ξ{. Although confined to be inside the tube, the polymer remains flexible with dynamics altered by the neighboring polymer network matrix.

One way for the probe polymer to move through the dense matrix is to be carried along with the tube as it is "renewed" by virtue of the dynamics of the confining chains. The "tube renewal time" is defined as Tr = L

2/Dtube

(deGennes, 1971, 1979), where D t u be is the associated diffusion coefficient. Assuming that the tube behaves as a "rigid cylinder", D t u be is a weak func-tion of the axial ratio and may therefore be given as proportional to 1/L [cf. Eq. (3.8.2)]. Since M p is proportional to L, it is concluded that

Another way for the probe polymer to move is by center-of-mass diffusion through a fixed matrix background. The internal relaxation modes of the probe polymer are altered by the "fixed" obstacles, i.e., topological contacts with the polymer matrix. The motion is described as the migration of "defects" along the polymer chain. The defects are assumed to: (1) represent a "stored length" bd; (2) migrate along the chain with a molecular-weight-independent diffusion constant A d; and (3) be conserved in number, nd, in the time course of relaxation. While the density of defects may vary within the chain, it was assumed to decrease to an average density at the ends of the chain. The relaxation times for a chain of ns subunits moving through a fixed matrix of obstacles is (de Gennes, 1971)

Since A d was assumed to be independent of M p , the longest relaxation time for the defect (n = 1), T d, is proportional to M p . It is clear that for this model Tr » Td in the limit of large M p . The de Gennes model is illustrated in Fig. 5.12.

(5.16.1)

5.16. Linear Polymers in Concentrated Solutions, Gels, and Melts 133

stationary

obstacle

matrix background

Fig. 5.12. Schematic representation of the de Gennes reptation model. A polymer in a con-

gested solution is envisioned as moving through a matrix of fixed obstacles. The polymer is

initially located in a "tube," which is "renewed" when the polymer moves a distance equivalent to

the length of the polymer. The internal relaxation modes of the polymer are defined in terms of the

decay of "defects" that are defined by a fixed number of subunits (b). It is assumed that the tube

renewal time is much longer that the defect relaxation time: TT » T d.

Let us freely associate the "stored length" bd as being an "effective Kuhn length", and assume that there are nd "defect" locations that are conserved through the decay process. The probability that any one of the ns subunits in a chain can be the "origin" of the statistical step is nd/ns. For the random-flight chain, one therefore has <r

2> = {nd/ns)bd. We retain the de Gennes definition

of A d and the relaxation time T d. The expression for D s e lf is

n <r2> nd bjAd pdbjAd

A e l f = = — 7 VÏ = 2 > (5.16.2) Td ns (nsa)

z n

z

sa where pd = nd/nsa is the linear density of defects along the chain. This is precisely the same as Eq. (III.7) of de Gennes's 1971 paper, viz, D s e lf =


pb\ AJN2a, where Ν = ns. If it is now assumed that Ad oc 1/ξ€ and bà oc ξ€,

and that Eq. (5.15.5) is valid, then for the reptation motion,

/ \ ( 2 - v ) / ( l - 3 v ) .

A e p ^ M -2

^ J (5.16.3)

A signature of reptation is D s e lf ~ M p

2. (The — 2 power law is not unique to

reptation, cf. Sections 5.19 and 5.21). Schaefer and co-workers (Schaefer et al., 1980; Schaefer, 1985) have spe-

cifically examined marginal solvents, with the result D s e lf ~ M p

2( c p / c * * ) ~

2-

5.

5.17. The Crossover Model for Congested Polymer Solutions

In the crossover model, a distinction is made between static (ξ0) and dy-namic (ξΌ) correlation lengths: - ( n s )

( 2 v G )^ / 6 - ( M p )

( 2 v o ); and A d =

ΙίΤ/6πη0ξΌ, where Ad is given in Eq. (5.16.1). Z) s e lf for the crossover model is

kT it y2 / v o)

Aelf =7^ , \ , 2 • (5.17.1) \8πη0 ζΌΜ

2

If ξ0 and ξΌ in the congested solution regime are assumed to have a concentration dependence similar to that given by Eq. (5.15.5), then D s e lf is (Callaghan and Pinder, 1984),

where y = y G - y O = [2/(3v G - 1)] - [v D/ (3v D - 1)], Β = (kT/lSn^) χ ( Î G * )

) G( C D * F

D' a n (

^ c

p * is a

critical concentration for the onset of congested solution behavior.

The de Gennes and crossover models are identical for theta and good solvents, but they differ in the rate of attaining these limits because of the Weill and des Cloizeaux (1979) exponents (cf. Section 3.12). Both models predict D s e lf ~ M -2

Ρ

Example 5.8. Crossover Region of the Self-Diffusion of Poly(styrene) in Carbon Tetrachloride and Deuteriobenzene

Callaghan and Pinder (1984) used pulsed field N M R techniques to obtain Aeif °f poly(styrene) (PS) in carbon tetrachloride. The basic principle behind these studies is that the chain statistics varied with the polymer concentration, as indicated by the reduced temperature-densi ty diagrams of Daoud and Jannink (1976) and Schaefer (1984). Data were obtained for six samples over the range 110,000 < < M p > w < 929,000, with < M p > w / < M p > n < 1.10 for all samples. The power law for D s e lf oc M~

ß was found to vary from β = 1.60

5.17. The Crossover Model for Congested Polymer Solutions 135

I 1 1 1 1 1.0 1.4 1.8 2 .2

log (c'p)

Fig. 5.13. Self-diffusion coefficient for poly (sty rene) in carbon tetrachloride as determined by pulsed N M R . The self-diffusion coefficient for poly(styrene) of molecular weights 110,000 (O),

233,000 ( · ) , and 350,000 ( χ ) appears to follow the predictions of the crossover model for flexi-

ble polymers [cf. Eq. (5.21.2)], provided the critical concentrat ion is a function of the molecular

weight (shown in the figure by arrows). The power dependence of the concentrat ion for the theta

( — 3.0), marginal ( — 2.5), and good (—1.75) solvents are shown in the figure. [Reproduced with

permission from Callaghan and Pinder (1984). Macromolecules. 17, 431-437 . Copyright 1984 by

the American Chemical Society.]

(63 k g / m3 solution) to β = 2.1 (200 k g / m

3 solution). D s e l f as a function of c" is

shown in Fig. 5.13. It was concluded that only those data pertaining to θ-like conditions ex-

hibited the M p

2 dependence as predicted by the reptation model. Callaghan and

Pinder emphasized that regions reflecting the power laws —1.75 and —3.0 are evident in the data, but that there is no clear region showing the —2.5 dependence predicted for marginal solvents by the Schaefer et al. (1980). Since the data in Fig. 5.13 are at concentrations where D s e lf does not display the Μ ρ

2 dependence required for reptation, the PS dynamics in these solvents may

not be reptation motions.

Example 5.9. Diffusion of PEO in Aqueous Solution Brown (1984) reported QELS and pulsed N M R studies of poly(ethylene oxide) (PEO) of low polydispersity in 0.1 M NaCl as a function of P E O concentration and M p (40 χ 10

3 < < M p > w < 6.1 χ 10

5). Dust was eliminated

by a continuous closed filtration method through 0.22 pm Millipore filters. The functions #

( 1 )(K, i) were obtained at θ = 90° and 25°C. Z) f a st and D s l ow were


determined for values of Δί that differed by factors of over 100. Plots of ln[D a p p(25°C)] vs. ln(c p) are reproduced in Fig. 5.14.

It is clear that for solutions of finite concentration, D s l ow (QELS) φ D s e l f(NMR) . It was reported that D s l ow - ( c p ) "

1 3 for M p = 1.5 χ 10

5 and

Aiow ~ (cp)

20 f °

r Mp = 6.61 χ 10

5. In regard to the M p dependence,

D f a st oc M p

0 , 5 8, whereas there was a high degree of curvature in the plot of

log(Aiow) v s

- !og(M P) at c p = 37 k g / m3.

The effective activation energy for the dynamic process (AED) was found to be ~ 15 kJ m o l

- 1 for D f a s t; ΔΕΌ was independent of the concentration. Since

ΑΕΌ was close to the activation energy for water, D f a st reflects solvent resistance to the polymer motion. In contrast, ΑΕΌ for D s l ow was ~ 24 kJ mol "

1

ο

- 1 0 -

- I I -

- 1 2 -

- 1 3 -

-14 —

Β

'fast

V Mp = 661,000

^Pslow

Ί I 1 1 ί 10 20 t 30 40 50

Cp (kg /m3 )

-II ι

-12-

-I4J

c

^fast

N. y r e p

Dslow)É

ι — I Γ 0 1 2

log(Cp)

Fig. 5.14. Diffusion coefficients for PEO in 0.1 M NaCI as a function of polymer concentra-tion. (A) The diffusion coefficients obtained from Q E L S (Z) f a st and D s l o w) and pulsed N M R (Dsel{)

techniques as shown above for M p = 148,000. (B) The diffusion coefficients obtained from Q E L S (D f a st and Aiow) and pulsed N M R (Ac l f) techniques as shown above for M p = 661,000. (C) Comparison of the experimental values with the predictions of the de Gennes reptation model. [Reproduced with permission from Brown (1984). Macromolecules. 17, 6 6 - 7 2 . Copyright 1984 by the American Chemical Society.]


and ~ 3 8 kJ m o l "1 for concentrations of 10.5 kg m ~

3 and 25.6 kg m~

3,

respectively. The ratio of the amplitudes for the slow and fast modes, As/Af, was reported

to be constant over a long time period, but AJA{ decreased with an increase in temperature. Brown chose not to interpret this observation as a "melting out" of the entangled species in which A{ increases at the expense of As, but rather followed the interpretation of Chu and Nose (1980) where the ratio changed due to expansion of the chain dimensions and concomitant increase in the "pseudogel" domain where " . . . the slower motions of entangled coils receive less emphasis". The precise meaning of this statement is not clear on the basis of the information given in the paper. That is, both the melting out and the "pseudogel" models should lead to a decrease in the relative amplitudes AJA{, depending on whether a two-state or three-state solution system is postulated. If a two-state model is proposed, as might be inferred from the discussion of only fast and slow modes, then the transfer of polymers from the fast mode to the slow mode must act to increase the ratio AJA{ at any fixed angle. On the other hand, if one has a third phase that was immobile, thus providing a "background" signal manifested as a baseline, then both Af and As would decrease in absolute value as more molecules enter into the "gel-like" phase. It would appear necessary, therefore, to examine the angle dependence and the "absolute" values of the amplitudes As and Af to assist in the interpretation of these quantities.

Example 5. JO. QELS Studies on Gelatin Solutions and Gels Yu and coworkers (Amis, Janmey, Ferry, and Yu, 1983; Chang and Yu, 1984) used QELS and forced Rayleigh scattering methods to study solutions of pigskin gelatin above and below the melting temperature. The solvents used in these studies were water and water-glycerol mixture containing 60, 73, and 9 1 % glycerol by weight. The water content of the "dry" gelatin (12%) was taken into consideration.

In the QELS studies, warm gelatin solutions were filtered through Gelman Metricel 0.45 pm filters into cylindrical scattering cells. Because of trapped dust and microscopic bubbles in the gels (T = 5°C, 12°C, and 20°C), the correlation functions were analyzed as heterodyne functions. Data were collected over the range 5000 ps > At > 0.5 ^s , with the result 2 χ 10

4 s "

1 < l /τ < 4 χ 10

4 s

- 1 (viz, a narrow relaxation distribution). In

contrast, ^( 1 )

(K, i) for the melted gelatin at 35°C exhibited two relaxation domains. Single exponential and second-order cumulant analyses were car-ried out, where both domains scaled as K

2. D s e lf for this gelatin system was

determined by forced Rayleigh scattering using fluorescein as a marker (Chang and Yu, 1984). These data are shown in Fig. 5.15.

The intrinsic viscosity for this gelatin preparation was measured to be


~ 64

-2.0 -1.5 -1.0 -0.5 0.01 0.1 1.0 10 100

log(Cp) Wt %

A Β

Fig. 5.15. Diffusion coefficients for gelatin. (A) Results of Q E L S studies on gelatin in the

melted form [ 3 5 C (#,0)] and the gel form [20°C( χ )]. [Reproduced with permission from

Amis et al. (1983). Macromolecules. 16, 441-446. Copyright 1983 by the American Chemical

Society.] B: Q E L S (O, and x ) and forced Rayleigh scattering ( · ) results for gelatin at

35 C. (O, · , and T). [Reproduced with permission from Chang, and Yu (1984). Macromole-

cules. 17, 115-117. Copyright 1984 by the American Chemical Society ( χ ) Reproduced with per-

mission from Amis et al. (1983). Macromolecules. 16, 441-446. Copyright 1983 by the American

Chemical Society.]

[rf\ = 22 mL/g, which gives c* ~ 1 Ι\_η]= 0.045 g/mL. The power law A a s t ^ C p

0 3 lies well outside the range +0.50 to +0.75 for Eq. (5.15.5).

Although D s l ow has no well-defined power law region, it is conceivable that the value —2.0 is the asymptotic limit. [The power laws of +0.30 and —2.0 were also reported for D f a st and / ) s l o w, respectively, by Hwang and Cohen (1984) in the QELS studies on poly(n-butyl methacrylate) (PBMA) in methyl ethyl ketone (MEK).] As in the case of P E O (cf. Example 5.9), Z) s l ow is smaller than Aeif- It

w a s noted by Chang and Yu (1984) that the c p dependence of D s l ow

parallels that of D s e l f. Attention lias thus focused on the dependence of the diffusion constants

on concentration in good or marginal solvents for dilute-to-gel regimes. The good-to-theta solvent conditions can be achieved by varying T, viz,

RO\x,cv) = RO\x){\ +k'mn'm\ (5.17.3)

where τ = (Τ — θ)/θ is the reduced temperature, n'm is the number concen-tration of monomer units, and kH is introduced to account for hydrodynamic interactions (Yamakawa, 1962);

k'm = 2mln'sA2 - kH - vpmp (5.17.4)


Example 5.11. Crossover Region for Poly(styrene) Solutions and Networks in Two Theta Solvents

Munch et al. (1983) reported studies on poly(styrene) (PS) of various mo-lecular weights and concentrations in cyclohexane and cyclopentane. Net-works were made by crosslinking with divinylbenzene (DVB) (Munch et al., 1977). Correlation functions were obtained over the range 20° < θ < 120° and analyzed by a second-order cumulant method. Dapp(T) was examined in the vicinity of the Θ temperature (34.5°C for cyclohexane, 20.4°C for cyclo-pentane). Plots of D c o ll vs. cp' are shown in Fig. 5.16.

As previously pointed out by Han and Akcasu (1981), Munch et al. noted that D c o ll decreases in the θ solvents upon an increase in cp. This observation was interpreted in terms of an expression similar to that of Eq. (5.17.3) but involving c p , in which it was assumed that the term — vpcp was small. It was therefore argued that at the θ temperature kHcp ~ a M j

/ 2c p , hence

DCon/D°p= 1 - a M j/ 2

c p . As shown in Fig. 5.17, DcoU/D^ vs. MLJ

2cp falls

on a universal curve. These authors point out that the network systems as well as the semidilute

systems fall on the same universal curve for Θ solvents, whereas the permanent network values of D c o ll consistently lie above the semidilute solution values of Dcoii - It was suggested that the restoring force associated with long-wavelength modes of gels in θ solvents is determined by the solvent-chain free energy interaction and that there is negligible contribution from the elasticity associated with the cross links.

cyclohexane M n = 130,000

A

-0-'

x 34.5°C ; ? 30.5 °C φ 25 °C Ό 22 °C

10

(mg/m L)

Ω ω Ο

CP

ο

cyc lopen t ane

- 1.8

1-6

1.4

1.2

ι.ο 5 7 0,000 ?N

20.4°C

130,000

Τ Τ 1.5 0.5 1.0

log (Cp)

2.0

Β

Fig. 5.16. Concentration dependence of D c o„ of poly (sty rene) in cyclohexane and cyclopent-ane. The solvent, temperatures, and molecular weights are indicated in the figure. The number of subunits is defined by ns = M p/ 104 , where the monomer molecular weight is 104 Daltons. (A) Cyclohexane; (B) cyclopentane. [Reproduced with permission from Munch et al. (1983). Macromolecules. 16, 71 -75 . Copyright 1983 by the American Chemical Society.]


uco\\

1.0-1

~i—ι—ι—ι—ι—ι—ι—Γ

Fig. 5.17. Normalized hydrodynamic diagram of poly(styrene) solutions at the θ temperature. Cyclohexane at 34 .5°C:(A)M p = 20,000, ( A ) M p = 130,000, ( • ) network system. Cyclopentane at 20.4' C: ( · ) Mp = 130,000, (O) M p = 570,000, ( • ) network system. The number of subunits ns

was given as M p/ 104 , where the monomer molecular weight is 104 Daltons. [Reproduced with permission from Munch et al. (1983). Macromolecules. 1 6 , 7 1 - 7 5 . Copyright 1983 by the American Chemical Society.]

Example 5.12. Temperature Dependence of ξΌ for Poly(styrene) in Cyclohexane

Adam and Delsanti (1980) examined D c o l l(T) for PS in cyclohexane over the range 0 ° C < Τ — Θ < 30°C. It was assumed that RG = n

v

s

Gz

aLK and RO =

nv

s

OT

hLK, where a = 2vG — 1, b = 2vD — 1. The reduced temperature was

defined as τ = ( Τ — θ)/θ. The sum k'm + vpmp was analyzed, where it was argued that fcH ~ (RG}RD/ns and A2 ~ Rl(T)/n

2 ~ τ

3 α, hence 2m\n'%A2 —

kH = Β2τ3α[\ — β 3 τ

( & _ α )] ~ —Bx(\ + ετ), where the Bt values are positive

and independent of τ. It was concluded that the difference 2m\n'%A2 — kH

must change sign at some positive temperature τ 0 if vH φ v G. They also argued that ξΏ = ( T / T * * )

V H / ( 3 V G _ 1 )T

_ 1L k , where τ** is associated with the onset of

chain overlap. The experimental values RD/RD θ were represented as a func-tion of θτ. If the scaling laws apply, then (ξΌ/^D )(6T)

b should be independent

of θτ for an appropriate value of b. The results are given in Fig. 5.18.

The best fit values of b were found to be —0.034 (dilute solution regime) and +0.33 (semidilute regime). They concluded that at Τ ~ θ: (1) ξΏ was insensitive to the distribution of segments within the polymer; (2) vH < v G; and (3) ξΌ is not proportional to ξα.

5.18. Universal Scaling Curves for the Correlation Lengths

As in the case of the reduced equation of state for gases, proper scaling of the solution properties of polymers should reduce all of the curves to a single universal curve. The question arises as to the proper reference parameter. For example, which of the following concentrations provides a universal definition of the overlap concentration c*: R p / M p , RG/Mp, 1 / M , or B^

1!

M/ D °

5.18. Universal Scaling Curves for the Correlat ion Lengths 141

LOCH di lute A

(ΘΖ) -.034

0 .95J

Q or

or

15 " Ί —

2 5

θ τ (°c)

1.21 .0.33

1.14-

1.1 -

semidi lute Β

~1

15 25 ΘΤ ( ° C )

Fig. 5.18 Verification of the temperature power laws of the dynamic correlation length of

poly(styrene) in cyclohexane. The reduced temperature is given by τ = (Τ — θ)/θ. The ratio

RD/RD0 is multiplied by (θτ)α. (A) Dilute solution conditions; (B) semidilute solution conditions.

[Reproduced with permission from Adam and Delsanti (1980). J. Physique. 41, 7 1 3 - 7 2 1 .

Copyright 1980 by Les Editions de Physique.]

Wiltzius et al. (1984) and Cannell et al. (1987) have examined in some detail various scaling relationships for the correlation lengths ζα and ξΌ for PS in methyl ethyl ketone (MEK) and toluene. These authors report that D/Dp vs. /e'OCp results in a remarkably universal curve. These authors also report that <W< D scales simply as c'p' and is therefore independent of the molecular weight. These plots are shown in Fig. 5.19.

The conclusion is that ξΌ is not proportional to £ G . It appears that ξ0 is more sensitive to changes in the polymer concentration than ξΌ since ξΌ/ξ0

increases with c'p\ This relative ordering of sensitivity of the static and dy-namic correlation length is also in agreement with the temperature studies of Adam and Delsanti (1980).


1.0

0.1

4 3 -

2 -

I —

A

Ί—Γτ~η—ι—ττη—ι—ΓΤ~η—ι—ι ι ι r ο.οι ο.ι ι ιο

k"c'n

1 1 I I I Mill 1 I I I Mill Γ Τ Τ

0.1 I 10

Β Cp (mg/mL)

Fig. 5.19. Scaling relationships for the dynamic correlation length. Solvent: Open symbols,

toluene; closed symbols, methyl ethyl ketone. Molecular weights (units of 106 Daltons): (O, · )

26; ( Δ, A ) 7.2; ( Ο, • ) 1.8; ( V, Τ ) 0.3; ( • , • ) 0.0175. (A) Compar ison of the dynamic correlation

length to the hydrodynamic radius. (B) Comparison of dynamic and static correlation lengths.

[Reproduced with permission from Cannell et al. (1987). In Physics of Complex and Sup-

ermolecular Fluids (S. A. Safran and N. A. Clark, eds.). Pp. 2 6 7 - 2 8 3 . Wiley Interscience. Copyright

1987 by John Wiley and Sons, Inc.]

5.19. Probe Diffusion in a Polymer Matrix

To better understand the diffusion process of polymers in the presence of other polymers, it is of value to follow the motion of a "probe" particle. This can in general be done by fluorescence techniques. Theoretical studies by Benmouna et al. (1987) on DLS by tertiary systems indicate that two relaxa-tion modes should be observed in correlation functions regardless of the rela-tive contrast factors. As in the model of Pusey et al. (1982) for a tertiary solu-tion of hard spheres summarized in Section (6.18), the two modes obtained by Benmouna et al. are (1) a collective diffusion mode, and (2) an interdiffusion mode representing the relative diffusion of the two species. It is important to note that these two diffusion modes are predicted even when one of the species is isorefractive with the solvent. Fortunately, the theory tends to indicate that two distinct decay processes are observed only under special conditions, and even then the two decay rates appear to be of comparable value (cf. Fig. 3 of Benmouna et al., 1987).

Forced Rayleigh scattering methods have been used to study D X r of labeled poly(styrene) (LPS) in a polymer matrix to determine the power law DTr ~ M~

ß. Antonietti et al. (1986) examined LPS in the range 1.8 χ

104 < M p < 1 χ 10

5 in a matrix composed of PS in the range 4.0 χ

103 < M p < 2.2 χ 10

5 and also a matrix of crosslinked PS. The result was

β ~ 2.5 for 3.0 χ 104 < M p < 1.5 χ 10

5 and β ~ 2.0 for M p > 1.5 χ 10

5.

Kim et al. (1986) examined D s e lf of LPS in the range 1.0 χ 104 <

M L PS < 1.8 χ 106 in a matrix composed of PS particles in the range

5.19. Probe Diffusion in a Polymer Matr ix 143

1.0 χ 104 < M P S < 8.4 χ 10

6. A major finding was that Dsei{ was dependent

upon M P S until M P S was 3-5 times larger than M L P S. Since the reptation model predicts that D s e lf is independent of M p above c*, this observation was interpreted as substantial tube renewal. It was also found that if M L PS > 1 χ 10

6 and c P S > 10 wt%, then D? r oc M ^ C ^ s , which differs from the

reptation model. In isorefractive tertiary systems, the index of refraction of the polymer in

excess concentration is matched with that of the solvent. Poly(styrene) is used as the probe particle because of its high scattering power (Lodge, 1983; Martin, 1984, 1986; Wheeler et a l , 1987). Lodge (1983) verified that the QELS data for these isorefractive systems does indeed give D s e lf by com-parison with pulsed N M R results.

Example 5.13. Isorefractive Study of the Diffusion of Poly(styrene) in Poly (vinyl methyl ether)/o-Fluor otoluene

Wheeler et al. (1987) examined the DseU for poly(styrene) (PS, M p = 6.5 χ 104,

1.79 χ 105, 4.22 χ 10

5, and 1.05 χ 10

6 Daltons) in polyvinyl methyl ether)/

o-fluorotoluene (PVME/o-FT) . (The PS and P V M E mix at the molecular level as required for these studies.) The temperature was 30.0 ± 0.1 °C, and dns/ôcPYME < 0.001 mL/g .

The early portions of the correlation functions were analyzed by a second-order cumulant method and the Provencher inversion method of analysis. The methods of analysis were checked with simulated data using a Schulz or log-normal distribution in molecular weight.

In view of the fact that P V M E / o - F T acts as a good solvent in regard to mixing with PS, a "universal curve" was anticipated in the treatment of the c p dependence of D p / D p . Plots of log(D p/D p) vs. log(c p/c*) are shown in Fig. 5.20 for "good" and "theta" solvent conditions.

These results were interpreted as a "collapse" of the PS in the presence of the excess concentration of P V M E to such an external that the θ conformation is attained. The power law at the higher concentration regime was — 1.9, which may reflect the onset of reptation.

Example 5.14. Probe Diffusion of Spheres through a Poly(acrylic acid)/Water Matrix

Implicit in the scaling relationship D p / D p = £ D / £ D *s the assumption that the

solvent viscosities cancel. One way to test this assumption is to examine the relationships between η0 and the apparent dynamic radius of hard spheres as a function of the matrix polymer concentration and size.

Lin and Phillies (1984a, 1984b) examined the diffusion of carboxylate-modified poly(styrene) latex spheres (PSLS) in aqueous solutions of poly-(acrylic acid) (PAA). This system is an "isorefractive" system since the scattering of the PAA was reported to be less than 1% of that for the PSLS.


H O Q.

Q on

S " M

- 2 -

- 3 -

good

οχ CA * t ? · χ # t Ä

I -

0-

-I -

- 2 _

- 3

t h e t a

Οχοχο · χ *°x ·ο 0 slope = -3

- 1 0 1 - 1 0 1

A l o g ( C p / c £ ) β

Fig. 5.20. Reduced diffusion coefficient for PS [ M p: (O) 6.5 χ 104, ( x ) 1.79 χ ΙΟ

5, (V)

4.22 χ 105, ( · ) 1.05 χ 1 0

6] in PVME/o-FT. The above curves were analyzed under the

assumption that C* reflected a good solvent (A), or a theta solvent (B). [Reproduced with

permission from Wheeler et al. (1987). Macromolecules. 20, 1120-1129. Copyright 1987 by the


The concentrations of PAA were over the range 1.3 χ 10 3

g/L < c P AA < 187 g/L for M P A A = 5 χ 10

4 Daltons and 4 χ 1 0 "

3g / L < c P ' A A < 19.1 g/L for

Aip AA = 1 χ 106 Daltons. The sizes of the PSLS particles in salt-free solu-

tions were determined by QELS methods to be 204 Â, 800 A, 6200 Λ, and 15,000 Â. The concentrations (w/v) were 5 χ 10~

4 for the 204 Â particle and

1.3 χ 10~5 for the other particles. The PSLS particles were introduced directly

into a prefiltered PAA solution.

The shapes of the \og(Dp/Dp) vs. log(c P' A A) curves for both molecular weight samples of PAA were similar to those in Fig. 5.20. The rate at which the downward curvature changed was dependent upon both the probe size and M P A A. The form Dp/Dp = exp[ — a(cp S L S)^] was assumed, where β was dependent on both M P AA and the size of the PSLS. For M P AA = 5 χ 10

4

Daltons, β = 0.9 ± 0.1 fit the data for the 204 A and 800 A particles if C

' P A A < 1 g/L. For the M P AA = 1 χ 106 Dalton system, the power law for the

204 A and 800 A spheres was β = 0.5, whereas β ~ 0.7 for the 6200 A spheres. The apparent radium was defined as r 0 = kT/6^sDapp, where η5 is the mea-

sured viscosity of the PAA solution. It was found that r 0 ~ 250 A for the 204 A spheres was concentration-independent in the low-molecular-weight PAA solution. This larger value of r0 was interpreted as adsorption of the PAA on the PSLS particles. The value of r 0 in the 1 χ 10

6 Dalton PAA solu-

tions, however, decreased as the PAA concentration was increased, as shown by the plots of log(r 0) vs. c P AA in Fig. 5.21.

The drastic reduction in r 0 to absurdly small values was interpreted in terms of differences in the macroscopic and microscopic viscosities. Using the value

5.20. Stretched Exponential Representation of Polymer Solutions and Melts 145

4

"3i 1 1 1 1 1 1 1 1 I I 0 4 8 12 16 20

Cp ( g / L ) Fig. 5.21. Apparent radius for PSLS in PAA/water. The PAA in this study was 1 χ 10

6

Daltons. Included are the results for PSLS coated with Tri ton X-100 to minimize adsorpt ion of

PAA. [Reproduced with permission from Lin and Phillies ( 1984a). J. Colloid and Interface Science.

100, 82-104. Copyright 1984 by Academic Press.]

of RH for PSLS in aqueous solutions, the microscopic viscosity. ηΜ, was defined as ηΜ = kT/6nRHDapp. It follows that ηΜ < ηΒ. Lin and Phillies inter-pret this result in terms of shear thinning on the distance and time scales of the QELS experiment. Note that this effect occurs for large probe particles in a matrix of high-molecular-weight polymers.

5.20. Stretched Exponential Representation of Polymer Solutions and Melts

The general form of the ln(D s e l f/Z)°) vs. ln(c^) curves resembles those shown in Fig. 5.13, i.e., continuous with no clear-cut linear regions. Phillies (1986) reports that the globular proteins hemoglobin and bovine serum albumin also


give ln(D s e l f/Dp) vs. ln(c'p') curves of similar shape. Since the globular proteins cannot undergo reptation motion, the question is raised as to the uniqueness of power laws as a signature of reptation motion.

Phillies and coworkers (Phillies et al. 1985; Phillies, 1986; Phillies and Peczak, 1988) proposed the following general form of a stretched exponential representation for the probe diffusion coefficient of polymers in dilute and semidilute solutions:

ß p r o b e = D%oheQxpl-a(c^)\RP)ô(MMyi (5.20.1)

where α, ν, δ, and γ are adjustable parameters, c'u and M M are the concentration and molecular weight of the matrix polymer, respectively, and RP is the radius of the probe particle. Phillies (1986) reported, for example, that the data of Callaghan and Pender (1984, cf. Example 5.8) could be fitted with the following parameters ( M M χ 10"

5; a, v): (3.5, 0.211, 0.627), (2.33, 0.192, 0.617), and (1.1,

0.0775, 0.75). Although this formulation fits the data to within 3 % , the stretched exponential representation itself is not a universal curve. Selected systems are given in Table 5.3.

Table 5.3

Stretched Exponential Parameters for Selected Sys tems0' '

Polymer ( M p) Probe (R) α V

P E Oc (7500) PSL (208) 0--350 0.035 0.81

P E Oc (7500) PSL (517) 0--350 0.034 0.83

P E Oc (7500) PSL (3200) 0--350 0.036 0.82

P E Oc (300,000) PSL (208) 0--3 0.14 0.88

P E Oc (300,000) PSL (517) 0--3 0.14 0.92

P E Oc (300,000) PSL (3200) 0--3 0.12 0.82

BSAC (67,000) PSL (517) 0--200 0.008 0.92

BSAC (67,000) PSL (3200) 0--200 0.0053 0.97

P S+ d

(271) Same — — 13.2 0.75 PS

++d (233) Same — — 0.19 0.62

a £ > Pr o b e = / ) ^ o b ee x p [ - a ( r ; ' )

v] , where a ~ R

ÔM;.

b P E O = polye thylene oxide): PSL = poly(styrene) latex: BSA =

bovine serum albumin. c Light scattering results in which the probe is in an "invisible" matrix,

with γ ~ 0.8 ± 0.1 and 0 < δ < - 0 . 1 . [Part ial listing from Phillies et al. (1985). J. Chem. Phys. 82, 5242.]

d Tracer diffusion measurements in which probe and polymer are the

same and the solvent is benzene ( + ) or carbon tetrachloride ( + + ). [Listed in Table I of Phillies (1986). Macromolecules. 19, 2367.]

5.22. Polymer Melts and the Glass Transit ion 147

5.21. Computer Simulation of Kolinski, Skolnick, and Yaris for Congested Solution Motion

Kolinski, Skolnick, and Yaris (1987) used Monte Carlo simulations of poly-mers diffusing on a diamond lattice as a function of the polymer concentra-tion and chain length. The power laws for center-of-mass diffusion and the terminal end-to-end vector relaxation were n~

21 and n

3A, respectively. From

an examination of the mean-square displacement perpendicular to its primitive path, it was concluded that no evidence of a tube was found if all of the chains were allowed to move. The authors were quick to point out that this observation does not preclude tube formation at much higher polymer concentrations. If some of the chains were "pinned down", evidence for reptation with tube leakage was observed. These calculations clearly indicate that extreme caution must be exercised in interpreting diffusion data in terms of reptation, since power-law dependences of D s e lf on cp and M p of the polymers are not sufficient to uniquely identify reptation motion.

5.22. Polymer Melts and the Glass Transition

As the glass transition temperature, T g, is approached, the thermal motions of polymers in a melt slow down tremendously, and the intensity of light also

CP

0.4

0 . 3 -

0.2 -

0 . 1 -

0 . 0 -

β = i.o

-6

ΙοςΠΓ,αα)

χχχχ*

Fig. 5.22. Correlation function for poly(styrene) near the glass transition temperature. The

solid curves are the Wi l l i ams-Wat t function [cf. Eq. (5.22.1)] for the values of β as indicated,

where β = 1 is a single exponential decay. The temperature was 115°C. [Reproduced with

permission from Patterson (1983a). Ann. Review Material Science. 13, 219-245 . Copyright 1983

by Annual Reviews, Inc.]


decreases. The decay of g(i\t) is a direct measure of the longitudinal relaxa-

tion that is governed by the modulus M = Kc + f G s, where Kc and Gs are frequency-dependent moduli of compression and shear, respectively. For a broad distribution of relaxation times of breadth /?, g

(1)(t) is approximated

by the Wil l iam-Wat ts function (cf. Patterson, 1983a, 1983b),

9{l)(t) = [ Λ ) ]

1 / 2 -

e xp [ - ( j J ] ' <

5-

2 Z 1)

which has an average relaxation time given by

W - j V ' W * - ^ ) . (5.22.2,

where F ( l /ß ) is the gamma function. A plot of g(2\t) vs. log(i) is shown in

Fig. 5.22 (Patterson, 1983a).

5.23. Rigid and Semiflexible Rods in Congested Solutions

Doi and Edwards (1978) examined the translational and rotational motion of rigid rods of length L and diameter d in the range 1/L

3 « cp « 1/dL

2. The

particles were assumed not to be statistically coupled (thermodynamically ideal), but their rotational motion was hindered (entanglement). The rota-tional freedom was confined to a solid angle (ΔΩ)

2 ~ ( a c/L )

2, where ac is

the nearest-neighbor distance. Hence De ~ (ac/L)2/x0, where τ 0 is the jump

time for the rotational motion. Estimating τ 0 ~ L2/D\\ and cp ~ \/acL

2,

one finds

(5.23.1)

where Eq. (3.8.2) was used for Dy. While there appears to be a reason-able agreement between experiment [Southwick et a l , 1981; Zero and Pecora, 1982; Mori et a l , 1982] and theory in regard to the cp and L dependence of D 0, the measured value of β was ~ 1000, whereas the theoretical value is β ~ \ .

Keep and Pecora (1985) suggested that a probable source of the discrep-ancies in measured and theoretical values of β lies in the estimate of τ 0 . If the neighboring rods diffuse only a fraction fL of their length to escape the cage, then τ 0 ~ (fLL)

2/D^. Treating the rotational motion of the central rod as

diffusion of a point on a spherical surface where the "intruding" rods that penetrated the sphere were projections on the spherical surface to serve as

5.23. Rigid and Semiflexible Rods in Congested Solutions 149

barriers to the diffusion of the point, they derived the ratio

De 200 no

Ml ( C pL ' 15π

( c PL3)

(5.23.2)

They reported that this expression, valid for De/D$ < 1, best agrees with experiment when fL ~

Maeda and Fujime (1984a, 1984b) examined the congested solution dynamics of a long semiflexible filament within the frame work of the D o i -Edwards (1978) model. They emphasized the importance of translation-rotation coupling as manifested in D a pp by the factors f^KL/2) and f2(KL/2):

KL r . KL + AD/ 2( — (5.23.3)

where AD = (D\\ — DL). In congested solutions, the diffusion coefficients are adjusted by the parameter β = (ac/L)

2, viz, D$ = ßDe, Df = ßD±, and

Df| = D| | . Maeda and Fujime chose for the overlap volume a2L = l / c p , or

ac = l / ( L c p )1 / 2

. Hence β = l / ( c pL3) and De - L~

6,

M . (5-23.4)

Du

1oL %

In the above models, the " trapped" particle escapes the "cage" by trans-lation along the long axis of the rod after achieving the proper orientation. Teraoka and Hayakawa (1988) proposed a model in which the neigboring particles serve as " temporal" hindrances, or perturbations, to the transverse translational motion of the probe particle. The obstruction is assumed to appear at time rR and disappear by time i Q such that the inequalities t > tQ > tR > t' hold, where t — t' is the time window of the observation. For a large number of perturbation elements (highly concentrated solutions), they obtained the ratio

i + 1 ' ^3 (5.23.5)

where γ is a numerical constant that differs from β. It is of interest to note the QELS results of Keep and Pecora (1988) on

the dynamics of poly(n-butyl isocyanate) (PBIC), poly(n-hexyl isocyanate) (PHIC), and poly(rc-octylisocyanate) (POIC) in organic solvents. The correla-tion functions, obtained at a fixed angle θ = 90°, were analyzed by C O N T I N and DISCRETE (cf. Section 4.11), and the relaxation times were expressed


as effective hydrodynamic radii. This analysis indicated that bimodal, then trimodal distributions evolved with an increase in c p . In all but the P H I C sample in cyclohexane, the secondary and tertiary peaks appeared at smaller radii than the primary (translational motion) peak. In all cases, the shift in relative amplitudes favored the secondary and tertiary peaks as cp was increased. These observations were interpreted as the onset of translational -rotational coupling as required to escape the "cage". It was also pointed out that the asymmetric molecules may be aligned in a "liquid crystal" structure.

5.24. Computer Simulations of Congested Solutions of Rods

As in the case of reptation, computer simulations of Odell et al. (1983) did not reveal significant caging, although rotational hindrance was in evidence. Of particular interest are the Brownian simulations of Fixman (1985a) for three levels of dynamic freedom for the system: (1) the probe particle can rotate within a cage of fixed rods; (2) the probe particle and the rods comprising the cage can rotate, thus attaining equilibrium; and (3) all of the particles can rotate and translate. The rods were approximated by three beads connected by two springs, which obeyed the Langevin equation with a memory term, viz,

^ = F . ( t ) = _ A ¥ ( t ) _ M(tf)\(t - t')dt\ (5.24.1)

ο

where ße is the effective friction factor of the bead and M(i') is the memory function. A conclusion of this study was that the cages achieved only "quasiequilibrium" with the probe rod in which the angular width of the cage is ΔΩ ~ ( l / c p L

3)

1 / 2, which is in agreement with the results of Maeda and

Fujime (1984b). Fixman (1985b) later pointed out that cages were not required for hindered rotation. The decrease in the rotational relaxation time can be effected through relatively long-range interactions of the probe rod with the surrounding lattice. Fixman noted that M(0) is the elastic modulus that measures the "stored" force in the surrounding "cage" that results when a probe rod undergoes an instantaneous fluctuation in its orientation. This "soft core" interaction may dissipate more rapidly through cooperative interac-tions of the cage rods with the surrounding external rods, rather than by longitudinal translation of the cage rods as in the Doi-Edwards model.

Summary

The rotational motion of rigid particles and internal decay modes of flex-ible and semiflexible particles can be examined by dynamic light scattering methods. In the limit KRG « 1, Z) a pp is associated with center-of-mass motion

Problems 151

that is governed by the flexing rigidity of the polymer. In the limit KRG » 1, D a pp is a measure of the segmental motion as governed by the torsional rigidity of the polymer. The decay rate for the intermediate K-range for flexible coils i.e., K

2(RG) < 1 and K

2b

2 < 1, exhibits either a X

4 (no hydrodynamic

interaction) or Κ3 (hydrodynamic interaction) dependence.

The statistical nature of a polymer is reflected in the power law dependence of RH and RG on the "correlation length" £ c . Although the power laws are the same for both radii at the asymptotic limits of solvent quality, the rate at which they attain these limits differs for the two characteristic lengths. Computer simulations using Brownian dynamics methods indicate that for congested solutions of coils and rods, there is no evidence of trapped configurations as postulated by the scaling law approach.

The diffusion coefficient of flexible and hard sphere probe particles through a matrix over the concentration regimes semidilute-congested-gel appear to be adequately described by the empirical "stretched exponentials."

Problems

5.1. The Ha l l e t -Nicke l -Cra ig model (1985) assumes that τ Γ Ο( is very slow during the data-collection interval. This assumption can be quan-titatively defined in terms of the translational motion perpendicular to the major axis of the rod, where T t r a ns = \/D±K

2. Neglecting end

effects, what is the extent of orientational relaxation for a rod of length 100 microns during the time i t r a n s? (Hint: because of the cylin-drical symmetry, the slowest mode is for j = 2.)

5.2. Estimates of the anticipated internal relaxation time can be made through the measured intrinsic viscosity and the molecular weight by using Eq. (5.8.7). Assume that you have a monodisperse sample of poly(acrylate), where the relationship between the intrinsic viscosity and the molecular weight obeys the power law [rf\ = 1.36 χ 10~

3

( M p )0'

8 9 cm

3/g .What is the anticipated value of τγ as obtained from

DLS data for a sample with [η] = 1616 c m3/ g ? Kitano et al. (1980)

reported that L p ~ 1 2 Ä and the mass/length ratio = 40 g/Â for poly(acrylate). What is the corresponding value of D p and (RG}

1/2

anticipated for this preparation? Using these estimates, at what angle might the P2(x, t) mode represent 5% of the total correction function? Is this angle reasonable?

5.3. The solubility of poly(ether ether ketone) (PEEK) appears to be limited to strong acids. Bishop et al. (1985) used static and dynamic light scattering and viscometric methods to study P E E K in 97.4% sulfuric acid. The parameters in the accompanying tabulation; cor-rected for sulfonation, were reported.


Dp°(30°C) ( 1 0 "

7 cm

2/ s ) (Â)

M (dL/g)

PEEK-1 P E E K - 2

1.86

1.40

155 204

0.72

0.93

31,800

52,000

Using η0 = 0.017 poise at 30°C and the knowledge that their Fig. 4 indicates that the double logarithmic plots of J R H , RG, and [rf\ as a a function of M p were linear, estimate values for the parameters v H, v G, and νη. Do these values support the expression of Weill and des Cloizeaux [1979, cf. Eq. (3.12.9)]? What conclusions can you draw from the ratio RH/RG in regard to the conformation of PEEK in sul-furic acid? Bishop et al. estimated the number of monomer units using the relationship ns = 1.5(M p)w/288.3, where the 1.5 corrects from the weight to the z-average molecular weight. Take into consideration that the backbone structure of PEEK contains rigid chemical bonds.

5.4. On the basis of your results in Problem 5.3, estimate the longest internal relaxation time τι anticipated in a DLS experiment for sam-ple PEEK-2. Compare this value with the free-draining random coil value. From the values of R H and R G and using the relevant informa-tion in Problem 5.3, estimate the "effective" persistence length for each sample.

5.5. Soda (1984) suggested that D p l at observed for the DNA systems may be due to a "crossover" from flexible-coil to "stiff-coil" regimes ob-served for internal relaxation modes as the scattering vector Κ is increased. To examine the feasibility of this suggestion, estimate the crossover mode number nc from Eq. (5.7.16) using the values L K = 2000 Â and L = 4 χ 10

5 Â. What is the value of τη for η = L/LK at

20°C? Use the value ξ = 4.5 χ 10~7 g/s in Eq. (5.7.18).

5.6. The mode number η can exceed the value of L/LK. Using the value L = 10

5 and L K = 10

3, estimate the reduced mode number n/NK for

which yb$/An = 0.9. If this value of η represents the fastest internal relaxation mode of the polymer, what is the number of segments ns

in the polymer?

5.7. Defining the quantities NK = L/LK, D p = /cT/(£L), and F(NK) = e x p ( - 2 N K ) + 2NK - 1, show that

3Ρ°τ, Η 4NK

8η2π

2ΛΓ,

F(NK) Κ

Problems 153

5.8. Kubota and Chu (1983) used photon correlation spectroscopy and viscoelastic methods to study poly(hexyl isocyanate) (PHIC) in hex-ane at 25°C. Zimm plots were constructed from the photon counts after calibration of the instrument with benzene and NBS 705 poly-styrene) in cyclohexane as references. The parameters obtained for sample PHIC-B are: M p = 2.02 χ 10

5 Daltons; D°p = 3.17 χ 10~

7 cm

2/ s ;

and (RG)1/2 = 51.4 nm. Recalling that the flexible coil limit is a

reasonable approximation for the η = 1 mode for long wormlike polymers, one may write τχ ~ 2{RG}

1/2/Dpn

2. Estimate the statistical

length L K for PHIC-B using the expression for 3Dpzn/L2 in Prob-

lem 5.8 and ρ = Mp/L = 715 g/nm.

5.9. Use the value ξ0 = 4.5 χ 10~7 g/s and calculate the ratio reff/b

(cf. Section 5.12) for an aqueous solution of DNA at 20°C. Is the Perico et al. (1975b) condition ( r e f f/b < 0.43) satisfied?

5.10. The study of poly(styrene) of molecular weight 5.50 χ 106 Daltons in

irans-decalin reported by Tsunashima et al. (1983a) gave the following parameters (cf. Example 5.6): τγ = 908 χ 1 0 "

6 s; and Dp = 1.84 χ

1 0- 7 c m

2/ s . Use the mass/length ratio M p / L of 39 g/À cyclo-

hexane (Huber et al. 1985b) to estimate L, and calculate the statistical length L K for poly(styrene) from the equation given in Problem 5.8. Compare with the value L K = 39 Â calculated for poly(styrene) in cyclohexane in Example 3.7.

5.11. The ratio (RG}i/2/RH for poly(styrene) in toluene appears to be

dependent upon the length of the polymer (Huber et al. 1985a, 1985b). The experimentally determined range of values for this ratio is 0.73 < < # Ο >

1 / 2/ # Η < 1-51 for the molecular weight range 1 . 2 x l 0

3< M p <

4.24 χ 105. Discuss this behavior in terms of the statistical properties

of the linear chains. Use as a guide the expressions for the expan-sion parameters a H and a G of Farroux (1976) and Farnoux et al. (1978), as presented in Example 3.8, and Weill and des Cloizeaux (1979). In the former case, assume the relationship < ^ 0 , Θ >

1 / 2/ ^ Η , Θ =

1.5. Use the relationships y = l/x = njns = Mc/Mp where M c is the crossover molecular weight, and set ν = 0.6. Can either, or both, of these models adequately explain the observed behavior? Explain. [The behavior was interpreted by Huber et al. (1985b) in terms of the wormlike coil model of Yamakawa and Fujii (1973).]

5.12. Compare the relaxation times τη for linear and circular chains of six friction units separated by a distance b. Assume the continuum limit for y and β.

5.13. Using the Zimm approximation for the intraparticle structure factor


P(K Ar) as given by Eq. (2.7.2), show that the reduced first cumulant KYjK

2 for a polydisperse system is (cf. Brehm and Bloomfield, 1975 and

Problem 4.3):

Note that P(K Ar) tends to correct for the distribution of scattering amplitudes, and its presence does not mean internal motions can be observed.

5.14. Cotts et al. (1987) used static and dynamic light scattering methods to study poly(di-n-hexylsilane), [ r c - ( C 6H 1 3) 2S i ] n , in rc-hexane at 25°C. These authors reported that over the range 0.05 g/L < c'p' < 0.9 g/L,

DcoU(K 0, c p) = 1.07 χ 10"7(1 + 290c p) cm

2/ s ,

with < M p > w = 6.1 χ 106 Daltons, RG = 108 nm, A2 = 8.0 χ 10~

5

mL/g-mole, M = 474 mL/g, and < M > w / < M > n = 2.3. The slope Β appears to be a constant for three of the four con-

centrations shown in their Fig. 9, and it is estimated to be Β ~ 1.2 χ 1 0

- 1 7 c m

4/ s . This functional dependence might be explained in terms

of polydispersity of the system (cf. Problem 5.13), since < M > w/ < M > n

was stated to be ~ 2 . 3 . The weight-average molecular weight used in this study, however, is much higher than that used in the sodium alginate study. Consider the possibility that the BK

2 term reflects a

subpopulation of Rouse chains, in which case Β is to be identified as ΙίΤο

2/\2ξ0 = Dpnsb

2/\2 in the free-draining limit. Using the reported

value for Dp, estimate the radius of gyration from B. Compare this with RG.Discuss the validity of this interpretation of B. What other alternative interpretation might be consistent with the observations?

5.15. From the log(D/Dp) vs. log(c p/c*) curves shown in Fig. 5.20, estimate the stretch exponential parameter ν as defined by Eq. (5.20.1).

5.16. Zero and Pecora (1982) used depolarized light scattering methods to obtain the rotational diffusion coefficient of poly(y-benzyl-L-glutamate) (PGLG) in 1,2-dichloroethane. The values of De were plotted as a function of l / (c p ' )

2. As shown in Fig. 5.23, the plots

appeared to be linear for the higher concentrations, as predicted by the D o i - Edwards theory. Deduce the power-law dependence from

K2 ~ <D°> Z + — « D ° > x < R à X - <D°R

2

G}Z).

^ 1 K

2 = DC0ll(O + BK2,

Problems

0 0.01 0 . 0 2 0 . 0 3 0 . 0 4

( I / C ρ ) 2 ( m L / m g )

2

Fig. 5.23. Rotational diffusion coefficient as a function of (l/c-J,')2 for PBGL in 1,3-

dichloroethane. The rotational diffusion coefficients were determined by depolarized light

scattering at an angle of 90° at 20°C. The molecular weights of the samples are shown in the figure.

[Reproduced with permission from Zero and Pecora (1982). Macromolecules. 15, 8 7 - 9 3 .


the information given in the figure. Which model, reptation or Do i -Edwards , best represents the data?

Additional Reading

McCammon, J. Α., and Harvey, S. C. (1987). Dynamics of Proteins and Nucleic Acids. Cambridge

University Press, New York.

Allen, M. P., and Tildesley, D. J. (1987). Computer Simulations of Liquids. Oxford University Press,

Oxford, UK.

CHAPTER 6

Hydrodynamic and Short-Range Interparticle Interactions

6.0. Introduction

Like Macavity in the T. S. Eliot poem, the hydrodynamic interaction between particles is long-range and the magnitude of its effect on Dm is not yet fully established. In contrast to electrostatic interactions, the velocity field gen-erated by one particle can be reflected by another particle before affecting the third particle, etc.

6.1. The Generalized Langevin Equation

Since this chapter launches the discussions on the role of interparticle interactions in the diffusion of "probe" macroparticles, the generalized Langevin expression to be utilized in the remainder of this book is now introduced. The force experienced by the "probe" particle ρ at time t in a system containing ΝΎ particles is given by

" Macavity's a Mystery Cat: he's called the Hidden Paw— For he's the master criminal who can defy the Law.

He's the bafflement of Scotland Yard, the Flying Squad's despair:

For when they reach the scene of crime—Macavity's not there /"

From Old Possum's Book of Practical Cats byT.S. Eliot (1888-1965)

+ V/ip(r, t) + ξ ρν ρ( ί ) + Σ [(UP · VjW - Z PeV0, p( t ) ]

= Ρ(ή-Ζρβ\Φ(τ,ή. (6.1.1)

The various terms are summarized in Table 6.1.

157

158 6. H Y D R O D Y N A M I C , S H O R T - R A N G E I N T E R P A R T I C L E I N T E R A C T I O N S

Table 6.

Classification of Forces in the Generalized Langevin Equat ion

Term Origin

n o

(Wjp'VpW

-Zpe\4>jp(t)

-Zpe\<t>(rj)

Newtonian force for viscous resistance of the yth particle with a friction factor ξ] and velocity v,(f)

Hydrodynamic coupling force between particles j and ρ Concentrat ion component to the chemical potential gradient of the pth species Force on particle ρ of charge Z p arising from the direct interparticle potential with particle j that, in general, depends upon time Force on the particle at location r that results from the application of an external potential that may be time-dependent Time-dependent fluctuating force that arises from the random collisions of solvent particles with the ;'th solute particle

6.2. The Stokes Solvent Flow Field Due to an Isolated Sphere

The objective of this section is to determine the nature of solvent flow about an isolated sphere of radius Rs. One can study such a pattern on a macro-scopic scale by introducing a dye marker upstream to the region of interest and describing the resulting pattern of the dye as it passes the target area. The pattern generated by the dye is referred to as a stream line, which is des-cribed mathematically by a stream function, φ. We borrow from Lamb (1932), Landau and Lifshitz (1959), Milne-Thomson (1960) and Brodkey (1967) in the derivation of a relationship between the solvent velocity and the stream function. In general, the stream line can be compressed or stretched along its vector components, or the stream line can undergo a rotational motion in its time course of development. The former motion is coupled with the equation of continuity for the solvent density p0(r, t) at position r and time t:

where v0(r, ή = [yx(r, t), vy(r, t), vz(r, t)~]T is the laboratory reference frame sol-

vent velocity. Eq. (6.2.1) ensures that matter is neither created nor destroyed within a volume element of the fluid. The volume element may be distorted in time and/or space; the degree of distortion depends upon the vectorial decomposition of the force at the surface of the volume element. Let us denote a stress tensor as Ρ whose elements are the force/unit area (P)y, where i, j represent a pair of spatial coordinates x, y, or z. The (P)yx com-ponent of the force along the x-axis is therefore [Pyx(x, y + Sy/2, ζ, t) — Pyx(

x, y — ày/2, ζ, ί)] δχδζ. Division by the volume element ôxôyôz gives

dp0(r, t) ^ dp0(r, t)

dt dt + [ V

r. p o( r , i ) v o( r , i ) ] = 0, (6.2.1)

6.2. The Stokes Solvent Flow Field Due to an Isolated Sphere 159

ôPyx/ôy, with similar expressions for the other components of the stress tensor. Newton's equation of motion for the fluid is thus

The viscosity assumption (Milne-Thomson, 1960) is that the viscous force acts in the direction of the changing velocity, i.e., the velocity components are normal to the volume surfaces. Denoting the separation distance be-tween two points in the solvent as or = r — r 0 = (ôx,ôy,ôz)

T, the first-order

change in velocities at two points is given by a Taylor series expansion, v 0( r 0 + (5r) — \0(r0) = (ôr

T · V)v 0. Using the rules of vector algebra, (Β χ C) χ

A = - [Α χ (Β χ C)] = - [B(A · C) - C(A . B)], one can write C(A - B) = ( A . B ) C = (<5r

r.V)v 0,

where Ω Γ Ο( = \ V χ v 0 is a vector of rotation, or vorticity, of a fluid element and has the components

dt = V

rP . (6.2.2)

(<5rr . V)v 0 = Ω Γ Οί χ or + R . <$r, (6.2.3)

d\z dvy

dy dz

2

dz dx (6.2.4)

2

dvy d \ x

dx dy

and R is a strain rate tensor,

d\v d \ x —

y- + — -

d \ x dx dy

dv„ dvx —t + — -

dx dz

R =

dx 2

dv^^dvy,

dy dx d\y

2

dy dz (6.2.5)

2 dy

dz dx dz dy dv:

~dz~ 2 2

The rotational motion can be expressed as an inner product,

Ω Γ 01 χδτ = \·δτ, (6.2.6)


where A is an antisymmetric matrix,

A =

d\x ÔVy

0 dy dx

0 2

dvx

dx dy 0

dvx <3vz dvy

dx dz dy dz

dx

2

dvy dvz

dz dy

0

(6.2.7)

It is clear from Eq. (6.2.7) that the eigenvalues of A are zero. Hence rotation does not contribute to the velocity normal to the fluid volume element. In terms of the scalar pressure P', the stress tensor is Ρ = — Ρ Ί + 2rç0R, or in terms of the total pressure, Ρ = — |T r (P ) ,

P = [ - i5' - ! ' i o ( V

r. v 0 ) ] I + 2, 7 oR.

Substitution into Eq. (6.2.2) gives the equation of motion,

dt = - V P ' - i j o : V ( V

r. v 0 ) + V

2v 0 + V ( V

r- v 0 )

(6.2.8)

(6.2.9)

which becomes for an incompressible fluid (V · v 0 = 0)

P ( A o

dt = — VP' + η0\

2v0 (Navier-Stokes Equation) (6.2.10)

A sphere is introduced into the fluid, and the origin of the coordinate axes is located at the center of the sphere. The boundary conditions are defined to be: v x = \ y = v z = 0 at r = Rs (stick boundary conditions); and \ x = — U® [cf. Eq. (3.7.1)], vy = v z = 0 at r = oo. Because of the symmetry, one employs a cylindrical coordinate system: ξ

2 = y

2 + ζ

2, y = £cos((/>), and ζ = £sin((/>).

The Laplacian operator is

d2 \_d^ 1 d

2

'dx~2+W

+ l ^ ^ W

(6.2.11)

Consider a circle of radius ξ whose origin is at x. The flux through the circle is 2πφ. Consider now a small extension ÔL from the point located at (χ, ξ) and in a direction parallel to either χ or ξ, with a corresponding change in flux of δφ. The velocity normal to the line ÔL is given by 2π δφ/2πξ ôL. These parameters are illustrated in Fig. 6.1.

6.2. The Stokes Solvent Flow Field Due to an Isolated Sphere

Fig. 6.1. Geometry for describing the solvent flow about a fixed sphere. A sphere of radius RH is

located at the origin of the coordinate system, and the solvent flows along the x-axis. The velocity

(v) components are given in Cartesian and cylindrical coordinate systems.

It follows that the velocity normal to each line SL is v x = —(\/ξ)δφ/δξ and \ ξ = (1/ξ)δφ/δχ. The angular velocity for rotation of the stream func-tion is ω = δνξ/δχ — δνχ/δξ,

1 δ2φ , δ

2φ 1 δψ

ω - ξ δχ

2 + 'δξ

2 (6.2.12)

The vorticity components are: ωι = 0; ω 2 = — cosin(</>); and ω 3 = cocos(<£). Let us now look at the operator V

2co 2:

" δ2 δ

2 l δ 1 δ

2 "

fa1 +

W +

~ξ ΊΓξ+ Τ

2 W

sin(0)

ξ

' δ^_

δ~χΎ +

δξ2

1 δ

ω sin(</>)

(ξω).

Hence from Eq. (6.2.12) and Eq. (6.2.13),

δχ2 +

δξ2

+ "sin(0)"

1 d~

sin(0) δθ

•A = 0,

δθ φ = 0.

(6.2.13)

(6.2.14)

(6.2.15)


The solution to Eq. (6.2.15) is given by φ = F(r)S [sin(0)], where F(r) is a power series in r" and S[sin(0)] is a power series in sin

f c(0). However, k = 2 is the

only term compatiable with Eq. (6.2.15). The equation for F(r) is thus [_(d

2/dr

2) - 2 / r

2]

2F ( r ) = 0, with the general solution F(r) = A/r + Br +

Cr2 + Dr

4. Taking the boundary conditions for v x and \ ξ into consideration

leads to the stream function

Φ = -

The radial velocity is

1 RS 1 fRs\3~ r

2 sin

2(0). (6.2.16)

~δθ

• sin(0) 1

3Rs Ι Α 2r 2V r

cos(0). (6.2.17)

The radia/ attenuation factor for the Stokes sphere is therefore [1

i(RsA-)3]-

To maintain the sphere at a fixed location as the solvent flows with velo-city — (7°, one must counteract the viscous force = p0d\0/dt = rç0V

2v0

[Eq. (6.2.10) with VP = 0] . The term η0

2^ο

c an be expressed as the curl

of the vorticity, V χ ß r o t = V χ (V χ v 0) = V ( Vr · v 0) - i | 0V

2v 0 - - , | 0 V

2v 0 .

The curl of the solvent velocity in spherical coordinates is V χ v 0 = (1/r) χ {d[r\o\ldr — d\r/d0}. The velocity ν θ is — [1/r sin(0)](d^/dr), or

v e = I/2sin(0) 1 3Rs

4 r (6.2.18)

Utilization of Eqs. (6.2.17) and (6.12.18) gives

3 V X V N - - " sin(0). (6.2.19)

Noting that v j · (p0d\0/dt) = d [ i p 0

vo - v 0] /d i , the rate of energy dissipation,

RE, is obtained by integration of v j · η0

2^ο

o v er the volume element dV=

r2 sin(0) d0 dr with the limits RS < r < oo and 0 < 0 < π, and 0 < φ < 2π,

i.e.:

(V χ v 0 )T . (V χ y0)dV = -6^0RS(U°X)

2 = FnV°. (6.2.20)

The viscous force is Fv = — £ s l / ° , hence one obtains the well-known Stokes friction factor for a sphere,

RE ξ* 0\2

6πη01ϊΒ (stick boundary). (6.2.21)

6.4. Transmission of the Indirect Interaction 163

If the solvent "slips" at the surface of the sphere, then the Stokes friction factor i s £ s = 4 π ^ 0Κ 5 . The mixed st ick-sl ip boundary expression of Felderhof (1977, 1978) is

ξ8 = 6πη0Κ8(\ — β) (mixed boundary), (6.2.22)

where 0 < β < y. After transformation of Eqs. (6.2.17) and (6.2.20) to Cartesian coordinates, the fluid velocity is (cf. Landau and Lifshitz, 1959)

I 3R

Ar \- r2 4

I 3rr

U° , (6.2.23)

where r2 = x

2 + y

2 + z

2 and rr is the dyadic product of the Cartesian

coordinate location r [cf. Appendix A].

6.3. The Stokes Friction Factor and Faxén's Theorem

Perturbations in the velocity field due to long-range hydrodynamic effects of other particles probably will not be evenly distributed over the surface of the probe particle. Faxen (1922) examined the case of a sphere at rest in a fluid of nonhomogeneous flow and obtained the result

/»

— Ρη = 6^0Rs(4nR2r

l v0(r)dS = 6^0Rs(v°0)s, (6.3.1)

Js

where <Vo>s is the average of the unperturbed velocity over the surface of the sphere. The viscous force acting on a hard sphere is therefore equal to 6^0RS

times the surface average of the velocity field that would have occurred if the sphere was not present (Faxén's Theorem).

6.4. Transmission of the Indirect Interaction

The question is whether the transmission of indirect interactions between macroparticles occurs on a time scale accessible to DLS methods. The Newtonian collision time between water particles at 20°C is on the order Tc o i i ~ (

mo/kT)

1/2IGQÏX'Q ~ 10~

1 2 s, where σ 0 = 3 χ 1 0

- 8 cm is the assumed

radius of the water molecule and a number density n'0 = 3.3 χ 1 02 2

parti-cles/cm

3. During this period of time, a macroparticle of radius 500 À

(Dp = 4.3 χ 10~8 cm

2/ s ) has moved ~ < Δ χ

2>

1 / 2 = ( 6 D T c o 1 1)

1 / 2, or

< Δ χ2>

1 /2 - (6 χ 4.3 χ 10~

8 χ 1 0 "

1 2)

1 /2 = 5.1 χ 1 0 "

1 0 cm.

In other words, the macroparticle is virtually stationary during the collision period of the solvent particles. This large separation in relaxation times of the solvent and macroparticle decay modes is the reason their velocities and forces are not correlated.


The time for the establishment of the velocity field created by a dif-fusing particle is assumed to be governed by the Navier -Stokes equation [Eq. (6.2.10)] with VP = 0. The equation to solve is the Fourier-transformed expression

^ φ ^ = -^Κ^0(ΚΛ (6.4.1) dt p0

The "apparent diffusion coefficient" is given by the kinematic viscosity η0/Ρο-Because of the macroparticle motion, the solvent velocity changes in time in accordance with Eq. (6.4.1), and a new velocity field is established. The "depth of penetration" is defined as < Δ Γ

2>

1 / 2 = [ 6 ( ί / 0 / ρ 0) Δ ί ]

1 / 2. [By comparison,

Landau and Lifshitz (1959) examined the system of an oscillating plane with frequency ω in a viscous liquid, with the result that the penetration depth is of the form δ = (2η0/ρ0ω)

1/2.~] If we assume that Δί = 10~

7 s is the min-

imum time interval of present-day correlators, then for water at 20°C, one has < Δ Γ

2>

1 / 2 to be [(6 χ 0.01/1) χ 1 0 "

7]

1 /2 ~ 7.8 χ 1 0 "

5 cm. The diffusing

solute particle has generated a new velocity field ~ 8000 Â from the site of origin in a time period of 10~

7 s. A macroparticle of radius Rs = 500 A

(Dp = 4.3 χ 10"8 cm

2/ s ) over the same time period has been displaced a

distance < Δ Γ2>

1 / 2 = (6 Χ 4.3 χ Ι Ο

8 χ 1 0 "

7)

1 /2 = 16 χ 1 0 "

8 cm, or about

3.2% of Rs. It is concluded that solvent-mediated interparticle interactions are instantaneous on the molecular time scale.

6.5. Time-Dependent Diffusion Coefficients

Since the friction factor is identified with the velocity correlation function, consideration must be given to the relative time scales of all processes that affect the motion of the probe particle. For pedagogical reasons, we examine the mutual diffusion of particles A and Β of masses m A and raB that are dynamically coupled to a "background" matrix. The coupled Langevin equations are

—TT~ = V

A (f> 0 V

B (f , 0 + + , (6.5.1)

at τΑ mA mA mA

dyB(r\t) 1 ? A B , fb(r ' , t) [ f

s(r ',f)

— - r — = - — vB(r ,0 vA(r, t) + + , (6.5.2) at τ Β m B m B m B

where τ, = mj/ξ*} is the molecular relaxation time, ξΑ (£B) is the isotropic friction factor for the initial flow field about particle A (B) in the absence of particle Β (Α), ξΒΑ (ξΑΒ) is the tensor that describes the hydrodynamic interaction between the particles A and B, f

b(r, t) is the fluctuating force at r

that couples the particles A and Β with the background matrix, and f s(r, i) is

6.5. Time-Dependent Diffusion Coefficients 1 6 5

the fluctuating force at r due to solvent collisions. Eqs. ( 6 . 5 . 1 ) and ( 6 .5 .2 ) are multiplied on the right by vA(r, 0) , and then the average of the dyadic products is taken. Thus, for Eq. (6 .5 .1) ,

% ^ = 1 YAAW - ^ · YBAW + Y„A(0, ( 6 . 5 .3 )

at τΑ raA

where \JL(t) = <v,(t) * v[(0)> (J, L = A or B), V b A( t ) = <[f b( t ) /m A] * vft0)> is

nonzero because the relaxation of the background matrix is on the order of τ Α, and <[f

s(t)/mA] * v A( 0 ) > = 0 since the relaxation time for the solvent is much

shorter than τ Α. A similar expression is obtained for Eq. ( 6 .5 .2 ) for V B A( r ) . The equations for V A A( r ) and V B A( i )

a r e inhomogeneous equations that can be

expressed in the form

- n - a ( i ) + F(t), ( 6 .5 .4 )

where Ω describes the initial rate of decay and F(t) the subsequent time evolution of a(i). The general solution is

a(0 = X ( 0 - a ( 0 ) + X(t-t')-F(t')dt\ ( 6 . 5 . 5 )

where X(i) must satisfy X(0) = I and dX(t)/dt = - Ω . X(i) (cf. Franklin, 1968) . The "propagator" X(i) has the general form X(i) = a(i) · a ( 0 )

_ 1 = e x p ( - H i ) .

Hence,

X A ( 0 = YAAW · EYAA(O)] -1 = exp(^j. ( 6 . 5 . 6 )

Solving Eq. ( 6 .5 .2 ) for VB A(f) and substituting into Eq. ( 6 . 5 . 1 ) leads to the

solution for ΥΑΑ(0>

YAAW = Y A A( 0 ) e x p ( ^ £ ) - ΥΑΑ(0Γ1 · £ Y A A( r - f ) · MBA(t')dt', ( 6 . 5 . 7 )

where Μ Β Α( ί ' ) represents a memory function defined as

ξ

MBA(Î') = — · XB(Î') · ΥΒΑ(Ο) + V b A( t ' )

- • Y B B ( O ) -1

m A

Ym(t'-t").mAB(t")dt", (6 .5 .8 )

where the secondary memory function m A B( i" ) is

"AB(Î") = · YAA(Î") - YbB(f"). (6-5 .9)


The first term in Eq. (6.5.8) is the time evolution of the initial hydrodynamic correlations between the particles A and B. The second term is the effect of the dynamics of the background matrix on the motion of particle A. By analogy with Eq. (6.5.7), the third term is a "memory feedback" term that "corrects" for the influence of particle A and the background motion on the velocity of particle B.

The time-dependent self-diffusion coefficient is

AeifW = ^ T r [ V A A( 0 ) ] T A 1 — exp

- - T r <YA A(0)>- < Υ Α Α ( 0 > · Μ ΒΑ ( 0 Λ (6.5.10)

To interpret the terms in Eq. (6.5.10), we first recall the Einstein expres-sion for the mean-square displacement of particle Α, <[Δχ Α(0) ] [Δχ Α(τ ) ]> ~ D s e l fT. At times t « τ Α, the exponential can be expanded with the result <[Δχ Α(0)] [Δχ Α(τ) ]> ~ i T r [ < v A( 0 ) * ν Α(0)>]τ

2. The mean-square displace-

ment during this time scale is governed by Newtonian mechanics, and the "effective" D s e lf increases linearly with time. For intermediate times τ » τ Α but

Ds

Self

Fig. 6.2. Schematic representation of the time dependence of < A J C2> / T . The effective diffusion

coefficient is given by the mean-square displacement < Δ χ2> divided by the elapsed time τ. At very

short times, the randomness caused by solvent collisions has not yet set in, and the particle moves in accordance with Newtonian physics. Eventually the motion of the test particle is governed by these collisions, thus giving rise to Z)fclf. As time progresses, the test particle becomes influenced by the dynamics of the lattice background of other solute particles. Eventually these effects average out, giving rise to the long-time self diffusion coefficient, D^cli.

6.6. K-Dependence of the Initial Decay Rate of GX(K, t) 167

t « τ Μ, where τ Μ is the decay time for the memory function, Dself(z) attains a relatively "constant" value that defines the short-time self-diffusion coefficient, use\f

DLf = -Tv ΥΑΑ(0)"*α

kT (6.5.11)

In the limit τ -> oo, one has D^ e lf = D T r, where

1 ^hlf — ^self

_ 3 ^

Γ [YAA(O)]-1-! yAA(t)-MBA(t)dt

ο

^s

Leif = D

s

nlf + A D m e m.

A schematic of DseU(r) vs. τ is given in Fig. 6.2.

(6.5.12)

6.6. K-Dependence of the Initial Decay Rate of (^(K, t)

The correlation function GX(K, r) is assumed to be due to center-of-mass motion only, and for Np hard spheres it is

ΟΛΚ,ί) = Σ Σ ^ ( Κ Κ ( Κ ) β χ ρ { - ϊ ΚΤ . [R,(0) - R f c(t)]}, (6.6.1)

j=l k=l

where a,(K) is the normalized relative scattering amplitude of component i. We follow Pusey and Tough (1982, 1985) by first taking the second derivative of Gj(K, t) with respect to f, defining the velocity at time t as dRj(t)/dt = Vj(r), ignoring terms containing the inertia term d\j(t)/dt, and then integrating once to the time τ:

dGj(K,i)

at

ο dtK

T • vj(0) * vj(r) · Κ α ? ( Κ ) ε χ ρ [ - ; Κ

Γ · AR / t ) ]

+ Σ Σ 7 = 1 Λ = 1

dtKT>Yj(0)*yl(t)^Kaj(K)ak(K)

exp{ -iKT . [R,-(0) - Rk(0)]} e x p [ - i K

T . AR k(t)] , (6.6.2)

where Rk(i) = Rfc(0) + ARk(f). Since the velocities (pre-exponential term) and positions (exponential term) of the particles are correlated, further simplifica-tion of Eq. (6.6.2) generally is not possible. In the following discussion, it is assumed that the particles are identical. If we further assume that τ is


sufficiently small that exp[ — iKT · AR f c(r)] ~ 1 over the entire range 0 < t < τ,

then dGi(K,t)

dt t = t Kr- . K

Gi(K,0) ^ (Κ ,Ο)

Κ Τ < 0 , * ( τ ) ε χ ρ ( - / ΚΤ· 4

G,(K,0) (6.6.3)

where (Ώη(τ)} and <D7-k(i)exp( — / Kr · AR^)> denote time-dependent am-

plitude weighted-ensemble average values for the self and pair diffusion co-efficient tensors, respectively:

<W> = Σ 7=1

< D j t ( T ) e x p ( - i Kr. A R J l k) >

dt<yj(0) * vJ(r)>a,2(K)

Np Np

= Σ Σ 7=1 k= 1

j*k

dt<vj(0) * v f c

T( i ) e x p ( - i K

r . AR,,)>a,(K)a,(K)

(6.6.4)

(6.6.5)

and the dynamic structure factor G^KÔ) is

G x(K,0) = 1 + Σ Σ ^ ( K ) a , ( K ) e x p { - i KT . [R,.(0) - R k(0)]}. (6.6.6)

j=l k=l

Comparison of Eq. (6.6.6) with Eqs. (2.8.1) and (2.10.2) gives the equality Gi(K,0) = S(K

T - AR). <D a p p(K , t )> over the range 0 < t < τ as defined by

Eq. (6.6.3) is

/ n or ϊ \ KT

' 1<W> + < e x p ( - / KT - AR,-,) - D , f c(t)>] - Κ

< Α Ρ Ρ( Κ , τ ) > = X ^ ( K ^ A R ) ' ( 6 A 7)

In the limit K - > 0 , < e x p ( - * Kr . ARjk)) -> 1 and l / G ^ O ) -+ (l/RT) χ

(dn/dCp)Ttß., hence < D a p p( K 0 , τ)> = D m [cf. Eq. (3.2.5)]. In the limit K

r» A R j 7 c» l , the exponential functions in G x(K,0) and the numerators

in Eqs. (6.6.7) and (6.6.8) undergo many oscillations. In this case, β^Κ,Ο) - • 1 and <exp( — / K

r · AR j 7c)> - • 0, and <D a p p(K,r)> is associated with the self-

diffusion coefficient. Over the intermediate range where ΚT · AR j 7c is on the

order of unity, both S(K T · AR) and <exp( — iK

T · AR j k)> undergo oscillations.

Example 6.1. < D a p p( K , τ)> vs. Κ for Poly(methyl methacrylate) in Hexane/Carbon Disulfide.

Pusey and van Megen (1983) reported QELS studies on poly (methyl methacrylate) (PMMA) at Τ = 19.5 ± 0.5°C as a function of the volume

6.6. K-Dependence of the Initial Decay Rate of G,(K, t) 169

fraction φρ over the range 0 < φρ < 0.50. The spherical latex particles were coated with poly(hydroxystearic acid). Because high concentrations of P M M A were to be used in these studies, multiple scattering was minimized by choosing hexane/carbon disulfide as the solvent, since its index of refraction is

Ε ^ 3 .

A

ο • •

0 p = 0 . 3 9

• • ,

0 6 x 0 p = 0 . 3 0

|°o° o0o ο <PD <* Xoß Cb α°Πΰ ρ pJQA •

0 p = 0 . 0 0 5

1 1 1 1 —

2 .0 2.5 3.0 3.5

K x l O "5 ( c m -

1)

0 2 4 6 8 10

K R S

Fig. 6.3. Reciprocal of D a pp for P M M A in hexane/carbon disulfide as a function of the scattering vector for three values of the volume fraction. (A) Experimental da ta for P M M A of

radius ~ 6 0 0 À with a particle density of 1.19 m g / m L . [Reproduced with permission from Pusey

and van Megen (1983). J. Physique. 44, 2 8 5 - 2 9 1 . Copyright 1983 by Les Editions de Physique.]

(B) B e e n a k k e r - M a z u r model predictions for the diffusion coefficient of hard spheres with

hydrodynamic interaction as a function of KRS and φρ (cf. Section 6.15). [Reproduced with

permission from Beenakker and Mazur (1984). Physica. 126A, 349-370 . Copyright 1984 by

North-Hol land Publishing Co.]


close to that of P M M A . The value of Κ was varied by collecting data over the angle range 60° < θ < 120° and by using four different wavelengths of incident radiation from a krypton ion laser [647 nm (red), 568 nm (yellow), 521 nm (green), and 476 nm (blue)]. The time window of the correlation functions was less than half of a decay time, and the correlation functions were analyzed by a second-order cumulant fit. Plots of 1/D a pp vs. Κ for φρ = 0.005, 0.30, and 0.39 are shown in Fig. 6.3. Also shown are the theoretical calculations of Beenakker and Mazur (1984).

Note that all three of the above plots exhibit oscillations that are virtually damped for Κ > 3 χ 10

5 c m '

1, and that the relative magnitude of these

oscillations increases with (/> P M M A. The former observation is consistent with the notion that D s e lf is obtained for large K. The latter observation indicates an increase in solution "order" as </>P M MA increases. The unexpected observation is that oscillations appear for φρ = 0.005, which was supposed to represent an infinite dilution solution. Pusey and van Megen attributed the oscillations in the φρ = 0.005 data as size polydispersity, which was later verified by numerical calculations for a system of spheres with a small degree of polydispersity (Pusey and van Megen, 1984).

6.7. Lowest-Order Correction to D m for Identical Spheres: the Batchelor and the Anderson-Reed Models

Exact solutions are known for the hydrodynamic interaction between two spheres in vertical alignment (Stimson and Jeffery, 1926) and horizontal alignment (Goldman et al., 1966). Since axial symmetry properties were employed, these methods cannot be extended to include three or more spheres. Smoluchowski (1911) was the first to examine the hydrodynamic interaction between two spheres whose line of centers did not exhibit any preferential direction of motion. To this first-order approximation, the hydrodynamic interactions increase the rate at which the spheres A and Β fall in the direction of gravity and that at which both spheres move sideways along their line of centers. The first correction term to the Smoluchowski result was obtained by Burgers (1942a, 1942b, 1942c); the direction of this correction term is to reduce the speed at which the spheres move along the line of centers. Kynch (1959) presented a general procedure for calculating two-sphere and three-sphere interactions in terms of surface harmonics to any order of the expansion parameter Rs/Rjm. The fundamental assumptions in the Kynch formalism are that the fluid velocity and the pressure at any point in space r were simply the superposition of the velocities and pressures generated by each sphere. The formalism was quite complicated, and analytic expressions were reported for only the lower-order two-body interaction terms.

6.7. Lowest-Order Correction to Dm for Identical Spheres 171

The Batchelor (1972, 1976, 1983) and Anderson and Reed (1976a) models are discussed in some detail in this section because of the fundamental differences in the treatment of direct and hydrodynamic forces in their calculations of Dm for two-sphere interactions.

The Batchelor Model

Batchelor (1972) examined the sedimentation of a sphere in an incompressible fluid that obeyed the Navier -Stokes expression [cf. Eq. (6.2.10)], where the inertia term p0d\0/dt was assumed to be negligible. The probe sphere A whose center is located at x 0 in a system of Np particles of a configurational density C(Np) moved with the velocity U A [ x 0 , C(JVp)] relative to the fixed solvent at r oo:

U A [ x 0 , C(NPJ] = U° + v ^ [ x 0 , C(JVP)] + W [ x 0 , C(NPJ], (6.7.1)

where: U A is the velocity the sphere A would have in the absence of other spheres (taken as that of a sphere falling under the force of gravity, U A = 2Ri(pp —Po)gl^0, where pp is the density of the particle); ν ό [ χ 0 , C(Np)~] is the perturbation due to the other particles; and W [ x 0 , C(NpJ] is the effect of image systems of all other spheres at the surface of sphere A. The solution to Eq. (6.7.1) was determined for a dilute suspension of spheres such that the fluid velocity at the reference location x 0 was uniform over the distance 2RS. Hence V

T ' \ ' 0 [ x 0 , C(NpJ] = 0. The center of test sphere A is now placed at x 0 , and the

velocity flow field at χ = Rs + x 0 due to sphere Β is, to the lowest-order series expansion in Rs = χ — x 0 ,

v 'o[x0,C(JVp)] = u [ x 0 , C ( N p) ] + hRl{V2u[x0,C(Npmx=X0, (6.7.2)

where u [ x 0 , C(iVp)] is the reference velocity at the surface of sphere A located at x 0 for the configuration C(Np).

The objective was to calculate the average velocity of sphere A, , as measured from the laboratory reference frame. Two types of distribution functions were employed: (1) a configuration density distribu-tion function P[C(N p) ] that describes the probability of the spheres being located simultaneously in the volume elements δτι, δτ2 \ and (2) a conditional probability P\_C(Np)\x0] for a system of Np + 1 spheres, where Np spheres are in a particular configuration where one sphere is at x 0 .

For a system of identical particles, one has |P [C(N p) ]dC(N p) = {Np)\. By requirement of a random distribution of spheres P[C(1)] = P(x + r) = n'p is the probability of finding any sphere anywhere in the solution. P[C(Np)] is used to calculate the average properties of the system as a whole. The conditional probability function P\_C{Np) | x 0 ] is used to calculate the average properties of the sphere located at x 0 . The two probability


functions are related by P[C(N p) ] = P ( x 0 + r )P [C(N p - 1) | x 0 + r] = n'pP[C(Np — 1) | x 0 + r ] , where the conditional probability is P[C(Np) | x 0 ] = P ( x 0 + r I x)P[C(Np — 1) | x, x 0 + r). The average velocity is now

 = J-j- J u A [ x 0 , C(iVp)]P[C(iVp) I x 0 ] dC(Np). (6.7.3)

Because of the long-range nature of solvent-mediated interactions, Eq. (6.7.3) cannot be integrated on the basis of negligible encounters of two or more spheres in a region of the solution. Batchelor (1972) partially circum-vents this problem by finding a quantity whose average value is known at x 0 and has the same long-range dependence as the velocity for a se-cond sphere located at x 0 + r. The difference between this quantity and can be treated by Eq. (6.7.3). If no long-range order exists and Rjk » 2RS, then one can approximate the conditional probabilities P [C(N P - 1) I x, x + r] and P [ C ( N p - 1) | x 0 + r ] by the configurational probability P [ C ( N p - 1)]. Hence, P[C(ATp) | x] - P [ C ( % ) ] ~ {P(x 0 + r | x ) - P[C(iVp - 1) I x 0 + r ] }P [C(N p - 1)], where the expression in the curly brackets { } is assumed to decrease rapidly with distance. The expres-sion for <v ' 0[x 0, C(7Vp)]> is

<v' 0[x 0,C(/V p)]> voL

xo>

xo +

r ] [ P ( x 0 + r | x 0 ) - P ( x 0 ) ] J r . (6.7.4)

We now turn to a system of two spheres interacting through a hard sphere potential. The probability of finding a second sphere at x 0 + r when the first sphere is at x 0 is: (1) P ( x 0 + r, x) = n'p, hence P ( x 0 + r, x) - P ( x 0 + r) = 0 when r > 2 # s ; and (2) P ( x 0 + r, x) = 0, hence P ( x 0 + r, x) - P ( x 0 + r) = — n'p when r < 2RS. The reference velocity on the surface of sphere A at χ due to the presence of sphere Β at x 0 + r, now denoted by u(x 0 + r, x), is assumed to be of the form given by Eq. (6.2.23). The average reference velocity is

<u(x 0 + r, x)> = - χ 2 R S

u(x 0 + r, x)dr = -5.5\]°Αφρ. (6.7.5) ο

In the evaluation < ^ P | V2u ( x 0 + r, x)>, the range of integration is divided

into two parts, x 0 + r — χ < Rs (inside and on the surface of the sphere), and x 0 + r — χ > Rs. The steady-state expression for the sphere velocity is when the total force on the surface of the sphere satisfies the relationship 77 0V

2u(x 0+ r, x) = VP — i r ç 0V [ V

r · u (x 0 + r, x)]. The mean value of the

drag due to viscous stress at the surface of the sphere is two thirds of the total drag force, or 4 π ^ 0 Ρ δ . Since integration over the range x 0 + r — χ > Rs is zero, the final result is

i P s

2< V

2u 0 ( x 0 , i ) > = i U A ( / ) p . (6.7.6)


<wy = n'p W(x0,x0 + r)dr= -1.55UA</> p. (6.7.7)

The inclusion of the hard sphere repulsive interaction term stems from the generalized Stokes-Einstein relationship between Dm and (dn/dCp)Ttfi> [cf. Eq. (3.2.4)], given as the virial expansion

dn = RT(\ + 2B 2n ' p + 3 B 3n p

2 + ···), (6.7.8)

with B 2 = 4VS for the hard sphere potential and Vs = (4π/3)Λ| for the volume of the sphere (cf. Example 2.4).

The terms in Eq. (6.7.1) are given in Eqs. (6.7.2), (6.7.5), (6.7.6), and (6.7.7), which combined with Eq. (6.7.8) give Dm to the lowest order in φρ (Batchelor, 1976):

kT °™ = ί 5 - ί

1 + ^ Ρ ) · (

6·

7·

9)

In a later treatment, Batchelor (1983) included long-range interparticle interactions for a distribution of spheres that differed slightly from the uniform distribution. Dm obtained in that study was

kT

6 7 W 7 0 & 5

Dm = T Z T - ^ - P + (1-45 - O . 5 6 a ) 0 P] (6.7.10)

where α = 3J 2°{exp[ —(/>(5)//cT] — \}s2ds, Φ(Ξ) is the pairwise interaction

potential, and s = r/Rs.

Anderson and Reed Model

Anderson and Reed (1976a) examined the one-dimensional diffusion problem for a dilute solution of hard spheres where the two spheres interact along their line of centers. The molar flux is

^ = - ö f ^ + ^ Y ^ C P , (6.7.11)

Because W(\0, x 0 + r) is relatively small, i.e., on the order of (# s/r)

4> it

can be approximated from Eq. (6.7.2), viz, W(x0, x 0 + r) = U A — U A — v'0

( x 0 , x 0 + r). Batchelor (1972) equated U A to a linear combination of the exact results of Stimson and Jeffery (1926) and of Goldman et al. (1966) for two spheres whose centers are aligned vertically and horizontally, respec-tively. The result for {W} is


where the local concentration CJr) is given by

Cp(r) ~ C P(Z) + ζ dCp(Z) z

2 d

2Cp(z)

dZ 2 dZ2 exp -Φ(τ)

kT (6.7.13)

where ζ is the relative coordinate and Ζ the laboratory-fixed coordinate, φ(τ) is the interaction potential, F(r) = [dc/>(r)/dr]

7 · e z , and / 2( r ) is the friction factor

for two identical spheres moving towards or away from each other. Substitution of Eq. (6.7.12) into Eq. (6.7.13) with a change to spherical coordinates and subsequent integration over the angles results in the survival of only the dCp(Z)/dZ term of the series expansion. This expression is then substituted into Eq. (6.7.11); a comparison with Fick's First Law of Diffusion [cf. Eq. (3.2.4)] then leads to the following expression for Dm/Dp:

Dm + 8 0 p / ,

/ =

r 2R<

dexp kT

λ{ν) dr dr

(6.7.14)

(6.7.15)

and X(r) = f2(r)/fp. In performing the integration, an approximate function

for À(r) based on the work of Rushton and Davies (1973) was employed for two equivalent spheres moving toward or away from each other at equal speed over the range ρ = (r/2Rs) — Ι,νίζ, X(r) ~ 1 + bp~

c. The reported accuracy for

the piecewise integration was ± 4% for the following (b, c, range of p): (0.2715, 0 .989,10"

6 to 0.013); (0.5510,0.812,0.013 to 2.3); and (0.6933,0.989,2.3 to oo).

The interaction potential was expressed as φ(ν) — φ Η δ + </>LR, where φΗ3 is the hard sphere potential, which is nonzero only when the spheres are in contact, and (/>LR is the long-range potential. The integral / was

Jo

d e x pV kT J

• w,

dp dp. (6.7.16)

It is noted that a result of this analysis is that hard sphere interactions do not contribute to the integral, and therefore hard sphere interactions do not contribute to the diffusion coefficient. The absence of the hard sphere interaction is a direct result of using X(r) to describe f2(r), which becomes infinite as the two spheres approach each other and cannot touch.

Solvent-mediated interactions were treated as a separate contribution and


estimated from the expression

where (Ul/Ui0)L and (Ul/Ul0)\\ are the ratios of the test particle velocities perpendicular and parallel to the line of centers relative to the isolated particle ( l / 1 0) . These ratios were obtained from hydrodynamic calculations reported in the literature. The final expression was

= 1 + 8φρ(Ι - A), (6.7.19)

where 4>LR/kT was of the forms 4>LR/kT = A(p + l ) "m and φ^/kT = A

e x p ( - m p ) / { [ l + ( m / 2 ) ]2( p + 1)}.

Comparison of the Batchelor and Anderson-Reed Models The major difference between the Batchelor and Anderson-Reed models is in the treatment of the hard sphere interaction. Batchelor assumed the full hard sphere interaction 8</>p, whereas Anderson and Reed argued that there should be no contribution from the hard sphere term. It is instructive, therefore, to compare the predictions of these two models with the experimental results of a well-examined system, bovine serum albumin. In this comparison, it is assumed that the BSA particles interact through a screened Coulomb potential [cf. Eq. (7.1.7)]. SANS studies indicate that this potential is adequate to describe S(K AR) even at high BSA concentrations (Nossal, et al., 1986).

Calculations based on the Batchelor model [Eq. (6.7.9)] were performed using parameter tailored to the BSA system for two spheres of charge Z p . The screened Coulomb potential [cf. Section 7.1 and Eq. (7.1.3)] was employed in the limit 0(s)//cT « 1, in which case α defined by Eq. (6.7.9) is

6Αλ2

ΟΗ

_A

D H

Ri

2

D H

(6.7.20)

Anderson and Reed reported calculations tailored to the BSA system in which Z p = 5, A = 2.5, and # s = 36 Λ for 0.1 M and 0.001 M monovalent salt, or λΌΗ = 9.6 Â and 96 Â, respectively. Since 0(s)/7cT ^ 0.11, the criterion for the validity of the above approximation of α is assumed to be met for the

exp - Φ \ Λ kT

dp, (6.7.18)


Batchelor model calculations. The results of the Anderson-Reed calculations and for the Batchelor model for Dm/Dp are summarized in Table 6.2.

There is a marked qualitative difference in regard to the sign of the first-order correction term for the high ionic strength solvent. The Anderson-Reed model allows for a sign change as the magnitude of the interaction between the two interacting spheres is diminished. This prediction is consistent with the QELS studies of Singh (1986) on chromatographically separated BSA monomers, in which D a pp was examined as a function of ionic strength and pH. ApP f ° r pH 6.5 and [KCl] = 1 m M found in his Fig. 16 can be expressed in the linear form D p = (1 + 840 p) . It is noted that Nossal et al. (1986) calculated an apparent charge | Z p | = 14 at pH 5.9, and that the value of the charge decreased as cp increased. Clearly | Z p | > 5 for the data obtained by Singh; hence the kO value of 84 is not out of line with the calculations based on the Batchelor and Anderson-Reed models given in Table 6.2. At [KCl] = 0.1 M and pH 6.5, however, D a pp was reported by Singh to be virtually independent of φ ρ . This observation is more in line with the Anderson/Reed model than with the Batchelor model. The crossover from positive to negative slopes upon changing the pH towards the isoelectric point of BSA has been reported by Raj and Flygare (1974) and Fair et al. (1978). Recall that at the isoelectric point Oh and Johnson (1981) obtained a slope of -0 .0166 dL/g. (cf. Section 2.14).

Aguirre and Murphy (1973) examined a model similar to that of Anderson and Reed, the primary difference being that the polyion was assumed to be a point charge. These authors obtained a value of kO = —2.625.

In their response to comments made by Phillies (1976) regarding the averaging procedures, Anderson and Reed (1976b) stated that the strongest support of their model is that it predicts the equivalence of the mutual and tracer diffusion coefficients; they cited the work of Keller et al. (1971) as experimental verification of this prediction. As later shown by Hall and

Table 6.2

Comparison of the Batchelor and Ande r son -Reed Results for D J D ^ for BSA

fl

A JD°P m ^DH(Â)

Anderson-Reed 1 + 750p 0.75 96 Batchelor 1 + 56.7φρ 0.75 96 Anderson-Reed 1 - \.5φρ 7.5 9.6 Batchelor 1 + 1.670p 7.5 9.6

a Anderson-Reed potential: (f)LR/kT = Aexp(-mp)/

{[1 + ( m / 2 ) ]2( p + 1)}; m = 2RS/λΌΗ, A = 2.5, Z p = 5,

Rs = 36 À.

6.8. Higher-Order Pairwise Interaction Terms: The Method of Reflections 177

Johnson (1980) (cf. Section 2.15) and others [viz., Brown (1984) in Fig. 5.14], the mutual and tracer diffusion coefficients are not equivalent. It does not necessarily follow, however, that the Anderson/Reed model is entirely in-correct. These authors base their calculations on the exact solution X(r) = f2(r)/f° obtained by Rushton and Davies (1973) as derived for a specific set of boundary conditions. The assumptions behind these calculations should be re-examined. As pointed out by Phillies (1976), Wolynes and Deutch (1976) showed that if slip rather than stick boundary conditions were used, then the divergence in f2(r) is much reduced. Within the context of the Anderson/Reed argument, the space-varying friction factor f2(r) may greatly reduce, but not prevent, the probability of hard sphere contacts. This supposition would lead to an "effective distance of closest approach", de{{, that may be greater than the hard sphere, or hydrodynamic, contact distance 2RH. Phillies (1984a) has argued that if de({ > 2 R H, then the hard sphere coefficient kHS would be more negative, i.e., less than 8 but not necessarily zero.

6.8. Higher-Order Pairwise Interaction Terms: The Method of Reflections

Smoluchowski (1911) first described the velocity field at the surface of the probe sphere in terms of a series expansion in the inverse interparticle separa-tion distance, which is referred to as the method of reflections.

Felderhof (1977) considered a sequence of hydrodynamic interactions involving only particles A and B. The initial velocity of sphere A is taken to be that of the isolated sphere, denoted by v A 0(r). This velocity field induces a force density on particle Β that generates a velocity field U B t( R B) with no net force on the solvent, i.e., F B Λ = 0. This field in turn induces a force den-sity on sphere A with a vanishing force, i.e., F A 2 = 0. The velocity field at any point r in the vicinity of the two spheres A and Β is given as the superposition of the velocity fields of the two spheres. The iterative procedure is sum-marized by

v(r)= f [vA , 2 , ( r ) + v B , 2 , + 1(r)], (6.8.1)

and at the surface of the two spheres,

U A ( R A ) = g U A, 2 J. (R A) (6.8.2)

J = O and

U B( R B) = f U B f 2, . + 1(R B) , (6.8.3)

J = O with the constraints F A 2 j = 0 and ¥B,2j-i = 0 for 7 > 1. The procedure is to expand the velocity field of one particle about the location point of the second


particle. The lowest-order reflection term v A 0(r) is, with r B = r — R B,

v A » = Σ - r r B : [ ( V S r . v A , 0 ( r ) ] r = R B, nô nl (6.8.4)

with a similar expression for v B 0(r). Felderhof (1977, 1978) examined the self and pair components of Dm to the order ( K s /

r)

7 f °

r the mixed slip-stick

boundary conditions. AC p was assumed to obey the equation of motion

^ = VT - D - [V ACp + ^ \ φ ) AC P] . (6.8.5)

The result for an incompressible solvent obeying the Navier -Stokes expres-sion for mixed boundary conditions was

VA.OM =

3R,

4 r A

I + • + 1 1 - 3j8A

4 1 -βΑ

P A ^3

I r A

• U A , (6.8.6)

where r A = r — R A, R A is the location of the center of particle A, and 0 < ß < i - [Note that Eq. (6.8.6) is the same as Eq. (6.2.24) with the solvent at infinity at rest.] Eqs. (6.8.1)—(6.8.4) are then employed to calculate U B a , U A 2, and U B 2, and hence D:

Ojm = ocjmqAB + ßjm(l (6.8.7)

where <rAB is the matrix generated by the dyadic product r A Br A B/ r A B, and the coefficients (xjm and ßjm depend upon r A B. The next step is to determine the linear and bilinear flow patterns. The linear flow pattern is determined from Eq. (6.2.3) with zero torque (ΩΓ ΟΙ χ δτ = 0) and a symmetric and tracer-less strain tensor R. The resulting expression is similar to Eq. (6.2.20), where corrections are made for the mixed boundary condition and choice of reference velocity. The bilinear flow pattern for the Cartesian coordinate α in the laboratory reference frame and relative coordinates β and γ was assumed to obey the equations v 0 α = ß yrßry and P0(r) = η0τ

τ · V

2v 0 . The third-rank

matrix ha ß y is symmetric and tracerless, and ha α y = 0 (incompressible fluid). In solving for U A 2, Felderhof (1977) divided the problem into three parts: (1) a symmetric component, V Q , that gives the unperturbed solvent flow at infinity; (2) a tangential component, v'0, that arises because Β exerts no force on the solvent; and (3) a symmetric force dipole component, v£, that exerts force on the sphere. Hence,

vs

0(r) = i v 0 + i r aV v 0 , a - . V

2v 0 - ^ r

2V

2v 0 ] , (6.8.8)

v i , « = | v 0 - i r aV v 0 , a + è r [ rT · V

2v 0 - è r

2V

2v 0 ] , (6.8.9)

vdoW = - ï V a V v 0 , a + i r

2V

2v 0 , (6.8.10)

6.8. Higher-Order Pairwise Interaction Terms: The Method of Reflections 179

— A b j 4 ί + 2β Β r A 3 4l+2ßB νΑΒ

| 5 1 - 3βΑ 1 - βΒ R \R

B ι (6 8 121 + 2 1 - βΑ 1 + 2j8B r A B

21 1 - 0 Β R g R A r i , Ί 3 1 - 4/?Β RBRA

r

1 1 - 6/?Β K B R A (

4 l - j 8 B r 6

-(Ι + 3σ Α Β) , (6.8.13)

and

9 1 - 3ft, J tgR A

4 1 + 2ft, r A B

D A B was written in terms of short-range (S) and long-range (L) components, P a b = B s ( r A B) + B° (r A B) + B<(rA B), where

kT 75 R3

ARi (1 - j?A)(l - ßB) P s (

rA B ) — 7 ~, 7 / , , τ ο w , , ι » ! ? A B . (6.8.15)

6πί70 4 r A B (1 + 2jffA)(l + 2ft,)

( Έ α α = - , , n n —— σ Α Β . (6.8.14)

kT 3 B2(r A B) = 7 — — (I + < ? a b ) (Oseen) (6.8.16)

6π^ 0 4 r A B

is the Oseen contribution, and the dipole contribution is,

?£(r A B) = kT 1 R

2

A(\ - 3 / y + Rl(l - 3βΒ) ( Ι - 3 σ Α Β) . (6.8.17) 6 ^ 0 4 r A BL l - j 8 A ' 1 - 0 B

An important result from this expansion is that V A · D A B φ 0 because of

Bs (rAß)- This places concentration limitations on the applicability of any

theory of DM that has DV 2 as a starting expression in the time evolution

operator. Felderhof (1978) evaluated the average values of the coefficients A (r A B) and

B(r A B) for a system of identical spheres. The equilibrium pairwise distribution was assumed to be <n 2(r A, r B)> = <n ' 0>^ (

0

2 )( r A B) , where the equilibrium dis-tribution function #

(

0

2 )(

γα β )

w a s assumed to be g (

0

2 )(r A B) = exp[ — </>(rAB)//cT]. The lowest-order expression for <D a p p> for a system of identical spheres was found to be

where W 0 = 0 = S72\L

0; r aVv s

0 ?a = 2v s

0; and r a W 0 a = - \ * 0 . After some algebra, the self part of D was found to be

P a a = DILL + ^HAA + <2>hAA + < 3>hA A], (6.8.11)

_ f 15 1 - ft, R3

BRA 9l-3ßB R>RA

< A p p > = £>p[l + (Κ + Ao + AD + + λΑ)φρΙ (6.8.18)


Table 6.3

Lowest-Order Volume Fraction Coefficients for Hard Sphere Diffusion with

Stick Boundary Condit ions

λ Reference

+ 8

+ 8

0

+ 8

- 6

- 6

0

- 6

0

0.0

0.28

0.0

0.29

0.0

- 1 . 8 3

- 1 . 8 3

- 1 . 7 3

+ 3.0

+ 1.45

- 1 . 8 3

+ 1.56

Altenberger and Deutch (1973)

Batchelor (1976)

Anderson and Reed (1976a)

Felderhof (1978)

where RA = RB = Rs, Ày is the second virial contribution, λ0 = —6(1 — /?) is the Oseen contribution, λΏ = \ — 3β is the long-range dipole component, / s = (75/256)(l - ßf/{\ + 2β)

2 is the short-range term, and the "self part"

A (r A B) is

In the case of stick boundary conditions (β = 0), one has

^ = 1 + ( 8 - 6 + 1 + 0.29 - \.Ί3)φρ = 1 + 1.560p. (6.8.20)

The difference in the Batchelor and Felderhof calculations lies in the parameter (W) = Às + λΑ = 0.28 - 1.83 = - 1 . 5 5 .

A comparison of the lowest order contributions to the mutual diffusion coefficient for four models is summarized in Table 6.3.

6.9. Effect of Divergent Terms on D m

Carter and Phillies (1985) employed a cluster expansion method to obtain Dm

up to the (Rs/r)

7 term in which the divergence terms are taken into consideration.

These authors expressed Dm as the product of two series expansions for: (1) the hydrodynamic interaction tensor, H; and (2) the inverse structure factor, 1/5(K). Two-, three-, and four-body interaction terms were included. Note also that these authors included the solvent backflow correction, 1 — φρ,

λΑ = (1-β)

91 l-β 1 1 - Aß

320 I + Aß ~ To 1 + β

1 1 - 6β

80 1 - β ' (6.8.19)

6.10. Concentrated Solutions: The Method of Induced Forces 181

in their equations. Dm with divergence terms was found to be

Dm = D°p(\ - O.8980p - 19.O10p

2 - 70</>

3), (6.9.1)

and when the divergence terms are ignored,

Dm = D°p(\ + 0.56</>p - 8.390p2 - 35(/>p

3), (6.9.2)

and to "correct" for the inclusion of solvent backflow Eqs. (6.9.1) and (6.9.2) should be multiplied by 1/(1 — φρ) ~ 1 4- φρ + φ

2

ρ, with the results, res-pectively:

Dm = D°p(\ + O.lO20p - 18.910p2 - 89.9</>p

3), (6.9.3)

Dm = D°p(\ + 1.560p - 6.830p2 - 42.8(/>p

3). (6.9.4)

6.10. Concentrated Solutions: The Method of Induced Forces

The method of reflections may in principle be used to examine multisphere interactions. The expressions can become quite complicated, since they require a detailed knowledge of the velocity field. It is to be realized, however, that only the immediate history of the velocity field is important, and not its total history from its point of genesis. The method of reflections views this boundary value problem from outside the sphere. The method of "induced forces" introduced by Mazur and Bedeaux (1974) views the boundary value problem from within the sphere.

The static Stokes equation for a collection of Np particles of radius Rs is written as

VT

. P = £ Fjn d

(r , i), (6.10.1) 7=1

where F}n d

(r, i) = 0 (|r — rj(t)\ > Rj). Because of the choice of boundary conditions, Ρ is defined as the negative of Eq. (6.2.8). Fourier transformation of Eq. (6.10.1) along with Eq. (6.2.8) gives the equation of motion,

η0Κ2\0(Κ,ή = - ί Κ Ρ ' ( Κ ) + Σ e x p [ - i K

T D. r 7 ( r ) ] · F}

n d(K). (6.10.2)

Given the incompressibility condition KT · v 0(K) = 0 and that P'(K) = 0 for

|r - Tj(t)\ < Rj, multiplication of both sides of Eq. (6.10.2) by I - (KK/K2) =

I — kk, where k = K/|K| is the unit vector in the direction of K, yields the expression valid for the region |r — Tj(t)\ < Rj,

η0Κ2ν0(Κ,ή = Σ e x p [ - / K

T . r,(r)](I - kk) . F f (K), (6.10.3)

7=1


with the formal solution

ν 0(Κ , ί ) = vg(K,i) + Σ fao*2)

-1 e x p [ - / K

7' . r 7 . ( 0 ] ( I - kk ) .F f

d(K) , (6.10.4)

where Vq(K , i) is the unperturbed fluid velocity and is taken as zero. Fjn d

(K) is expanded as a series in k,

00

Fjn d

(K) = Σ (-iKRj)pk

pQM

{j

p+l\ (6.10.5)

p = 0

where Μγ+i] = (i/Rj)

p(\/p\)ld

p¥f

d(K)/dK

p^0 and the notation in-

dicates a p-fold product of k that results in a tensor of rank p, i.e., k 5 = kkkkk. The pth force multipole Μγ

+ί) is a tensor of rank ρ + 1. F^

n d(K), however,

is a vector quantity, or a tensor of rank 1. Since the final tensors on both sides of this equation must be of the same rank, the symbol Ο represents a p-fold contraction between the tensor k p

and M * .p + 1 )

, i.e.,

[ k 3 Θ M«*']„ = ZEZWX4i.y./),a- <6-10-6) α β y

F}n d(K) can be expanded in terms of irreducible multipoles:

00 / i V IF s'm(KR)

F}"d(K) = Xo(2p + 1)!!^J - ~ ~i^R~

J^

Θ [ F f ( K ) ]

( p+ (6.10.10)

where (2p + 1)! ! = 1 - 3 - 5 — (lp — 1) · (2p + 1) and bp denotes an irreduc-

ible tensor of rank p. [Irreducible means that the tensor is traceless and symmetr ic] The irreducible tensors to the third order for the vector b are:

K = ba\ babß = babß-\oaßb2\ and babßby = babßby-\{öaßby + öaybß + ößyba)b

2.

These authors also showed that

(6.10.8)

The surface moments of the fluid velocity field are

( - ) ' c -<Φο(Φ*=^ψ L K ^ ^ ^ e x p O - K ^ R X i K ) , (6.10.9)

(2π) J dKp KRi

where nt is a unit vector normal to the surface and d/dK = kd/dK + (\/K) (I — kk) · d/dk. The zeroth-order moment is related to the average velocity of the sphere, <v 0(r)> s. The velocity of the ith sphere is

1 1 ivp oo

= [ F i n d ( K ) ] < 1 > + £ ^ Θ [ Ρ ^ ( Κ ) ] η ( 6 · ι α ι 0 )

6.10. Concentrated Solutions: The Method of Induced Forces 183

where

3R,{2m- 1)!! f i V "_ 1 )

R dK dK(l - kk)

KRj yy ,J

'dKm-

1 KRj

Ry is the distance between the spheres i and j , and

[Fj"d(K)] (i) _

1 f« . . . s i n i K ^ ) „

Ί ^ ΓΡ

' ( κ )

·

(6.10.11)

(6.10.12)

The quantity A(?, m)

is a dimensionless tensor of rank m + 1 and is referred to as a connector because it connects the force mul t ipoles[Fj

n d(K)

( 1) to

[F}n d

(K) ]( m )

. The first-order moment is related to the rotational motion of the sphere.

The formal solution is not of any concern in the present development, except for the introduction of the tensors B

( 2'

2 ) and B

( 3-

3 ):

B<2.2) = _

]}<3,3)

3 - 3 ! !

8π

9 - 5 Ü

8π

dkk(I - kk)k,

dkkk(l - kk)k.

(6.10.13)

(6.10.14)

Although higher-order moments are zero, they lead to the general expressions

|j(m,nt) _ "(2n - 1)!!

R"-1

3R, dk dK

d"'1 sin(KRt)

δΚ" KR, ÔKn

KR:

™ij — 3Ri(2n- l ) ! ! ( 2 m - 1)!!

8 π2

Ri

χ (I - kk)

R

dK"

dk

s'm{KRj)

dK - 1

1

sm(KRi)

dû"'1 KR(

τ e x p ( - î K ' .R 0. ) , (6.10.16)

and B( m , I)

= 0 if m φ η. It is noted that B( m m)

are independent of the particle positions, whereas the tensors A | "

m ) depend upon the interparticle distances.

For m > 3, one finds that B( m m)

= - A( m

'm )

(K 0 = 0). The general expression


for the mobility tensor for the ith sphere is

oo oo oo oo Ν ρ Ν ρ

Η % = ί ^ Γ + Σ Σ Σ - Σ Σ Σ - Σ s = 1 m ι = 2 m2 = 2 ms = 2 ji = 1 ji - 1 Js - 1

(6.10.17)

The convention is such that the rightmost superscript index of the tensor on the left contracts with the leftmost superscript of the tensor on the right. We complete this introduction by noting that if Eq. (6.10.8) is used in Eq. (6.10.16), then the connectors can be written as

3R dkk

n r( I - k k ) k

2π dKU -

R2i

An + 2 4m + 2 K

2}K

n + m-

2Qxp(-iK

T .R 0.).

(6.10.18)

Integration of Eq. (6.10.18) over Κ results in two terms that are proportional to R'u

(m + "~

1). A general conclusion from this formalism is that a sequence of

5 4- 1 connectors has contributions to the mobility proportional to R^M,

where M = 1 or 3 for 5 = 0 and

M = £ 2mt + 2 + j

i= 1 (s> l ; m i > 2 ; 7 = -s - 1, - s + 1, - s + 3 , . . . s + 1). (6.10.19)

One conclusion from this set of equations is that it is not possible to obtain M = 2 or M = 5 for translational mobility. Another result obtained by Mazur and van Saarloos (1982) is that the dominant contributions for a cluster of η spheres are of the order R 7 .

( 3 n~

5) for translational motion.

6.11. Evaluation of <Df e l f>

The generalized Einstein relationship is

<pself> = (6.11.1)

where μ„ are defined by Eq. (6.10.17). In the virial expansion approach, Beenakker and Mazur limited their analysis to two- and three-sphere inter-

6.11. Evaluation of <D^ e l f> 185

r 1 / r \3

(6.11.3)

for 2RS < r < 4RS. Direct integration yielded

- ^ f = I [ l - \.73φρ - 0.93φ2

ρ + 0(φ3

ρ)-] (two-body), (6.11.4)

where only the Rjk

7 term in the mobility was evaluated:

67ttioRsftu = Σ Σ ζ21(3Λ2(3Λ3(3Λ2

Λ M 16 \RikJ \RkmJ \RmiJ

x Ri*Rim[(l - 3α2)(1 - 3 a

2) + 6a f c

2a

2 -h 6 a f a f ca w] , (6.11.5)

where the direction cosines are ott- — R l fe · R i m, ock — Rfcl- · R^m, and ocm — Rmi - R m Ä, and g(rl2, r 2 3, r 3 1) = 1 if none of the spheres overlapped and zero otherwise. The three-body contribution was found to be D f e l f/ D p = I1.80</)

2.

Hence the scalar value for the two- and three-body reduced diffusion coefficient is (Beenakker and Mazur, 1982, 1983)

^self £ = l - 1 . 7 3 t f p + 0 . 8 8 # . (6.11.6)

In comparison, Batchelor (1976) obtained for a probe particle submerged in a sea of identical particles.

^ = ^ = 1 - 1 - 8 3 < V (6.11.7) u

v Ρ

[It is noted that Batchelor did not distinguish between long-time and short-time values of D s e l f. D T r is thus equated with Dlel{ since memory effects were not considered.]

The many-body case was carried out by numerical evaluation of the con-nector representation of the problem (Beenakker and Mazur, 1983). The self diffusion tensor was expressed in terms of a density fluctuation expansion,

0 0

Dfeif = X D( 2 p )

, (6.11.8) p = 0

actions. In the two-sphere case, the expression for D f e l f/ D p was found to be

^ = I + < n ; > „ j ^ - ( r ) [ - f (hjn + - 171)].

(6.11.2) The function g

{2)(r) was piecewise defined: (1) g

(2)(r) = 0 for r < 2RS;

(2) g{2\r) = 1 for r > 4RS; and (3)


where D( 2 p)

contains terms on the order of ({δη')2ρ}, The expressions

associated with the unperturbed component are

dK \m{KRs)

[1 l Ü f X y -1

, (6.11.9)

S(x) = ^{Si(2x)x 1

+ ^cos(2x)x 2

+ ^ s i n ( 2 x ) x 3

- [sin(x)]2.x~

4 - 4[sin(x) - x c o s ( x ) ]

2x ~

6} , (6.11.10)

and

Si(x) = 2x sin(t)(dt)

(6.11.11)

The numerical results to the first expansion term, D ^ e lf = D ^ f + D ^ , given in Table I of Beenakker and Mazur (1983), is fit by the least-square expression

1 - 2 . 1 2 7 2 0 p+ 1.735c/)2. (6.11.12)

Beenakker and Mazur (1984) later improved their calculations to include all orders of a certain class of correlations, the so-called ring self correlations. The ring self correlations are the connectors A<ü>°(

r = 0). Instead of an expansion

in the density n'p(r\ the expansion was carried out with the "renormalized density" γ = ηο(ηρ}ΰ

1ηρ(

τ)· Calculations were carried out to the first per-

turbation term D j ^ / D p . The numbers in their Table III give the form )

' P '

^ s e l f

D° = 1 - 2 . 1 9 9 0 P + 1.21670

2. (6.11.13)

The largest percentage error in using this expression is at φρ = 0.45, where Eq. (6.11.13) gives a value of 0.26 compared to the reported value of 0.28. All other computed values are within +0.01 of the reported values.

Example 6.2. Concentration Dependence of D^ e lf for Coated Silica Particles. Van Veluwen et al. (1987) reported DLS measurements on doubly coated silica particles [y-methacryloxypropyltrimethoxysilane (TPM) was the first coat, and then trimethydroxysilane (TMHS)] . The resulting particles were un-charged in tetrahydrofurfuryl alcohol (THFA) and had an index of refraction close to that of THFA in the range 30°C < Τ < 35°C (to minimize multi-ple scattering). Values of φρ were determined from weight fractions and the densities, pp = 1.58 and p 0 = 1.04 g /cm

3. D a p p was obtained by second-order

cumulant analysis methods. D f e l f/ D p vs. φρ is shown in Fig. 6.4. Also shown are the virial expansion results of Beenakker and Mazur [cf. Eq. (6.11.6)], the lowest-order correction term result of Batchelor (1976) [cf. Eq. (6.11.7)], and the many-body results of Beenakker and Mazur (1983, 1984).

6.12. Evaluation of <D s

L

e l f> 187

0.2-\

4 1 1 1 1 1

0.1 0 . 2 0 . 3 0 .4 0 .5

0P

Fig. 6.4. Normalized self diffusion coefficient as a function of volume fraction of coated silica

particles: ( χ ) 30°C; ( · ) 35°C; ( ) l - 1 . 8 3 0 p (Batchelor, 1976); (---) 1-1.73φ ρ + 0 .880\

(Beenakker and Mazur , 1983); ( ) (Beenakker and Mazur, 1983). [Reproduced with permission

from van Veluwen et al. (1987). Faraday Discussions. 83 . 59 -67 . Copyright 1987 by The Royal

Society of Chemistry.]

An important feature of these data is that they support the hypothesis that the velocity field is established on a time scale much less than the macroparticle diffusion time.

6.12. Evaluation of < D ÊLF >

<D^ e l f> is defined on a time scale in which the solute lattice has relaxed. The probe particle experiences nonhydrodynamic interactions that are contained in the "memory function" M(K, t — t') associated with the relaxation of ACp(K,t) (cf. Section 6.5). The expression for DêU (Ackerson, 1978; Hanna et al., 1982; Hess and Klein, 1981, 1983a; Klein and Hess, 1983) is

(6.12.1) z - 0

where M(K, z) is the Laplace transform of M (K, t). Separation of mean field contributions gives

Z>s

L

e,f = D°p + ADmean + ADmeJK) = Dfe lf + ADmem(K). (6.12.2)

Hanna et al. (1982) examined the hard sphere situation for two particles with-out hydrodynamic interactions and obtained the exact solution

£>s

Le.f = D°p(l - 2φρ) (hard sphere). (6.12.3)

^ S E L F — ^ S E L F +

M ( X , Z )


They also examined the effect of (linearized) hydrodynamic interactions to the Oseen level using Felderhofs reflection series up to ( # s /

r)

7 (1977). The

solution to this operator, the density distribution function p ( r 1 2, z | r 0) , was represented as a series expansion in spherical harmonics. Evaluation of A D m em = AD^lf + AD™{, where AD^\{ is the zero-order and A D ^ f is the second-order correction term, gave A D m em = 0.09</>p for the Oseen model and ADmem = O.15980 p for the Felderhof model. Noting that A D m e an - 0 for the Oseen model and A D m e an = — \.Ί3φρ for the Felderhof model, Hanna et al. (1982) obtained

uself

1 - 0.09φ ρ (Oseen level) (6.12.5)

and

= l - 1.890p (Felderhof). (6.12.6)

Yoshida (1983) improved upon these calculations and found that A D m em = - 0 . 2 6 2 0 p , or

^ £ = 1 - 1.9960p. (6.12.7)

Example 6.3. The Concentration Dependence of Df e lf and Dê]{. Van Megen et al. (1987) examined poly(methylmethacrylate) (PMMA) in a host dispersion of poly(vinylacetane) (PVA) particles that were index-of-refraction-matched in a mixture of ds-decalin (n0 ~ 1.481) and irans-decalin (n0 ~ 1.469). The diameter of these particles as determined by QELS mea-surements was reported to be 170 nm and 660 nm for samples P M M A / 1 and P M M A / 2 , respectively, and 166 nm for PVA. System Tl was at the ratio PMMA/1 :PVA = 1:43. The system T2 was composed of one silica particle for every 500 P M M A / 2 host particles, and the solvent system was a mixture of decalin (eis and trans with n0 ~ 1.48) and carbon disulfide (n0 ~ 1.63). The particles were stabilized by a comb polymer of P M M A with "teeth" of poly(12-hydroxystearic acid). The effective volume fraction was determined from the weight fraction using assumed stabilization layer of thickness 9 nm. 0 e ff was estimated from the core volume fraction at the crystallization point, with the result </>eff = 1.1250c.

Values for Df e lf and Dêl{ were estimated from first cuumulant and single exponential analyses, respectively, of the correlation function. The angle used in these studies was below the first peak maximum in the static structure factor. The reduced diffusion coefficient D^cl{/Dp for samples T l and T2 vs. </>eff are presented in Fig. 6.5.

6.12. Evaluation of <D^ e l f> 189

D s e l f / D p

1 . 0 4

0.6 J

0.2-J

Δ N

Vshort time Δ \ Ν (first cumulant)

S * Δ 0

f ΟΝ

\ Δ

long time ^r>^

A . (exponential)

v ^ ξ

T 0.2 0 . 4 0.6

A 0 e f f

H 1 1 1 1 1 0 0.1 0 .2 0.3 0.4 0.5

B Fig. 6.5. Reduced self diffusion coefficients of P M M A and silica as a function of the effective

volume fraction. (A) Δ , Sample T l ( P M M A ) ; · , sample T2 (silica); O , da ta taken from Kops-Werkhoven and Fijnaut (1981). The effective volume fraction is 1.125 times larger than the core volume fraction. [Reproduced with permission from van Megen et al. (1987). Faraday Discussions.

83, 4 7 - 5 7 . Copyright 1987 by The Royal Society of Chemistry.] (B) M o d e - m o d e coupling approximation calculations of Medina-Noyola using a potential of interaction φ{χ) composed of two parts: (1) a hard sphere repulsive potential 0(x)//cT = OO for 1 > x; and (2) a long-range attractive Yukawa tail, φ{χ) = X ' e x p [ — z(x — l ) ] /x for 1 < x. The parameters are χ = r/2Rs; K'

is a "strength parameter"; and ζ is a "screening parameter". [Reproduced with permission from Medina-Noyola (1987). Faraday Discussions. 83, 2 1 - 3 1 . Copyright 1987 by the Royal Society of Chemistry.]


Medina-Noyola (1987) calculated D^ e lf in which the interaction potential consisted of a hard sphere part with a long-range attractive Yukawa tail. The tracer friction factor was calculated by the Laplace transformation of a con-tracted description of the Markov process (cf. Appendix E). These results are also shown in Fig. 6.5.

The memory function as defined by Eq. (6.12.2), i.e., <£>SEIF> ~~ <^hif>> appears to be a nonlinear function of the volume fraction. The value of the memory function, valid up to 0EFF = 0.10, is estimated from the above data to be A D m em ~ —2.2. This difference is an order of magnitude larger than the theoretical calculations of Yoshida (1983) [cf. Eq. (6.12.7)]. Caution must be exercised in accepting this analysis, however, because of the effects of polydispersity (the reported polydispersity was 4%, 15%, and 25% for P M M A / 2 . P M M A / 1 , and PVA, respectively; cf. Chapter 4). The Medina-Noyola simulations shown in Fig. 6.5 clearly indicate that values of A D m em

on the order of unity can be attained with attractive interparticle interactions between spheres.

6.13. D s e lf and the Glass Transition for Hard Spheres

One consequence of the hard sphere potential is that the diffusion process may cease at sufficiently high volume fractions and both D^ e lf and Dfe lf become zero. Before this happens, however, the suspension of hard spheres undergoes a series of structural transitions, as can be inferred from discontinuous changes in the value of Dê]{. Less sensitive to the local solution structure is Z>SE]F since, by definition, the diffusion distance is much less than the particle diameter.

Example 6.4. The Glass Transition of PMMA. In their studies on PMMA, van Megen et al. (1987, cf. Example 6.3 for details of the solution and sample preparation methods) also examined the hard sphere "glass transition". The sample used in these studies was designated P M M A / 3 , which had a reported radius of 125 nm and a polydispersity of 15%. A plot of ^ ln [ / l ^

( 2 )( i ) ] vs. t is shown in Fig. 6.6 for four core volume

fractions φ0. [Our notation differs from that of van Megen et al., where they defined g

i2)(t) as the intensity correlation function.]

There are two important features of the curves shown in Fig. 6.6. First, two relaxation processes appear with only a small change in the volume fraction in going from 0C = 0.331 to φ0 = 0.349. The second feature is that both the ordinate intercept and the longest decay rate decreases upon increasing the volume fraction. These two observations are indicative of a "phase transition" as the concentration of the particles increases above some critical value. The particles become "trapped" in cages of surrounding solute particles. Appar-ently the particles are "free" to move on the fast time scale, which corresponds

6.13. Z) s e lf and the Glass Transi t ion for Hard Spheres 191

0 .2 0 . 4 0 .6 0.8 1.0

DELAY TIME (S) Fig. 6.6. Logarithm of the intensity correlation function as a function of time for P M M A / 3 .

The volume fraction φοί{ is that for the P M M A core. [Reproduced with permission from van

Megen et al. (1987). Faraday Discussions. 83, 4 7 - 5 7 . Copyright 1987 by The Royal Society of

Chemistry.]

to motion within the cage. The fact that the transition to a "frozen state" is not achieved was assumed by these authors to be due to the polydispersity of the sample.

Pusey and van Megen (1987) reported studies on larger P M M A parti-cles (diameter of 340 nm) that were also stabilized by a 10-Â layer of poly-(12-hydroxystearic acid). However, there were no host particles of PVA. The nearly isorefractive solvent, with a weight ratio of 2.52:1 of decalin and carbon disulfide, respectively, was used to minimize multiple scattering. The DLS measurements were obtained at a scattering angle near the Bragg diffraction peak. The volume fractions were normalized to the theoretical value for the onset of crystallization as determined in computer simulation studies on hard spheres (Hoover and Ree, 1968; Woodcock, 1981; Angel et al. Woodcock, 1981), i.e., φρ = 0.494 = φ{, where the subscript "f" de-notes "freezing". From plots similar to those shown in Fig. 6.6, these authors identified four solution structure regimes: (1) for </)eff < 0.49, the P M M A particles diffused as a fluid, as inferred from a relatively fast rate of decay; (2) for 0.49 < </>eff < 0.542, there coexisted liquid and crystalline structures, as indicated by two relaxation modes; (3) the crystalline state prevailed over the range 0.542 < 0EFF < 0.565, as evidenced by a very slow, but measurable, re-laxation rate and well-defined Bragg diffraction peaks; and (4) the glass state


was characterized by a virtual baseline, i.e., infinitely slow decay rate, which dominated in the highest volume fraction range 0.565 < </>eff < 0.614. The randomness of the glass state precludes the occurrence of Bragg peaks. The four solution regimes for hard spheres are schematically represented in Fig. 6.7.

0{

oooy&oo

°&οοο

oooooo oooooo ° ° ο 0 ο o ° ° ο ο ° ο

° ο

closest packed

glass

(random closest packed)

crystalline

crystalline/ liquid

liquid

Fig. 6.7. Schematic representation of the solution regimes for hard spheres. The liquid regime ((/>eff < 0.49) exhibits one relaxation time characteristic of freely diffusing particles. The crystalline/liquid regime is bounded by the "freezing point" volume fraction (φί ~~ 0.494) and the "crystalline point" volume fraction ( $ c ~ 0.542). The crystalline/liquid regime is envisioned as having "solute cages" that hinder the diffusion of particles on the long time scale. Both liquid-like and crystalline-like regions coexist. The crystalline regime, characterized by highly ordered arrays, prevails for 0.542 < 0 e ff < 0.565. The "glass transit ion" occurs at φα = 0.565, and the "glassy regime" extends over the range 0.565 < </>eff < 0.74. The glassy state is characterized as having a random-order packed array. The hard spheres are virtually immobile at this stage. (The above diagram may be subtitled: A physicist's model of Escher's image "Butterflies".)

6.13. Z>self and the Glass Transi t ion for Hard Spheres 193

Within the context of DLS nomenclature, the "glass state" for a suspension of hard spheres is defined when D^ e lf = 0, but it is not necessarily required also to have D*eU = 0. Let us define a relative volume fraction φρ = φρ/φ0, where φβ is defined when all motion ceases. We assume that Eq. (6.11.12) is identified with D^ e l f. Hence,

^ = 1 - (2 .19910 G)0 P + [1 .2167 (0 G)2]«> p)

2 - F ( 0 P) (6.13.1)

where ¥(φχ

ρ) is an unspecified power series in φρ whose value is determined by φ0 at D^ e lf = 0. It is curious to note that if φ(3 = 0.556, then F((/>p) = 1 - (2.199 χ 0.556) + 1.2167(0.556)

2 = 1 - 1.2227 + 0.3761 - 0.1534. This

number can be expressed as the sum (Jolley, 1961)

Σ η un\n - L η =

° 1 5 3 4

' ( 6 1 3 2)

„ = i (2n — \)(2n)(2n + 1) Hence one might infer that F(</>p) can be approximated by a series of the form given by Eq. (6.13.2). The motion of the hard spheres does not cease at φρ = 0.556, however, since this is in the crystalline regime of the solution structures. Let us choose the value φα = 0.64 as the volume fraction at which the spheres become immobile, which corresponds to "random close-packing" for hard spheres (Batchelor and O'Brien, 1977) and which was used by Rallison (1988) in a study of Brownian motion in concentrated suspensions of hard spheres. The reduced diffusion coefficient D

r

sum for </>G = 0.64 is defined with Eq. (6.13.2) as

where Dr

BM = 1 - \Λ0Ίφρ + O.4984(0 p)2 is the corresponding Beenakker-

Mazur expression. The ratio of the coefficients in the last two terms in the Beenakker -Mazur (1984) expression is 1.4070/0.4984 = 2.8, which is close to the ratio 3!/2! = 3. This hints at the possibility of expressing the diffusion coefficient for hard spheres in closed form as a series m( — x)

n/nl with χ = φρ.

Noting that GO ( — χ)

η

Σ L

- r - = e x p ( - x ) - l , (6.13.4)

one obtains after some algebraic manipulation

D^= 1 - ^ [ e x p ( - 0r

p ) - 1 + 0r

P ] , (6.13.5)

where e = 2.71828.. . is the Napierian base number. Rallison (1988) examined the Brownian diffusion of interacting systems of

rods and spheres. Neglect of all interparticle interactions except the hard


Table 6.4

Comparison of the Reduced Diffusion Coefficient as a

Function of the Reduced Volume Fraction"

»m Di D EXP

0.064 0.100 0.864 0.810 0.864 0.869

0.128 0.200 0.739 0.640 0.738 0.745

0.192 0.300 0.623 0.490 0.620 0.630

0.256 0.400 0.517 0.360 0.511 0.522

0.320 0.500 0.421 0.250 0.409 0.421

0.384 0.600 0.335 0.160 0.315 0.326

0.448 0.700 0.259 0.090 0.227 0.237

0.512 0.800 0.193 0.040 0.146 0.153

0.576 0.900 0.137 0.010 0.070 0.074

0.64 1.000 0.091 0.000 0.000 0.000

Dr

BM = 1 - 1 .407^ + 0.4984(φ ρ)2;

£ > R = ( 1 - < / > P )2

1 0 0 ( _ Λ >Γ

)Η _1

D ™ = D ™ - a 5 9 5 7 W ) 3 X i ( 2 y i_ l x ^ K 2 y | + 1 );

£ > E X P= l - - ^ [ e x p ( - < / >r

p) + ΦΡ- 1].

sphere repulsion leads to the asymptotic limit ΦΡ -> </>G for the Rallison reduced diffusion coefficient, D

T

R = A(\ — ΦΡ)2, where A = 2 7 π 3 ( φ 0 )

5 / 1 2 8 . It is noted that A = 0.7023 for Φ0 = 0.64, hence this expression is not valid as ΦΡ 0. The functions D

T

BM, Dlum, Dr

exp, and Dr

R with A = 1 are compared in Table 6.4 over the range 0.1 < ΦΤ

Ρ< 1.

6.14. Are Hydrodynamic Interactions Screened?

It was stressed by Beenakker and Mazur (1984) that the hydrodynamic inter-actions between two solute particles were not screened by the presence of the other solute particles. On the other hand, Snook et al. (1983) used a screened hydrodynamic interaction that was of the Yukawa form (cf. Exam-ple 6.3 and Fig. 6.5). Although both of these approaches appear to give sat-isfactory results, Carter and Phillies (1985) pointed out that the physical basis for hydrodynamic screening is not clear. They argued that momentum transferred from the solvent to a solute particle is virtually instantaneously

6.16. Tertiary Solutions of Hard Spheres

transferred back to the solvent because of the recoil of the solute particle. They also pointed out that the magnitude of the hydrodynamic interaction between two particles is affected by other particles, but the range of the hydrodynamic interaction is not diminished as would be the case for the screened interaction. If this were not the case, then one could not partition hydrodynamic interactions according to their power laws. By citing the work of Muthukumar (1982), both Beenakker and Mazur (1984) and Carter and Phillies (1985) were quick to point out that screened hydrodynamic interac-tions are important if the systems contains immobile particles. A macroscopic analogue may illustrate the difference between mobile and immobile objects in regard to screening hydrodynamic interactions. A breakwater protects boats moored at a marina from the impact of waves, whereas the presence of other boats does not offer such protection.

6.15. Evaluation of </> a p p(K, τ)>

Using their formalism as introduced in the previous section, Beenakker and Mazur (1984) examined the cumulant expression for <D a p p(X, i )> as given by Eq. (6.6.7). To the lowest order, the ratio S(K

T - AR) D{K)/D^ was

^ • y > W _ , + k , . A a > - a , . t

Ρ

r i / rexp( /K

r . r ) k

r · Af t^r) - k [#

( 2 )( r ) - 1], (6.15.1)

where k = K/K and r = ARjk. Selected results are plotted in Fig. 6.3, where solution structure is evident for φρ > 0.25.


Tertiary solutions (polymer-polymer-solvent) were introduced in Exam-ples 5.13 and 5.14 in the context of isorefractive studies of probe diffusion through a polymer matrix. Pusey et al. (1982) proposed a coupled-mode model for two polymers that are not isorefractive with the solvent, where AC(K, t) obeys the equation dAC(K, t)/dt = Κ

2Ό · AC(X, i), where AC(K) =

[ACA(K\ ACB(K)y and

D D A DA

& Β Α ^ B B .

Diagonalization of D, X 1

· D · X = ΛΙ, gives the eigenvalues

λ± = £ > A A + £ B B ± [ΦΑΑ - £ > B B )

2 + 4 D A BD B A]

1'

2

(6.16.1)

(6.16.2)


and the matrix Χ,

ΦΑ λ+)

D~

ΦΑΑ λ-) (6.16.3)

where = [(£>AB)2 + ΦΑΑ

— ^ ± )

2]

1 / 2· The vector of normal modes is given

by the expression Δη = Χ 1

· AC, which has the normalized components, with Αλ = λ+ — 1 _ ,

Δη, =

Δη =

(2λ+ - DBB)D'

2DABAX

DAA)D-

ACA + Pi ΔΤ'

-ACn

(λ.

ϋΑΒΑλ ACA

D~

ΔΙ ACB.

(6.16.4)

(6.16.5)

Note that the coefficients of ACA(K) and ACB(K) are of the same sign in Eq. (6.16.4), but of opposite sign in Eq. (6.16.5). Pusey et al. interpret the + mode as the collective mode since the coefficients of ACA and ACB are the same sign. The collective diffusion coefficient is

DAA + DBB + [ (D AA - D B B)2 + 4 Z ) A BD B A]

1/2

(6.16.6)

Pusey et al interpret the — mode as an exchange mode with

ft, {DAA + D n E(#A + 4 D A BZ ) B A]1/

The normalized function g(l)(t) for this model is

g{1\t) = AQxp(-DcoUK

2t) + (1 - / l ) e x p ( - D e xX

2i ) ,

where A = An+(K)/lAn+(K) + An.(K)].

(6.16.7)

(6.16.8)

Example 6.5. DLS Measurements in the Dextran-Dextran-Solvent System

Daivis et al. (1984) used QELS techniques to examine a mixture of two dif-ferent sizes of dextran particles in 0.1 M NaCl at 25°C. The starting material was designated as dextran T20 and T500. Dextran T500 was fractionated on a Sephacryl S-400 column, and a fraction designated as dextran F600 was used in the QELS experiments. The values of < M p > w of T20 and F600 were 20.4 χ 10

3 and 864 χ 10

3 Daltons, respectively. Dextran F600 (component

B) was at a fixed concentration c B = 0.005 g/mL, whereas 0.04 g/mL < cA< 0.17 g/mL for dextran T20. Estimation of the overlap concentration yielded the value c* ~ 0.225 g/mL, hence these concentrations were assumed to be in the dilute solution regime. QELS data were collected at Θ = 90° at dif-


O I

197

•2 J

-3 J

Ί Γ

0.2 0.4 0.6

D E L A Y T I M E ( M S E C )

0.8

Fig. 6.8. Time profile of the logarithm of the correlation function of a mixture of dextran

particles. The correlation function shown above was for 0.104 g /mL of dextran T20 and 0.005

g /mL of dextran F600 in 0.1 M NaCl at 25°C and a scattering angle of 90°. [Reproduced with

permission from Daivis et al. (1984). Macromolecules. 17, 2376-2380. Copyright 1980 by the


Table 6.5

Calculated Values of the Tertiary Diffusion Coefficients of Dextran"

(g/L) £>AA DAB £>BA £>BB DC

B DS

B

50 7.64 51.7 0.0162 1.31 7.50 1.44

75 7.65 57.7 0.0142 1.122 7.52 1.25

100 7.66 59.2 0.0124 0.840 7.55 0.95

125 7.66 58.7 0.0104 0.644 7.57 0.73

150 7.67 57.1 0.0083 0.386 7.60 0.45

a Reproduced with permission from Daivis et al. (1984). Macromolecules. 17, 2376-2380.

Copyright 1984 by the American Chemical Society.

* Units of 1 0- 11

m2/ s . Sample A is dextran T20 with D A(25°C) - 7.54 χ 1 0 "

11 m

2/ s ; Sample

Β is dextran F600 with £>B(25°C) = 1.08 χ 1 0 "11

m2/ s . Sample Β was at a fixed concentrat ion of

c'i = 5 g/L.


ferent delay-time intervals. The function y(t) was defined in relation to the correlation function C(t) by y(t) = [C(f) - 1 ]

1 / 2. Plots of ln[y(f)] vs. t clearly

indicate two relaxation modes, as shown in Fig. 6.8. The data were analyzed as a two-exponential decay function as defined by

Eq. (6.16.9) with an adjustable baseline Δ. They reported that the most con-sistent results were obtained when Δ = 0. This was because of polydispersity of the sample preparations ( M w / M n > 1.24), in which the slower decay modes became absorbed in the parameter Δ (cf. Chapter 4). The results are sum-marized in Table 6.5.

It is important to note that D A B φ D B A in these calculations, and that these coefficients are affected differently with an increase in C A.

6.17. Is Macavity There?

As in the case of Macavity in T. S. Eliot's book "Old Possum's Book of Practical Cats" referred to at the beginning of this chapter, proper assessment of the role of hydrodynamic interactions between solute particles in either D m

or D s e lf appears to be an elusive commodity. One of the difficulties in obtaining reliable experimental data at high

concentrations is the effect of multiple scattering on the correlation function. To assess this effect, Mos et al. (1986) used conventional QELS methods and cross-correlation techniques to study the concentration dependence of the mutual diffusion coefficient of poly(styrene) latex particles in water and silica particles coated with an aliphatic hydrocarbon to render them soluble in toluene and xylene. Since the primary focus of their study was to determine the effect of multiple scattering on the QELS correlation function, no effort was made to index-match the particles. The cross-correlation experiment, in which multiple scattering effects are minimal, was carried out with two photo-multiplier tubes at scattering angles of + 90° and — 90° to the incident laser beam. Comparison of the cross-correlation results with the conventional QELS results indicates that multiple scattering gives the appearance of a positive concentration coefficient for Dm for all three systems as determined by QELS methods, as shown in Fig. 6.9.

The experimental result of the cross-correlation study that the coefficient of φρ is slightly negative poses an interesting theoretical problem. Recall that in the Batchelor model (cf. Section 6.7), the introduction of the hard sphere repulsion interaction 80p cancelled most of the lowest-order hydrodynamic interaction contribution leading, nonetheless, to a positive volume fraction coefficient kO. Inclusion of higher-order cluster terms reduces the value of /cD, perhaps to the extent that they virtually cancel the hard sphere repulsion term. At the other extreme, the model of Anderson and Reed (cf. Section 6.7) indicates that the two spheres do not "experience" the hard sphere potential, hence kHS = %φρ should not appear in kD. Absence of the kHS term necessarily

6.17. Is Macavity There? 199

14

10

ο . 6

Q 2

A 16

SILICA IN XYLENE

—Ί 1 I 0 . 0 4 0.12 0 .20

Β Ο Ο Ο ΟΟ

* ·

SILICA IN TOLUENE

—Ι 1 1 0 . 0 4 0.12 0 . 2 0

C

8 ·

LATEX IN WATER

— ! 1 1 0 . 0 4 0 .08 0.12

C P ( g / m L )

Fig. 6.9. D a pp determined from autocorrelation ( O ) and cross-correlation ( · ) experiments.

D a pp determined by autocorrelat ion of the scattered light intensity at θ = 90° gives rise to an

apparent positive concentration coefficient, whereas cross-correlation analysis gives rise to a

negative concentration coefficient. This difference in behavior is at tr ibuted to multiple scattering

effects. (A) Silica in xylene; (B) silica in toluene; (C) latex spheres in water. [Reproduced with

permission from Mos et al. (1986). J. Chem. Phys. 84, 4 5 - 4 9 . Copyright 1985 by the American

Institute of Physics.]

requires the omission of hydrodynamic interaction terms if the experimental

value of kO is ~ 0 . There is some ambiguity as to the interpretation of the cross-correlation

results. Carter and Phillies (1985) interpret the slightly negative slope for the silica particles in terms of hydrodynamic interactions, whereas Mos et al (1986) interpret these data as aggregation.

The focus is now given to the φρ dependence of D^ e lf and D j ; l f. Van Veluwen et al. (1987, cf. Example 6.2) report that the initial slope of D*eU vs. φρ for coated silica beads was in good agreement with the value —1.83 obtained by Batchelor (1976) for the tracer particle (A) diffusing in a sea of host par-ticles (B), i.e.,

οΑ = ο°Αι-φΒ CWl (6.17.1)

1 +λ

2λ

where λ = RB/RA - Batchelor reports that C(l) = 1.83. It is to be emphasized that Eq. (6.17.1) was derived for a very dilute solution of particles 1 diffusing in a solution of particles 2 at a considerably higher concentration.

On the basis of these comparisons, it may be concluded that for hard spheres at low volume fractions, the hydrodynamic interactions, alias "Macavity", may be indeed "hidden" in Dm by the virial expansion terms. However, the magnitude of the contribution of the hard sphere potential part may be in question. The hydrodynamic interactions are more apparent,


however, in Dfe lf and Dêlf. The situation is not as clear, however, for vol-ume fractions φρ > 0.25, where the Beenakker -Mazur formulation deviates from the experimental measurements. It may be that the spheres are suffic-iently close that higher-order clusters ae required to describe the data. On the other hand, computer simulations that contain only the hard sphere potential and ignore hydrodynamic interactions appear to be adequate in the description of the various "phase transitions" in hard sphere systems. It remains to be seen if hydrodynamic interactions in this regime are simply a minor correction to the hard sphere potental effects.

Summary

A velocity flow field generated by a particle is established over a distance of several thousand angstroms on a time scale much smaller than the relaxa-tion time of the macroparticles. Hydrodynamic interactions are therefore "instantaneous" on the time scale of the diffusing solute particles and thus tend to decrease the value of D a pp because the test particle must diffuse against a velocity gradient. D s e lf has been described by the methods of reflec-tions and induced moments. In the method of reflections, the hydrodynamic interaction in Z) s e lf is expanded as a power series in Rs/r. It was shown by Felderhof (1977) that V D A B Φ 0 for the higher-order (short-range) terms because of multiple-body interactions. Inclusion of these divergent terms has a significant effect on the mathematical form of the diffusion coefficient, as shown by Carter and Phillies (1985). The method of induced forces par-tially overcomes the requirement of historical knowledge of the velocity field by partitioning the interactions according to their multipole moments. The method has been developed to the cubic term in the solute volume frac-tion for a system of identical particles. The model appears to fail for φρ > 0.3, since the solution of hard spheres tends to "freeze" for φρ ~ 0.5. A suspen-sion of hard spheres has at least four solution states: liquid (φρ < 0.5),crys-talline (0.55 < φρ < 0.6), glass (0.6 < φρ < 0.64), and closest-packed (limiting value at φρ = 0.74).

The precise expressions for Dm and D s e lf are not yet resolved. The math-ematical form of Dm depends on whether or not the thermodynamic and kinetic forces are separable in performing the average, and also whether or not the effective distance of closest approach is greater than or equal to the hydrodynamic diameter. On the experimental side, care must be taken to eliminate multiple scattering effects that tend to increase the measured value for the mutual diffusion coefficient. The multiple scattering effects can be reduced by using a solvent to "pseudo"-match the index of refraction of the particle. Care must also be taken in choosing a system that does not aggregate.

Problems 201

Problems

6.1. Assume an incompressible fluid and the existence of a velocity potential φ such that

Show that the fluid does no rotate in the flow process.

6.2. Use the Batchelor expression for D s e lf [Eq. (6.11.7)] and calculate the volume fraction necessary to reduce the self diffusion coefficient by 10%. From this value of φρ, estimate the ratio K S / < J R 0 > , where {R^} is the average distance between the spheres in a cubic packed array. What is the magnitude of the attenuation of the radial velocity at this distance? Use Eq. (6.2.17).

6.3. There are two requirements to be met if D s e lf is to be measured by DLS techniques. Κ must be sufficiently large that S(K AR) = 1 and <DyF C(T)exp( — / K

r · Rjk)y = 0 [cf. Eq. (6.6.3)]. These conditions are

simultaneouly met if one can write the second condition as a product of averages, viz, <D j 7 c(i)><exp( — iK

T · AR j f c>. Using the information in

Example 6.1, estimate the minimum value of the scattering angle if you wanted to study the self diffusion coefficient for P M M A of radius 500 Â using the 488 nm line of an argon ion laser.

6.4. The conceptual difference between the long-time and short-time values of D s e lf is that the former includes memory effects of the solution "lattice". Theoretically one can obtain a value for Dêl{ if the particles have moved a distance considerably less than their own diameter. Since the "probe length" of light is on the order of 1/iC, there must be some minimum value of the scattering angle, 0 m i n, to accommo-date the diffusion distance requirement. That is, the light is sensitive to differences in the index of refraction over distances equal to or greater than \/K. Let us assume particles of diameter 2000 Â in cyclo-hexane at 20°C (η0 = 0.66 cp, n0 = 1.445). What is the value of 0 m in if λ0 = 488 nm? What is the value of DêU if hydrodynamic interac-tions are neglected? What is the theoretical value of the homodyne relaxation time?

6.5. Assume a hard sphere of radius 2000 Â in cyclohexane at 20°C (cf. Problem 6.4). If the normalized correlation function must decay to the value 0.90 and at least 10 points in the correlation function must span this range of decay, estimate the minimum value of the data col-lection interval time Δί at the 90° scattering angle if λ0 = 632.8 nm.

-δφ

dx

-δφ -δφ

δζ


How far has the particle diffused during the time interval 10Δί? To assure that the short-time diffusion coefficient is being measured during the time interval 10Δί, the concentration of the particles can-not exceed a certain value or else the particle will encounter other particles. Assuming that the maximum concentration is characterized by twice the diffusion distance, what is the maximum value of φρ

that can be used in this study? If hard sphere interactions are the only component of ADmem (neglect of hydrodynamic interactions), calculate a value for Dê]{. What data collection interval Δί must be used for a single exponential analysis if the correlation function decays to the \/e value?

6.6. Bohidar and Geissler (1984) examined the diffusion coefficient of insulin solutions over a period of seven days (cf. Example 2.1). The data were obtained at 21°C and a scattering angle of 90° and ana-lyzed by the second-order cumulant method of analysis. The concen-

0 . 3 0 . 6 0 . 9

C p x I O4 ( g / m L )

Fig. 6.10. D a pp versus cp for insulin over a one-week period. The numbers in the above figure indicate the number of days after the sample preparat ion. The weight-average molecular weights obtained by static light scattering were reported to be (in Daltons): 7410 (1 day); 7,920 (2 days); 30,610 (5 days); 46,900 (6 days); 56,200 (7 days). [Reproduced with permission from Bohidar and Geissler (1984). Biopolymers. 23, 2407-2417. Copyright 1984 by John Wiley and Sons, Inc.]

Problems

tration dependence of {Ds

app} for five different days is shown in Fig. 6.10, along with the weight-average molecular weight obtained from static light scattering. Using the partial specific volume value of vp = 0.695, represent the above data by (D

s

app) = D°(l - fcD0P). How does the value of kO compare with the theoretical values for spherical particles? Decompose fcD into hydrodynamic and excluded volume components, kO = kH — 2 B 2 N A / M p . How do the values of kH

compare with the hard sphere models? Using the molecular weights given in Fig. 6.10 and the values of Z) p, deduce the thickness of the "hydration shell". Is this a reasonable value?

6.7. Using the data given in Example 6.5 and Table 6.4, calculate the ratio of amplitudes for the + and — modes, Δ π + / Δ π _ , as a function of CA(K) using Eqs. (6.16.4) and (6.16.5). How does this compare with the experimental value of A/(\ — A\ where A varied from ~ 0 . 4 to ~ 0 . 6 over the concentration range 0.05 g /mL to 0.1 g /mL and was virtually constant at the higher concentrations?

6.8. What is the expression for ga\t) in a tertiary system if two different

molecular species had identical values for their radius, i.e., D A A = D B B? This is the situation suggested by Weissman (1980) for the poly dispersity interpretation of two relaxation modes and the impetus for the study by Pusey et al. (1982).

6.9. In view of the discussion in Section 6.19, discuss the relative merits of monitoring a conformational change in a flexible polymer through D m , Dseif,

a nd HIF- Assume that the conclusions drawn from the

hard sphere system also applies to flexible polymers.

6.10. Show that Dm = D s

L

e lf = Dfe If + A D m e m to the linear term in φρ9 where the hydrodynamic interaction is terminated at the Oseen level of approximation. [Hint: use Eq. (6.12.5).]

Chapter 7 Polyelectrolyte Solutions

"Multitudes of stars veiled behind clouded sky.

The lesser light had sailed through neighbor-clouds nearby "

From The Light of Night (Anonymous)

7.0. Introduction

The opening lines of the poem The Light of Night appropriately describe the dual roles of small ions in the description of polyion diffusion. The first effect of a "cloud" of small ions surrounding each polyion in solution is that of a "veil" that screens the surface charge of the polyion. This long-range effect is to reduce the apparent charge on the polyion and thereby reduce the magnitude of the direct polyion-polyion interactions. As passive components, the small ions are assumed to instantaneously relax to their equilibrium distribution and therefore act to screen the interactions between the polyions. The second effect is an indirect coupling of the polyions as they ".. sailed through neighbor-clouds nearby". That is, the dynamics of the polyions become statistically coupled by way of overlapping ion clouds. As active components, the small-ion dynamics can have a significant influence on the dynamics of the polyions.

7.1. The Poisson-Boltzmann Equation and the Debye-Hiickel Screening Length

The Poisson-Bol tzmann equation relates the distribution of charge in the solution to the electrical potential φ(τ):

τι2Μ< λ - 4 π ρ ( Γ ) 4π (-ΖΛβφ(χ)\

where ε is the dielectric constant of the medium, Z a is the magnitude (with sign) of the charge on ion a, and e is the absolute value of the electron charge. It is φ(τ) that couples all of the electrical species in the solution.

205

206 7. P O L Y E L E C T R O L Y T E S O L U T I O N S

The Bjerrum length is defined as λΒ = e2/ekT. The physical significance of

the Bjerrum length is that it represents the "minimum" distance between charges on a polyion that can be supported by the solvent. A second length scale of importance is the Debye-Hückel screening length, λΌΗ. The physical significance of x D H lies in the solution to the linearized Poisson-Boltzmann equation (Debye and Hückel, 1923a, 1923b), viz, exp[ — Zae</>(r)//eT] ~ 1 — Zae</>(r)/fcT. Hence, the Poisson-Bol tzmann equation becomes

Ν2#Τ) = -Γ-Σ An „ Να{Ζαβ)

2φ(ν)

kT (7.1.2)

where Na is the number of particles a. Assuming spherical symmetry about the "test" particle and substituting the average <<£(r)> for the actual potential φ(τ), one has

δ ·δ<Φ(φ'

δν 1

or \λΟΗ

4ne2\

<Φ(Φ,

2 _ 8ηΝΑλΒΙ5

1000

(7.1.3)

(7.1.4)

n'a = NJV, Is = (1000/2NA)XN;Z^ = IXC«Z« i s t he i o n ic strength of the

solution, and Q is in moles per liter. The general solution to Eq. (7.1.3) is

<Φ(Φ = -exp(V-^ ) + - e x p ) r VdhJ r UDl

(7.1.5)

The positive solution is unphysical since exp(r/ADH) OO as r -> OO, hence Β = 0. To solve for A, it is assumed that only the test particle α is in a cavity of radius a. By equating the electric field inside (Zae/sr

2) and outside

[— K<0(r)>] the cavity at r = a, one obtains

(Z ae)exp A =•

a ε 1

λΌΗ_

(7.1.6)

The Debye-Hückel potential is therefore

7.3. Small I o n - P o l y i o n Coupled Modes: General Framework 207

7.2. Statistical Properties of Electrolyte Systems

Fluctuations in the concentration of ionic species cannot in general be considered as independent events because of the condition of charge neutrality in the solution. Following Hermans (1949), it is assumed that the volume element under consideration is so large that fluctuations in neighbor-ing volume elements can be ignored. The free energy for a concentration fluctuation, AF C, is

A F c = IIGSRA«:A«i, (7.2.1) m j t/nmunj

and with a charge fluctuation, A F Z , is

Σ^ΑΗ)

AFY = ' (7.2.2)

Substitution of d2F/drimdrij ~ kT/n' = constant, where ri = rim + n'r

gives for the condition for independent fluctuations in the charged species, AFZ/AFC ~ ν

2/3/λΙΗ « 1. For a 1:1 salt at 0.1 M, ri = 2 χ 0.1 χ 6.02 χ

10 2 3/1000 = 1.20 χ 1 0 2 0 part icles/cm 3 and λΌΗ ~ 9.6 χ ΙΟ" 8 cm. Hence Κ « / 1 3

) Η = 8.8 χ Ι Ο " 2 2 c m 3 . The number of particles in λ^Η is 1.20 χ Ι Ο 2 0 χ 8.8 χ 10 2 2 ~ 0.1, which has no meaning in fluctuation theory. Hence, fluctuations in neighboring volume elements are correlated.

7.3. Small Ion-Polyion Coupled Modes: General Framework

It is assumed that the spherical polycation has a charge Zpe, radius a p , and diffusion coefficient Dp. The cations and anions have charge and diffusion coefficients Zce, D c, Z ae , and Z)a, respectively. The Poisson equation is

4-Tie V 20(r) = -lZpArip(r,t) + Z a A < ( r , i ) + Z cAn ' c(r , i)] , (7.3.1)

ε

where Aw}(r,i) is the number fluctuation of species j at r and time t. The diffusion equation for the jth species is

Ô-M^ = D,* A„i(r,t) + '^φ&νίφΜ. (7.3.2)

tit kT Combining Eqs. (7.3.1) and (7.3.2) and subsequent Fourier transformation [ V 2 A n } ( r , i ) ^ - Χ 2 Δ η } ( Κ , ί ) ] gives

^ M = - n ( K ) . A « ' ( K , t ) , (7.3.3) dt


where An ' (K , i )r = [Δη ρ(Κ,ί) , An'a(K,t), Δη^Κ, ί ) ] , and

DP(K2 + X;

2 DPZC

Ω(Κ) = Α ( κ 2 + Κ 2 (7.3.4)

DC{K2 + X C

2

where ÀJ2 = 4ne

2(rii}uZf/ekT is the partial contribution of the ith ion to the

Debye-Hückel screening parameter, \/λ^Η = λ~2 + λρ

2 + λ~

2. It is noted

that V2(/>(r) as defined by Eq. (7.3.1) is the linear term that was omitted in the

expression for V2</>(r) given by Eq. (7.1.2). Eq. (7.3.1) is therefore a statement of

charge neutrality within a region of solution and does not relate directly to inter particle interactions. The difference between these two definitions of V

2(/>

is underlined upon removal of equal numbers of cations and anions: Eq. (7.3.1) remains unchanged, whereas Eq. (7.1.4) is affected because of the change in the ionic strength. Thus the polyion-polyion interaction should increase in accordance with Eq. (7.1.5) and thereby affect the polyion diffusion process. The situation is similar to that examined by Hermans (1949) (cf. Section 7.2). It is concluded that the D, values are associated with the mutual diffusion process (cf. Section 7.6).

The roots of the characteristic determinant |Ω(Κ) — λ\\ = 0 are the normal mode decay rates. The secular equation is χ

3 + aix

2 + a2x + a3 = 0, where

the coefficients are

a, = ~[Dp(K2 + λρ

2) + Da(K

2 + λ~

2) + DC(K

2 + Λ "

2) ] , (7.3.5)

a2 = K2[DaDp(K

2 + λρ

2 + λ~

2) + DcDp(K

2 + λρ

2 + λ~

2)

Substitution of χ = y — (a1/3) into the secular equation yields the reduced cubic equation, y

3 + ay + b = 0, where a = a2 — (fl

2/3) and b = (2a x/27) —

(ala2/3) + a3. Substitution of y = ζ + ν gives a quadratic equation for z3

(or ν3), ( z

3)

2 4- bz

3 — (a

3111) = 0, which has the solutions

+ DaDc(K2 + λ;

2 + λ-

2)\

α3 = -DaDcDpK\K2 + λ;

2 + λρ

2 + λ~

2).

(7.3.6)

(7.3.7)

y = ζ -h ν =

(7.3.8)

7.3. Small I o n - P o l y i o n Coupled Modes: General Framework 209

The solutions given by Eq. (7.3.8) are referred to as Cardan's solutions to the reduced cubic equation. Numerical evaluation indicates that (b

2/4) +

(a3/'27) < 0, hence all of the roots are real. The roots can be expressed in a

more convenient form upon defining iq = [(b2/4) + ( a

3/ 2 7 ) ]

1 /2 and ρ = —\b

and, where q and ρ are real,

y = (ρ + ty)i/3 + ( P - I QY I \ (7.3.9)

Writing ρ = rcos(0) and q = rsin(0), where r2 = — (a/3)

3 and cos(0) =

- f t / [ 2 ( - a3/ 2 7 )

1 / 2] , one has

3 \ 1/6

27 COS

"(0 + 2kn)

3 (fc = 0,1,2). (7.3.10)

In order to establish the physical significance of these roots, Eq. (7.3.9) is expanded to the fourth order:

,1/3 1 + Ϊ 1 /3

1/3 ^ . « ^ 42 . 1 0 î

- i T ^ ^ (7-3-11) 3 p2 / 3

9 p5 / 3

162 p8 / 3

'

which defines, after rearrangement, the quantity A:

A=pl 13 1 + 3 1

9p2

3p 2/3 1

I 0 Q2

5 4 p

A similar expansion of the second term in Eq. (7.3.9) gives

I0q2 "

Β = ρ113

The three roots are now

1 + i l l · Γ 1

9p 2 J +

iql3p

2li \62p8l\

(7.3.12)

(7.3.13)

λι = A + Β

λ2 =

3 '

(A + Β) Α-Β

2 K

' 3 '

(Α + Β) Α-B α,

2 2 1

' 3 '

(7.3.14)

(7.3.15)

(7.3.16)

In the limit Κ -»• 0, ^ is found to be independent of Κ and is identified with the Debye mode, or plasmon mode. The other two roots for Κ -> 0 are proportional to X

2, where λ2 is associated with polyion diffusion,

£ U ( K ) = K2'

(7.3.17)


Table 7.1

Eigenvalues for Coupled Mode Model"

Κ Χ 1 ( Γ5

Method' ' Q X 100 λχ X 10 7

λ2 Χ 1 0 "4

/ 3 X 1 0 "5

1.0' 2 500 1.00 20.1 0.699 2.00

4 20.1 0.695 2.00

T 20.1 0.695 2.00

\.od

2 500 0.01 0.354 5.85 2.01

4 0.354 5.72 2.00

T 0.354 5.72 2.00

2.5'' 2 500 1.00 20.2 4.52 12.5

4 20.2 4.52 12.5

T 20.2 4.34 12.5

2.5d

2 1000 1.00 20.6 3.22 12.5 4 20.6 3.04 12.5 T 20.6 3.04 12.5

2.5d

2 500 1.00 20.3 1.43 12.5

4 20.3 1.24 12.5

T 20.3 1.24 12.5

« = = 2 x 10~5 c m

2/ s ; Z c = - Z a = 1.

b 2: two-term expansion of Eqs. (7.3.12), (7.3.13), and (7.3.15); 4: four-term expansion of

Eqs. (7.3.12), (7.3.13), and (7.3.15); T: tr igonometric solution [cf. Eq. (7.3.10)]. C D P = 6.2 Χ 1 0 "

7 c m

2/ s ; C p = 1 Χ 1 0 "

8 M .

d D°= \ χ 1 0 "

7 c m

2/ s ; C p = 8 X 10""

8 M .

and λ3 is associated with small ion diffusion. Evaluations of the three roots using the trigonometric and expansion expressions are presented in Table 7.1. In this calculation, C a = CcZc/\Za\ + C p Z p / | Z a | to ensure electrical neutrality. As indicated in Table 7.1, the two-term expansion completely fails for large values of K. Errors in the four-term expansion become significant for very large values of Z p and low ionic strengths, i.e., extreme polyelectrolyte conditions.

7.4. Small Ion-Polyion Coupled Modes: Κ = 0 Limit

Lin et al. (1978) and Tivant et al. (1983) obtained analytical expressions for the polyion-small ion coupled diffusion modes in the limit Κ = 0. Tivant et al. (1983) extended this theory to include Z a # - Z c and Da Φ Dc:

Dapp(K =0) = i{D p( l - i p) + Dc(l - i c) + Da(l - i j - (A)1 / 2

} , (7.4.1)

A = Dp(i - g + DS(I - g + z)a(i - g

+ 2DpDc(tptc - g + 2DcDa(tcta - g + 2 D p D a ( g a - g , (7.4.2)

7.4. Small I o n - P o l y i o n Coupled Modes: Κ = 0 Limit 211

l o g ( C s ) (M)

Fig. 7.1. Dapp(K = 0) as a function of the electrolyte concentration. The diffusion coefficients

for the univalent anions and cations are Da = Dc = 2 χ 1 0 "5 c m

2/ s .

and tv = (DJX2)/^ (Dj/Àj) are transport numbers. Eq. (7.4.1) reduces to that of

Lin et al. (1978),

Dapp(K = 0) = i[Z) p(l — Ω) + D s(l + Ω)],

where Da = Dc = D s, Z c = —Z a = 1, and

Ω =

DpZp - Ds 1 + !

-7 C

DpZp + Ds 1 +

(7.4.3)

(7.4.4)

Substituting 2C S/C p = [ 2 ( C s ) a d d ed + Z p C p ] / C p , one obtains the asymptotic limits

D a p p(X = 0) = Dp (ψ

Dapp(K = 0) DpDs(2 + zp;

DpZp + 2DS

• oo

2 Q

(7.4.5)

(7.4.6)

Note that if DpZp » D s and (2C S/C p Z p) , then Eq. (7.4.6) predicts that Dapp(K = 0) ~ D s. In other words, /r^/z/y charged polyions diffuse at the same rate as their counterions under very low ionic-strength conditions. Simulation plots of Dapp(K = 0) versus ln(C s) for BSA are shown in Fig. 7.1.

Example 7.1. BSA Study of Doherty and Benedek Revisited. In their pioneering work, Doherty and Benedek (1974, cf. Example 4.2) used BSA as a test system for the predictions of Stephen's model (1971) for low-


ionic-strength solvents. Stephen's result can be obtained from Eq. (7.4.3) if one allows 2CJZpCp -+ 0 and DJZpDp » 1:

App = flpO + Zp) (Stephen's result). (7.4.7)

They estimated the actual charge on the BSA from isoionic solutions using the charge balance equation and pH measurements, — Z p[ B S A ] — [ O H ] = [ N a

+] + [ H

+] . Since the pH measurements indicated that the hydroxyl

and hydrogen ion concentrations were negligible, the charge was given as — Z p = [ N a

+] / [ B S A ] . The ionic strength was then varied by addition of

NaCl. Using Stephen's expression, the effective charge was determined from D a p p, i.e., Z e ff = (D a p p/D p) — 1, where Dp is the monomer diffusion coefficient obtained from their polydispersity analysis of the data, as discussed in Example 4.2. These authors found that Z e ff < Z p for all low-ionic-strength data and thus concluded that the Stephen model was, at best, incomplete. As indicated above, Stephen's result obtains from the coupled-mode theory only for the unphysical limit 2C S/C p -> 0. One can rearrange Eqs. (7.4.3) and (7.4.4) to obtain a quadratic equation for Z p in terms of the experimental parameters r = D p/D 8, R = Dapp(K = 0)/D p, and y = 2 ( C s ) a d d e d/ C p :

* « ' - * » . ( I + f lZ . - , . ( 7 . « ) R - 1

A plot of (1 — Rr)/{R — 1) vs. y has a slope Z p and an intercept —(1 + δ)Ζρ. By comparing the values of Z p from the slope and intercept one can deter-mine <5, and thus whether or not counterions should be included in the calcu-lation of 2C S. It is assumed that r = Dp/Ds = 6.5 χ 10

7/ 2 χ 1 0 "

5 - 0.0325

is constant and that the following identities hold between the two notations: R = Dapp/Dp = α + 1, and ( C s ) a d d ed = / s . Limiting the calculations to low-ionic-strength results, we have from their Table I the following two sets of data: R1 = 3.2, yx = 3.5 χ ΙΟ"

3 M; and R2 = 2.9, y2 = 5.0 χ 10~

3 M. Solv-

ing the simultaneous equations, one obtains for the slope and intercept of Eq. (7.4.8) the values 69.47 and 18.55, respectively. The apparent charge cal-culated from the slope is therefore Z p ~ 8.3, hence one has from the intercept (1 + δ) ~ 18.55/8.3 = 2.2. In view of the approximations made in these cal-culations, the small ion-polyion coupled-mode theory adequately describes the experimental data. In view of the result that δ ~ 1, it is concluded that the Stephen limit cannot be experimentally attained.

7.5. D a p p( K ) / D a p p( K = 0) vs. Κ for Weakly Coupled Polyelectrolytes

The expressions for Dapp(K) [Eq. (7.3.7)] and Dapp(K = 0) [Eq. (7.4.3)] were obtained from the linear expansion of the Boltzmann weighting factor; hence the ionic species are said to be weakly coupled. In contrast, the X-dependence

7.5. Dapp(K)/Dapp{K = 0) vs. Κ for Weakly Coupled Polyelectrolytes 213

'APP W / D A P P( 0 )

1 . 0 - I C P ( I 0 "9M )

Π 2 0 0

5 0 0 I 0 0

2 0 0 I 0

0 . 5

I 0 "7C M

2/ S

2 X L O "5 C M

2/ S

I 0 "4

M

2 0 0 I 0 0

0 I 2

Κ χ Ι Ο "5 ( cm" ' )

Fig. 7.2. Dapp(K) I Dapp(K = 0) vs. Κ for small ion-polyion coupled-mode theory.

for the small ion-poly ion coupled-mode model results for ionic strengths that give 1/λΌΗ ~ K. It is of value, therefore, to assess the conditions under which Eq. (7.4.3) can be used to interpret D a pp obtained for polyelectrolytes. Plots of Dapp(K)/Dapp(K = 0) vs. Κ are shown in Fig. 7.2.

It is clear from these results that for dilute solutions of highly charged polyelectrolytes, the use of Eq. (7.4.3) can lead to erroneous conclusions. It is of interest to examine the dependence of Dapp(K)/Dapp(K = 0) on Cp at the scattering angle θ = 90°, which is routinely chosen in "single angle" studies reported in the literature. This value of θ corresponds to Κ ~ 2.5 χ 10

5 c m

- 1

for aqueous solutions and λ0 = 488 nm. D a p p(K) /D a p p(K = 0) vs. C p at C s = 1.2 χ 10~

4 M for several values of Z p is given in Fig. 7.3A.

There are three important features of the data shown in Fig. 7.3. First, Dapp(K)/Dapp(K = 0) -> 1 at the extremes of the concentration range. This means that Dapp(K = 0) given by Eqs. (7.4. l)-(7.4.4) cannot be indiscriminately applied at finite concentrations. Second, the rate of change increases with the charge of the polyion. The third feature is that the curves for any two charges cross at a particular value of C p . The ratios Dapp(K)/Dapp(K = 0) for these "isoconcentration" points exhibit identical X-dependent curves (cf. Fig. 7.3B).

An error may be introduced in the charge Z a p p if Eq. (7.4.3) is applied to data taken at finite K. To examine the magnitude of this error, values of D a p p(X) were generated using the charge Z p and analyzed by the Dapp(K = 0) expres-sion to obtain the value of Z a p p. A plot of Z a p p vs. Z p is shown in Fig. 7.4 for selected values of X, C p , and Is that are chosen so that Dapp(K)/Dapp(K = 0) Φ 1.

It is concluded that Z p may be underestimated by more than a factor of two for data obtained at high scattering angles (90°) and / s < 1 (Γ

3 M. The error

introduced in the use of Eq. (7.4.3) to the computat ion of Z p will increase as the ionic strength is lowered. A corollary is: one should not combine data obtained under different ionic conditions to estimate Z a p p for highly charged polyions.


Dapp<K)/CW0)

1.0

0.8

0.6

K = 2 .5x 10 cm Cs= 1 0 - 4 M

Dg= 2 x l 0 "5 cm2/s

Dp= I 0 ~ 7 cm2/s

60

o \ \\ \

î; \ioo N

^

350

? r = - = 9- 7

"Ί 10 20

A ( I0 "

ÖM)

all charges

10 10

D a pp( K ) / [W 0 )

8

B

2

( l05/cm)

Fig. 7.3. Polyion concentration dependence of Dapp(K) I Dapp(0) for various polyion charges.

(A) The scattering vector, electrolyte, and infinite dilution diffusion coefficients are fixed and

given in the figure. The concentrat ions at which two or more lines cross are referred to as

"isoconcentrations". (B) The Κ-dependence of D a p p(K) /D a p p(0 ) for two isoconcentrations: ( • )

Z p = 500, C p = 21 χ ΙΟ8; (Δ) Z p = 20, C p = 21 χ 10

8; ( · ) Z p = 200, C p = 4 χ 10

8; (O) Z p =

350, C D = 4 χ 108.

7.6. Inclusion of Hydrodynamic Interaction 215

7.6. Inclusion of Hydrodynamic Interaction: The Belloni-Drifford Model for Polyelectrolyte Solutions

Hydrodynamic interaction and memory effects have thus far been neglected in describing the dynamics of polyion-small ion coupled modes. Belloni and Drifford (1985) examined the initial rate of decay for the special case in which hydrodynamic interactions provide an additional coupling mechanism be-tween the polyions but are neglected in describing the small-ion dynamics. Following Akcasu and coworkers (Akcasu et al., 1984a, 1984b), the time course of AC(K, t) is governed by a "Liouville-like" operator, defined by the generalized frequency matrix Ω(Κ),

Ω(Κ) = < [ A C ( K , 0 ) * ]r - & - AC(K,0)> - S (K) -

1, (7.6.1)

that governs the initial decay rate for the dynamic structure factor, i.e., dS(K,t)/dt = -0 (K)-S(K , i ) . The elements of S(K) are referred to as the partial structure factor [S(K)] j fc (cf. Appendix B),

[S(K)], fc = öjk + « Q U * < Q > J1 / 2 J[0g>(r) - 1] e x p ( - l T . r) Ar. (7.6.2)

The matrix Ω(Κ) is diagonalized by Q_ 1

· O(K) · Q = A, where the eigen-values Xi are the decay rates for the coupled modes. The eigenmatrix M(K, i) is defined by

M ( K , r ) = J nijCxpi-Ajtl (7.6.3) j= ι

where m, = Q · d, · Q1

, and d7 is a matrix whose elements are all zero except the jth diagonal element, which is equal to one. S(K, i) can be written as S(K, t) = M(K, t) - S(K). The eigenvalue λί is associated with the Debye mode and is subtracted from the general expression, since it relaxes at a rate too fast for present-day correlators. The intermediate time range in the DLS experiment is thus defined as \/λί « t < 1/At with i > 1, and the initial decay of S(K, t) is

aS(K, r)

dt = - K

2[ M ( K , 0 ) - λ - ^ m j · S(K). (7.6.4)

i = 0

Restriction of i f to hydrodynamic interactions alone gives < [ A C ( K , 0 ) * ]r •

5e - AC(K,0)> = X2H(K), where

[H(K)] Jk = D]b,k + kTdCjXiC^)1 12

x ( T ) z Z e x p ( - KT. r ) d r , (7.6.5)


and (T) zz is the hydrodynamic interaction tensor component along the scattering vector K. One now has the identity Ω(Κ) · S(K) = —K

2H(K), which

results in the equality /lym,-S(K) = K2m,-H(K) , and thus the initial rate

of decay

dS(K,t)\

dt = - K

2[ M ( K , 0 ) = - m , ] - H ( K ) . (7.6.8)

Belloni et al. (1985) examined the Κ -» 0 limit of Eq. (7.6.8), in which limit the only contributor to the frequency matrix is the Debye mode. One then has m ! = Ω(Κ = 0)/λι = H(K = 0) · QDHMI> where Q D H is composed only of the elements qu = (4ne

2/ekT)((C^)u(Cj')u)

1/2ZiZj • D a pp is now

A app(K - 0)

Γ ,

Q Μ(Κ ,0 ) ·Η(Κ)

H ( K ) - Q D H- H ( K ) )

7 S(K)-

22

(7.6.7)

where the subscript 22 denotes the eigenvalue λ2. In applying Eq. (7.6.7) to the special case in which hydrodynamic interactions occur only between the polyions, Belloni and Drifford (1985) obtained for Dapp(K -> 0) = D B D

1 -

^ B D = Hn-Su

(7.6.8)

where the subscripts jj for qn now refer to the solution components of the polyions, cations, and anions.

Example 7.2. Small-Ion Dynamic Effects on D a pp for BSA. Neal et al. (1984) examined the osmotic susceptibility (from total intensity measurements) and diffusion coefficient of charged BSA. The data were presented as the ratio DmS(K)/Dp, which was assumed to be equivalent to f°plfm. Data were collected for C B SA - 4 χ 1 (Γ

5 M over the range of 0.00019

M < / s < 0.116 M at 20°C.

The experimental ratio DmS(K)/Dp was compared with the theoretical ratio Hn/Dp. Choice of the appropriate g

(2\r) for the calculation of Hn in

Eq. (7.6.5) was determined from the analysis of S(K) obtained from total intensity measurements. It was concluded that the hypernetted chain (HNC, cf. Appendix C) calculations were more accurate in this respect than calcu-lations involving a virial expansion or the Boltzmann factor using the screened Coulomb potential. Even with this care in the calculations, the theoretical curve of DmS(K)/Dp vs. 1/S(K) was significantly higher that the experimental curve, where the discrepancy increased with a decrease in Is.

7.7. Dynamic Attenuat ion 217

Ι Ο Ί

S ( K ) D ,

+ ' F V - ^ A S = L . 5

0 . 5

A S = 2 . 0 ^

J L

3 5 7

I / S ( K )

Fig. 7.5. Small ion-polyion coupled-mode correction to the BSA mutual diffusion coefficient.

The da ta points and the solid line (hydrodynamic correction calculations) are from Neal et al.

(1984): ( · ) Ζ ρ = - 1 3 , ( 0 ) Z p = - 9 , ( + ) Z p = - 6 . The dashed lines are the calculated curves of

Belloni and Drifford in which dynamic at tenuat ion is included, using the small-ion radius as

indicated in the figure. [Reproduced with permission from Belloni and Drifford (1985).

J. Physique Lett. 46, LI 1 8 3 - L I 189. Copyright 1985 by Les Editions de Physique.]

Belloni and Drifford (1985) reanalyzed these data using Eq. (7.6.8). As-suming that D2 = D 3 , defining D2 = D s, and noting that Z a = — Z c = 1, the experimental ratio is now

where / s = C s + [Z^CJl) is the ionic strength. In this analysis, the values of the parameters were the same as those used by Neal et al. (1984), with the best fit radius of the small ions being 1.5 Â. These results and those of Neal et al. (1984) are plotted in Fig. 7.5. It is concluded that both small-ion dynamics and hydrodynamic interactions have significant effects on the diffusion of BSA.

7.7. Dynamic Attenuation

For a screened Coulomb interaction in the limit C p / C s « 1, Sn can be expressed as

S(K)Dt

/ / 1 1Z 1 C p + 2 / sD s° J ' (7.6.9)

7. P O L Y E L E C T R O L Y T E S O L U T I O N S

Eq. (7.6.9) in the limit αρ/λΌΗ - • 0 is, to the linear term,

72C { 2D° V

(7.7.2)

The term {1 — [2Dp/(Da + D°J]} differs from unity because of the finite speed at which the small ions can move and acts to reduce the magnitude of the static component Z p C p / 2 C s obtained from Sn. This term is referred to as "dynamic attenuation" (Schurr and Schmitz, 1986).

7.8. Electrolyte Dissipation

Envision a positively charged particle with an associated ion cloud moving through a solvent. As the charged sphere sediments or diffuses, the ionic atmosphere "lags" behind the motion of the sphere. The instantaneous non-spherical distortion of the ion cloud leads to a separation of positive and negative charge centers that is described in terms of spherical harmonics, with the dipole term being the leading term. The velocity of the positive probe particle is therefore "retarded" by the negative charge center of the ion cloud, which is manifested as an additive electrolyte dissipation term, £ e l, to the friction factor of the polyion, viz, ζρ = ξϊϊ + £el.

The point of embarcation is the Medina-Noyola description of fTr

(cf. Appendix E) that relies on solutions to the Orns te in-Zernike relation (cf. Appendix C). It is assumed that the transport coefficient matrix is diag-onal and isotropic, L0- = D f < /<n}(r)> eq ~ ££<η}>Μ<50·, where D? is the free diffusion coefficient and is the uniform (bulk) concentration of the jth species.

POINT CHARGE APPROXIMATION. Medina-Noyola and Vizcara-Rendon (1985) considered polyion diffusion with two species of small ions of charge Z 2 = — Zt=q having equal transport coefficients (D^n^ = D\n'2 = Dsn'). ÎThe direct correlation function is c<-

2 )[r,r',n'] = (qe)

2(-\)

(i+j)/(skT\R - r'|), and

the equilibrium cross correlation function is

Medina-Noyola and Vizcarra-Rendon (1985) obtained the result first ob-tained by Schurr (1980), who used a fluctuation-dissipation model for the electrolyte dissipation component £ e l for a spherical bead of radius ap and charge Z D :

σ0·(Γ,Γ') = n'AGIR - r') -(n'qe)

2(-l)

ii+j)

fcTe|r-r'| e x p ( - / c | r - r ' | ) . (7.8.1)

£.i = (Zpe)

2

12apsDs

G(KaJ iSchurr (7.8.2)

G(Kapf kSchurr [1 - (1 + 2κα ρ)βχρ( -2κ :α ρ) ] . (7.8.3) καρ

7.8. Electrolyte Dissipation 219

SMALL IONS OF FINITE SIZE. To examine the effect of the finite size of the small ion on the electrolyte dissipation term, Vizcarra-Rendon et al. (1987) expressed £ e l in terms of the small-ion radial distribution functions gp

2j\r) = ηρ]\τ) — 1

(cf. Appendix Β), with the result

[ rÄ#( r ) ]2dr , (7.8.4)

J - „ ο

where D'} = Dp + D° is the relative diffusion coefficient. For small ions of finite size, the hard sphere potential </>hs(r) = + oo dominates in the region 0 < r < a', where a' = ap + a-. Hence g

{

p

2/(r < a') = 0 and hp

2](r < a') = — 1

in this region, and £ e l = ξ'^ + ξ'^, where

Éel = Σ NJDJ

4nkT * (ap + a}f

Chs ^ L

& 1

AnkT * η)

(7.8.5)

(7.8.6)

The hard sphere component ξ'^ is present even in the absence of charge on the solute particles. Because ^ s is present even in the absence of charge, Vizcarra-Rendon et al. (1987) argued that ^ s should in fact be associated with the purely hydrodynamic contribution ξ° = kT/Dp. The basis of their argument lies in the limit aâp. Direct substitution of a, = ap into Eq. (7.8.6) led these authors to the conclusion that ^ h s ^ f ^ T ^ h ^ where φτ = η'Ύπ(2αργ/6 is the volume fraction of all of the solute components with the total concentration riT = Ση). Let us consider an alternative inter-pretation, where ξ'^ is assumed to be valid for all values of the solute radius. If aj = as for all f then = (K p/3)[1 + (α8/αρ)γη'Ύξ°, where Vp = 4πα ρ

3/3

is the volume of the probe particle. This result clearly indicates that ^ s is proportional to η'Ίξ® for ajap « 1 and η'ΊαΙ only for ajap » 1 and not when as = ap. One can interpret ξ'^ as the additional friction that results from the requirement of having NT = n'TAV solute particles in the volume AV that diffuse along with the probe particle. That is, by the choice of the center of the probe particle as the center of the reference coordinate system and the neglect of fluctuations of solute particles within the volume AV, the diffusing unit was defined as the probe particle and the associated solute particles. It is for this reason, therefore, that one may associate ξ'^ with ^ as an additive term in the spirit of the Kirkwood formulation for a structure of point friction centers. Imai and Mandel (1982) used a cell model in solv-ing the Onsage r -Nav ie r -S tokes equation and conclude that the effective friction factor had an additional hydrodynamic contribution due to the counterions, ξ0.


Debye-Hückel limit. The Debye-Hückel limit is obtained by linearization of the Boltzmann weighting factor in the pair correlation function for par-ticle j in a mean potential <</>(r)>, viz, n}(r) ~ 1 — [Z /e<(/>(r)>//cT]}. Vizcarra-Rendon et al. (1987) used the Debye-Hückel potential for <(/>(r)> as given by Eq. (7.1.7) to calculate £'el in Eq. (7.8.5) in the limit ap » α,, with the result

and

l2eDqap

G(Kap)DH

G(Kap

2θρ

DH

Σ

Σ n

'k(zk

(7.8.7)

(7.8.8)

{7.8.9)

Nonlinear dependence of the polyion charge. Thus far the polyion charge has entered into the problem only as a linear term. Vizcarra-Rendon et al. (1987) also considered the case in which n'j(r) = <«}>„ exp[ —Z /e<(/>(r)>//cT], where <0(r)> is again the Debye-Hückel potential. Assuming q = \ZYe\ and Ζγ = — Z 2 , these authors obtained

(Zpe)2

\2sDqap

G(KÜ. \EXPDH

2y

G(Kap)EXPDt1 lDH

D H .

-/(x),

(7.8.10)

(7.8.11)

I(x) = [(x + Xo)2{[exp(y) - l ]

2 + [ e x p ( - y ) - l ]

2} ] dx. (7.8.12)

y = [exp( - x /x 0) ] / (x + x0)> * = D + ( « P/ ^ D H ) ] / ( V 0pMDH)> and 7 = |Z peq|/efc7a p.

Hydrodynamic interactions. The above development has ignored the effect of the hydrodynamic flow field on the distribution of small ions about the probe polyion. Booth (1954) examined the sedimentation velocity of a charged particle in a Stokes flow field. Envision a positively charged sphere whose center lies in a plane perpendicular to the Stokes flow field. As the sphere


sediments, the ion cloud attains a new equilibrium configuration with more anions above the plane than below. The velocity of the particle described by a series expansion in Z p is, to the first correction term for a 1:1 salt of equal mobilities,

U = U0 1 + 6Zle

2D°pV3(Kap)

apD°s

(7.8.13)

Geigenmuller (1984) corrected a small error in Booth's expression for K3(x), which in Geigenmuller's formulation is expressed as FB(x) = — 72V3(x), and which is denoted in this text as G(/ca p)

B to be consistent with the above

expressions:

FB(x) = G(x)B = - 7 2 F 3 ( x )

1 2 1 3

2X +

2

21 39

+ 16 x

3(x

2 12)T(x) + - ( l + x " ' )

+ Ϊ 5

( 4*

5

where T(x) = — exp(x)£/(x) + χ 1

— χ 2

+ 2x 3

— 6x 4

and

Ei(x) =

90x + 180)x2T(2x)Wl + x )~

2, (7.8.14)

(7.8.15) t *exp( — t)dt.

The value of the function V3(KÜP) in the asymptotic limits of high and low salt is zero, passing through a minimum value of approximately —0.0017 at καρ = 0.25. It must be emphasized that Booth mentioned in his paper that fluctuations were not taken into consideration and assumed that such fluctuations would be small compared to the effect of a redistribution of the equilibrium configuration. On the other hand, as pointed out by Geigenmuller (1984), Booth included hydrodynamic interactions between the small ions and the probe particle that were neglected in the fluctuation-dissipation models. G(KÜP) for four models are compared in Fig. 7.6.

In his doctoral thesis, Belloni (1987) incorporated dipole effects in the friction factor in the description of the apparent diffusion coefficient, which in the linear (high salt) region is of the form

D a pp = Z)p* + D ^ / c H - ^ s -9Z p

2 S*3(Kap)UKap)

C 1 + καη

(7.8.16)

D°p{l + ΙΖ2

ρλΒΩ0

ρ/αρΟ^ν3(καρ)}, Λ(καρ) is Henry's function, (καρ) is a series expansion in the zeta potential that exhibits a sigmoidal

where D*


ι 1 1 1 1 1 1 1 1 Γ

1 2 3 4 5 6 7 8 9

Fig. 7.6. G(KÜP) VS. καρ for four types of small ion-polyion interactions. Booth [Eqs. (7.7.5)-

(7.7.7)]; Schurr [Eq. (7.9.5); Linearized D e b y e - H ü c k e l [Eq. (7.9.14)]; Exponential D e b y e -

Hückel (y) [Eqs. (7.9.16)-(7.9.21)].

shape as a function of \η(καρ) with the asymptotic limits Sf (καρ - • 0) yV and Ξ%(καρ oo) -^ji (Booth, 1954: cf. Section 9.3), kH represents the in-direct hydrodynamic interaction term, and ks accounts for the direct poly ion-polyion interactions as manifested in the second virial coefficient.

Example 7.3. Effect of Ion Size on the Electrolyte Dissipation Term for Colloidal Particles.

Schumacher and van de Ven (1987) reported PCS studies on the diffusion of colloidal gold and poly(styrene) latex spheres at 25°C as a function of ionic strength and the size of the small ions. A plot of D a pp vs. /cap for the latex spheres in the presence of three different types of small ions is reproduced in Fig. 7.7.

The presence of a minimum in Z) a pp around καρ ~ 1 is the proposed signature of electrolyte dissipation. The depth of the minimum in D a pp

depends upon the size of the small ion. Since the latex spheres are negatively charged, it may be anticipated that the added cations would have a greater influence on fp than the added anions, where it is observed that ( C H 3 C H 2 ) 4 N

+

has the largest friction increment.

Example 7.4. Electrolyte Dissipation Effect for Charged Poly(styrene) Spheres.

Gorti et al. (1984) used fluorescence recovery after photobleaching (FRAP), QELS, and Doppler shift spectroscopy (DSS, cf. Chapter 9) techniques to


12.5

ο il I |2·°_

CL CL σ Ο

11.5

I 2 3 4 5 6

Fig. 7.7. Effect of ion size on electrolyte dissipation of poly (sty rene) latex spheres. The salt type

is given in the figure. [Reproduced with permission from Schumacher and van de Ven (1987).

Faraday Discuss. Chem. Soc. 83, 7 5 - 8 5 . Copyright 1987 by the Royal Society of Chemistry.]

study the tracer and mutual diffusion coefficients and electrophoretic mobility, respectively, for charged poly(styrene) spheres as a function of ionic strength. The poly(styrene) spheres were labeled with fluorescein isothiocyanate. The QELS measurements were carried out at a scattering angle of 90°. The tracer and mutual diffusion coefficients are plotted as a function of καρ in Fig. 7.8, where ap= 184 Â. The solid line is the calculated value of Dp = kT/fp, where /p = 6πη03,ρ + £ e l and £ e l was computed from Eq. (7.8.2) with Z p = 110.

l0-\ , , Γ-

Ο 5 10 15

Fig. 7.8. Electrolyte dissipation correction for charged poly (sty rene): ( Δ ) D m, ( • ) Dx.

[Reproduced with permission from Gort i et al. (1984). J. Chem. Phys. 81, 909-914 . Copyright

1984 by the American Institute of Physics.]


7.9. Polyelectrolyte Dissipation

Phillies (1981) examined the effect of direct interactions with the polyion matrix on the friction factor of the probe macroion in which #

(

p

2) (r: AR, t)

expanded as a series in a( i )cos(Kr · AR) + fc(i)sin(K

r · AR), where a(t) and

b(t) are implicit functions of AR, the probe polyion- j th polyion separation distance. In the dual limits (/>pj(r)//cT = - {exp[-</>P 7(r)//cT] - 1} « 1 and AR = v ps with 0 < s < t, Phillies obtained for the force (spherical symmetry)

ν n' kT f00

<F

> = -TT4- dK ( RT

' A R ) W ( K ) ]2. C

7-

9-

1)

( 2 π )3 Jo

Upon obtaining Dm from Kx without hydrodynamic interactions and as-suming a harmonic potential for the weak interaction limit, viz, gv

2}(f) =

1 + ( £ 0 / 8 π3 / 2

) e x

P ( — r2/4rl% the following result for fm to the linear term in

the thermodynamic volume fraction φτ = cpvp was obtained:

f 03917 1

/ - = / S { l + * T 2 i ^ 7 i| 0

S) ( O )

-1 1

} - ( ?

·9·

2 )

Example 7.5. Effect of the Concentration of 204 Â Spheres on Dêl{ for 815 Â Poly(styrene) Latex Spheres

Phillies (1984b) examined the effect of polyion concentration on D s e lf for poly(styrene) latex spheres in the presence and absence of supporting elec-trolyte. Deionized water conditions were achieved by means of a specially treated ion exchanger over a period of weeks.

The radii of the carboxylate-modified poly(styrene) spheres were deter-mined by QELS methods to be 204 Â and 815 Â. The correlation function of the mixture exhiuited two widely separated relaxation times. The fast mode decayed to zero for times larger than 1 ms, hence data for t > 1 ms were analyzed as a single exponential decay function.

The concentration of the smaller species was fixed at 6 χ 1 0 "5 by volume,

and the larger species varied over the range 0 -0 .6%. The self diffusion co-efficient for the larger particle under the two solvent conditions as a function of the volume fraction of the smaller particle is given in Fig. 7.9.

These data indicate that Dêl{ associated with the larger particle is vir-tually independent of the smaller particles for φρ < 0.003 in the presence of counterions. In the deionized solvent, however, Dê]{ monotonically decreases over the entire range of φρ examined. Phillies (1984b) compared the experi-mental results with several theories on hard sphere systems based on the Smoluchowski, Kirkwood, and Fokke r -P l anck equations. The data appear to follow the predictions of Hess and Klein (1983b) in which D s e lf approaches Dp/2 at large volume fractions.

7.10. Counter ion Condensa t ion—the Manning Theory 225

0p ( 2 0 4 A particle )

Fig. 7.9. Dêif °f poly (styrene) latex spheres as a function of the volume fraction of smaller

spheres. The equivalent hydrodynamic radius of the larger spheres was determined by Q E L S

methods to be 815 Â, and that of the small spheres to be 204 Â. The open circles are data taken

with counterions present, whereas the closed circles represent samples that were in the presence of

a deionizer over a period of several weeks. [Reproduced with permission from Phillies, (1984b).

J. Chem. Phys. 81, 1487-1492. Copyright 1984 by the American Institute of Physics.]

7.10. Counterion Condensation—the Manning Theory

Polyions exhibit a relatively high charge density in the solution, hence the equilibrium distribution of the small ions will be greatly affected by the polyion electric field. Manning (1978, 1979) used a simple two-state model for the distribution of small ions around a linear array of charges, where the overlapping electric fields were assumed to be generated by a Debye-Hiickel potential. The small ions were assumed to be in one of two states, either free in the bulk medium or associated in the vicinity of the polyion. The latter state results in the definition of an "effective unit charge", z e f f, for each charged group on the polyion, zeff/zp = 1 — |Z C| 0 C, where the lower-case zp

indicates a single charged group on the polyion, |Z C| is the magnitude of the charge of the counterion, and ö c is the number of counterions associated with the charge z p . The charges along the polyion are assumed to contribute a pairwise repulsive interaction term of the Debye-Hiickel form, leading to the electrical contribution to the free energy

<<7c> = Σ —d* -JW1 e x p ( - K | i -j\b), (7.10.1)

where b is the distance between charges. For \i —j\ -» oo,

<#e,> = - ( 1 - | Z c | 0 c )2^ l n [ l - e x p ( - K f e ) ] . (7.10.2)


The electrostatic free energy is counterbalanced by the free energy of mixing, <0mix> = ö c l n « n ; > I O C/ < n

,

c > J , where « > l o c is the "local" concentration of the counterions (particles/cm

3) in the vicinity of the charged group. To

proceed in minimizing the total free energy <# e l> + (gmix) with respect to 0C, the concentration (ric)loc is written as 9JV'P, where V'p is the volume (cm

3)

associated with the charged group to which the counterion is bound. In setting dK#ei> + <0mix»/d0c = 0, one obtains the condition for minimization of the free energy,

1 + In = -2\Ζ0\ξΜ(ί - | Z c | 0 c ) l n [ l - exp(-Kfc)], (7.10.3)

where ξΜ = XBjb is referred to as the Manning parameter. If we now allow Wc}u 0, then we have the untenable situation that ln(oc/<né>UKP) + oo. It is noted, however, that in this limit exp(/cb) side of Eq. (7.10.3) is

1 — Kb, hence the right-hand

|Z C| £ M(1 - |Z c|0 c) ln[(*&)2] = | Z J U 1 - |Z c | 0 c) ln

1000

ZnNAXBb2Is

(7.10.4)

Expressing 2N A/ S/1000 as ( Z2< r c a> u + Z

2< n ' c > J and collecting the two

terms on the left-hand side of Eq. (7.10.3), one has from the arguments of the logarithmic terms the equality <M' C> UK p/^0 c = 4 π £ Μ / ?

3( Ζ

2« \ + Z C

2« > J ,

where e is the base of natural logarithms. From the prefactors, one has the relationship |ZC|0C = 1 — ( | Z C | £ M)

_ 1, which leads to

Z2

( Î M - I Z J "1

) *3

(7.10.5)

It is clear from Eq. (7.10.5) that 1 < | Z C | £ M . This means that the minimum charge separation distance that a solvent can support is the Bjerrum distance, λΒ. If b < λΒ\Ζ0\, counterions will condense onto the linear structure until <6> e ff = λΒ. A similar conclusion was reached by Russel (1982) for the non-linear Poisson-Bol tzmann equation.

Example 7.6. Experimental Verification of Counterion Condensation: Electrophoretic Mobility of a Polyion.

Klein and Ware (1984) used electrophoretic light scattering (ELS) methods to test the prediction that counterions condense onto a linear structure when ξΜ = 1. The polyion used in these studies was 6,6-ionene, which on the basis of assumed tetrahedral geometry has a charge spacing of b = 8.7 Â. By varying ε of the solvent by addition of methanol to water, the Bjerrum length was adjusted over a range such that 0.82 < ξΜ < 1.85. The molecular ex-tension of the polyion was monitored by intrinsic viscosity measurements,

7.10. Counter ion Condensa t ion—the Manning Theory 227

ID

>

Ε υ

χ 3

5-1

44

3H

2H

6 , 6 - i o n e n e

4 m M KBr + M e O H

2 0 ° C

• I

0.8 Τ -Ι.4

Fig. 7.10. Effect of counterion condensation on the electrophoretic mobility of 6,6-ionene.

[Reproduced with permission from Klein and Ware (1984). J. Chem. Phys. 80, 1334-1339.

Copyright 1984 by the American Institute of Physics.]

where it was established that in an aqueous solution of 0.025 M KBr the 6,6-ionene was in an extended form, and that its extension increased as metha-nol was added. The electrophoretic mobility for 6,6-ionene with 0.004 M KBr is plotted as a function of ξΜ in Fig. 7.10. These data clearly indicate that counterions condense onto the polyion structure as ε is decreased.

Example 7.7. Experimental Verification of Counterion Condensation: Tracer Diffusion of Small Ions.

Ander and Kardan (1984) used radioactive 2 2

N a+ and

4 5C a

2 + to study

the effect of polyion charge density on the diffusion of the small ions. Data were obtained for various values of the ratio X = [N p] / [ iV s] , where [ N p] is the normality of the polyion and [ N s ] is the normality of the salt. Plots of D N a +/ D N a+ vs. \/ξΜ are shown in Fig. 7.11.

The solid curves in Fig. 7.11 are computed from Manning's theory: DNa+/D°Na+ = 0Μ6/ξΜ for ξΜ > 1 and DNa+/D^ = 1 - (0.55ξΜ)/(ξΜ + π) for ξΜ < 1. These data indicate that counterion condensation does occur at ξΜ = 1 in accordance with the prediction, and that Manning's theory ac-curately reproduces the data for ξΜ > 1. These data also indicate that the Manning theory must be modified in the description of small-ion diffusion for ξΜ < 1.


0 . 0 0 . 5 1 . 0 1 .5 2 . 5 5 . 0

Fig. 7.11. Effect of polyion concentration on the diffusion of counterions. X is the ratio of

normalities, polyion/salt . [Reproduced with permission from Ander and Kardan (1984).

Macromolecules. 17, 2431-2436. Copyright 1984 by the American Chemical Society.]

Example 7.8. Experimental Verification of Counterion Condensation: Structure Factor of Condensed Counterions.

Derian et al. (1987) reported X-ray scattering studies on a solution of octyl-trimethylammonium bromide (OTAB) micelles. An important observation was that the scattering cross-section for the bromide ions is of comparable magnitude to that of the micelle, as evidenced by significant scattered intensity at high scattering angles. The coefficients for the interference factor were therefore different for the micelle and the bromide ions. Theoretical calcula-tions of the structure factor using the HNCA were performed on two models. The "structural model" was calculated on the basis of separate contributions by the condensed bromide ions and the central micelle. The charge used in these calculations was the structural charge, and the radius was the radius of the micelle. The "effective model" calculation treated the condensed bromide ions and the micelle as a single unit, where the effective radius was larger than the actual radius and the effective charge was less than the actual charge to account for the presence of the bromide ions. Aside from the low Κ region, the theoretical structure factors were identical.

The X-ray scattering intensity from the OTAB solution was placed on an absolute scale using the method of Zemb and Charpin (1985). The scattered intensity of water at the instrument settings used in the experiment was

7.11. S m a l l - I o n Distribution about Charged Rods, Planes, and Spheres 229

i_

IW o; Q'

Q Q B

q' Ο ^9r

Ο a

0 0 0.6

Fig. 7.12. X-ray scattering intensity by the O T A B system. (O) indicates an experimental value,

and the solid line is the theoretical curve with Z p = 27. [Reproduced with permission from Derian

et al. (1987). J. Chem. Phys. 86, 5708-5715. Copyright 1987 by the American Institute of Physics.]

/ w = 4 x 1 0- 8

, where the intensity at any particular scattering angle is I(K) = C(K)/T Αίφ0, where C(K) is the total photons counts over a time period Δί with a transmission Τ of the incident flux φ0. The scattering in-tensity of OTAB, expressed as / / / w , is given in Fig. 7.12. Also shown is the theoretical curve for the structural model using a charge Z p = 27, which is also the aggregation number N. Assuming that the scattering is due largely to the hydrocarbon core, Derian et al. obtained for the volume of the core the value Vcore = Zpvm = 21 χ 216 Â

3 = 5832 Â

3, where vm = 216Â

3 is the

volume of one hydrocarbon tail. This volume leads to a core diameter of 2α ρ ~ 23.2 Â. The values of Z p and ap were stated as being in good agreement with dynamic light scattering studies by DrifTord et al. (1987).

7.11. Small-Ion Distribution about Charged Rods, Planes, and Spheres

Zimm and Le Bret (1983) used the Poisson-Bol tzmann equation to examine the effect of geometry on the counterion condensation phenomenon. Their results indicated that there is a finite number of counterions that cannot be "diluted away" from the rod geometry, regardless of the volume of the solution. This fraction of residual counterions is the same as predicted by the Manning model; hence they are said to be condensed. Using the criterion that "undilutable" counterions are condensed, Zimm and Le Bret concluded that all of the counterion are condensed for the plane geometry and that none of the counterions are condensed for the spherical geometry. The calculations of Zimm and Le Bret (1984) for a charged cylinder and the hypernetted chain (HNC) calculations (cf. Appendix B) of Belloni (1985) for the pair distribution functions for asymmetric charged spheres are shown in Fig. 7.13.

230 7. POLYELECTROLYTE SOLUTIONS

A R = c y l i n d e r r a d i u s

R. = i o n r a d i u s

C P ο

o -

- M

<iPcae R c = 9

I I I I I 2 0 4 0 6 0 8 0 100

I -

o -

-I -

- 2 -

81 - 4 -

Β 12 m o n o v a l e n t c o u n t e r i o n s

— < · · · . . .

2 m o n o v a l e n t c o i o n s

Ί 1 1 1 1 1 1 1 Γ 10 4 0 7 0 100

~ l ι T e a I p c

^ A f c

\\/ r f :

χ ; ·

1 ; p a ; X '

r f :

χ ; ·

1 ; p a ; X '

1 : : χ ' Γ

/ ; 1 1

0 100 2 0 0

r ( Â )

Fig. 7.13. Small-ion distribution about charged cylinders and spheres. (A) Monte Carlo

calculations of counterions about a cylinder. [Reproduced with permission from Zimm and Le

Bret ( 1984). Biopolymers. 23, 271-285. Copyright 1984 by John Wiley and Sons.] (B) Monte Carlo

calculations of small ions about a cylinder. Note that the coions are excluded from the region near

the surface, whereas the counterions are at a concentrat ion larger than the bulk concentration.

[Reproduced with permission from Zimm and Le Bret (1984). Biopolymers. 23, 271-285 .

Copyright 1984 by John Wiley and Sons.] (C) Partial distribution functions for small ions and

polyions about a test polyion. The parameters in the hypernetted chain calculations are

2a{ = 5/5/50, Z, = + 1/— 1/ —40, C, = 0.2/0.1/0.0025M. Note that these calculations indicate

two small-ion shells. The first is the cations (counterions) near the surface of the charged sphere,

as also predicted in the Monte Carlo calculations. The second is an anion shell, indicated by the

arrow in the figure. [Reproduced with permission from Belloni (1985). Chem. Phys. 99, 4 3 - 5 4 .

Copyright 1985 by Elsevier Science Publishers.]

In all three calculations, the counterion concentration in the vicinity of the charged surface is greater than the bulk counterion concentration. It is of interest to note that the H N C calculations indicate a "shell" of anions im-mediately outside of the "shell" of counterions, as indicated by the arrow in the figure.

7.12. The Electrostatic Contr ibut ion 231

Example 7.9. Titration of NAN with Mg2 +

. Native adenovirus nucleoprotein (NAN) is composed of duplex D N A ( M p = 23 χ 10

6 Daltons), a molecule of "terminal protein" covalently

linked to the 5' end of each strand, ~ 1 8 0 molecules of polypeptide V ( M p = 48,500 Daltons), and - 1 0 7 0 molecules of polypeptide VIII ( M p = 18,500 Daltons). Polypeptides V and VII are not covalently bound to the DNA and can be selectively removed. Electron micrographs indicate that the N A N particle has a compact core structure with extended arms and loops of DNA, with an overall diameter ~ 2 1 5 ± 2 nm (Ennever et al., 1985).

The titration of N A N with M g2 +

in 0.005 M Tris buffer (pH 7) at 20°C was monitored by QELS methods (Schmitz et al., 1987). D a pp obtained from a single exponential fit was found to be independent of both Δί and θ (20° < θ < 120°). Although internal decay modes should be observed on the basis of the 216 nm diameter, the insensitivity of D a pp to θ was interpreted as the D N A loops and arms being "invisible" relative to the central core region. Hence any change in D a pp upon addition of M g

2 + was directly attributed to

the radius, without the potential complications that would result from segmental motion.

The data for θ = 40° are illustrated in Fig. 7.14, along with a pictorial representation of the N A N particles at the two extremes of the titration curve. These data clearly indicate that displacement of N a

+ by M g

2 + results in a

decrease in the flexing rigidity of the D N A in the loops and arms region with an eventual collapse of these structures onto the central core region of the N A N particle.

7.12. The Electrostatic Contribution to the Persistence Length of Flexible Polyelectrolytes

The average energy of bending is related to the elastic bending constant Y by Eq. (5.7.12). If Y is the sum of an intrinsic part ( y i n) and an electrostatic part ( y e l) ,

then

<Eh} = Tin

2

" 3 V

ds (7.12.1)

where y e l may depend upon d2r/ds

2 since the electrostatic energy does not act

along the contour length of the polyion. Retaining the relationship between Y and L p , one can write (Yamakawa, 1971) L p = L i n + L e l . For chains of finite lengths, the analytical expression for L e l ( L ^ , the analytical model) is (Odijk, 1977; Odijk and Houwaart , 1978; Hagerman, 1983; Weill et al. 1984)

A U A IΛ LJ


I ^ flexible arms and loops \(weak scattering)

core (strong scattering)

[Mg++Lock = 8.16 mM

20°C

—ι 1 1 I I I 0 0.1 0.2 0.3 0.4 0.5 0.6

mL added Mg Fig. 7.14. Titration of NAN by M g

2 + as monitored through the apparent diffusion coefficient.

<feeff > is an average effective spacing between charges, which depends upon the extent of counterion condensation and

u \ ι 8 ι ^Ρί-^) *ω = ι-3>> + ^ —

Note that h(co) = 1 and Eq. (7.12.2) reduces to that derived by Skolnick and Fixman (1977) and Odijk (1977) for small deviations from the rodlike con-figuration where b'u ~ \i - j\b{l - - j\b)

2/l2R

2]}

1/2, and s = \i - j\b

and Rc is the curvature of the rod. The above expression is valid if ξΜ < 1, where counterion condensation of the Manning type does not occur. Odijk and Houwaart (1978) later suggested in an ad hoc manner that for ξΜ > 1,

L e i = ^ (£M>1). (7.12.4)

Manning (1969, 1978, 1979) proposed that the intervening subunits be evenly spaced along the chord connecting the subunits 1 and Ν + 1, with an "effective" charge separation between each subunit of b' = bsm(6)/6.

y + 5 + • (7.12.3)


Manning's theory results in L e l = — a 2 l o g 1 0( C s ) , where a2 is a constant. Schurr and Allison (1981) pointed out the geometric error in Manning's approach in that Θ/Ν should be used for the angle increment for each sub-unit, with the resulting expression for the nearest-neighbor spacing, b' = 6sin[ö/(2N)]/[ö/(2A0]. Restricting their development to nearest neighbors only, and then following the procedure of Skolnick and Fixman (1977), Schurr and Allison obtained for an upper limit L e l < b(l — 0 c)

2£M^DH / [4Î>

2] .

In deriving this expression, it was argued that: (1) counterions condense onto the unbent rod; (2) subsequent relaxation occurs in the bent configuration that results when the stochiometric charge is reduced by the counterion condensation; and (3) 0C was unchanged in the bending process.

Le Bret (1982) numerically solved the appropriate Poisson-Bol tzmann equation for a toroid with either a conducting or a nonconducting surface. The solutions for small surface charges were obtained by a multipole expan-sion, whereas a two-dimensional grid was used for the higher surface charges. Fixman (1982) used a power series expansion in the parameter 2 D H/ # C

a nd

cos(0) for deviations from rodlike structure. Both of these models gave com-parable results for cylinders of similar properties and show a slight deviation from the —1 power dependence of L e l on C s for conducting cylinders. For the nonconducting cylinder, the slope of — l n ( L e l) vs. — ln(C s) depends upon the value of C s, varying from an asymptotic limit of — 1 for C s « 1 M to the value of — | for C s ~ 1 M.

Calculations of the C s of L e l based on these models are shown in Fig. 7.15. The Le Bret (LB) and Fixman (F) calculations are based on the dimensions of DNA, whereas the Skolnick-Fixman/Odi jk (SF/O) calculations employed b = 1.2 k to avoid Manning condensation.

The models discussed thus far are based on small deviations from rodlike behavior, where the distances between charged groups i and j differ slightly from the value |i — j\b. It is emphasized, however, that the Coulombic inter-action does not act along the contour length of the chain but is transmitted directly through the medium. Hence y i n and y e l cannot in principle be treated on equal footing for very flexible chains. To illustrate this difference, consider a set of flexible coils with the same curvature but variable lengths Ν + 1. For those configurations in which 0 < 90°, the electrostatic interaction between these sites 1 and Ν + 1 will decrease with an increase in the value of Ν until the bending angle 0 = 90° is attained, after which the electrostatic interaction energy between these two sites increases with an increase in N.

We can computer-sum exactly the pairwise interaction energy for a flex-ible linear polyion that undergoes a continuous bend about a central point (Schmitz and Yu, 1988). A line connecting beads 1 and Ν + 1 is the chord for an arc of angle 0 = 2φ associated with a circle of radius Rc = Nb/Θ. The spatial distance between the ith and jth bead is the chord length for an

7. P O L Y E L E C T R O L Y T E S O L U T I O N S

SF/O

\6A

14 J

I2j

8H

LB/F conducting

LB/F

nonconducting''··

non conducting \ LB/F

conducting Ί 1 1 1 1 1 2 4 6 8 10 12

-ln(Cs) Fig. 7.15. Comparison of persistence-length calculations. (LB/F) Le Bret /Fixman; (OH)

O d i j k - H o u w a a r t (values of the contour length are given in the figure.) Also shown are lines with

slopes of — i, — \, and — 1 for comparison purposes.

angle |z — ]\φ/Ν, b'u = bsin(\i — ]\φ/Ν)/(φ/Ν). For the screened Coulomb potential,

EA0) = A^- f (TV + 1 -j) Φ M

exp - Kb sin Ν

b sin

(7.12.5)

where A = [_(ezef{)2/s]. It is assumed that the curvature is constant and re-

d2r

lated to the radius of the circle by \/Rc = — H e n c e , y e l and L e l for the as

numerical model are

2[E e l(fl) - E e l(0)] 2[E e l(fl) - E e l(0)]R c

2

1

0 \K

(7.12.6)

ds

7.12. The Electrostatic Contribution 235

, „ _ 7 e t _ 2 [ £ e l( f l ) - E e l( 0 ) ] K c

2

L el ~kT~ LKF ·

{ 1 A 2 J)

Several values of L ^ / L ^ are listed in Table 7.2. This comparison clearly indicates that L" x ~ over a wide range of

values of C s. The largest percentage difference is when λΌΗ < b, which is to be expected since the Odijk formulation is based on a continuous charge density. It is also noted that there is a dependence of on Θ. This is most significant for the smaller chain lengths, where λΌΗ approaches the contour length of the chain. It is concluded that even though the Odijk formulation was based on small deviations from rodlike behavior, the Odijk formula works very well even for large deviations from rodlike behavior.

We now address the question of "chain stiffening" upon lowering the ionic strength. Since the focus is on L e l , it is assumed that L i n « L e l under all ionic-strength conditions. The probability for a polyion bent through an angle θ

Table 7.2

LLI/LÎ for Various Values of L, ÀDH, and 0

θ

L (A) (A) (A) 10 40 100 130

4.4 64.8 0.555 1.03 1.03 1.02 1.01 11.8 2.65 0.90 0.89 0.86 0.83

237.8 8.25 0.78 0.78 0.75 0.74 645.8 8.32 0.78 0.77 0.75 0.73

4.4 352.8 0.654 1.13 1.13 1.13 1.13 11.8 4.55 1.05 1.05 1.04 1.04

237.8 203 1.00 0.99 0.97 0.95 645.8 238 0.99 0.99 0.97 0.95

4.4 712.8 0.665 1.14 1.14 1.14 1.14 11.8 4.77 1.06 1.06 1.06 1.06

237.8 581 1.03 1.03 0.99 0.97 645.8 892 1.03 1.02 0.99 0.97

4.4 3592.8 0.674 1.15 1.15 1.15 1.15 11.8 4.95 1.07 1.07 1.07 1.07

237.8 1,660 1.05 1.05 1.04 1.03 645.8 7,980 1.05 1.05 1.02 0.99

4.4 21,593 0.676 1.16 1.15 1.15 1.15

11.8 4.99 1.07 1.07 1.07 1.07 237.8 1,960 1.06 1.06 1.06 1.06 645.8 13,700 1.06 1.06 1.05 1.05


Ρ ( θ ) / Ρ ( 0 )

0 20 40 60 80 100 θ

A

1 I I I I I I I I I I I Γ -2 0 2 4 6 8

!n(L/ADH) Β

Fig. 7.16. Statistical properties of linear polyelectrolytes. (A) Relative probability of a continuous bend through an angle 0. The contour length and the D e b y e - H ü c k e l screening length are given in the figure. It is assumed in the calculation summarized above is that only the electrostatic interaction between pairs of charges along the polyion backbone contributes to the persistence length. This is equivalent to the assumptions that the intrinsic persistence length is zero and that the entropy for both configurations is identical. (B) \n{ÀDH/L

n

el) vs. ln(L/ADH) for several contour lengths. The minimum exhibited in the numerical calculations of L e l occurs at L//tDH = 4.1155 in accordance with the analytical solution L

a

el.


relative to that of a rod configuration (Θ = 0) is

Ρ(θ)_ i - [£e l ( f l ) -£e l (0 ) ] W) = Q x p

\ kf (7.12.8)

Calculations of Ρ(θ)/Ρ(0) were carried out for b = 1.2 Â and plotted as a function of λΌΗ and θ in Fig. 7.16. Also shown in this figure is a plot of

I n ( W ^ e i ) vs. ln (L /A D H) . As might be intuitively expected, these calculations indicate that for a chain

of fixed length, Ρ(θ) for large angles decreases as C s is lowered. On the other hand, these calculations also show that the screened Coulombic interactions do not cause the polyion to become rodlike at the lower ionic strength, even if /DH > L. Ρ(θ)/Ρ(0) appears to attain a minimum breadth as L is increased (cf. the λΌΗ = 646 calculations). The "turnaround point" is manifested as a minimum in the l n (A D H/ê i )

v s-

m( ^ M ö H ) P ^

o t shown in the insert. According

to the Odijk model of for finite chain lengths, the minimum in the above plot occurs when ymin = (L/ÀOH)min = 4.1155. The corresponding value for the ratio ^ D H/ ^ e i is

This prediction of the Odijk formulation is verified in the quantitative evaluation of P(40) /P(0) listed in Table 7.3.

The physical significance of the " turnaround" point is related to the response of the polyion to a change in the ionic strength. The effect of the screened Coulombic interaction is to greatly reduce the range of the interaction between two groups. Starting from high salt conditions, a decrease in C s first acts to "stiffen" the polyion for all chain lengths. This result is simply due to the fact that the interaction energy between nearby groups exceeds the thermal energy kT. Since Eqs. (7.12.2) and (7.12.4) predict that l n ( / l D H/ L e l) vs. l n ( l / A D H) has a positive slope, the region to the right of ymin in Fig. 7.16 is said to be "rodlike". As C s is further reduced, the situation is eventually reached where the additional interaction energy becomes less than kT. When this situation obtains, L e l becomes somewhat insensitive to λΌΗ and hence is referred to as

(7.12.9)

Table 7.3

P(40)/P(0) as a function of L for λΌΗ = 645.8 Â

L(Â)

P(40)/P(0)

65

0.960

353

0.847

2153

0.589

2513 0.584

2657°

0.584fl

2873

0.585

21,592

0.864

a Predicted minimum in accordance with Eq. (7.12.8).


the "coillike" response to changes in C s. The double logarithmic plot at this extreme condition takes on the form ln(/ l D H) vs. ln ( l / / l D H) , which has a slope of — 1. This region is to the left of ymin in Fig. 7.16.

Example 7.10. L E L for Poly(styrene sulfonate) as a Function of the Molecular Weight and Ionic Strength.

Weill and Maret (1982) reported magnetic birefringence studies on poly(styrene sulfonate) (PSS) as a function of M p at several polyion concentrations with no added supporting electrolyte. These data were subsequently analyzed by Weill et al. (1984) in terms of the finite chain model.

Fig. 7.17. L*, /L^p as a function of the Debye-Hiickel screening length for P S S . The degree

of polymerization is shown in the figure along with the theoretical value of the counterion concentration. The experimental data were those of Weill and Maret (1982). Since the analytical expression for L e l gives the correct values of Lel for a wide range of bending angles as judged from exact numerical results, the discrepancy between the experimental and theoretical values of L e l is attributed to a "counterion sheath" of the condensed counterions in the vicinity of the charged surface. Since this sheath may extend 10 -20 Â from the surface (cf. Fig. 7.13), nearby charged groups on the polyion are more highly screened than those pairs further apart , as illustrated above. The discrepancy between L" t and L "

p is partially rectified by a " two-state" distribution of

counterions, i.e., sheath and bulk solution regions. The calculations were performed with an ionic s t rength-dependent radius r s for the sheath, i.e., r s = a

0 + bÀDH, where a

0 is a constant thickness

that represents the condensed ions, and λΌΗ represents the diffuse cloud. It is suggested that a continuous decrease in the counterion concentrat ion within the sheath as indicated by the numerical results in Fig. 7.13 is required to match the experimental points. (K. S. Schmitz, unpublished calculations.)

7.13. Composi te Diffusion Coefficient for Flexible Polyion in the D e b y e - H ü c k e l Limit 239

The contour lengths (in Â) used in these calculations were ( L ; M p) : (180; 15,000); (480; 40,000); and (1680; 140,000). Two methods were used to estimate the equivalent ionic strength: (1) [C s] = 0 .18[C p] (Manning condensation): and (2) [C s] = 0 .36[C p] . They reported that method 1 gave good values for L e l for the lower-molecular-weight samples, whereas method 2 resulted in better values for the higher-molecular-weight samples.

It seems highly unlikely that the Manning condensation phenomenon can discriminate between high and low molecular weights. Since the numerical calculations make no approximations other than that of a pairwise screened Coulombic interaction, the failure of the analytical expressions to reproduce the experimental data accurately is not traced to the mathematical approxi-mations involved. One possible source of error in the application of the theory is the use of a constant value for λΌΗ based on the bulk ionic strength rather than a "distance-dependent" screening parameter. Counterion condensation in the vicinity of the polyion would necessarily reduce the importance of the nearby charged groups if calculations based on the bulk ionic strength are employed. The ratio L

a

el/L^s is plotted as a function of ln(C p) in Fig. 7.17. Also

shown in this figure is a visualization of the effect of counterion condensation on the calculation of L e l .

7.13. Composite Diffusion Coefficient for Flexible Polyions in the Debye-Hückel Limit

There is at present no complete theory that takes into consideration all possible contributions to the diffusion modes in complex systems, as given by Eq. (6.1.1). We therefore resort to the construction of a "composite" diffusion coefficient that employs in an ad hoc manner the essential features of the isolated theoretical developments.

The starting point is the generalized small ion-polyion coupled-mode theory in which Dapp(K) is defined by Eq. (7.3.17), where Dp = D°pH(K)/S(K) in accordance with the discussion in Sections 7.3 and 7.6 and in Example 7.2, and D p = κΤ/{ξ^ + (Tei ) as given in Sections 7.8 and 7.9.

For simplicity of discussion, ξ' is limited to electrolyte dissipation effects as discussed in Section 7.8. Note that in this approximation both ξ° and £ e l de-pend upon the radius a p , which in turn is dependent upon / s through L e l for flexible polyions. For linear polyelectrolytes, ap = Fl{R

2

3}v = F 2 [ L ( L e l +

L i n) ]v, where F l 5 F 2 , and ν are parameters that depend upon the statistical

nature of the polyion and Is. For example, theta conditions are defined by ν = 0.5, F, = f, and F2 = ( M Y ' 2 -

We describe to find an expression for the ratio D^/D™ as some power law in the parameter s < 1 (depending upon the ionic strength, 5 = L e l/ L i n or

2 4 0 7. P O L Y E L E C T R O L Y T E S O L U T I O N S

5 = L i n/ L e l) . The REFERENCE RADIUS AP is defined by the expression

ap = a°pl\ +SY = A0

P(\ + u), ( 7 . 1 3 . 1 )

where U = VS + v(v — l ) s2/ 2 + · · · .

Returning now to £ e l, it is noted that depending upon the size, charge, and ionic strength conditions, the ratio £ e i / £ h can become significantly larger than unity. It is assumed in the following that £ e l/ £ h = ocG(Ap/ÀOH) = ν < 1 , where α = Z p ( a s / t B / 1 2 ö p ) . Hence, in combination with Eq. ( 7 . 1 3 . 1 ) and defining D p

0 = ΙίΤ/6πη0ΑΡ as the diffusion coefficient for the REFERENCE CONFORMATION,

" Î - ^ i ' - ï T ï H ' - r h } - ( 7

" -2 )

There are four characteristic ratios that must be considered for flexible linear POLYION SYSTEMS: ( 1 ) ΑΡ/ΛΌΗ< 1 and L e l/ L i n< 1 ; ( 2 ) ΑΡ/ΛΌΗ< 1 and L e l/ L i n> 1 ;

( 3 ) ΑΡ/ΛΌΗ > 1 and L e l/ L i n < 1 ; and ( 4 ) a p / A DH > 1 and L e l/ L i n > 1 .

Example 7.11. Flexible Coil in the Debye-Hiickel Limit. It is of interest to examine the dependence of D p on Is when both electrolyte dissipation and polyion expansion are considered. The polyion is modeled after poly(lysine) or poly(acrylate) in which L i n ~ 1 0 Â . It is assumed that L / L

p » 1 and L e l/ L i n » 1 , hence L e l = Λ^Η/4λΒ and S = Lin/Lel. The par-

ticle obeys random coil statistics with ap = 0 . 3 8 ( L L i n)0 , 5

. In the D e b y e -Hiickel limit [cf. Eq. ( 7 . 8 . 8 ) ] , ν = oc2x/(\ + x)

2, where χ = a°(l + ύ)/ΛΌΗ.

Hence,

D°p = D?-±- - j - = D™-±-, ( 7 . 1 3 . 3 )

ι + w ι + y ι + γ

where 7 = 2oc°ΛΌΗ/{apu2\_\ + ( / D H/ ß p W ) ]

2} , and the reference value a

0 =

Ζ2

ρα,ΛΒΙ\2{α°ρ)2. It is noted that the derivative dD°p/dXOH = - [ / ) £ / ( 1 + Y )

2]

(dY/dÀDH) is zero when

4 - « p 4 - 4 v L

^ »a

* =Q, ( 7 . 1 3 . 4 )

;2

where flp = ap[\ + ( 4 L i n / l B / / l D H ) ] v . It is to be emphasized that A D H is a function of the charge of the polyion by

- i - = 4 π Λ Β [ ( 2 « > Μ + Z p<n' p>M) + Z2

< n p > J , ( 7 . 1 3 . 5 )

^DH

where Zp(rip}u is the number of univalent counterions released by the polyion. The minimum in D p vs. ΛΌΗ depends upon v, L i n , L, and Z p .


Example 7.12. Effect of Ionic Strength on the Internal Modes of Linear and Supercoiled pUC8 DNA.

Langowski and coworkers (Langowski et a l , 1986; Langowski, 1987) have reported QELS studies on the linear and supercoiled forms of plasmid pUC8 DNA. The plasmid DNA was prepared from E. coli cells, extracted by means of a RNAse/proteinase Κ treatment, and purified by CsCl/ethidium density gradient centrifugation. The linear form was obtained by incubation with EcoRl in 0.1 M NaCl, 0.01 M Tr i s -Hc l (pH 7.5), and 0.02 M M g C l 2 . The DNA was then extracted with phenol, precipitated with ethanol, and redis-solved in the appropriate buffer. The pUC8 D N A has 2717 base pairs, which corresponds to L ~ 9238 Λ for the B-form. Internal modes should be detect-able by QELS.

Langowski et al. (1986) characterized the correlation functions as a biexponential function. The assignment reliability of internal modes was tested through the Berg-Soda model for circular DNA, where the Gj(K, t) were generated by Eq. (5.11.8) up to the fifth eigenvalue. The parameters were: L = 924 nm, L p = 65 nm, and h = 12.5 nm. The results are shown in Fig. 7.18.

The above curves indicate that the biexponential analysis is reliable for the analysis of G^K, f ) up to a value of K

2 ~ 2.2 χ 10

1 0 c m

- 2, or θ ~ 52°

for incident light of wavelength λ0 = 488 nm. C O N T I N and DISCRETE

H 1 1 1 1 r -

Ο Ζ 4 6 8 10

K2 x i o '

1 4 ( m

- 2)

Fig. 7.18. DISCRETE analysis of simulated correlation functions for circular polymers. The

correlation functions were generated up to the fifth eigenvalue of the B e r g - S o d a model for cyclic

chains with hydrodynamic interaction [cf. Eq. (5.11.8)]. The correlation functions were then

analyzed as a biexponential function using the program D I S C R E T E (cf. Section 4.11) (solid lines)

and compared with the fast and slow mode values of the B e r g - S o d a model (dashed lines). The

biexponential function appears to be a reliable representation of the da ta for Θ < 52°.

[Reproduced with permission from Langowski et al. (1986). Biophysical Chemistry. 25, 191-200.

Copyright 1986 by Elsevier Science Publishers.]


\ 2 ( s_ l

)

4 0 0 0 0 - 1

20000H

0 2 4 6 8 1 0 12

Κ 2

χ 1 0 - 1 4

Fig. 7.19. Κ2 dependence of the relaxation rates of the fast component for supercoiled and

linear pUC8 plasmid DNA. The experimental values for the fast decay rates of supercoiled ( · ,

upper solid line) and linear ( O , lower solid line) pUC8 plasmid D N A are compared with the

theoretical predictions for the rigid rod ( ) and Berg-Soda ( ) models. [Reproduced with

permission from Langowski et al. (1986). Biophysical Chemistry. 25, 191-200. Copyright 1986 by

Elsevier Science Publishers.]

(cf. Section 4.11) were used to demonstrate that two exponentials were present in the biexponential analysis curves, rather than a continuous distribution of decay rates.

Internal flexibility of the linear and supercoiled DNA is inferred from comparison of the K-dependence of the fast decay rate with the theoretical expressions for rotation and the Berg-Soda model. The results are shown in Fig. 7.19.

It is clear that a purely rotational motion cannot account for the K-dependence of the internal relaxation time. These authors interpreted the discrepancy between the Berg-Soda model, which considers only bending modes, and their observations as evidence for one or more additional internal motions, such as torsional relaxation.

Langowski (1987) examined the salt dependence of superhelical and linear pUC8 DNA over the range 0.0011 M < / s < 1.0 M at 22

ÜC. The homodyne

correlation functions were analyzed as the square of a biexponential function with decay rates λχ and λ2. The decay rate of the slow mode, λΐ9 was associated with the center-of-mass diffusion. The "pure" translational diffusion coeffi-cient was determined for each sample by extrapolation of λ{/Κ

2 = D s l ow to


0.2-\

0 . H

" Ί

Ι Ο

K 2( I O l 4m - 2)

0 . 2 .

0 . 1

" Ί Γ

4 6

2

π -

ι ο

κ " (IO'V2)

Fig. 7.20. Effect of salt on the internal relaxation modes of supercoiled and linear pUC8 plasmid DNA: D2 for the fast mode of linear (A) and supercoiled (B) D N A , and a2 for the fast mode

of linear (C) and supercoiled (D) p U C 8 D N A . The values of D2 for the supercoiled D N A were

averaged over the four highest scattering angles and given as a function of the counterions. [ N a+] :

( ) ι M; ( ) 0.1 M; ( ) 0.01 M; ( ) 0.001 M. [Reproduced with permission

from Langowski (1987). Biophysical Chemistry. 27, 2 6 3 - 2 7 1 . Copyright 1986 by Elsevier Science

Publishers.]

Κ = 0. D S ] 0 W(X = 0) was found to be virtually independent of Is for the supercoiled preparation. This value of Dslow(K = 0) was reported to be equi-valent to that of a cylinder of length 350 nm. In contrast, A i o w ( ^ = 0) for the linear form was reported to be 3.0, 3.2, 3.28, and 3.4 picoficks for the ionic strength of 0.001 M, 0.01 M, 0.1 M, and 1.0 M, respectively. This slight dependence on / s was assumed to be a result of changes in the excluded volume parameter and L e l .

The K2 profile of λ2 for the linear form was reported to have a zero

intercept, whereas the intercept for the corresponding plot of the super-helical form was nonzero and independent of Is. The nonzero intercept was interpreted as rotational relaxation with a decay time t r ot = 130 /is. This value of τ Γ Οί was reported to correspond to a cylinder of length 220 nm with a hydrodynamic radius of 1.75 nm. The discrepancy in the cylinder length for τ Γ ΟΙ compared to the value of 350 nm obtained from D s l o w(K = 0) was


attributed to internal decay processes. The observation that Ds]ow(K = 0) and τ Γ ΟΙ were independent of Is for the supercoiled DNA was interpreted as evidence that no shape change occurred.

The decay rates for linear and superhelical DNA appeared to be indepen-dent of the ionic strength (cf. Fig. 7.20) and proportional to K

2, hence one

can assign a D a pp to the decay process. The K2 profile of Af for the linear

and supercoiled forms at the four ionic strengths are also shown in Fig. 7.20. A{ for the linear form undergoes a rapid change in value around K

2 = 1 χ

1 01 4 m ~

2 and is virtually independent of Is. In contrast, A{ for the super-

coiled DNA gradually increases with K2 and shows a definite ionic strength

dependence. It was suggested that the ionic strength dependence of A{ for the supercoiled DNA was due to a balance between the Coulombic inter-actions of the intertwined regions of the D N A and the internal tension of each strand. Under low-ionic-strength conditions, the electrostatic repulsion between the intertwined regions supresses the segmental mobility perpen-dicular to the superhelical axis. As 7S is increased, the repulsive electrostatic interactions are reduced, thus resulting in an increase in the amplitude asso-ciated with this motion.

It should be noted that the contribution of the counterions from the plasmid DNA to the ionic strength is negligible. Using the plasma D N A concentration of 100 jug/mL results in an increment increase of ~ 5 χ 1 0

- 5 M as compared

to 1 χ 1 0 "3 M due to the added salt.

7.14. Semidilute Regime for Polyelectrolytes: Scaling Laws

The description of polyelectrolyte dynamics can become very complicated, even in the dilute solution regime, if one were to include all possible coupling schemes between the polyions and small ions. Odijk (1979) extended the scaling law concept to include polyelectrolyte systems, starting from the Flory radius for an infinitely dilute solution of a flexible polymer (Flory, 1953):

where Lc is a characteristic length, and constant factors have been ignored. We denote a correlation length associated with L c by λ0.

Odijk defines the semidilute regime by the concentration range b~3 »

C p » C*, where b is the average distance between charged groups, and C* is the concentration at which the chains begin to overlap (Odijk originally used the notation cp" for this regime):

where L c = L in the computation of RF. There are constraints on Àsd in the semidilute (sd) regime: (1) λ8ά ~ RF at C p = C*; (2) Àsd must scale as C p to some

(7.14.1)

(7.14.2)

7.15. Effect of Small Ions on the Scattering Power of Polyelectrolytes 245

power in the semidilute regime, i.e., A S D = RF(C*/Cp)x; and (3) Àsd must be

independent of L. Under these conditions one has

/ r * \ 3 / 4

4 , ~ RF(^j = Lp

m^{bCp)^\ (7.14.3)

The correlation length, or radius, for a "blob" is denoted by Àb (not to be confused with the Bjerrum length λΒ). By definition, there is no interference from other chains within a blob containing g monomer units. L c is now gLp, which upon equating Àb = RF gives

^ - g ^ L ^ k h l (7.14.4)

The number of blob units is given as L/gLp, hence the radius of a single chain is R(Cp) ~ (L/gLp)

l/2Àsd. The quantity g can be eliminated by Eqs. (7.14.3) and

(7.14.4) by virtue of the proportionalities Àh ~ RF ~ Àsd,

R{Cp)~L^L^X^{bCp)^. (7.14.5)

Eqs. (7.14.2), (7.14.3), and (7.14.5) constitute the fundamental expressions for the description of solutions of semiflexible polyelectrolytes, where the various regimes are determined by the form of L p .

In the case in which L e l » L i n in the absence of added salt, Lp ~ L e l ~ λ^Η, hence

R(Cp) ~ Ü^*k*i*(bCv)-v\ (7.14.6)

and since the only free counterions must come from the polyion, λ^Η ~ l/bCp

and

R{Cp)~L'i2X)i\bCpyV. (7.14.7)

Various "transitions" can be defined by the power law relationships between C p and specific lengths. In the absence of added salt and very dilute solutions, one might expect the flexible coil to be in an extended configuration since L e l » L, i.e., L e l ~ ADHMB ~ 1 Μ Β ^ Ρ · AS C p is increased, there exists a con-centration C** such that L = L** = L*,*. The extended polyelectrolyte begins to contract as the charges become screened by the free counterions.

7.15. Effect of Small Ions on the Scattering Power of Polyelectrolytes

Considerable work was done in the 1950s and 1960s on the role of small ions in determining the scattering power of the polyelectrolytes (Mysels, 1955; Stigter, 1960a, 1960b; Vrij and Overbeek, 1962). These studies emphasized that poly-electrolyte systems are tertiary systems, viz, the solvent, the polyion, and the small ions.

We slightly modify the arguments proposed by Overbeek (1956) and Vrij and Overbeek (1962). Consider a unit cell containing one polyion and a sufficient number of coions and counterions to provide electrical neutrality.


Let us further stipulate that the unit cell is of sufficient size that the neutrality condition is realized at some distance r N from the polyion that is well before the boundaries of the unit cell are reached. It is assumed that the polyion is positively charged. Because of the neutrality condition,

| Z p | + |Z c |<rc c> e q - | Z a | < n a > e q = 0 (r = r N) . (7.15.1)

For convenience, it is assumed that the small ions are symmetric, i.e., — Z a = + Z C = + Z S . The distribution of small ions is determined by the Boltzmann weighting factor, exp(±(/> r), where φ

τ = Zse</>°//cT > 0 is the reduced inter-action potential, and the surface potential φ° is positive. The Boltzmann distribution of counterions and coions about a charged surface is similar to that shown in panel Β of Fig. 7.13. If φ" « 1 and the bulk concentra-tions of counterions and anions and equal, viz, <n a> u = < r c c > u = < X > U >

t n en

< " c ) e q —

< ^ a > e q ~ —

2 < X > u 0r. What this result means is that half of the

polyion charge Z p in Eq. (7.15.1) is compensated for by a deficit in the coion concentration and half by an excess of the counterion concentration within the volume defined by the radius r N . For φτ » 1, the Boltzmann weighting factor can no longer be linearized. The coions are effectively excluded from the volume defined by r N . The polyion charge is therefore compensated for exclusively by the counterions. From another point of view, an increase in the surface charge density can be related to an increase in the apparent binding constant of the counterion to the surface of the polyion. The fraction of the polyion charge compensated for by a deficit of coions, a, is defined for a symmetric salt by (Overbeek, 1956)

expl -γ- I - expl —

(7.15.2)

Note that 0 < α < \ , where α = 0 for a neutral polymer. The thermodynamic unit composed of the polyion with associated small

ions is in osmotic equilibrium with the remaining solution. For the ratio of index of refraction increments measured at constant chemical potential of the salt Kdnjdgp)^ and constant concentration \_(dnjdgp)g, Vrij and Overbeek (1962) have derived

Δ~ΗΛ «Z„MM δβν)μ_ , p V a dnA l δη,

M 1 / A - Ν . (7.15.3)

where M s ( M p ) is the molecular weight of the added salt (polymer), and (dnjdg^g is the index of refraction increment for component i at constant

7.15. Effect of Small Ions on the Scattering Power of Polyelectrolytes 247

weight concentration g of all other components. The intensity of light scat-tered by a charged particle (a > 0) is reduced relative to the neutral particle (a = 0). This effect can become very important in the interpretation of 7 t i l s. Kratohvil (1984), for example, has discussed in detail the pitfalls that may be encountered in the interpretation of light scattering data of bile salt micelles.

Example 7.13. The Role of Ions in Determination of the Effective Molecular Weight of BSA from Light Scattering Data.

Stigter (1960b) has reexamined the light scattering data of Doty and Steiner (1952) in terms of the thermodynamic expression for light scattered by multicomponent systems. The components of the tertiary system are the solvent (component 1), the electroneutral polyions (component 2), and a 1:1 salt where one ion is common to that of the counterion of the polyion (component 3). The general expression analogous to Eq. (2.8.6) derived by Stigter is, in the pèsent notation,

where: c2 is in g /mL and p° in particles/mL; / = (dns/dc3)T P,C'/(SnJdc2)T PjC\ and B'3 = 3B° + 2 (3B 2/ 3pJ ) ° (3p? /Sp 2)

0. The term 2(dB2/dpî)°(dpî/dp2)

0

is the first expansion term in p2 of the second virial coefficient to account for the variation in pf (the superscript "0" denotes solution properties at infinite dilution), p * is the "effective" salt concentration, and A° is the first virial coefficient in the expansion of p * in p 2 , i.e., p * = p°(l — A®p2). The parameter H* in this text is similar to H defined by Eq. (2.5.14), except that the index of refraction increment in taken at constant Τ and Ρ instead of Τ and μ', and is not to be confused with H* defined by Stigter.

Doty and Steiner (1952) reported turbidity data on solutions of B S A - H C l , pH 3.30, in the solvent HC1-0.0054 M NaCl, pH 3.30. In identifying the components 2 and 3, Stigter defined component 2 as the isoionic BSA with bound HCl with an effective molecular weight M2 = (1 + a ) M B S A, where M B SA is the molecular weight of isoionic BSA and α is the additional con-tribution due to the bound HCl. The concentration was also adjusted to the form c2 = (1 + oe)cB S A. The component 3 was defined as the "mixed salt" con-centration of HCl and NaCl, where in this case the HCl was not bound to the BSA. The effective mixed salt concentration was therefore cf = c$cl + cftaci- Rearrangement of Eq. (7.15.4) gives the operational equation

H*i (7.15.4)

Θ M 2(l + fA°lP°3)

2


W h e re Γ1 = „ „ , — \ 2 / 1 , / · , 0 . . 0 ^

T2 =

: 2A/»B?

M B S A( 1 + α )2( 1 + / Λ ? ρ ° )

2' '

ζ~ M

2 N A / ö ß

P

3NlB°, 2Nl /dB Ï2

and T 3 = ^ p - - i ^ [ ^ ) ( / l ? c a a Cl + 6 x 1 0 -4

M ) .

According to this formulation, a plot of the data given in the form of Eq. (7.15.5) has an intercept T{T2 and a limiting slope of TiT 3. The value of B2 is determined from the limiting slope of the Debye plot, i.e., H*c2/Re

vs. c2. Stigter (1960b) pointed out that B3 can be evaluated from B2 since B3jB\ ~ constant, where the value of the constant depends upon the model (Stigter and Hill, 1959; Stigter, 1960a). A value for A\can be obtained from p3

or calculated from a model (Stigter, 1960a), i.e.,

Λ, - - f t < > . • « >

for the Donnan model and

Ζρνί , Zle2vl

Xn«iekT

(7.15.7)

for the Debye-Hückel model, where m* is the number of moles of added salt for each mole of solvent. The derivative (dB2/dp3)° is obtained from the plot of l n B 2 versus \np3 (cf. Stigter and Hill, 1959). Hence T3 can be estimated from T2. The theoretical limiting slope is compared with the slope that re-sults from the choice of T x. The values of H*cBSA/Re were obtained from the data of Doty and Steiner (1952) and re-expressed by Stigter in the form of Eq. (7.15.5). The results are shown in Fig. 7.21 for two choices of Tx and the values / = 0.74 χ 10"

3, p°3 = c°3NJM3 = 5.9 χ 10~

6 NA particles/mL, and

p ^ a C1 = 5 . 4 χ 1 0 6

NA particles/mL. Also shown in this figure are the theo-retical values for the limiting slopes for the two different curves.

It is apparent from the below curves that the choice of Tx = 1.377 χ 1 0- 5

better represents the data. Based on the assumption that ^ of the adsorbed H

+ were neutralized by Cl~ and 70 HCl molecules associated with one iso-

tonic BSA molecule, Stigter estimated α to be on the order of 0.036. Using this estimated value for a, the calculated value for (1 + fA^p^)

2, and the "best fit"

value for \/Tx, Stigter obtained the value M B SA = 68,500. This value calculated from the three-component model is compared to the value calculated from the intercept of the Debye plot for BSA at pH 3.30 given in Fig. 4 of Doty and Steiner (1952). The estimated intercept is 1.4 χ 10~

5, which yields for the two-

component system an apparent molecular weight of 1/1.4 χ 1.0"5 = 71,429.

This value is significantly higher than that obtained from the three-component model.

Summary 249

α>

ο

ε

_ 5

r o

φ

r r 4

/ m

/

Ν .

ι 1 1 1 1 1 1 1 r

2 4 6 8

c ' b s a (

*/ L)

Fig. 7.21. [ ( / / * c B S A/ / ? 0 - T , ] / c B SA vs. c B' SA in aqueous solutions at pH 3.30. Values of

H*cBSA/Re from experiments of Doty and Steiner (1952). ( Ο ) Τλ = 1.377 χ Ι Ο "5; ( · ) Τγ = 1.32

χ ΙΟ"5. [Reproduced with permission from Stigter (1960b). J. Phys. Chem. 64, 842-846 .


It is concluded that omission of the small-ion effect on / t i ls by charged molecules leads to significant errors in the calculation of molecular parameters.

Summary

The small ions and polyions are statistically coupled in aqueous solutions because of the long-range Coulombic interactions. The role of the small ions can be classified as passive or active, depending on whether they affect the equilibrium or dynamic properties of the polyion. The system of polyions, anions, and cations coupled through a mean spherical potential result in three relaxation modes: (1) an angle-independent Debye, or plasmon, relaxation time; (2) an angle-dependent relaxation time that monitors the relaxation of


the polyions as influenced by the small ions; and (3) an angle-dependent relaxation time that describes the decay of small-ion fluctuations. Under the special conditions of very low ionic strength and intermediate charge on the polyion, Z) a pp for the polyion can also be angle-dependent. The angle dependence, however, only affects the rate at which Dapp(K) approaches its asymptotic limits for C p / C s » 1 and C p / C s « 1 and not the limiting values.

Asymmetric fluctuations in the charge distribution about the polyion in-crease the value of the drag coefficient of the polyion. In the case of electro-lyte dissipation, the additional friction drag exhibits a maximum value around αρ/λΌΗ — 1, if hydrodynamic interactions are neglected, or αρ/λΌΗ ~ 0.25, if hydrodynamic interactions are included. The magnitude of the effect is like-wise considerably smaller if hydrodynamic interactions are included. Hence the fit to experimental data results in a larger value for the effective polyion charge if hydrodynamic interactions are included. Unless electrolyte dissipa-tion effects are taken into consideration, an increase in the apparent friction factor upon lowering the ionic strength may be erroneously interpreted as an expansion in the dimension of the polyion.

Numerical computations indicate that counterions condense onto highly charged surfaces, and that the extent to which they do this depends upon the geometry of the surface. The condensed ions cannot be diluted away from plane and cylindrical surfaces, but they can be diluted from spherical surfaces.

Numerical evaluation of pairwise screened Coulombic interactions for L e l

shows that the analytical expressions remain valid for bending angles as large as 120°. Mathematical limitations of these theories cannot therefore be the source of failure to describe quantitatively the experimental results on polyion studies in very low ionic strength solvents. These failures must be ascribed to phenomena not considered in these theories, such as local and distant dependent (nonuniform) ionic strength effects.

Solutions of polyelectrolytes may be categorized on the basis of relative length scales such as / D H> RG, L e l, and the average distance between the centers of mass of neighboring polyions.

The scattering power of polyions is affected by the distribution of small ions in the vicinity of the polyion surface. In the presence of coions and counterions, a decrease in the ionic strength tends to reduce the scattering power of the polyion. Total intensity measurements under low-ionic-strength conditions must be carried out under constant chemical potential conditions.

Problems

7.1. In Example 7.2 it was demonstrated that if the concentration of the counterions from the BSA was included in the calculation of the ionic strength, the apparent charge on BSA obtained from QELS data was

Problems

in better agreement with the titration charge. What are the necessary conditions in which the omission of the conterion contribution results in less than 5% error in the estimated value of Z p ? (cf. Section 7.4).

7.2. Assume a hypothetical polyion system in which D p = 1 χ 10~7 c m

2/ s

and Z p = 100. Two samples of this polyion are exhaustively dialyzed against two different symmetric salt solutions of concentration C s = 10~

3 M, one being monovalent and the other being divalent. In both

cases D® = 1 χ 10~5 c m

2/ s . Calculate the value of Dapp(K = 0) antici-

pated for each solution. What is the percentage difference between these two values? Can QELS methods be used to detect this differ-ence in the apparent diffusion coefficient?

7.3. Hantz et al. (1981) reported QELS studies on a monodisperse prep-aration of chromatin core particles (cf. Example 3.6) as a function of ionic strength. The QELS data were obtained at 25°C using the 514.5 nm line of an argon ion laser. The reciprocal relaxation time was found to be proportional to Κ

2 over the angle range 40° < Θ < 120°.

The concentration of the core particles was in the range 0.1 g/L < c p < 2 g/L. In addition to the added salt (KCl or NaCl), the aqueous solvent contained 0.1 m M EDTA. The data in the accompanying tabulation were estimated from their Figs. 1 and 2.

D p( l ( T7 c m

2/ s ) [ C s] (M, added)

1.6 2.0

1.8 1.5

2.5 1.0

3.0 0.5

3.2 0.1

3.2 0.01

2.1 0.001

1.9 0.008

1.9 0.006

These data were interpreted in terms of an unfolding of the compact core particle structure in the high and low salt regimes. The low salt unfolding presumably is due to an increase in the electrostatic repulsion of like charges that results from a decrease in screening between the charges. It is possible, however, that electrolyte dissipation effects may also contribute to the increase in apparent friction. Recalling that the added EDTA in the Hantz et al. study places a lower limit on the ionic strength, estimate Z p on the basis of electrolyte


dissipation effects under the assumption that the conformation of the core particle does not change. The electrophoretic mobility charge was reported to be Ζ

μ

ρ = 104, which was stated to be in good agreement with the structure charge Z p

t r uc = 124, based on the core histones with

M p = 201,000 Daltons. How does your calculated value of Z p compare with Z£?

7.4. Assume that the high salt unfolded form of the core particle in Problem 7.3 is a rigid cylinder. Using a diameter of 30 Â, estimate the length of the cylinder. Is this value consistent with that for 146 base pairs of B-form DNA? If not, what is the necessary cylindrical diameter to be consistent with a cylinder of length corresponding to 146 base pairs of B-form DNA? Does this diameter seem reasonable? How do the hydrodynamic dimensions compare with the theoretical volume based on vp = 0.69 and M p = 201,000?

7.5. Hirakawa et al. (1987) studied the Is and pH dependence of the translational diffusion coefficient of bovine mercaptalbumin (BMA). The titration charge (Z p

ü) of the BMA in different salt solutions (NaCl)

was determined by

-m _ 7 H C H C I - ^ H + I P "5'

2

where y H, aH, C p , and C H C1 are, respectively, the activity coefficient and activity of the hydrogen ions, and the molar concentrations of protein and added HCl. Use of this expression with the measured values of the pH indicated that Zp

l depends upon [ s a l t ] a d d ed as well as the solu-

tion pH.

These authors reported that l /τ vs. K2 plots were linear, hence most

of the reported QELS data were taken at θ = 90°. The pH profile of D a p p(25°C) for 0.09% (w/w) solutions of BMA in their Fig. 4 exhibited a maximum around pH 3.8 for [ N a C l ] a d d ed = 0 and 0.0005 M, was virtually flat for [ N a C l ] a d d ed = 0.002 M, and slightly decreased with a decrease in the pH for [NaCl ] = 0.005 M and 0.1 M. Representative values of D a p p(25°C) χ 10

7 (cm

2/s) for [ N a C l ] a d d ed = 0 and 0.0005 M

for c p = 0.09 g/dL are listed in the accompanying tabulation, along with values of Z p

h from their Fig. 1.

pH 5.2 4.5 4.0 3.8 3.5

κ 0 8 16 40 —

[NaCl ] 0 6.4 13.8 17.5 18.0 12.5 0.0005 6.4 7.6 10.0 11.0 10.0

Problems

Using the value M p = 69,000 Daltons to calculate C p , the titration charge to estimate 2C S = 2 [ N a C l ] a d d ed + Z p C p , and as = 1.5 Â for the small ion radius, calculate the theoretical value for Dapp(K = 0) from Eqs. (7.4.3) and (7.4.4). Compare these values with those given in the above summary. What conclusions can be drawn from this com-parison? If good agreement is not found, estimate the value of Z p

needed to bring the theoretical values of Dapp(K = 0) in concert with the measured values.

7.6. Kowblansky and Zema (1982) reported QELS studies on the 7S and c'p

dependence of Dapp for sodium acrylamide/acrylic acid copolymers (PAM) for different charge densities. Assuming the functional form

D a p p(25°C) = Dp + Bc'p9

where c'p is in g/dL (weight percent), the tabulated parameters were estimated from their Fig. 4 and Table II for their preparation NaPAM-65 ( M p = 1.5 χ 10

6 and ξΜ = 1.85).

[ N a B r ] B2

0.10 3.83 104

0.25 4.7 58

0.50 4.95 45

1.0 6.0 14

Calculate the effective polyion radius and second virial coefficient from these data. Interpret Rapp and B 2 in terms of: (1) the linearized form for the small ion-poly ion coupled-mode theory with electrolyte dis-sipation effects, and (2) polyelectrolyte expansion by changes in the electrostatic component to the persistence length and the hard sphere expression for B 2 . Are these interpretations internally consistent? If not, what adjustments may be made to Rapp and B 2 to give consistent results?

7.7. Meullenet et al. (1978) used QELS methods to study Dp of the co-polymer maleic acid/ethyl vinyl ether as a function of the degree of neutralization a. The neutral repeat unit is of molecular weight 188 Daltons and has two titratable acid groups. M p of the fraction-ated samples was determined by static light scattering. Values of Dp

of the copolymer in 0.1 M NaBr were found to be independent of Θ over the range 0.5 <K

2 χ 10~

1 0 c m

- 2 < 3.5. The ratio Dp(oc)/Dp(oc= 1)

is shown in Fig. 7.22 for two different weight fractions.


0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9

Fig. 7.22. Normalized diffusion coefficient for maleic acid/ethyl vinyl ether copolymer as a function of the degree of neutralization. ( D ) M p = 75,000, C'p = 0 . 1 6 ; ( O ) M p = 75,000, C p = 0.12;

( • ) M p = 600,000, C p = 0.10; ( · ) M p = 600,000, C'p = 0.07. [Reproduced with permission from

Meullenet, Schmitt, and Candau (1978). Chem. Phys. Lett. 55, 523-526 . Copyright 1978 by

Elsevier Scientific Publishers.]

These data indicate that the change in Rapp for the lower molecular weight sample occurs at a lower value of a, exhibits a larger degree of cooperativity, and has a larger percentage change than the higher molecular weight sample. Disucss this behavior in terms of L e l given by Eq. (7.12.2). Support your conclusions with quantitative calculations whenever possible.

7.8. Newman (1984) reported QELS data on superhelical pBR322 plas-mid DNA (4363 bp) titrated with 2-hydroxyethanethiolate (2,2',2" terpyridine) platinum(III) (PtTS). The diffusion coefficients were mea-sured at 20°C in 0.2 M NaCl, 2 m M sodium phosphate, and 0.2 mM EDTA at pH 7.0. There was a clear angle dependence of D a pp that could not be fit with a rigid rod model for K

2 > 4 χ 1 0

1 0 c m

- 2. The

translational diffusion coefficient was therefore computed for θ < 30° and was found to be virtually independent of the D N A concentration for c p < 0.5 g/L. The titration profile of D a pp vs. [ P t T S ] / [ b p D N A ] was found to have an asymmetric shape about a minimum at [ P t T S ] / [ b p D N A ] ~ 0.11. These observations were interpreted in terms of an unwinding of the supercoiled DNA to the minimum point

255

corresponding to the relaxed state, which was followed by overwind the D N A with the opposite twist. Selected data taken from his Fig. 3 are tabulated.

[ P t T S ] / [ b p D N A ] 0.05 0.11 0.20

D a p p(20°C) x 108 ( cm

2/ s ) 3.70 3.10 4.30

Assume that the transitions supercoil relaxed -» supercoil can be modeled by cylinder open circle -> cylinder. Estimate the equi-valent hydrodynamic dimensions for the three states of pBR322 plasmid DNA on the basis that the volume of the particle remains constant. To calculate the volume use the values vp = 0.55 m L / g and 660 Dal tons/bp. Assume the Kirkwood bead model for the open circular form, i.e., a toroid, and the Tirado and Garcia de la Torre ex-pression for the rigid cylinder [cf. Eqs. (3.8.3) and (3.8.4)]. Hint: Use Example 3.5 as a guide in the toroid calculations, where the num-ber of Kirkwood beads and their diameter are adjustable parameters.

Langowski et al. (1985) used QELS methods to determine the effect of single-strand binding (SSB) protein on the dynamics of supercoiled DNA (pBR322 and pUC8). The incident wavelengths used in these studies were 632.8 nm and 351.1 nm to cover a wide range of X-values. Even with the 351.1 nm line, the plateau region for D a pp vs. K

2 was not

attained. These authors averaged the values of D a pp at low angles (30°, 40°, and 50° using the 632.8 nm data) and high angles (100°, 110°, 120°, and 130° using the 351.1 nm data) to obtain, respectively, Dp and D p I a t. These values are given in Fig. 7.23 as a function of the stoichiometry of the added SSB protein to the pBR322 D N A system.

These data indicate that the binding of the SSB protein to the superhelical D N A increases the internal flexibility of the DNA. In contrast to the binding of intercalating molecules such as ethidium, the binding profile of the SSB protein does not possess a minimum. The SSB protein thus does not overwind the D N A upon binding beyond the relaxed state. They argued that the observed change in Dp must be due to a conformational change in the supercoiled DNA.

Assume that the conformational change for supercoiled D N A is from a cylinder to an open circle. Estimate the equivalent subunit bead diameter and circular radius necessary to reproduce the observed change in D p at the saturation level of SSB protein binding. Estimate the segmental friction factor ξι from D p l at using the Soda model for circular chains [cf. Eq. (5.11.8)], assuming that the slowest internal


2I°C

12

Ε οοϋ I I

10

351.1 nm

- I /*

û

4.0

-i 1 1 î Γ

3.5

\ 632.8nm

4 ~i 1 1 1 1 8 12 16

SSB/pBR322 Fig. 7.23. Translational and segmentai diffusion coefficient of SSB/pBR322. (A) The values for

the apparent diffusion coefficient are averages of da ta taken at 351.1 nm, at the angles 100°, 110°,

120°, and 130°. (B) Data were taken at 632.8 nm at scattering angles of 30°, 40°, 50°. [Reproduced

with permission from Langowski et al. (1985). Biochemistry. 24, 4022-4028 . Copyright 1985 by

the American Chemical Society.]

relaxation mode is observed at the higher scattering angles. How does this value compare with the theoretical value given by Eq. (5.11.7)? Use the estimated Kirkwood bead diameter, hence number of beads, obtained from Dp*. What conclusions can be drawn from this analysis?

7.10. Given a sample of NaPAA of M p = 105 that is fully charged. Ac-

cording to the theory of Odijk (1979), at what concentration would the highly extended rod begin to bend in the absence of added salt? Use M p / L = 40 g/Â for PAA (Kitano et al., 1980) and a monomer weight of 72 g/mol.

Problems

7.11. Weill and Maret (1982) used magnetic birefringence methods to obtain the functional dependence of the L p of poly(styrene sulfonate) as a function of 7S. The measured specific C o t t o n - M o u t o n constant CM/cp

is related to L p by

g M = ^ "A A a A y L p i l - — cp 45 nsÀ0kTm0L0

p [ 3L

1 — exp

where Δα = 4 χ 1 0 ~2 4 c m

3 is the optical anisotropy, Ay = 6.6 χ

1 0 "2 9 erg/0e is the magnetic anisotropy, m 0 = 209 is the monomer

molecular weight, ns = 1.333, λ0 = 546 nm, L 0 = 2.5 Λ is the mono-mer length, and Τ = 20°C.

CM/Cp [ N a C l ] C p (monomer)

( 1 0 "13 cm

2) / (0e )

2g (M) (M)

7 0 0.096 9.2 0 0.048

39 0 0.0048 4.1 0.08 0.096 2.85 0.5 0.096

Calculate L e l from the data summarized in the accompanying table for M p = 140,000 Daltons. Compare these values with the Odijk and Skolnick-Fixman theories [Eqs. (7.12.2) and (7.12.4)]. In estimating 7S, assume Manning condensation for calculation of the small ions contributed by the polyion. Using Is as an adjustable parameter, what is the "effective" ionic strength necessary for the experiment and theory to be inconcert? What conclusions can you draw from this analysis?

7.12. Tivant et al. (1983) reported QELS results on the effect of Is on Dp of chondroitin sulfate ( M p = 14,000 Daltons as determined from sedimentation equilibrium measurements) and heparin (Mp = 17,000 Daltons as provided by the supplier). These molecules are long-chain mucopolysaccharides in which the repeat unit is a disac-charide. The experiments were carried out in both systems at 25°C and a scattering angle of 90° with a fixed polyion concentration of


15 g/L. (The values of D a pp are not the infinite dilution values.) Selected results corrected to water conditions are presented in the table.

App( ιο-7 cm

2/ s )

[NaCl ] ί.o 0.5 0.25 0.1

Chondroi t in

sulfate 5.8 6.65 8.0 11.15

Heparin 7.8 8.7 10.25 12.7

On the basis of these data, deduce the functional dependence of D a pp

on [NaCl] under the assumption that the polyion dimensions are independent of the ionic strength. From an appropriate plot, estimate the high salt limit for D a p p, and therefore the equivalent hydrodynamic radius. Using the value of this radius, determine the relative im-portance of the electrolyte dissipation term, where the Debye-Hückel form is assumed. How does this value compare with the hard sphere value based on a partial specific volume vp = 0.56 mL/g? Linearize Eq. (7.13.2) and estimate the magnitude of Z p .

7.13. Briggs et al. (1982) examined Dm for micelles as a function of ionic strength and surfactant concentration at 40°C. The data given in their Fig. 1 for myristyltrimethylammonium bromide (MyTAB) are summarized by the expression Dm(40°C) = 1.4 χ 1 0

- 6 + tfCMyXAB, in

which the tabulated values apply.

[ N a B r ] 0.02 0.05 0.10

a 260 100 30

Obtain an expression for Dm(40°C) as a function of C p and C s, where C s

includes the contribution of the counterions from MyTAB. What is the dynamic charge on the MyTAB, assuming it to be independent of the ionic strength?

7.14. Assume Manning condensation for poly(acrylate) in which L = 2000 Â, L i n = 10 À, and the reference conformation is that of a random coil. Using the linearized form of Eq. (7.13.2), D p ~ D p° ( l — u — v\ estimate the relative contributions of coil expan-sion (u) and electrolyte dissipation (v) to Z)p when the total ionic strength is 0.001 M. [Note that υ also takes into consideration coil expansion.]

Problems 259

7.15. Following the procedure in Example 7.11, obtain an expression for D® for the case in which L i n/ L e l » 1 and αρ/λΌΗ > 1. Use as a reference conformation the rigid rod with ν = 1. [This situation obtains for mononucleosomal D N A under low-ionic-strength conditions.]

7.16. Use the linearized form Dp ~ Dp°(\ — u — v) and compare the power laws for the C s dependence for the flexible-coil (cf. Example 7.11) and the rigid-rod (cf. Problem 7.15) reference conformations.

7.17. Expansion of the Boltzmann weighting factor leads to the small i o n -polyion coupled-mode potential [first expansion term as defined by Eq. (7.3.1)] and the Debye-Hiickel potential [second expansion term as defined by Eq. (7.1.2)]. Both of these potentials lead, in the high salt limit, to contributions of Z p C p / 2 C s to the second virial coefficient. Discuss how the interpretation of Dp vs. C p/ C s is affected if both terms are included. (Hint: Linearize the appropriate expressions.)

CHAPTER 8

Colloids

"Ί am not going anywhere. We monks are always on the way, except during the rainy season. We always move from place to place, live

according to the rule, preach the gospel, collect alms and then move on. It is always the same. But where are you going, Siddhartha?"

Siddhartha said: "It is the same with me as it is with you, my friend. I am not going anywhere. I am only on the way. I am making a

pilgrimage.'"

From Siddhartha by Hermann Hesse ( 1877-1962)

8.0. Introduction

Colloidal particles are classified as lyophobic or lyophilic. This division is based on three important characteristics of these systems. First, lyophobic particles have little, if any, affinity for the solvent. Hence their stability is governed solely by their charge. Lyophile particles do have a strong affinity for the solvent. The stability of these systems is governed by their charge and solvation properties. Second, the lyophilic colloids form reversible struc-tures, whereas the lyophobic particles form irreversible structures. Third, a lyophobic colloid may be flocculated upon addition of a small amount of electrolyte. Examples of lyophobic colloids are dispersed paints and gold and silver iodide sols. Examples of lyophilic colloids are gelatin and micelles.

Because of their high charge density, colloidal suspensions may exhibit unusual solution properties. Shown in Fig. 8.1 are digital enhanced photo-productions of a videotape of latex particle suspensions. As in the case of the monks in the excerpt from Siddhartha, the particles in Fig. 8.1 "move from place to place" but sometimes dwell in well-defined lattice locations. Within the ordered domain there also appear "holes" in the lattice structure. [Where perhaps the collection of alms is not so profitable?]

261

Fig. 8.1. Solution structure and dynamics of latex particles. Upper panel: 4.0% (v/v) of a suspension of N400 latex particles of diameter 4000 A and surface charge density 6.9 μC|cm

2

(scale on the photograph). Lower panel: Time-lapse photograph of a 2.0% (v/v) suspension of N300 latex particles of diameter 3200 Â and surface charge density 1.3 μ Ο / α η

2 (scale not shown).

The photographs were taken from a videotape (Rembrandt 3500C, Nippon A S C O M , Tokyo, Japan) obtained with a reversed-type reflecting microscope (AXIOMAT IAC, Carl Zeiss, Oberkochen, West Germany) and enhanced by a digital processing system (SEMIPS, Carl Zeiss, Oberkochen, West Germany). The system was at room temperature. The photographs were taken by K. Ito, H. Okumura , and N. Ise. (The photographs were kindly provided by N. Ise.)

8.0. Introduct ion 263

Fig. 8.2. Multidetector analysis of light scattered by a suspension of poly(styrene) latex spheres. (A) Schematic of the experiment. The sample was a monolayer to submonolayer of

charged polymer spheres in aqueous colloidal suspension, contained in a thin, wedge-shaped gap

between flat quar tz plates. The monolayer was crystalline (2D hexagonal crystal packing, hcp) in

the thickest part of the wedge (t ~~ 10 μπι), which was separated by a well-defined interface from

the liquid region. Measurements were made in the liquid phase away from the l iquid-crystal

interface. (B) Debye-Scher re r diffuse ring, with the incident beam waist away from the sample

plane (illuminated area of diameter d ~ 50 μπι). The speckle size, K c o h, is comparable to \/d. The

Debye-Scherrer ring radius, KDS, in the liquid is identical with Kw in the crystal at the l iqu id -

crystal interface line and decreases as one moves further into the liquid phase. (C) A 30 ms video

frame of < / t i l s(X)> for the liquid {2n/Km = 0.87 μπι and ~ 2 5 particles illuminated) showing

strong sixfold correlations and anticorrelations. (D) Measured normalized zero-time intensity

cross-correlation function C(^ ,0) obtained in the 2D colloidal liquid. The strong correlations

at φ = 0°, 60°, 1 2 0 , and 180° display directly the 2D-hcp nature of the local liquid structure.

[Reproduced with permission from Clark et al. (1983). Phys. Rev. Lett. 50, 1459-1462. Copyright

1983 by The American Physical Society.]

Clark, Ackerson, and coworkers (Ackerson and Clark, 1983a; Clark et al., 1983; Ackerson et al., 1985) reported multidetector laser light scattering studies for monolayer suspensions of poly(styrene) spheres in deionized water (cf. Fig. 8.2). The scattered pattern in the plane perpendicular to the incident light (observation plane) was recorded with a video recorder using fiber optic

264 8. C O L L O I D S

probes, where the fixed probe defined the angle φ = 0 and the other probe was moved to various angles φ in the observation plane. In the upper photograph, the laser beam was out of the sample plane, showing the Debye-Scherrer diffuse ring. The lower photograph was taken with the laser beam in the sample plane. The hexagonal array indicates bbc crystalline structure for the poly(styrene). These arrays of six spots persist over a period of tens of milli-seconds, then reappear with random orientation.

8.1. The Electric Double Layer

The electric double layer is a result of the unequal distribution of cations and anions in the vicinity of the charged surface, where the distribution of small ions is determined by the surface potential φ° (cf. panels Β and C of Fig. 7.13). This region is referred to as the electric double layer. The surface concen-tration of counterions, < C c > s u r f, can become quite large. For example, if Z ce0° / /cT = - 10 and <CC>„ = 0.005 M, then the Boltzmann weighting factor gives < C c> s u rf ~ 110 M. Because of the finite size and mutual repulsive inter-actions, however, there is an upper limit to the number of counterions that can condense onto the surface. The electric double layer may be partitioned into two different statistical regions. The region of "saturation" is referred to as the Stern layer (Stern, 1924), whereas the diffuse region governed by Boltzmann statistics is referred to as the Gouy-Chapman layer (Gouy 1910; Chapman, 1913), or Gouy region.

8.2. The Stern Layer

It is assumed that there are no well-defined lattice locations on the surface and that the repulsive nearest-neighbor interactions tend to distribute the counterions uniformly over the charged surface. The reciprocal average dis-tance between nearest neighbors is approximated by (l/rjm) = < X ' >

1 / 2,

where (n'é) is the average number of adsorbed counterions per unit area. We now draw a circle of radius l/<rc' c'>

1 /2 from a central particle. Since the

distribution is uniform, the distance between any two particles on the circle must also be 1/<η^>

1 / 2. Therefore the average number of nearest neighbors

is six, and the average nearest-neighbor interaction energy is proportional to 6 < l / r / w> / 2 = 3 < Χ ' >

1 / 2, where the factor of two corrects for the double

count between pairs. Hence Epair/kT = [ ( Z ce )23 < X ' )

1 / 2/ s k T ] . We take into

consideration the finite size of the ions by defining the average reduced surface-counterion association parameter = ea°/easkT, where σ° is the surface charge density. We introduce the layer-independent, reduced nearest-

8.3. The G o u y Region and the D e b y e - H ü c k e l Region 265

neighbor parameter β as β = 3e2/ekT = 3λΒ. The total energy of interaction

of the adsorbed particle with the first layer is

^ = Z ca, + Z« 2««;') ,) 1' 2/?, (8.2.1)

where < τ Ο ι is the average number of adsorbed counterions in the first layer. Saturation occurs when El/kT = 0 with a density at saturation of

If the second layer is formed on top of the first at a distance 3as from the surface, then the surface interaction parameter for finite-size counterions is a 2 = eo°/saskT. This interaction, however, is modified because of the presence of the first layer. Since only the nearest particles contribute to the interlayer interaction, there will be three such interactions with the foundation hexagonal lattice of the first layer. Denoting the average contribution per bound species as Z c a l 5 2, and recalling that β is independent of the layer number, one has

^ = Z c a 2 + 3ZCX 2 + Ζ2«Κ)2γ'

2β. (8.2.2)

At saturation in the second layer E2 = 0, hence, < Χ ' > 2 = [ (a 2 + 3Zcoci,2)/Zc/T|2. Since | a 2 | < l a j because of the finite size of the counterions and there are additional repulsive interactions due to the presence of the first layer, it follows that < « c > 2 < < Χ > ι · This trend continues for successive layers, hence the number of particles at saturation in each layer decreases as the layer number increases. There is a distance from the surface in which the small-ion distribution can be described as an exponential decreases. This distance thus defines the "thickness" of the Stern layer, denoted by ôs.

8.3. The Gouy Region and the Debye-Hückel Region

The "Gouy potential" φ° is defined as that potential at which the Boltzmann weighting factor first can be applied to the description of the counterion distribution (Verwey and Overbeek, 1948). The "Gouy" concentration is therefore <C C> G = <C c> uexp( + \Ζ^φ°/kT\). As one proceeds further into the solution from the charged surface, the Debye-Hücke l linearization solu-tion to the Poisson-Bol tzmann equation becomes a valid approximation. The Debye-Hückel "surface potential" φ

ΌΗ deduced from this expression may be considerably less than the actual surface conditions for highly charged systems. The relationship between φ°, φ*

3, and φ

ΌΗ is illustrated in Fig. 8.3.

266 8. C O L L O I D S

A

surface distribution

ο d.

0°

0E

0D

Stern slayer Β

\ Gouy \ region

Debye- Hückel -__________region

DISTANCE Fig. 8.3. Representation of the electric potential in the vicinity of a charged surface. (A) The

Stern layer of thickness os (shaded) accounts for the finite size of the adsorbed ions, which prevents the concentration from exceeding the closest-packed density. Also indicated is the hypothetical packing pattern for the primary and secondary layers. The G o u y - C h a p m a n region is composed of mobile ions in the solution phase. (B) The magnitude of the potential associated with the charge of a particle is dependent upon the mathematical form used to describe the interparticle interaction. Shown are 0 ° , the actual potential on the surface of the particle that may lead to a Stern layer if unrealistically large concentrat ions of "condensed counter ions" are used directly in the Boltzmann weighting factor; φ°, the Gouy potential, which is less than φ° by an amount sufficient to allow correct Po i s son-Bol t zmann statistics; and φ

ΟΗ, the Debye-Hi icke l potential,

which is correct for the linearized Po i s son-Bol t zmann expression.

8.5. The Double Layer above Spherical and Planar Surfaces 267

8.4. General Features of the Problem of the Interaction between Charged Surfaces

The solutions to the Poisson-Bol tzmann equation are dependent upon the values of the functions φ(ν) and άφ{ν)Ιάν at the boundary values of r. The objective is to calculate the work required to bring two surfaces from infinity to the separation distance R l 2 . This can be done by integration of the force between the two surfaces over the desired distance or by calculating the free energy involved for the process. These methods can be carried out by choosing either a constant surface potential with variable charge or a constant charge with variable surface potential.

Our point of departure is the classical Der jaguin-Landau-Verwey -Overbeek (DLVO) theory for identical spheres (Derjaguin, 1934, 1939, 1940a, 1940b; Derjaguin and Landau, 1941a, 1941b, 1945; Verwey and Overbeek, 1948). The historical development of the long-range repulsive Coulombic and short-range attractive van der Waals interaction terms as presented by Verwey and Overbeek (1948) is reviewed. These sections are then followed by discus-sions of effects not considered in the DLVO theory.

8.5. The Double Layer above Spherical and Planar Surfaces

Our interest at this stage of development is the double layer above spherical and planar surfaces for the purpose of modeling the interaction between spheres in the asymptotic limits αρ/λΌΗ « 1 and αρ/λΌΗ » 1, respectively. It is assumed that a symmetric supporting electrolyte is present in sufficient quantity that the counterions contributed by the surface make a negligible contribution to the ionic strength. Upon defining Z s = Z c = — Z a > 0, one can write the average charge density (p(r)} as

<p(r)> = Zse(<n'+}cq - <«'~> e q) = 2 Z s K < > u sinh(y), (8.5.1)

where (n's}u is the uniform concentration of small ions, y = Ze^(r)}/kT and sinh(y) = [exp(y) - e x p ( - y ) ] / 2 .

The Double Layer above a Spherical Surface

The spherical Poisson-Bol tzmann equation for symmetric salts is

- SnZse(n'syusmh(y), (8.5.2)

which has no analytical solution. In the limit αρ/λΌΗ « 1, the Po i sson-Boltzmann equation can be linearized, which leads to the Debye-Hiickel results given by Eqs. (7.1.1)-(7.1.7). Differentiating Eq. (7.1.7) and evaluating

268 8. C O L L O I D S

at r = ap gives surface charge density for the polyion p, [d<( /> p( r )>/dr] r = up = — 4πσ°/ε = — Ζρ/εα

2. The relationship between the surface potential and the

charge is therefore

Z p = αρε Φ°Ρ (8.5.3)

According to Gorin (1939), correction for the finite size of the counterions in an adsorbed layer on the spherical surface is substitution of ap + a s for ap in Eq. (8.5.3).

The Double Layer above a Planar Surface

If there are no inhomogeneities in the planar surface structure, then V2 =

d2/dz

2. Upon defining ξ = ζ/λΌΗ and then multiplying Eq. (7.1.1) by

ldy/άξ, one obtains

άξ dy J = [ e x p ( 3 0 - e x p ( - 3 0 ] - ^ (8.5.4)

where 2{ay I άξ)(ά2 y / άξ2) = ά(άγ/άξ)2)/άξ. Integration with the boundary conditions dy/άξ = 0 and y = 0 at ξ = oo gives

^ 2

, exp(y) + e x p ( - y ) + Cl = 2cosh(>;) - 2. (8.5.5)

Choosing the negative root of Eq. (8.5.5) (to ensure that ay/άξ < 0 for y > 0) and taking the boundary condition y = y° for ξ = 0, one finds the exact expression for y:

y = 2 In 1 + 7 ° e x p ( - £ ) "

1 - y ° e x p ( - £ ) _

where Y° = [exp(y°/2) - l ] / [exp(y° /2) + 1] = tanh(y°/4).

(8.5.6)

Chew and Sen (1982a) examined the potential about a sphere in the thin layer approximation using a series expansion in the parameter λΏΗ/αρ. Inside the sphere (r < λΌΗ\ they started with Eq. (8.5.2) and introduced r = ap + λΌΗξ, which led to an expression equivalent to Eq. (8.5.6) for the nonperturbation component. To the first-order correction for a spherical chargea surface.

v0{Y°2V - e x p ( - 2 g ) ] - 2 g } e x p ( - g )

y i n s i de = y + 2 i D „ Y flp[1_y«2exp(_2g)] · <8·

5·

7)

Eqs. (8.5.6) and (8.5.7) at ξ = z / / D H « 1 give

<Map + z) = < T - 2 / c T s i n h ( V ) - ^ (8-5.8) V 2 / Λ Π Η

8.6. Classical D L V O 269

for planar surfaces and

γ Z y (8.5.9)

for curved surfaces. The initial decrease in the potential with distance for the curved surface in the thin layer approximation is twice that for the planar surface.

8.6. Classical DLVO: Repulsive Interaction between Spheres in the Asymptotic Limits of αρ/λΌΗ

Following Verwey and Overbeek (1948), the total repulsive potential energy of interaction between two identical spheres of fixed surface potential is

= 0 ° [ ^ p (0 0

) —

Z p ( R l t 2) ] , where R12 = R2 — Ri- Simplifying features in the calculation obtain at: (1) αρ/λΌΗ « 1, where surface features can be ignored and only Rl2 enters into the calculation; and (2) αρ/λΌΗ » 1, where the inter-action between the two spheres is treated as interactions between concentric parallel plates. Both approaches are illustrated in Fig. 8.4.

«ρΜοΗ « 1

Let p0 be a point in the solution located a distance r 1 and r2 from the centers of sphere 1 and sphere 2, respectively. The angles 6t are defined as those subtended by the lines Ri2 and rt (cf. Fig. 8.4). The functional relationship

Fig. 8.4. Schematic representation of sphere-sphere interaction. (A) In the limit αρ/λΌΗ « 1 (thick layer), the detailed distribution of charge on the surface is of no consequence. Hence all of the charge can be condensed at one point, and only the distance between the centers of the spheres is required in the calculation of the interaction energy. (B) Curvature of the particle is impor tant in the limit ap/ÀDU » 1 (thin-layer limit). This is taken into account by approximat ing the intersphere interaction as a series of concentric p l a n e - p l a n e interactions. One shortcoming of this model is that there is no interaction between planes at different values of h. Another shortcoming is that the integration is carried out to infinity, which is partially justified by the fact that the planes at large values of h contribute considerably less to the interaction energy than planes near h = 0.

A Β

270 8. C O L L O I D S

between the charge and the interparticle spacing must be symmetric about the interparticle axis and the exchange of particles. The angle symmetry allows one to express the interaction potential </>(r() in terms of the sum over Legendre polynomials of the form P,,[cos(öf)] F„(rf), where the functions Fn(r) are found by the requirement that each term in the series solution in-dependently must satisfy the Poisson-Bol tzmann equation. Because of the interchange symmetry (Verwey and Overbeek, 1948),

<t>(rur2JM = A

e x pl ~ 7 ~ ADH

exp

^ [ c o s ^ f l F ^ J e x p /,i [ cos (0 2) ]F 1( r 2) exp

^DH

P2[cos(eiUF2(ri)Qxp P 2[cos(c? 2)]F 2(r 2)exp

+ • +

(8.6.1)

where F^) = 1 + (ADHA\X F2(ri) = 1 + (3>W>\) + 3 ( / D H/ r I )2, and A, Àu

and λ2 are chosen so that φ(ν = ap) = φ° at the surface of each particle, viz, for particle 1,

Φΐ=- Φ(?ι = ap)sin(01)i/(/)1. (8.6.2)

The charge Z, as a function of R12 is determined from Gauss's theorem, which for sphere 1 becomes

εα: sin(01)ii(/)1, (8.6.3)

where one has the general expression

θφ _ οφ οφ r\ + r\

dr1 drx dr2 2rxr2

(8.6.4)

The integration over the surface of the sphere 1 is achieved by making the substitutions: s i n ^ ) ^ = r2dr2/Rl2ap; c o s ^ ) = (R\2 + a\ - r

2

2)/2Rl2ap;

8.6. Classical D L V O 271

and cos(ö 2) = (R2

n + r\ - a\)j2Rh2r2. By combining the results of the integration of Eqs. (8.6.2) and (8.6.3) to eliminate the constant A, one obtains for the charges for sphere 1

1 + α •4 [1 - (5(1 + α)], (8.6.5)

where α = / t [ l + (1/ftx)] + / 2 [ 1 + (3/bx) + (3 / f t 2x 2) ] , j8 = (1 + α)/{1 + (l /2fec)exp[-fc(x - 1)][1 - exp(-f>)](l + α)}, δ = {exp[-fc(x - l)]/2ftx} χ {[(b - 2)/(b + 2)] + exp(-f t )} , 6 = 2 a p / / D H, and χ = RU2/2ap. The interac-tion potential at constant φ° is

φ02εα β

VR = 2 Χ

Ρ e x p [ - f t ( x - 1)] (constant φ°). (8.6.6)

If Z p is to remain constant, then the repulsive potential energy of interaction

is Z p[0 (oo ) — </>(Klt2)]> which leads to

VR = — J F —c x

P t ~b (

<x -

LK (constant Z p) , (8.6.7)

where β' = (1 + α)/[1 - δ(l Η- α)]. Since 0.6 < β, β' < 1, one has to a good approximation (Verwey and Overbeek, 1948)

_ ? L Z ^ Z . e x p ( - & x ) , (8.6.8) κ Τ χ

where y = (φ02εαρ/21ίΤ) exp(fc). Note that Eq. (8.6.8) obtains when either the

potential or the charge is held constant.

«PMDH » 1

Derjaguin (1934,1939) treated the case of two large interacting spheres as a series of concentric flat surfaces perpendicular to the line of centres. The inter-action between paired planes separated by a distance H is assumed to be independent of the other paired surfaces (cf. Fig. 8.5). For two parallel planes separated by a distance 2Z, d

2y/dç

2 = sinh(y), where ξ = ζ/λΌΗ with bound-

ary conditions at ζ = Ζ : (1) y = Ζ5βφ(ζ = Z)/kT = u; and (2) άγ/άξ\ζ = ζ = 0. The constant Cx in Eq. (8.5.5) has the value — 2cosh(w) because of boundary condition (2), hence at ζ = Ζ,

Ζ <ty{2[cosh (y) - c o s h ( u ) ] } - 1 / 2. (8.6.9)

The integral can be numerically evaluated from mathematics tables for elliptical integrals of the first kind.

272 8. C O L L O I D S

The free energy associated with the charging process is the integration of φ(ζ)ρ(ζ) over the volume for the ion charge Xe, where 0 < λ < 1. The free energy difference between planes separated at an infinite distance [G(oo)] and the finite distance H \_G(H)~\ gives the contribution of that particular pair of rings to the free energy difference for the sphere-sphere interaction, i.e., 4nh[G(H) — G(oo)] dh. The total repulsive interaction potential is

4nh[G(H) - G(oo)] dh = 2nap [G{H)-G{oo)]dH. (8.6.10) JHO

For small potentials, 2[G(H) - G(oo)] can be approximated by (εφ02/4πλΌΗ)

[1 - t a n h ( t f / 2 i D H) ] (Verwey and Overbeek, 1948):

ε α ώ02

VR ~ ' 2 l nD + e x p ( - b x ) ] (8.6.11)

As pointed out by Verwey and Overbeek (1948), Eq. (8.6.11) should not be used if b < 1.25 (τ < 2.5) by choice of the upper integation limit of oo instead of the more realistic value of Η + 2ap. The physical significance of this limitation is that the interaction between the more widely separated "plates" is highly screened by the intervening small ions. Breakdown therefore occurs when the "range of visibility" of the electrostatic interactions exceeds that of the greatest separation of two interacting "plates" of the sphere. From Eq. (8.6.11) for αρ/λΌΗ » 1, one has

VR(x) εαΌφ02

~ - ^ - e x p ( - f c x ) . (8.6.12) kT 2kT

HV ; v }

Comparison with Eq. (8.6.8) clearly indicates that the decay of VR with distance is slower for the planar than for the spherical surfaces. The geometry of the particle therefore influences the form of the repulsive interaction potential.

8.7. Classical DLVO: Hamaker Expression for the van der Waals-London Attractive Interaction

To complete the D L V O theory, attention is now given to attractive van der Waals, or London, forces. These forces arise because of an "instantaneous" asymmetry in the electron distribution in the valence shell of atoms in one molecule that "induces" an asymmetry in the distribution of electrons in a neighboring molecule. Hamaker (1937) obtained the following expression for the attractive interaction between identical spheres of radius ap:

K = "if {(*2 - D"1 + x~2 + 21n( ~9}' (8,7'1)

where AH is the Hamaker "constant" that depends strongly on the nature of the particles and the medium. (See Section 8.11.) The range of AH is

8.9. Applications of the D L V O Potential to Real Systems 273

10 21

J < AH < 10 19

J at room temperature. The asymptotic limits of Eq. (8.7.1.) are: VA AH/24x for χ -> 0; and VA~ -AH/36x

6 for χ -+ oo.

It is emphasized that the Hamaker model is a measure of the internal energy of the interaction between the two spheres and not the free energy of the interaction.

8.8. Properties of the Classical DLVO Potential

Summarizing the above results, the total pairwise interaction potential in the D L V O model is of the form

U(x) = - A A xn + AR(j)°e{f

2x-

mexp{-bx\ (8.8.1)

where AA and AR are, respectively, the proportionality "constants" for the attractive and repulsive parts of the potential, 1 < η < 6,0 < m < 1, and 0 ° f f

is the "effective surface potential". For small values of x, the attractive part decreases as 1/x, whereas the repulsive part in the planar approximation decays as exp( — bx). For χ » 1, the attractive part of the potential decays as 1/x

6, which is a much more rapid rate than the spherical approximation for

the repulsive part, i.e., ( l /x)exp( — bx). If the prefactors are of comparable magnitude, then the attractive part of the potential should dominate for small values of x, whereas the repulsive part should dominate for χ » 1.

The relative values of the prefactors may be altered by varying the ionic strength. For highly charged systems at low concentrations of added salt, the repulsive part of the potential dominates over all distances. An increase in the concentration of added salt results in the dominance of the attractive part of the potential at short distances, thus leading to phenomena such as floc-culation. The D L V O potential provides a working model for the inter-pretation of colloidal properties. Hypothetical distance profiles of U(Rl2) are represented in Fig. 8.5, where the repulsive hard sphere potential is included for completeness.

8.9. Applications of the DLVO Potential to Real Systems

The generalized dilute solution form for Dm is

A n = ^"If^ = D

°Al +

k»c'n) = ß p d + *D0P), (8-9.1)

where /cD = k'{ — k'f\ k'{ is the static interaction part, k\' includes hydro-dynamic interactions, and kO = k'n/vp. The relationship between the solution structure factor S(K AR) and the Laplace transforms of the total and direct correlation function, h

{2)(K) and c

( 2 )(K) , respectively, is

S(K AR) = 1 + <n'p}h^(K) = i _ < n^ e m { Ky ί8·

9·

2»

274 8. C O L L O I D S

hard \ sphere \ repulsion

\

\ Coulombic \ repulsion

\ \

Coulombic repulsion

LOW IONIC - STRENGTH

HIGH IONIC STRENGTH van der Waals attraction

I R | , 2/ 2 R

S

Fig. 8.5. The radial dependence of the DLVO interaction. The D L V O model for the

interaction between two charged particles has two terms, a long-range, repulsive Coulombic

interaction and a short-range, attractive van der Waals interaction. Adjustment of the Coulombic

interaction changes the nature of the system. At low ionic strength, for example, the Coulombic

interaction dominates at all distances, thus keeping the particles from coming near to each other.

As the ionic strength is increased, the Coulombic interaction no longer "masks" the attractive part,

which may lead to aggregation of the particles.

where /i( 2 )

(r) = gi2)(r) — 1 and g

i2)(r) is the pair distribution function

(cf. Appendices B, C, and D). The theoretical description of g{2)(r) based on the

Ornstein-Zernike (OZ) relationship, and several approximate solutions to the OZ relationship are discussed in some detail in Appendix C.

The mean spherical approximation (MSA) and renormalized mean spheri-cal approximation (RMSA) of Hayter and coworkers (Hayter and Penfold, 1981; Hansen and Hayter, 1982) are discussed in detail in Appendix D. The basic premise of their approach is that the highly charged particles "experience" neither the hard sphere repulsive interaction potential nor the attractive van der Waals interaction, hence these portions of the potential are not included in the MSA and RMSA calculations.

Example 8.1. Intermicellar Interactions in Aqueous Sodium Dodecyl Sulfate Solutions and the DLVO Potential.

Micelles are aggregates that are formed spontaneously and reversibly in aqueous solution once a critical concentration of amphiphilic monomers is


reached, referred to as the cmc, i.e., the critical micelle concentration. Con-centration studies on micellar systems are generally done above the cmc and reported as a function of the difference in concentration relative to the cmc, viz, c'n = Cp — c'émc in g/L. The micelle is characterized by its aggregation number and its surface charge density. Part of the difficulty of interpreting data on these systems is to account for the aggregation that occurs with a change in the monomer concentration. Other pitfalls in the interpretation of micellar systems have been discussed by Kratohvil (1984).

Corti and Degiorgio (1981) reported static and dynamic light scattering studies on sodium dodecyl sulfate (SDS) as a function of monomer concen-tration (2 < Cp < 30 g/L) and added NaCl (0.1 M < [NaCl] < 0.55 M) at 25°C and 40°C. The QELS data were obtained at a fixed angle of 90° with λ0 = 514.5 nm (argon ion laser). They assumed that under their solution conditions, aggregation was a secondary effect compared to surface charge effects. Dm(25°) vs. cp — c'^mc is plotted in Fig. 8.6.

The value of /c, in Eq. (8.9.1) was determined from 1 / M a pp vs. cp — c'émc and was of the assumed form

/c, = 8 -f- 24 J x2| l - e x p ^ ~ ^ * ^ j r f x . (8.9.3)

[The parameter χ used by Corti and Degiorgio was defined as (R12 — 2α ρ) /2α ρ, which in the present notation is χ — 1.] The attractive part of the potential U(x) is given by Eq. (8.7.1), and the repulsive part is given by either Eq. (8.6.8),

Ε ο

1.5

ο Ι Ό -

0 . 5

! __ ·~0.Ι M - - - -# 0.2 M

.__ - · 0.3 M

Φ 0.4 M

- - -· 0,45 M

- -· 0.5 M

-1 1 1 1 0 10 2 0 3 0

Cp-C;m c(g/L) Fig. 8.6. Mutual diffusion coefficient plotted as a function of the amphiphilic concentration at

various salt concentrations at 25°C. [Reproduced with permission from Corti and Degiorgio,

(1981). J. Phys. Chem. 85, 711-717. Copyright 1981 by The American Chemical Society.]

276 8. C O L L O I D S

Table 8.1

D L V O Parameters for the SDS Micelle system at 2 5 ° Ca fc

(mL/g) (mL/g)

[ N a C l ] Dp x 106

(M) (g/L) (cm2/s) (A) Exp Theo Exp Theo

0.10 0.43 0.96 25.3 16.1 16.2 40.2 29.3 0.20 0.27 0.95 25.5 8.3 6.0 21.1 14.3 0.30 0.24 0.92 26.0 4.4 2.2 13.7 8.2

0.40 0.17 0.88 26.9 - 0 . 5 - 0 . 6 3.0 3.9

0.45 0.16 0.88 26.9 - 7 . 2 - 1 . 4 - 5 . 1 2.5

0.50 0.15 0.84 27.9 - 1 3 . 9 - 2 . 4 - 1 9 . 5 0.7

a Z p = 37, AH = 1 UkT - 4.5 χ 1 0 "

20 J.

b Data taken from Corti and Degiorgio (1981). J. Phys. Chem. 85, 711.

in which case φ° = 2Zpe/[sap(2 + b)\ or by Eq. (8.6.11), in which case 0 ° = (2kT/ e) sinh~

1(Zpe

2 / 2a

2, ekT). The theoretical expression for k{ was of

the form

kf — kHS + dxF(x){\ -QXp[—j-±l)^ (8.9.4)

where for the Batchelor model, kHS = 6.55 and F(x) = 11.89x + 0.706 -1.69x

_ 1, or for the Felderhof model, kHS = 6.44 and F(x) = 12x - (15/8x

2) +

(27/64x4) + (75/64x

5) . The extrapolated values for D°9 hence RH9 and the

experimental and theoretical values of kD and /c, with Z p = 37, assumed in-dependent of ionic strength, are summarized in Table 8.1.

Versmold and coworkers have reported light scattering studies on suspen-sions of poly(styrene) latex spheres as a function of ionic strength (Hartl et al., 1983), temperature (Hartl and Versmold, 1984a), volume fraction (Hartl and Versmold, 1987), and structure forming/breaking salts (Hartl and Versmold, 1984b). In these studies the solution structure factor was identified as S(K AR) = </(K)>/<7(K)> 0, where (I(K))0 is the measured scattered light intensity for a highly diluted solution.

Example 8.2. RMSA Calculations on Poly(styrene) Spheres under a Variety of Solution Conditions.

Hartl and Versmold (1987) examined S(K AR) for two values of Is and cal-culated g

(2)(x) = g(r/2Rs) for PS spheres (Rs = 54.5 nm). The results are shown

in Fig. 8.7.


2.0·

1.2

0.4

< ]

CO

A 2.58χΙθ"6Μ

2.0-j

1.2

0.4

IC

Ί 1 1 1 1

-5 Ie= 1.37x10 M

ι 1 1 1 1 0.5 1.5 2.5

K ( i o5 c m '

1)

2.5

0.5J

Β

CP 2.5-] D

0.5-J

1.5-

~ι 1 1 1 1 10 20 30 40 50

r/2R e

Fig. 8.7. Solution structure factors and pair distribution functions for latex particles. The

diameter of the particles was 109 nm at a concentrat ion of 1.42 χ 1 018 par t ic les /m

3. Fo r S ( K ) , ( # )

are experimental values, and ( χ ) represent RMSA calculations. (A) At / s = 2.56 χ 10~6 M, the

solution structure factor exhibits well-defined oscillations that reflect ordered arrangements of particles, as indicated by the pair distribution (B). (C) At the higher ionic strength of Is =

1.37 χ 1 0 "5 M, the solution structure factor exhibits no oscillations, hence r andom distribu-

tion of solute particles as indicated by the pair distribution function (D). [Reproduced with permission from Hart l , and Versmold, (1987). Ζ. Physikalische Chemie Neue Folge. 1 5 3 , 1-14. Copyright 1987 by R. Oldenbourg Verlag, München. ]

These data indicate that S(K AR) depends strongly on Is as expected for a repulsive screened Coulombic potential. It is noted that the value of χ at which g(x) = 0 decreases as Is is increased. (Not shown are the calculations on the deionized sample, where r/2Rs ~ 6.6.) Calculations of g(x) for the smaller PS spheres (Rs = 37.5 nm) in deionized water were done for différent particle concentrations, np. The results are given in Fig. 8.8 for np' = 2.9, 4.4, and 9.0 (units of 10

1 8 part icles/m

3) .

In contrast to the results of Gruner and Lehmann (1979), the height at the maximum in g(x) appears to be independent of np. Hartl and Versmold suggested that this was due to a decrease in AR that was compensated for by an increase in λΌΗ upon an increase in np. It is noted that χ for which g(x) = 0 decreases as np increases.

Example 8.3. Temperature Dependence of Ordered Structures of Poly(styrene) Spheres in Deionized Water

Hartl and Versmold (1984a) reported the temperature dependence of S(KAR) over the range 21°C < Τ < 70°C for PS particles ( 2 # s = 109 nm)

278 8. C O L L O I D S

G ( R / 2 R S )

2.5H

0 . 5 J

0 5 1 0 1 5 2 0

R / 2 R S

Fig. 8.8. Pair distribution functions for latex particles at different concentrations: ( · )

9.0 x 1 018 par t ic les /m

3; ( χ ) 4.4 χ 10

18 par t ic les /m

3; ( Ο ) 2.9 χ ΙΟ

18 par t ic les /m

3. The diameter

of the particles was 75 nm. [Reproduced with permission from Hart l and Versmold (1987). Ζ. Physikalische Chemie Neue Folge. 153, 1-14. Copyright 1987 by R. Oldenbourg Verlag, München.]

at Hp = 1.62 χ 101 8 part icles/m

3 in deionized water. Their results indicated

that ordered domains exist over the entire temperature range. In contrast, Schaefer (1977) reported a "melting" of the ordered structure such that for Τ > 55°C, no structure remained. Hartl and Versmold suggested that the apparent discrepancy stemmed from the fact that they separated the ion exchanger from the sample prior to the taking of data. Hence the variation of / s with Τ in their study arose solely from the PS particles, whereas the ion exchanger could have contributed to 7S in the Schaefer study.

These studies underline the importance of complete removal of excess ions to generate and maintain the colloidal "crystalline" state.

Example 8.4. Effect of Structure Making/Breaking Salts on the Solution Structure Factor for Poly(styrene).

Hartl and Versmold (1984b) also examined D a pp of PS particles as a func-tion of [LiCl] (structure forming) and [CsCl] (structure breaking) at pH 6.5 (isolectric point) and pH 2.0. To test if there were interactions between these particles, they superimposed plots of g

(1)(t) vs. K

2t for θ = 30°, 90°, and 120°

for PS in deionized water (pH 6.5). These plots all exhibited the same slope, yielding a value of D a pp ^ 1.6 χ 10~

8 c m

2/ s with 2RS = 260 nm, which is

identical with that determined by electron microscopy. According to their

8.11. Beyond the Classical D L V O Potential 279

Fig. 3, Z) a pp goes from 1.6 picoficks to 0.75 picoficks in going from [LiCl] = 0 to [LiCl] = 6 M. By comparison, D a pp reaches a plateau value of ~ 1.75 picoficks at [CsCl] — 3 M. These results obtain for pH 6.5 or 2.0.

The functional dependence of D a pp on the concentration of added salt, [salt], paralleled the solvent viscosity dependence, thus D a pp = kT/ [6TW/0(1 + B ' [ sa l t ] )Ä s] ~ D°p(\ - ß ' [sal t ]) , and η8 = η0(\ + ß"[sal t ] ) . The values of B' obtained from the QELS studies were 0.13 M 1 and — 0.06 M " 1 for LiCl and CsCl, respectively. These values compared favor-ably with B" = 0.143 M " 1 and - 0 . 0 5 2 M " 1 for LiCl and CsCl, respectively, (Robinson and Stokes, 1970; Endom et al., 1967). They concluded that aggre-gation and changes in Rs do not influence the behavior of Z) a pp as a function of [salt] , but rather the observed dependence can be accounted for by the effect of salt on the structure of the solvent.

8.10. Analytical Expression for S(K AR) for the Hard Sphere Potential in the Perçus- Yevick Approximation

It is of value to have an analytical expression for S{K AR) without resorting to numerical methods of analysis. To this end we use the direct correlation function in the Perçus-Yevick approximation for hard spheres (Ashcroft and Lekner, 1966; van Helden and Vrij, 1980): c(r) = 0 for r > 2RS; c(r) = Αγ + A2(r/2RS) + A3(r/2Rsf for 0 < r < 2RS, where A, = - ( 1 + 2φρ)

2(\ - φρ)~\

A2 = 6φρίΙ + (0 P/2 ) ] 2(1 - φρ)~\ and A3 = -(φρ/2)(\ + 2φ ρ)2(1 - φρ)'\

Expansion to the two lowest-order terms yields the desired result,

(1 + 2φ}

Γ 1 + (\+2φΡΓ

2(^8)2Λ (8.10.1)

8.11. Beyond the Classical DLVO Potential

The classical D L V O potential is composed of a van der Waals attraction term and a Coulombic repulsion term for two particles moving in a continuum medium. We discuss in this section some features of interparticle interactions that are not included in the D L V O potential.

ΦΡ-Dependent Repulsive Coulomb Potential

Beresford-Smith et al. (1985) treated the colloid particles and small ions on the same footing in the Ornste in-Zernike (1914) equation which interact through an unscreened Coulomb potential, Uu(r)/kT = —Ζ{Ζρ

2/rekT (cf. Appen-

dix C), and ionic strength is a function of the colloid concentration, viz, 1 / / 2 H = ( i / ; w2 ) + ( ! / ; 2 ) + ( j / ^ ( c f S e c t i on 7 3) T h e se a u th o r s obtained

280 8. C O L L O I D S

for the effective interaction potential between two colloid particles

Μ ± = ( Ζ ρ β)2 ( 1 + φ ρ ) 2

exp(fr) exp

rs 1 +

(8.11.1)

where b = 2αρ/λΌΗ. As pointed out by Beresford-Smith et al., Upp(r) given by Eq. (8.11.1) is such that a decrease in φρ actually promotes solution structure. This prediction is contrary to the classical D L V O model.

Geometry- and Frequency-Dependent van der Waals Interaction

Hamaker (1937) assumed a pairwise addition of the oscillator interactions in the derivation of Eq. (8.7.1). The geometric and frequency dependence of the van der Waals interaction could be factored, which is not generally valid.

Because of the finite time of motion of electrons in their orbits, all material bodies have "instantaneous" fluctuations in the electron distribution, which gives rise to transient electric and magnetic fields. Because of the finite time for the fluctuating signal to reach other oscillators in the material, there is a loss of correlation between the oscillators that is referred to as "retardation damping" and characterized by a damping constant y.

Parsegian (1973) considered retardation effects for the case of two inter-acting hydrocarbon planes (h) separated by a water (w) gap of thickness L, with the result

AH(L,T) 3/cT

+ Σ A

w h ( l + r „ )exp( - r„ ) (8.11.2)

where rm = 2Le^ / 2/ccüm and Δ0· = (ε ; — ε7·)/(ε( + ε,·) with frequency-dependent dielectric susceptibilities (Parsegian and Ninham, 1969; Ninham and Par-segian, 1970; Parsegian and Ninham, 1971; Parsegian, 1973):

E(<U)= ι +Σ- + Σ- ω

ω,

(8.11.3)

- iœyj

Mitchell and Ninham (1972) used the Lifshitz theory (1956) to calculate van der Waals interactions for spheres of radii ai and a2 with the neglect of retardation effects. For the planar approximation in the limit of close approach,

^ m = 0 q= 1 ([ (8.11.4)

8.11. Beyond the Classical D L V O Potential

In the asymptotic limits 1 « ap/H0 and 1 » ap/RU2, three different frequency ranges were identified: (1) the microwave frequency, m = 0; (2) the infrared frequency, 1 < m < 15; and (3) the ultraviolet frequency, m > 15. The relative contributions of these ranges are proportional to the absorption character-istics of the system.

Interaction between Two Unequal Spheres

Hogg, Healy, and Fuerstenau (1966) (HHF) employed methods for the problem of unequal spheres similar to those of Derjaguin (1934, 1939) for treating large identical spheres as discussed in Section 8.6. The interaction energy obtained by these authors was of the form

U(H0

_ ζπαχα2

+ (Φϊ

(φϊ + φ°2)2\η 1 + exp

DH

</>°)2ln exp (8.11.5)

As pointed out by Barouch and Matijevic (1985), U{H0)HHF

overestimates the interaction energy. Reexamination of the double layer interaction between charged planar and spherical surfaces led to the result that under certain circumstances, the two surfaces may attract each other at very short distances but repel each other at larger distances. Consider the simpler case of two planar surfaces separated by a distance L and with surface potentials φ°γ and φ2 - The solution to the linear Poisson-Bol tzmann equation is

φ(χ)=-

+ φΐ sinh LDH

sinhf - —

(8.11.6)

The induced surface charge densities p(y) = [άφ(χ)Ιάχ]χ=γ for these sur-faces are

— (/>?cosh

P(0) LDH.

and

p(L)=-

sinh

- 0 ? c o s h ( — ) + φ°2

sinh

(8.11.7)

(8.11.8)

282 8. C O L L O I D S

The product p(0)p(L) indicates the strength of interaction,

φ\φ« l + c o s h2( - ^ ) -(</>?

2 + <^

2)

(8.11.9)

If φ°{ > 0, φ°2 > 0, and φ°2 > φϊ, then p(0)p(L) > 0 for L/ÀOH « 1 and p(0)p(L) < 0 for L/ÀDH » 1, where the crossover from repulsive to attractive interactions occurs at cosh(L c/ / l D H) = $ 2 / 0 ? > 1-Extending these concepts to curved surfaces, the portions of the two surfaces that are closer than a critical distance of separation Rc are attractive, whereas those portions separated at R > Rc are repulsive.

Effect of Finite Size of the Solvent

Israelachvili (1985) reported that the force between two crossed mica cylinders in a fluid, with or without an electrolyte, exhibited oscillations at separation distances of a few angstroms, i.e., a "hydration force". The hard sphere nature of the solvent and the alignment of solvent dipoles could explain the oscillatory behavior (Henderson and Lozada-Cassou , 1986; Henderson, 1988). The van der Waals attraction, solvent dipole orientation, and the solvent hard sphere repulsion forces dominate at small separation distances. As the two surfaces are brought closer together, the solvent particles become "more packed" and the force curve increases. Upon attaining a closest packed state, some solvent particles must be expelled in order to force the plates closer together, and the force curve again drops.

Fluctuations in the Net Charge

Kirk wood and Shumaker (1952a) examined the effect of charge fluctuations on the solution polarization Ρ = (4π /3)π ρ[α ρ + (<μ ρ>/3/(Τ)] and <μ ρ> = < μ Ρ >

2 + Αμ

2, where Δ μ

2 = <(μ ρ — <μ ρ>)

2> is the mean-square dipole

moment and a p is the induced polarizability. They showed that the dielectric constant increment measured for ovalbumin, horse hemoglobin, and human serum albumin could be interpreted in terms of fluctuations in the dipole moment without the need to postulate a permanent dipole moment. These authors also showed that fluctuations in the surface charge of the particle re-sults in an attractive interparticle potential (Kirkwood and Shumaker, 1952b):

kT

K S

(8.11.10)

2s2a

2a

2R

2kT 1 +

where ai2 = ax + a2 and Δ μ2 = (AZj)

2a

2.

8.11. Beyond the Classical D L V O Potential 283

Distribution of Small Ions of Finite Size

Medina-Noyola and McQuarrie (1980) solved the Ornste in-Zernike equa-tion for small ions of finite size using the MSA. These authors found that the hard sphere potential acts as an attractive term in the potential of mean force. The D L V O model was recovered in the limit a s - > 0 . Patey (1980) solved the Ornste in-Zernike equation for spherical particles and found a long-range attraction in the colloid—colloid potential profile. H N C A calcu-lations indicate an attractive interaction also occurs between parallel plates (Lozada-Cassou, 1984; Lozada-Cassou and D' iaz-Herrera , 1988). However, it was only when the calculations were extended to include the three-particle distribution function that the D L V O potential was recovered in the limit as -> 0.

Sogami (1988) proposed a model for the interaction between two negatively charged spheres in the presence of small ions of unit charge, which were exluded from the volume of the macroparticles and assumed to be in constant equilibrium with the polyions. Sogami invoked the weak gauge invariance property of the Helmholtz free energy by adding a constant to the reduced potential (/>r(r) = e<j)(r)/kT = </3u + φ » , where 0 U = ( « > u - « > « . ) / (<Xc>u + < O J - The adjusted reduced potential 4>

r'(r) was assumed to obey

the linearized Poisson-Bol tzmann equation, with the surface boundary con-dition [ n j · V 0 r ( r ) ] r = fl = — 4 π / Β Ζ ρ σ ° , where n 1 is the unit vector normal to the surface of the sphere. The reduced potential shift φη may be interpreted as being a measure of the asymmetry of cations and anions, under conditions of a uniform distribution arising from the release of counterions from the polyion. Because of the nonuniform distribution of small ions, Sogami dif-ferentiated between two "screening lengths": (1) a bulk screening length / s , given by 1/Λ| = 4nÀB((n'c)u + (n'a}u); and (2) the local screening length / s , given by \/λ2 = 4nÀB(Nc + Na)/V. The pair potential was determined from the difference in the Helmholtz free energy for the macroions in a given configuration relative to that at infinite separation, where the boundary problem of the linearized Poisson-Bol tzmann equation with a screened Coulomb potential was solved by Green's integral theorem. Sogami then obtained the following expression for the pair interaction potential:

(8.11.11)

where y = (Nc - ΝΛ)/(Κ + Nc),

s inh(f l p/ / l s)1 2

KM*) J (8.11.12)

284 8. C O L L O I D S

and ν = (y2/4)[sinh(ap/Às)/(ap/Às)~]

2. It is noted that the potential has a

minimum at

Km i» = £ - [ « + («2 + 4ut ; )

1 / 2] (8.11.13)

2v

Fluctuations in the Ion Cloud Distribution

Oosawa (1968) examined the interaction between two parallel rods, where an attractive interaction was shown to arise if the fluctuations in the small-ion distribution about one rod [<5ρ(>Ί)] was correlated with the fluctuations in the small-ion distribution about the other rod \ßp{y2)~\- ^

e interaction

hard sphere solvent layers

net charge fluctuations Θ

ΘΘ

θ

Θ

θ θ J Θ

Ό' · dipole fluctuations in the ion cloud distribution (temporal)

induced surface potential Q G θ

θ θ θ

=> θ θ — Θ Θ

Ω _

Θ

_ θ _ Q Q Θ Α , θ Q θ θ |

charge density cross correlations

asymmetric distribution of small ions (equilJibrium)

Β

D

F

Fig. 8.9. Schematic representation of correction terms to the classical DLVO model. (A) Finite solvent size gives rise to oscillations in the force-dis tance profile. (Β) Attractive interaction region induced on a finite region of the smaller of two spheres. (C) Net charge fluctuations on the surface of the charged particle that result in attractive interactions between two spheres. (D) Counter ion fluctuations along two rods give rise to attractive interactions if the fluctuations are correlated (becomes effective at distances less than 7 Â in water). E: Attractive d ipo le -d ipo le interactions caused by the temporal asymmetric fluctuations in the associated ion clouds about two charged particles. (F) A asymmetric equilibrium distribution of small ions between two charged polyions.

8.12. Diffusion in Structured Colloidal Suspensions 285

energy between the two points on the rod is

U(R)° = ( » ( h ) 2 ) ( ¥ h ) 2 )

s2R

2kT

(8.11.14)

Fulton (1978a, 1978b) examined long-range correlations due to fluctuating dipoles in his analysis of dielectric relaxation where the charge imbalance decays on the order of the Maxwell relaxation time τ Μ = ε 0/ 4 π σ δ , where σ 8 is the conductivity of the solution. Fulton (1978a) noted that for t > τ Μ , there were both long-time and short-time particle correlations, and that the short-time correlations were modified by the long-time correlations. The physical argument of this coupling was that at large distances there was a "backflow of charges" that affected the short range correlation. Geigenmuller et al. (1983) showed that the Onsager symmetry relationships lead naturally to the coupling of relaxation domain for systems exhibiting relaxation modes on two different time scales.

The D L V O correction terms are shown in Fig. 8.9.

8.12. Diffusion in Structured Colloidal Suspensions

Ackerson (1976, 1978) has provided a general framework for the N-particle Smoluchowski-Einstein equation for interacting Brownian parti-cles, Anp(r, t) = exp(0r)Anp(r,0), where Anp(r, t) is the excess concentration at

It is recognized that n'p(r, t) can be written in terms of Dirac delta func-tions δ(τ' — r), which in turn can be written in terms of volume integrals of exp (K

r- r ) . We associate An p(r ,0) with aK as defined by Ackerson (1978),

aK = ( l / A / p )1 / 2

^ e x p ( i KT · r,). The static and dynamic structure factors are,

respectively, ϋ^Κ,Ο) = < α _ κ α κ > and G^Kj) = <a_ Kexp(0i )%>. One may write dGx(K, t)/dt = <a_Kexp(Ôi)ÔaK>. Ackerson used the Mori projection operator scheme (1965) in defining the projection operator ρ for the intermediate scattering function, ρ A = (a _KA}aK/G ^Κ,Ο). One now writes ^GX(K, t)/dt as the sum of two parts,

r and 0 is

dG^K, t)

di = <a_ Kexp(0i )p0a K> + <a_ Kexp(0i)( l - ρ ) 0 α κ> . (8.12.2)

Using the intermediate scattering function and defining Kx = < α _ κ 0 α κ > / G x(K,0), one has <a_Kexp(Ôt)p(OaK)> = — Χ ^ ^ Κ , ί ) . To evaluate the second

286 8. C O L L O I D S

term, Ackerson used the identity

exp(Or) = exp[(l - p)0i] + i/iexp[0(r - τ)]ρ0βχρ[(1 - ρ)0τ] (8.12.3) ο

and defined the force correlation function as

H(Kj) = <F*(K,0)exp[(l - p)0i]F(K,0)>.

In the weak coupling limit F(K, 0) = ( 1 - ρ )0aK = iK T . L(K, 0) +

( / v x — D°pK2)aK, where L(K,0) is the additional direct interaction force,

L(K,0) = ( - N "1 / 2

) Σ X C V ^ K T ^ e x p O ' K ^ r , ) , (8.12.4)

and Τ is the hydrodynamic interaction tensor. Hence,

<a_Kexp(0r)(l - p)0aK} = ' , ff(K, t — T)GJ(K , τ)

0 * G I (K | O ) · , 8

·, 1 5)

The expression for the decay of G t(K, i) thus becomes

' , H(K, Γ - τ ^ Κ , τ )

* C I( K ! O ) ' ( 8

·1 1 6)

which shows that direct interparticle interactions affect K{ only through g

(1\r), whereas forces exerted on other particles enter as a "memory effect". The nth cumulant is obtained by using the operator 0 η times on

ô(r — r{N}) = S(r — r j ^ r — r2)ô(r — Γ 3 ) · · · ( 5 ( Γ — rN) and then performing the desired ensemble average. The time course of <ΔΗΡ(Γ ,0)Δ/Ί'Ρ(Γ , f)> is given by

P \ n

P " = <An p(r ,0)0"An p(r ,0)>, (8.12.8)

where An p = £ < HR — r,) — <rc p> u. Repeated integrations are carried out by

parts until the operator V no longer operates on the delta function, and the averages are obtained by assuming the joint probability of observing the y'th and ith particle in the volume as being < Π Ρ >

2#

( 2 )( Γ ) .

Example 8.5. Hydrodynamic Interactions of Organophilic Silica Particles in Nonpolar Solvents.

Vrij and coworkers have reported several studies on coated silica particles in nonpolar solvents (van Helden and Vrij, 1980; Vrij et al., 1983; Dhont et al., 1985; Philipse and Vrij, 1988). The coat around the core silica particle con-sists of short aliphatic chain molecules so that the particles are soluble in organic solvents.

8.12. Diffusion in Structured Colloidal Suspensions 287

Philipse and Vrij (1988) reported static and dynamic light scattering studies on silica particles coated with a thin layer of y-methyacryloxypropyltrimeth-oxysilane (TPM) in ethanol and e thanol- to luene solvents. For silica parti-cles in ethanol, RH = 83 ± 1 nm was obtained from D a p p, which agreed well with Rs = 79 + 1 nm obtained from J t i ls data.

The refractive index and dielectric constant of a 70% ethanol- to luene mixture was reported to be n0 = 1.4578 and ε 0 = 10, respectively, where np = 1.453 for λ0 = 546 nm. Multiple scattering effects were minimal since n0 ~ np. The coated silica particles were charged in the e thanol- to luene sol-vent, as determined by electrophoresis measurements. Specific conductivity measurements of the silica solutions resulted in the value λΌΗ ~ 50 nm.

The static light scattering measurements were obtained for λ0 = 546 nm or 436 nm, whereas the DLS measurements were performed at λ0 = 488 nm. At a concentration of 161 g/L, it was reported that the "color" of the solution changed with angle, going from faint yellow at θ = 0 to bluish at θ = 90. Since the contrast between the silica particles and the e thanol- to luene mixture varied with temperature, the temperature of 33°C was chosen from a study of 7 t i ls of the 161 g/L solution as a function of Τ at θ = 90°.

The solution structure factor S(K AR) was defined as S(K AR, c'p) ~ A i i s (

cp ) / A i i s ( l - 5 g/L), where the most dilute solution was 1.5 g/L. Oscilla-

tions in this ratio did not settle down to the value of 1 at the higher scatter-ing angles, but rather approached values in the range 1.5-2.5. This was attri-buted to differences in the contrast at the various concentrations, hence the curves were internally normalized to unity at the high scattering angles. It is noted that at the higher concentrations, Kmax(cp)

_1/3 was a constant,

where Kmax is the location of the peak maximum of S(K AR). The ratio Dm(K)/Dp was likewise determined from the low concentration results, where Dp ~ 4.5 ± 0.2 picoficks was the average value of Dm(K) for high-angle data of the 1.5 g/L sample. Dm(K)/Dp vs. Κ are shown in Fig. 8.10.

Philipse and Vrij (1988) used the RMS A method (cf. Appendix D) to fit S(K AR). The volume fractions φρ and the ratio 2RS/ÀOH were determined from independent measurements of the weight fraction, Z)p, and the conductivity, where the potential φ° was adjustable. The ratio Dp(K)S(K)/Dp was assumed to reflect solely the hydrodynamic interaction H(K). Using the same param-eters for the best fit of S(K\ Philipse and Vrij (1988) tried to fit the func-tion H(K) with the Beenakker -Mazur theory for hard spheres (1984, cf. Section 6.11) using RH = 80 nm and the volume fraction determined from the actual solution concentration. The results, shown as the dotted lines in Fig. 8.11, did not have good agreement with the experimental curves. On the other hand, a much better fit is obtained if effective hard sphere diameters and volume fractions obtained from the Percus-Yevick calculation of S(KAR) are used in the Beenakker -Mazur theory. The PY parameters are

0 1 2 3 4

K x l O "5 ( c m " ' )

Fig. 8.10. Ratio of Dv{K)jD® vs. Κ forTPM-coated silica particles methanol-toluene. cp(g/L):

Δ, 161; • , 126.6; O, 69.3. [Reproduced with permission from Philipse and Vrij (1988). J. Chem.

Phys. 88, 6459. Copyright 1988 by The American Physical Society.]

CO

A

i.o 0.5

1 I

1 I

1 I

1 I

1

12 3 4

KxlO (cm )

ooo ? C" = 14.8 g/L

Fig. 8.11. Structure factors and hydrodynamic interaction terms for TPM-coated silica particles in ethanol-toluene. The RMSA calculations of the structure factors (solid lines) were carried out with a particle diameter of 160 nm, a screening length of λΌΗ = 550 nm, and a con-tact potential of 60 kT. The hydrodynamic interaction parameter H(K) was determined as DmS(KAR)/Dp. The dashed lines are the Beenakker -Mazur model based on parameters used in the structure factor calculations. The solid lines are the B e e n a k k e r - M a z u r calculations using an effective volume fraction (to account for the effect of charge) obtained from the PYA calculations of the structure factor. The primary objective was to fit the peak maximum. [Reproduced with permission from Philipse and Vrij (1988). J. Chem. Phys. 88, 6459-6470. Copyright 1988 by The American Physical Society.]

8.13. Fractal Objects 289

Table 8.2

PY Parameters for T P M - C o a t e d Silica Beads

in E thano l -To luene"

(g/L) ΦΡ

2Rf

(A)

A K a ve

(Â)

161.0 10.1 42.0 265 280

126.6 7.9 40.5 270 304

69.3 4.3 25.5 360 371

14.8 0.9 - 4 . 5 < 4 2 0 475

a Taken from Philips and Vrij (1988). J. Chem. Phys.

88, 6459-6470.

compared in Table 8.2, where Rs = 80 nm for the RMSA calculations, and ARave is the average distance between neighboring spheres.

One conclusion drawn by Philipse and Vrij was that the charge on the particles enhanced the hydrodynamic interaction between particles, thereby leading to larger values for the effective volume fraction and particle radius. This conclusion is not in concert with the calculations of Stigter (1980), who showed that the fluid velocity is decreased by the presence of charge on the particles (cf. Section 9.3). The RMSA calculations in Fig. 8.11 at the higher volume fraction also fall below the experimental structure factor S(KAR). Recall that Belloni and Drifford (1985) showed that dynamic attenuation could account for the discrepancy between the experimental and theoretical values of DmS(KAR)/D°p reported by Neal et al. (1984) (cf. Example 7.2). Dynamic attenuation may likewise account for the present observations for which H(K) > 1.

8.13. Fractal Objects

Plato proposed in the "Timaeus" (ca. 380 B. C.) that there were two orders of reality, Being and Becoming. The reality of Being was perceived to be based on the truths of mathematics and logic, where things "exist unto themselves". The reality of Being was governed by three factors: Forms, Copy, and Receptacle. Forms can be thought of as the "blueprint" of an object, Copy as the physical existence of the object, and Receptacle as the space occupied by the object. Plato relates the basic elements of Forms to the geometric shapes of four regular solids, representing earth, air, fire, and water. The four regular solids were, in turn, constructed from two types of right triangles: (1) a right triangle with two equal sides; and (2) a right triangle with unequal sides.

290 8. C O L L O I D S

Borrowing on these ideas, let us construct an equilateral triangle from a 30°-60°-90° triangle referred to as the basic triangle. One now has the option of forming a larger equilateral triangle by proper alignment of either three or four of these basic triangles. This new triangle contains the "blueprint" of higher-order structures, and is therefore called the "Form" triangle. This process can be repeated to ever-increasing size of the equilateral triangle, as illustrated in Fig. 8.12.

Clearly, the view remains the same for these types of structures regardless of the level of magnification or development of the equilateral triangles. This property is called self similarity and is the basis of scaling laws.

In his book Chaos-Making a New Science, Gleick (1987) describes how Mandelbrot discovered some form of "scaling" law in regard to cotton prices, electronic transmission noise, and river floods. For example, in the case of electronic transmission noise Mandelbrot noted that the transmission

A

Β

Fig. 8.12. Plato's Forms and fractal objects. Plato's Forms are based on geometric shapes from which all objects are constructed. Shown is how the 3 0 - 6 0 - 9 0 triangle can be arranged to form a "primary particle", i.e., an equilateral triangle. The equilateral triangle at the branch point (indicated by the arrow) is composed of either three or four "primary particles". (A) The branch point composed of three equilateral triangles (center region is open) contains the blueprint for an open structure called the "Sierpinski gasket". (B) The branch point composed of four equilateral triangles (center region is an equilateral triangle) contains the blueprint for a solid structure.

8.15. Diffusion versus Activation Control in Bimolecular Solution Kinetics 291

of random errors appeared to be the same percentage of the signal regardless of whether the observation period was seconds, minutes, or hours. The prac-tical outcome of this observation was that it would be of no value to increase the power of the signal to eliminate random errors, but rather that the signals should be duplicated to detect random errors.

If the "scaling" concept associated with the occurrence of random noise could be applied to the shape of irregular aggregates, then the power law "d f" represented the fractal dimension of the object. [The term "fractal" was coined by Mandelbrot in 1975 from the Latin word fr anger e, meaning "to break" (Gleick, 1987).]

8.14. "How Long Is the Coast of Britain?"

The concept of a fractal dimension can be applied to a large number of ob-jects of study. These include: surface area (monolayer adsorption); pore dis-tribution (gels); reaction sites (aggregation); and mass distribution. It is

Fig. 8.13. Colloidal gold as a fractal object. [Reproduced with permission from Weitz et al.

(1985c). Surface Science. 158, 147-164. Copyright 1985 by Elsevier Science Publishers.]

292 8. C O L L O I D S

emphasized that df of irregular objects does not have to be an integer. A discussion of the characterization of surface irregularities is found in a paper by Pfeifer (1987). A system that has been extensively examined is colloidal gold, as shown in Fig. 8.13 at various levels of magnification.

In regard to light scattering measurements, the fractal dimension is related

8.15. Diffusion versus Activation Control in Bimolecular Solution Kinetics

The kinetics mechanism of aggregation is crudely partitioned into two regimes: (1) diffusion-limited, in which the particles "stick" on contact; and (2) activation-limited, in which the particles "stick" only after multiple collisions. The realization that colloidal systems may exhibit fractal dimen-sions has aided considerably in the understanding of the nature of the inter-particle interaction potential that governs the rate of the aggregation process.

To provide some insight as to the nature of diffusion- and activation-controlled kinetics, the bimolecular solution kinetics model of uniformly reactive spheres as developed by Schurr (1970) is reviewed. The reaction is viewed from the average view of particle A, which is achieved by superposition of all of the particles A onto the origin of the reference coordinate system. From this superposition position, particles Β move with a relative mobility D/kT = (DA + DB)/kT, where DA and DB are the Stokes-Einstein diffusion coefficients for particles A and B, respectively. The flux J(r) for any spherical surface a distance r from the origin is governed by both random diffusion \_dCB(r)/dr~\ and direct intersphere interactions [ ö ( 7 ( r ) / ö r ] . The reaction rate is equated with the net flux of particles φΜ through the surface at R T , viz, ΦΜ = ^ C A C B ( Ä t ) — / c 2C £ , where k1 and k2 are the intrinsic forward and reverse rate constants, respectively. Solving the differential equation with appropriate boundary conditions leads to the expressions

to D s e i f and < ^ >1 / 2

by

< K è >1 / 2

~ M p ^

(8.14.1)

(8.14.2)

k{ = —-——-—— (forward rate), 0 D * 1 + # c * D

(8.15.1)

Κ = — , P 2 C

, (reverse rate). 0 D * I + GckD

(8.15.2)

The long-range intersphere interaction parameter gD is

(8.15.3)

8.16. Equilibrium and Kinetics Distribution of Cluster Sizes 293

gc = exp[ l / (K T) / /cT] is the contact interaction parameter, and fcD = 4nRTD is the diffusion-limited rate constant. If the association reaction occurs instantaneously upon contact, then gDkx » gckD and fcf ~ kO/gO (diffusion-limited). The Smoluchowski limit is obtained for identical spheres of uniform surface reactivity with U(r) = 0, thus gO = 1 :

2kT 8fcT kf = 4n(2RA)- — = — = ί*. (8.15.4)

6πη0ΚΑ 3η0

If OC^D » £ / D ^ I > then fcf = kl/gc (activation-limited, or chemical-limited). In general, aggregation reactions may not be in either asymptotic regime. Con-sider the ratio

# Df c i Ιηο^ι F(Rj)

F(Rj) = exp

gckO 2kT Rß

U(Rj

(8.15.5)

kT e x P ™ . ( I I « )

where Rß = RARB/(RA + RB) is the "reduced radius". For fixed RA and RB, the dominance of activation- or diffusion-limiting mechanisms can be controlled through adjustment of either the long-range interaction potential or short-range surface potential. Depending upon the conditions, aggregate structures are referred to as "diffusion-limited aggregates" (DLA, or DL aggregates) or "activation-limited aggregates" (ALA, or AL aggregates).

8.16. Equilibrium and Kinetics Distribution of Cluster Sizes for Diffusion-Limited Aggregation

The Wit ten-Sander model (Witten and Sander, 1981, 1983; Meakin, 1983a, 1983b) was an early attempt to simulate aggregate growth. The aggregate size was propagated by addition of one primary particle at a time. [The monomer units for colloid aggregates are referred to as primary particles (Forrest and Witten, 1979; Weitz and Huang, 1984).] The model was extended to include aggregation of previously formed clusters and hence is referred to as the cluster-cluster model. The point of departure in examining the kinetics of irreversible aggregation in the papers of interest (Cohen and Benedek, 1982; Botet and Jullien, 1984; Weitz et al., 1985b, 1985c; Meakin et al., 1985a) is the Smoluchowski (1916a, 1916b) equation,

Α Γ 1 oo

1Γ = -, Σ Q Σ KkJCj. (8.16.1) at z i+j=k j= ι

where the coefficients KtJ and KkJ are the kinetic kernels, or rate constants, for the reaction. The first term represents the generation of clusters of size k from

294 8. C O L L O I D S

smaller clusters j and /, whereas the second term represents the disappearance of clusters of size k to form higher-order clusters of size k + j .

We follow closely the paper of Cohen and Benedek (1982) because they examined both the equilibrium and kinetics distribution of cluster sizes and have discussed in detail the sol-gel transition. The monomer types examined by these authors were ARB, A R B ^ _ l 5 and RAf , where R is a nonreactive central group and / ' is the functionality of the reacting group A or B (cf. Sec-tion 3.15). In the first two cases A reacts only with B, whereas A reacts with A in the last case. Our interest focuses on the limit of high functionality, which virtually corresponds to a uniformly reactive surface, as might be the case for colloids interacting through the D V L O potential.

The starting point of their investigation was the Flory-Stockmayer theory for multifunctional polymer condensation (Flory, 1941a, 1941b, 1941c, 1953; Stockmayer, 1943, 1944). The central feature of the Cohen-Benedek model is the calculation of the canonical partition function Z(K, T, {N s}, N0) for a system of N0 solvent molecules and a distribution of NM monomer units amongst the Ns clusters of size s subject to the constraint NM = £s iV s. The number of ways of combining the Nm distinguishable units is Ω({Ν5}) = (Λ

Γ

Μ)!Π[(ω 5)Ν ί ί

/(Λ^)!], where œs = WJs\ is the characteristic degenerary factor and Ws is the statistical weighting factor that depends upon the func-tionality f and the type of the monomer. The partition function for the sys-tem of solute and solvent particles for the dilute solution, Z(V, T, {JVS}, N0), was then used to calculate the chemical potential for solvent and solute particles, from which one obtains Xs = NJN0 = X\œsexp[ — (s — 1)ε//ίΤ], where ε is the binding energy of one monomer unit in the s-mer.

In order to calculate Xs, one needs to relate Χγ to the total concentration of monomer units XM:

00 00

XM = X sX, = Σ sX\a)sy-o-l>, (8.16.2)

s = 1 s = 1

where y = exp( — ε/kT). Clearly s - > o o is not a realistic estimate of the upper limit to the cluster size in a finite system. This limit can be used for sim-plification of the mathematics if the series converges well before unphysical cluster sizes are formed. The key to whether or not the series converges for large values of s lies in the form of the degeneracy œs, and hence the type of monomer unit. That is, X\œsy~

{s~

l) -> 0 for 5 » 1. The degeneracies for finite

5 are: œs = 1 (ARB); oos = ( / ' - s ) ! / [ ( / ' s - 2s + l ) !s! ] (ARB r _ J ; and ω, = (f'nf's - s)\/lf's - 2s + 2)!s!] ( A R B r ) . The asymptotic limit of all three systems can be written in general as ω χ > >1 ~ Aq

ss~\ where the functional

forms of the parameters A, q, and τ are identified in Table 8.3.

We now digress slightly to introduce the conditions necessary for the sol-gel transition. The summation in Eq. (8.16.2) converges for the ARB

8.16. Equilibrium and Kinetics Distribution of Cluster Sizes 295

Asymptotic Limit Expression for Dn = Aq"n

Monomer type A" τ

(Ύ RAf 5/2 RAf

(z - 1 )5 /2 5/2

A R B r _ ! z

z

3/2 A R B r _ ! ( z - 1 )

3'

2

( z - i r 1 3/2

ARB 1 1 0

fl Taken from Cohen and Benedek (1982). J. Phys. Chem.

86, 3696-3714. b ζ = f — 1; / ' is the functionality of the reacting group.

and A R B ^ - ! monomers but not for the R A / r monomer. One rewrites the product sXs for the R A / r system for s » 1 in the form sXs = Ay(Xl/X

c

1)ss~

3/2,

where X\ = y/q is a critical mole fraction of free monomeric units. It is now transparent that the summation in Eq. (8.16.2) will converge only as long as Xx < X\. Physically, this means that the sol state can only support a certain number of free monomer units. At free monomer concentrations above the critical concentration, a portion of the monomers will leave the sol state to form a coexisting gel state.

For identical and independent sites, the probabilty that any one func-tional group has reacted is p x = (1 — α )

/, where a is the a priori probability

that any given functional site has reacted. Hence, X{ = XUP\ - Since the sites any independent, any reaction between two functional groups is simply a dimerization reaction. One can then write α = 2ξ2/(ξι + 2ξ2\ where ξι + 2ξ2 = f'XM, and ξί and ξ2 are the mole fractions of the monomer and dimer reactive sites, respectively. From Eq. (8.16.2) one has ξ2 = £\j2y. Com-bining these results leads to a/(l — a )

2 = f'XM/y, from which it follows

X{ - XM{[2(1 + 4XMf'y~l)

l/2 - 2~]y/4XMf'}

f'. The high functionality

limits (f'»s,s» 1) are obtained using Stirling's approximation for the degeneracies. The resulting expressions are given in Table 8.4.

Cohen and Benedek (1982) considered the following kinetics kernels of the form (A, ß, and C are constants): (1) K i tj = A; (2) K u = B(i + j); and (3) K i j = Ci). The first expression obtains for monomers of the type ARB since there is only one reactive Β site on the polymer. It also obtains for diffusion-limited aggregation of spheres of equal size and uniform reac-tive surface. Hence Ki} = (ks/4)(Ri + Rj)(Ri

l + Rj

1) where fes is defined

Table 8.3

296 8. C O L L O I D S

Table 8.4

Equilibrium Distribution Functions in the

High-Functionali ty Limit0

Monomer type Distribution function^

RAf

exp(-2M(2MB"1

nn\

ARBf

exp(-nb)(nb)n~

l

Xn = (l b) (0 < < 1)

" Taken from Cohen and Benedek (1982). J. Phys. Chem.

86, 3696-3714. b h is the average number of bonds per monomer unit.

by Eq. (8.15.4). In the case of fractal objects, the relationship is Ktj ~ [2 + (/ /y)

( 1 / d /) + ( v / 0

( 1 / d / )] - There is no analytic solution if the clusters differ

so greatly in size that i/j » 1 or i/j « 1. However, if the units are of com-parable size, the kinetic kernels are relatively independent of the cluster sizes. If this is the case, then

Xs(t) = * M [ 1 - &( i ) ]26( i )

s _1 (ARB), (8.16.3)

where = (ί/τΑ)/|^1 Η - ( ί /τΑ)] and \/τΑ = ΑΧΜ/2. Kernels of the form B(i + j) may be applied to polymerization of units of type A R B y . j . Since an A unit reacts with a Β unit in order to form an aggregate, a y-mer has only one free A unit and j(f — 1) — j + 1 = j(f — 2) + 1 potential Β bonds. In the limit / ' » 1, an A unit on a y-mer can react with any one of the if Β units on an /-mer, and the A unit on the /-mer can react with any one of the jf Β units on the y'-mer. The rate is therefore enhanced (/ + y)-fold over the reaction of one site. The time evolution of Xs(t) is

X.(t) = X M[ 1 - K t f l ^ - ^ W " "1 ( A R B , . , ) , (8.16.4)

s!

where b(t) = 1 — exp( —ί/τΒ) and 1/τΒ = BXM. The third reaction case pertains to the reaction unit RAf . Here a y'-mer has

fj — j + 1 = j(f — 1) + 1 reactive sites, and an /-mer has i(f — 1) + 1 reactive sites. For / ' » 1, the number of ways of forming an aggregate between these two units is (f)

2ij times that of one site. Thus

Xs(t) = ^ ) e x p [ - 2 s f e ( i ) ] [ 2 s f e ( i ) ]s-

1 (RAf), (8.16.5)

8.17. Time Evolution of the Number of Clusters for Irreversible Aggregation 297

where b(t) = t/zc [fe(i)<i] and \ / T C = CXm/2, where b(t)<^ limits the aggre-gation to the sol region. It is concluded that the form of the kinetic kernel determines the form of the kinetic distribution of the aggregates. Note the similarities between Eqs. (8.16.3), (8.16.4), and (8.16.5) with the equilibrium distributions for high functionality given in Table 8.4. It is tempting to draw the conclusion that the equilibrium and kinetics distributions are equivalent. One is reminded, however, that the kinetics expressions do not include the reverse reaction in this study, whereas both forward and reverse must be (implicitly) included in the equilbrium distribution. As noted by Cohen and Benedek (1982), diffusion-limited processes for the RAf unit are not neces-sarily proportional to the product of the number of reactive sites on each aggregate.

8.17. Time Evolution of the Number of Clusters for Irreversible Aggregation

Botet and Jullien (1984) investigated the kinetics development for the most probable configuration of an equilibrium distribution of cluster sizes for systems sufficiently dilute that gelation does not occur. These results may be applied to D L aggregation. Van Dongen and Ernst (1985) examined the general solution to the Smoluchowski equation for all concentration con-ditions in which one of the associating aggregates is much larger than the other. These results may describe AL aggregation.

Botet and Jullien obtained the most probable configuration by maximizing the entropy function and assuming a Maxwel l -Bol tzmann distribution of discrete energies. Since these assumptions are also the basis of the C o h e n -Benedek approach for the equilibrium distribution, one may directly com-pare the results of these two models. Botet and Jullien introduced a fractal character by assuming a scaling relationship between the kinetics kernels, Κλ.λ. = Κ°αλ.λ. = Κ°λ

2ωα^, where K° is a constant. The kinetics expression

is rearranged to

i+j=k j=l

Kk/2,k/2Nk/2(Nk/2 — 1) 2X f c > fc 1) 2Kk,kNk(Nk - 1)

(8.17.1) 2

The last two terms contribute only when k is even. For a total number of monomer units NM, the time course for the total number of clusters Nc is

298 8. C O L L O I D S

The result in the limit 1 « Nc « NM was found to be Nc(t)/NM ~ αωΐ y

, where (χω = (ΑΚ

0/ν){\ - 2 ω )

1~

2 ω/ Γ ( 1 - 2 ω ) , Γ ( χ ) is the G a m m a function, and A is

a constant. These authors assumed general scaling in space and time. The kinetic exponent α is associated with the velocity (v) for a particle of j units, ν ~ j

a. The trajectory of movement has the dimensionality dO. The time

for moving a distance AR is

t ~ ra(AR)

dO ~ ( d

c) -

( d G a )( A K )

d D, (8.17.3)

where the characteristic radius of the aggregate is d° with fractal dimension dG. It follows that a change in the length scale by a factor b requires a change in the time scale, viz, t ~ [b

idD~

dca)]t'9 if Eq. (8.17.3) is to remain invariant.

The kinetic kernels of dimension volume/time are rescaled to Ki%j ~ [b^s-dv+^^K'i where ds is the dimensionality of the space. Hence the general scaling law,

Κλ,-,λ,· ~ Aa + I ( d s

"d D ) / d G l

/ C i f < /. (8.17.4)

Equating Λ = 2ω and noting the restriction ω < we have

a < a c = ι _ ^ ç i p _ ? ( 8 1 7 5)

where a c is the critical value of α for which ω = \ . It is noted that α is a negative quantity. One therefore has the identity for γ in the asymptotic limit of Eq. (8.17.2),

y = (1 - 2ω)~' = — ^— — > 0. (8.17.6) « G ( 1 - a) - (ds - dO)

a is related to the most probable configuration Nf ~ Fœ(x), where 0 < χ = NJ/NM< l , a n d

il - 2ωΫ~2ω

F J X) = Γ ( 1 - 2 ω )

x"

2"

e xP Î - (

1 -

2 ω)

χ] ' (

8·

1 7·

7)

which has a maximum at x m ax = — 2ω/(1 — 2ω) = — α/(1 — α). Botet and Jullien (1984) emphasize that the above results are good only in the sol state of the solution, where the concentration of monomers is such that gelation does not occur. Also, the particles must be sufficiently large that they do not move very far, i.e., the scaling law for the velocity (ν ~ j*) must be small. Under these conditions scaling laws hold for the time course of the aggregation process, and the power law is related to the kinetics and spatial dimensions of the system under examination.

Van Dongen and Ernst (1985) classified the solutions to the Smoluchowski equation for homogeneous kernels of the form Ktj ~ i

ßj

x, where i « j and the

8.18. Static Scattering Properties of Colloidal Aggregates 299

parameter λ is defined as λ = μ + v. As noted by these authors, the large particles cannot react at a rate faster than their size. Hence ν < 1 and λ < 2, with no restrictions on the exponent μ. Three classifications were defined by the sign of the parameter μ: μ > 0 (class I); μ = 0 (class II); and μ < 0 (class III). The significance of the parameter λ, which is equivalent to 2ω, is that it separates gelling (λ > 1) from non-gelling (λ < 1) processes. These authors considered two methods of solution for these three classifications. The general solution found from the recursion relation method [rrm, which is finding a solution of the form is correct for all homogeneous

kernels of class I systems (gelling and nongelling) and gives the exact solution as t - • oo. On the other hand, the rrm is not valid for class II and class III systems. The scaling functional method (sfm, which is finding a solution of the form Nk(t) ~ s~

2f(k/s), where s is the mean cluster size] is valid for non-

gelling class I systems. It is convenient to summarize their findings for the case when ν = 1, hence λ = μ + 1 and the kinetic kernel is of the form Ktj ~ For Λ.< 1, the solution Nk(t) goes to zero for small and large values of k with a maximum, Nmax(t) ~ i

1 / ( 1 _ A ), for λ < 1. For λ = 1, the

solution Nk(t) varies with k as a power law, Nk(t) ~ fc~3/2. The cutoff value

of k in the power law distribution is an exponential function of time, i.e., fcmaxW ~

e xP (

i/

T\ i ) for A = 1, where τ Μ is a sample-dependent relaxation time.

For λ > 1, the solution Nk(t) also varies with k as a power law, Nk(t) ~ k~\ where τ = (3 + λ)/2 > 2. Gelation occurs at the time tg and the cutoff value of k diverges according to the expression fcmax(i) ~ [1 — ( ί / ί 8 ) ] ~

2 / ( λ~

υ for λ > 1.

8.18. Static Scattering Properties of Colloidal Aggregates

For colloidal aggregates, there are at least two levels of structure that can be probed by light, neutron, and X-ray scattering techniques: (1) the distribution of the primary particles; and (2) the substructure of the primary particle.

Schaefer et al. (1984) reported that for colloidal silica aggregates / t i ls ~ K~

dG for K(Rl)

l/2 > 1 and was independent of Κ for K(Rç\y

/2 « 1. To

include both regimes, Lindsay et al. (1987) proposed that

™=

r i | ϊ^γτ- (8.18.1)

3d{

where <M> is the average weight of the aggregate. The probe length of visible light is too long to probe the local dimensions of

the primary particle. This regime falls in the realm of the neutrons and X-rays. At this length scale, the probe can distinguish the sharp boundaries between the primary particle and the solvent. Porod (1951, 1982) (see also Guinier and Fournet, 1955) examined the scattering from a particle composed of a random

300 8. C O L L O I D S

distribution of mass and empty space at a constant mass density. Using the present nomenclature, the average density of the aggregate is therefore given as (ρ) = ρθ, where ρ is the mass density of the primary particle and θ is the fraction of the space of the aggregate that is occupied by the primary particles. In this analysis the mass correlation function y(r) introduced by Debye and Bueche (1949) was

y(r) = (p{r)p(r')} = p2y0(r\ (8.18.2)

where y0(r) was originally called the characteristic function of the macro-particle (Porod, 1951). The dimensionless function y0(r) indicates the proba-bility of finding mass at a distance r from a point that is known to have mass. Clearly, y0(r oo) = 0 and y0(r 0) = 1. Hence,

P(K) = P0(K)p2

>G0 dr

y(r) sin(Kr)47ur2 — , ο Kr

(8.18.3)

where P0(K) is the structure factor for the primary particle. In its simplest form, the characteristic function is expanded as y0(r) ~ 1 — ar + br

2 + · · ·. If the aggregate is modeled as a sphere with holes, then (Porod, 1982) y0(r) = 1 — (3r/2d c) + i ( r / d c ) 3 , where d

c is the characteristic distance for which y(r > d

c) = 0. Substitution into Eq. (8.18.3) leads, for the tail of the intensity

function, to the lowest-order correction,

3 r

2 dr 1

Jo 2dc

sin(Xr) _ 1

Kr ~ Κ3 χ sin(x) dx ax

Κ

where χ = Kr. Integration by parts gives for the major term

P(K) ~ P 0 ( K ) W pp Ä ,

, (8.18.5)

(8.18.6)

where the macroparticle is composed of np primary particles of surface area sp. This result indicates that for K ( R Q }

1 /2 » 1, the scattered intensity is due only to the structural details of the primary particle. That is, interference between the primary particles is neglible. It is noted that the tail of the spectral density profile scales as Κ

4 and hence is referred to as the Porod tail. The Porod tail always appears for any scattering geometry where there is a sharp bound-ary between two scattering regions.

Sinha et al. (1984) have proposed the following expression for P(K) for colloidal aggregates:

9 M 2

P(K) ~ ^Τ^ΓΟ + QrA/2lsm(q) - ^ cos (^ ) ] 2s in [ / l a r c t an (ß ) ] , (8.18.7)

8.18. Static Scattering Properties of Colloidal Aggregates 301

Κ

Fig. 8.14. Schematic representation of P(K) vs. Κ for fractal objects. There are four regimes

for P{K) that can be identified for fractal objects: (1) the center-of-mass (c.o.m.) diffusion region,

where P{K) ~ 1 and is independent of Κ since KR « 1; (2) the Guinier region, in which internal

interference between the primary particles starts to become important and P(K) ~ 1 — ^{Rl)K2;

(3) the fractal regime, in which P(K) ~ K~df; and (4) the Porod tail, in which P{K) decays as K~

A

due to the sharp interface between the primary particles and the solvent, and hence there is no

internal interference between the primary particles.

where q = Kap, Q = KR, A = d{ — 1, ap is the radius of the primary particle, and R is the size of the cluster. The scaling laws for P(K) are: K° (center-of-mass diffusion region, K(Rl}

1/2 « 1); K~

dr [fractal regime, K(Rl)

l/2 » 1

(Martin and Ackerson, 1985)]; and K ~4 (Porod tail).

The various regions of P(K) are shown in Fig. 8.14.

Example 8.6. P(K) for Silica Aggregates. Schaefer et al. (1984) reported light and X-ray scattering studies on aggregates of colloidal silica particles. Aggregation of the primary particles, of radius ap = 27'Â, was achieved at a pH of 5.5 and salt concentrations greater than 0.5 M. A power law was not obtained until more than 37 hours had elapsed.

The intensity vs. Κ profile obtained from light and X-ray scattering mea-surements are reproduced in Fig. 1.2. The fractal nature of these aggregates is confirmed from the long-wavelength data (slope = —2.1) and the Porod tail for the short-wavelength data (slope = —4). The break in the slope occurs at \/K ~ 27 Â, the radius of the primary particle, which shows that the integrity of the primary particle is retained upon cluster formation.

302 8. C O L L O I D S

8.19. Dynamic Light Scattering by Colloidal Aggregates

Long-range interaggregate interactions are neglible under conditions neces-sary to cause DL aggregation of colloidal particles. / t i ls may therefore be con-sidered to be a function of the particle size and the polydispersity of the sample, and Kx for colloidal aggregates in general is represented as (Lindsay et al., 1987)

κγ(Κ) = 1

/ t i l s (K) M

2N(M)P{K, M)ID(M)K

2 + Γ ] dM, (8.19.1)

where Γ denotes the decay rate for rotation or internal motion. The kinetics of slow aggregation can in principle be followed by monitoring K^K) as a function of time. One must exercise caution, however, in the interpretation of K^K) for data taken at a fixed scattering angle. At early times one has the inequality K { R Q }

1 /2 « 1, whereas at the later stages of aggregation one

attains the inequality K ( R Q )1 /2

» 1. Interparticle interactions may be mani-fested in Dm at the lower scattering angles, especially under ALA conditions.

Κ

Fig. 8.15. Hypothetical profile of Kx(K)jK2 vs. Κ during aggregate growth. Shown to the right

of each profile is a schematic representation of the solution conditions at a fixed wave vector (probe length shown by double-pointed arrow) at different time intervals such that tx < r 2 < r 3. Top : The initial stages of aggregation, in which K(RQ}

1/2 « 1 and Κ AR < 1, where AR is the

average center-to-center spacing. Under these conditions, the repulsive interparticle interactions are manifested in the mutual diffusion coefficient at the low scattering angles. As the angle increases, the pairwise interaction component to the mutual diffusion coefficient decreases in magnitude so that the self diffusion coefficient is measured at the higher scattering angles. Middle: As the particles aggregate, the profile decreases overall. The average distance between particles increases, however, such that Κ AR > 1 and the repulsive interactions between particles are no longer evident in the profile. Orientat ion relaxation becomes important , however, at the larger scattering angles. Bottom: The aggregates have now grown to a size that greatly exceeds the dimensions of a particle. The profile is anticipated to reach a plateau region as in the case of rods and random coils.

303

As the particle size increases so that K ( R Q }1 /2

~ 1 at the fixed scattering angle, internal modes start to come into play, as well as size polydispersity. The effect of internal modes is to increase the value of Dapp(K). Recall, for example, the T-even phage system reported by Wilson and Bloomfield (1979b, cf. Example 5.3). As the aggregate grows to the extent that X < R G >

1 / 2 » 1,

one would expect Dapp(K) to become mass-independent as the internal motions of the aggregate are probed. It is also possible that the center-of-mass diffusion coefficient may cross over from mutual to self diffusion in the time course of the aggregation, since AR increases as the particles grow in size. The time dependence of Dapp(K) = K^fy/K

2 vs. Κ is schematically repre-

sented in Fig. 8.15.

Example 8.7. Aggregation Kinetics of Colloidal Gold Weitz and coworkers reported aggregation studies on colloidal gold, with a parimary particle of radius ap = 7.5 nm (Weitz and Oliveria, 1984; Weitz et al., 1984, 1985a, 1985b, 1985c). The reaction conditions were controlled by the addition of pyridine, which displaces the negatively charged citrate ions on the colloid particles, resulting in the repulsive electrostatic interaction potential between the aggregating particles. Transmission electron microscopy (TEM) studies were used to determine df. For example, ln(M) vs. ln(R) plots for AL aggregates of colloidal gold, reported by Weitz et al, (1985b), are shown in Fig. 8.16 where df = 2.02.

QELS studies on colloidal gold were reported by Weitz and coworkers (Weitz, Huang, Lin, and Song, 1984, and Weitz, et al. 1985b, 1985c, 1987) using the 632.8 nm line of a He-Ne laser. A plot of Kx vs. K

2 for data taken at the

same elapsed time was deemed linear with a zero intercept within the error

7H

3H d f = 2 . 0 2

~ Ί —

9

L N ( R ] ( A )

Fig. 8.16. Logarithmic plot of the mass of the cluster vs. the size of the cluster for TEM measurements of the AL aggregates. [Reproduced with permission from Weitz et al. (1985b). In Scaling Phenomena in Disordered Systems (R. Pynn and A. Skjeltorp, eds.). Plenum Publishing Co., New York. Copyright 1985 by Plenum Publishing Co.]

304 8. C O L L O I D S

limits of the experiment. Hence K1 was of the assumed form

—y ~ [ M( d f

"1 ) / d f

] N ( M ) J M . (8.19.2)

Representative plots of R vs. t under D L and AL aggregation conditions are shown in Fig. 8.17. The power-law under assumed DL conditions, viz.

gave the fractal dimension d{ ~ 1.75 (insert in Fig. 8.17). The DL aggregation data was interpreted in the following manner: As the

aggregates grow, AR increases whereas Kl/K2 decreases. The particles must

therefore diffuse over a greater distance at a slower diffusion rate, hence the overall reaction rate must decrease. To account for the AL results, these authors suggested that the mechanism changes from AL to DL during the time course of the reaction. They did not explain why this crossover occurs.

Wilcoxon et al., (1989), pointed out that the He-Ne line ( / 0 = 632.8 nm) is close to the optical resonance peak at 680 nm for colloidal gold. The strong interaction of light with the metallic sol may therefore contribute signifi-cantly to the experimental observations. The authors examined the / 0 and 0 dependence of light scattered by colloidal gold where aggregation was in-duced by changes in pH, addition of salt, and neutralization by pyridine. The fractal dimension obtained from / t i] S(C M) was reported to be df ~ 1.7 for / 0 = 632.8 nm and d{ ~ 2.0 for / 0 = 457 nm. The increase in d{ as one moves away from optical resonance was attributed to multiple scattering effects. The relaxation times obtained from polarized ( τ ν ν, translation) and depolar-ized ( T V H , orientation) light scattering were reported to obey the respective relationships, τ ν ν ~~ (CMt)

x and i v h ~ (C Mf )

3* where χ = 0.38 ± 0.04. The com-

plementary nature of the power-laws indicated that the aggregates are nearly spherical and internal motions do not contribute to τ ν ν. Since the expected value for χ based on computer simulations of Brownian kinetics is JC = 1/1.78 = 0.56 is significantly larger than the observed value, Wilcoxon et al. concluded that the "fast" aggregation kinetics process was not diffusion-limited.

Attention is now focused on the AL results in Fig. 8.17. The crossover from AL to DL kinetics is possible in accordance with Equation (8.15.7) if F(RT) increases at a faster rate than does i.e. the interparticle interactions become less repulsive as the particle size increases. On the other hand, it could be that there is no crossover from AL to DL kinetics at all and the observed behavior may simply be a consequence of the kinetic kernel for the reaction. Ball et al. (1987) proposed a model of AL aggregation based on the results of van Dongen and Ernst ( 1985) (cf. Section 8.17). The smaller particle "samples" the accessible nooks and crannies of the surface of the larger particle by Brownian motion. If the smaller particle comes within a distance w of the surface of the larger particle, a reaction may occur. The "reaction volume" for spherical

(8.19.3)

8.19. Dynamic Light Scattering by Colloidal Aggregates 305

(Λ

" T 1 1

- 6 - 4 -2 l n ( t ) (hrs)

Fig. 8.17. Aggregate radius of colloidal gold as a function of time. (A) The apparent

hydrodynamic radius for aggregates of colloidal gold was estimated from the first cumulant as a

function of time under diffusion-limited and activation-limited conditions. (B) Double logar-

ithmic plot showing a power-law dependence of the later stages of diffusion-limited aggregation.

[Reproduced with permission from Weitz et al. (1985c). Surface Science. 158, 147-164. Copyright

1985 by Elsevier Science Publishers.]

306 8. C O L L O I D S

particles thus can be written as 4 π ( # Α + # B ) 2 w = 4 π ( Μ Α

/ 3 + M B

/ 3) 2 w for ds = 3. The fraction of space accessible for reaction is therefore φ0 = VJV, where V is the total available solution volume. Hence Ktj = Κ°φον — ( M A

/ 3 + M B

/ 3) 2 . If M A = Μ β = M, then Ki%j ~ Μλ where λ = (ds - \)/ds.

For fractal objects the inequalities are wR{ds~

l) < Vc < R

d% or (ds - \)/df <

λ < djd(. Given that AfB » M A , the kinetics kernel is Kid ~ Μ ΑΜ (

Β

λ~ u . The AL kinetics shown in Fig. 8.17 therefore may be interpreted in terms of the results of Ball et al. (1987) in which the maximum aggregate size increases exponentially with time, kmax ~ εχρ(ί /τ Μ) . This places restrictions on the power-law for the size distribution to be τ = 1.5 and on the value of / = 1. It was noted by Ball et al. that the case with / = 1 is unique in that the system is stabilized by adjustment of the fractal dimension d{. An increase in λ from unity causes an increase in the number of smaller particles, which in turn can penetrate deeper into the larger fractal objects, thereby increasing d{ and reducing the value of / back to unity (a "feedback" mechanism). The Ball et al. (1987) model for the AL aggregation process is shown in Fig. 8.18. The spherical units represent a "blob" of mass M A = / M M, where M M is the mass of a primary particle. A "blob" in this context is not the same as was em-ployed in Section 5.15 for semidilute solutions of flexible polymers. According to Hentschel (1984a, 1984b), a"colloidal blob" represents a "limit growth" of primary particles whose size is determined by a "screening length" that is the distance a particle can penetrate a growing structure. Once the limiting size has been attained, the "blobs" themselves aggregate to form larger aggregates. The larger aggregate in this illustration therefore is composed of M B / M A = j "blob" units. The model allows the /-mer to penetrate the overall (fractal) structure of the j blobs but does not allow interpénétration of the blob units.

Recent studies on DLA and ALA of gold, silica, and poly(styrene) indicate that orientational modes may make a significant contribution to the first cumulant (Lindsay et a l , 1988a, 1988b). The Z) a p p(K)/D a p p(0) vs. KR plots appeared to have the same general profile as those predicted for long thin rods (Wilcoxon and Schurr, 1983a; Hallet et al., 1985) and Z i m m - R o u s e chains (Lin and Schurr, 1978) (cf. Sections 5.9 and 5.10).

Thus, from the discussions in Example 8.7, one concludes that the aggregation kinetics of colloidal particles can be controlled by altering the surface charge of the particles. It may not be possible, however, to identify uniquely the DL or AL kinetics solely on the basis of the power laws obtained from computer simulations or specific reaction kernals. This is because the required asymptotic limits may not have been reached, hence the kinetics may have both activation and diffusion character. As indicated in Section 8.15, the solvent viscosity dependence of the rate constant should be determined to ascertain the degree of diffusion and activation control of the aggregation processes.

8.20. Orientat ion Constraints on DLA Kinetics 307

Fig. 8.18. Model for activation-limited aggregation of colloidal system. In the model for ALA

of Ball et al. (1987), the reaction occurs between two aggregates of different size. The "blob",

representing the smaller reactant, samples several regions of the reactant surface of the larger

particle, indicated above by the dotted envelope. The reaction volume about each large aggregate

is schematically represented by the spherical particle in the upper right. The reaction volume is

given as 4n{RA + RB)2w-

8.20. Orientation Constraints on DLA Kinetics

Because the aggregate particle is quite open, RH/RG ~ 0.8 (comment made by Pusey on the paper of Lindsay et al, 1987). If the reaction occurs at the outer regions of the two aggregates of equal size, then RT > 2RH. Since the material is not uniformly distributed in the aggregates but rather in "clumps" on the surface, it may be assumed that the equivalent hydrodynamic sphere has several "holes" (recall Problem 3.1). It is possible that two aggregates may approach each other with an orientation such that a reactive surface of one particle is aligned with an open region of the other particle. Rotation of the smaller particles is sufficiently rapid relative to the translational motion that the proper orientation can occur before the two particles diffuse apart, and

308 8. C O L L O I D S

Fig. 8.19. Orientation effects on the diffusion-limited rate of aggregation of fractal objects. The

dashed circles above indicate the effective collision radius of the fractal objects. In both cases, the

orientation of the fractal objects do not permit association of the particles. The linear and curved

double-arrow lines indicate the relative mean-square translational and orientational displace-

ments per unit time. (A) The orientational displacement is of similar magni tude as the transla-

tional displacement for fractal objects of smaller dimensions. (B) In contrast , the orientational

displacement for fractal objects of smaller dimensions is considerably less than the translational

displacement of the larger fractal object. This means that orientational relaxation may assist in

the rate of aggregation for the smaller particles but not the larger particles.

the reaction proceeds. The situation is different for two large particles with unfavorable alignment. The orientation relaxation of larger particles is much slower than the translational diffusion, hence the two large particles would diffuse apart without reacting. This hypothesis is based on the bimolecular solution studies of Schmitz and Schurr (1972), who examined the effect of rotational diffusion on the reaction of a mobile sphere with a localized reaction area with a fixed reaction site on a plane. They found that the reaction rate for the rotationally mobile spheres was twice that for the fixed-orientation ensemble. To adapt this result to the reaction of colloidal aggregates, assume that the aggregates are encompassed by a hypothetical sphere that defines the "contact" radius of the aggregates. For convenience, the contact distance is

8.21. Compute r Simulation of Colloidal Aggregation 309

associated with the hydrodynamic radius, i.e., 2RH. Two aggregates are now brought into "contact" in such a way that a portion of the surface of one sphere lies directly over a "hole" in the second sphere. In order for the aggregation reaction to occur while in this initial configuration, one sphere must rotate an angle δθ relative to the other sphere. The question now arises as to whether the required angular displacement occurs during the time period it takes for the two aggregates to diffuse away from each other. It is noted that the distance that the two aggregates must move away from each other to prevent contact is independent of the aggregate size. This translational distance is denoted as δΗ0, where H0 = r — 2RH (cf. Fig. 8.6). The transla-tional and rotational diffusion coefficients for the sphere are Dt = kT/6^0RH

and De = kTfôn^Ru, respectively. The ratio of the mean-square displace-ments in the center of mass (6D ti/2) and the angle [2π(2ϋ θί ) /2] is simply

<OHl)_ 3D, _ 3 / ? â i g 2 0n

Μ " 2 4 " ^ · ( · 0 )

Since δΗ0 is independent of the size of the spheres, Eq. (8.20.1) indicates that {o0

2}R

2

i = constant. The larger spheres therefore undergo smaller angular displacement per unit translational displacement than do the smaller spheres. The "elbow" in the kinetics profile for the DL aggregation in Fig. 8.17 may therefore reflect an increase in reaction rate of the smaller-sized aggregates due to rotation of the aggregates to the proper alignment, as shown in Fig. 8.19.

8.21. Computer Simulation of Colloidal Aggregation

Computer simulations of colloidal aggregation kinetics have been carried out in two-dimensional (Kolb et al., 1983; Meakin, 1983c; Vicsek and Family, 1984; Meakin et al., 1985b), three-dimensional (Meakin et al., 1985a, 1985b; Meakin, 1988), and multidimensional (Meakin, 1983b) space. In general, the calculations were performed on a specified lattice structure (dimensions and type) with periodic boundary conditions. Since the minimum step size is equal to the distance between adjacent lattice points, the choice of a time interval and particle mobilities is crucial to the generation of DL structures. One must also keep in mind that in the D L process the step represents the net move-ment of a random motion process.

Vicsek and Family (1984) used a two-dimensional square lattice. It was assumed that the particle mobilities were independent of their sizes and that the time interval scaled with the number of primary particles s in the aggregate—i.e., Δί = s/NM, where NM is the total number of primary units in the system. Clusters were picked at random from the set of aggregates, and clusters that touched in the diffusion simulation were assumed to stick to form a larger cluster. The cluster, or clusters if no aggregation occurred, was then

310 8. C O L L O I D S

Table 8.5

Computer Simulation of Aggregation

Space dimension

d. Fractal dimension

df

2

3

4

5

6

1.70a, 1.38", 1.67

c, 1.80

d

2.50°, 1.84e

3.32*

4.20°

5.3α

a Meakin (1987). [Average values in Table 2.]

" K o l b et al. (1983). c Witten and Sander (1981). (Single particle

growth.) d Stanley (1977). (Percolation.)

e Meakin et al. (1985a). (Hydrodynamic inter-

actions included.)

returned to the stock list of particles. These authors found for this model the following scaling law for the number of clusters with 5 primary particles, Ns(t):

where f(s/tz) is a cutoff function with the limits f(x)~ 1 for χ « 1 and

f(x) « 1 for χ » 1, w = (2 — τ)ζ, τ < 2, and ζ is the power law for the mean cluster size in the limit t oo. These results indicate that the number of clus-ters of size s was scaled in both time and size. The results of D L aggregation simulations are summarized in Table 8.5.

The simulation studies of Meakin et al. (1985a) for DL aggregation and Meakin (1988) for AL aggregation included hydrodynamic interaction be-tween primary particles that was approximated by the Ro tne -Prage r (1969)/ Yamakawa (1970) hydrodynamic interaction tensor (cf. Section 3.7). In the study of Meakin et al. (1985a), pairs of clusters were randomly selected, rotated to randomly determined orientations, and then allowed to aggregate by means of a random walk through a three-dimensional lattice (six adjacent lattice points). [Since the orientation of the clusters did not change in the diffusion simulation, these calculations should be valid only in the limit of large particles.] The clusters in this study contained from 50 to 350 primary particles. The time interval was chosen at random from a uniformly distri-buted set of numbers 0 < χ < 1, where a move is made to the neighboring lattice point if the criterion χ < DJD0 is met, where Ds and D0 are the diffu-sion coefficients for the cluster of s primary particles and isolated primary

(8.21.1)

8.21. Compute r Simulation of Colloidal Aggregation 311

particles, respectively. The natural unit of time is therefore the number of attempted moves. It was found that RG ~ s

0 5 5 4, whereas RH ~ s

0 - 5 4 4, re-

sulting in fractal dimensions of dG = 1.81 and dO = 1.84, respectively. They pointed out that uncertainties in the calculations did not allow definite con-clusions regarding the possible difference in rate at which RH and RG ap-proach their asymptotic limiting values as in the linear polymer case. [Recall the model of Weill and des Cloizeaux (1979) summarized in Section 3.12.] It was then assumed that Ds = D 0s~°

5 45 held for all aggregate sizes for the

simulations of 5000 and 8000 primary particles on a lattice of (133)3 sites,

using periodic boundary conditions. The relationship between Ns(t) and the parameters s and t was found to be

m = 5~2ν(βτ)- (8·21·2)

Knowledge of the parameters ζ = y, ds = 3, d D, and dG allows one to calcu-late the parameter α from Eq. (8.17.5). The result for the present system is α = —0.55, and thus the velocity relationship ν ~ s

- 0 , 5 5.

The results of computer simulations of Meakin et al. (1985a,1985b) are schematically represented in Fig. 8.20. It appears from A and Β in this figure

l n ( s f2)

Fig. 8.20. Schematic representation of the distribution of aggregate sizes as a function of time.

(A) The higher-power diffusion law Ds ~ s~2 shows well-defined time zones in which certain sizes

dominate the distribution. (B) The lower-power diffusion law Ds ~ s " °5 shows a much longer

persistence of the particle sizes as time evolves. The slope of the envolops of both plots is — 2z.

(C) Universal curve for diffusion-limited aggregation.

312 8. C O L L O I D S

that the breadth of the distribution Ns(t) depends on the power law for the diffusion coefficient, Ds ~ s~

m. The slope of the tangent line is 2z, where ζ is

the power law for the time coordinate (Meakin et al., 1985b). Part C of this figure is a "universal curve" for the functional form given by Eq. (8.21.1), where the data in the upper two plots fall on the single curve.

Summary

The asymmetric distribution of cations and anions about a charged surface gives rise to the concept of a "double layer". The double layer is conceptually partitioned into a Stern layer, which accounts for the finite size of the small ions in the presence of highly charged surfaces, and the G o u y - C h a p m a n region. Because of the mathematical complexity of describing the interaction between two charged surfaces that interact through their double layers, two asymptotic models are adopted for spherical particles: point charges for particles sufficiently separated that the details of the surface character of the particles are not important; and concentric parallel plates whose interactions occur along the direct line of action.

The classical DLVO theory assumes that attractive van der Waals and repulsive Coulombic interactions are the primary interactions between two particles. This potential explains the stability of highly charged colloidal systems in which the Coulombic repulsion term dominates at all distances of separation, and flocculation in which the attractive van der Waals interaction is dominant at very close distances of approach. The DLVO potential of interaction appears to describe quite adequately the diffusion data of charged micelles and poly(styrene), and also the static structure factor in the vicinity of the peak maximum. One must, however, take into consideration possible effects not included in the D L V O potential. These additional effects include the finite size of the solvent and small ions, fluctuations in the charge distribution of small ions (net charge and ion cloud) and electrons (frequency-dependent van der Waals interaction), and geometric differences in the interacting particles.

Colloidal aggregates appear to exhibit self similarity, thus qualifying as fractal objects. The structures of these aggregates are controlled by their association kinetics as defined by the kernel for the Smoluchowski equation. Large repulsive interactions tend to slow down the reaction and lead to more compact structures (higher fractal dimension) compared to those of systems with small repulsive interaction potentials. The slower kinetics process is described as "activaton-limited", whereas the faster kinetics process is described as "diffusion-limited".

Problems

8.1. Given the surface charge density of σ° = 6.9 pC/cm2 for the N400

latex particles in Fig. 8.1, estimate the density of adsorbed ions at

Problems

saturation in the first layer of the Stern layer from the information given in Section 8.2. Compare this with the Boltzmann weighting factor prediction. Calculate Is on the basis of the released counterions.

8.2. In the RMS approximation (cf. Appendix D), an "effective" particle radius is a parameter that is adjustable to fit the data. The additional "effective" volume presumably represents a "barrier" of closest approach for the two macroions. From g

(2)(x) for latex particles in

deionized water reported by Hartl and Versmold (1987) (cf. Fig. 8.8), obtain a value for the scaling ratio 5 ' = ap/ap

{ for the three concen-

trations. Assuming that the diameter of the particle is constant, give a physical explanation of the change in 5 ' with particle concentration. [Consider / s as a function of C p and also the Beresford-Smith et al. (1985) potential given by Eq. (8.11.1).]

8.3. Determine from the classical D L V O potential the distance that two identical particles must approach each other in deionized water at 25°C in order to have the repulsive and attractive parts cancel each other, i.e., R[U(R) = 0 ] . Assume the charge to be fixed at Z p = 100 and that the particles can have a radius ap of 100 Â or 1000 Â. Represent your results as the ratio R[U(R) = 0]/2ap. In performing these calculations use Eq. (8.6.8) for VR(x)/kT and VA(x) = 1 0 ~

2 0 J/24x, where χ = r/2ap.

To calculate / s , assume np = 101 2 particles/mL. Once the values of

R[U(R) = 0] have been determined, check the validity of these re-sults by comparing the values of VA calculated from Eq. (8.7.1). Use Eq. (8.5.3) to relate Z p to φ°.

8.4. Given a polyion of radius ap = 100 Â and charge Z p with [KCl] = 0.001 M at 25°C, calculate φρ in which the Beresford-Smith et al. (1985) potential [cf. Eq. (8.11.1)] differs by 10% of the D L V O repulsive potential [cf. Eq. (8.6.8)]. [Ignore the b/2 term in Eq. (8.11.1).]

8.5. Kirkwood and Schumaker (1952b) showed that a fluctuation in the net charge of a polyion leads to an attractive interaction between two particles. Assume a solution of identical charged spheres of radius ap = 100 Â and charge Z p = 200 in a 0.001 M aqueous NaCl solution at 25°C. Use Eq. (8.6.8) to evaluate the calssical D L V O repulsive interaction energy for a separation distance of 1000 Â. What is the required magnitude of the charge fluctuation to balance this repulsive energy exactly? [Hint: ignore the α12/λΌΗ term in the denominator of Eq. (8.11.10).] Is this a reasonable value on the basis of the analysis of Hermans (1949) as summarized in Section 7.2?

8.6. Given a 0.001 M aqueous solution of NaCl at 25°C and identical spheres of radius ap = 100 Λ and charge Z p = 200, obtain a value of y to obtain a Sogami minimum at Rmin = 1000 Â [cf. Eq. (8.11.13)].

314 8. C O L L O I D S

Using this value of Rmin for the interparticle separation distance, calculate the value of the Sogami potential U(R)

S given by

Eq. (8.11.11). How does this compare with the K i r k w o o d -Schumaker value in Problem 8.5?

8.7. From the information given for the N400 latex particles in Section 8.0 and in Fig. 8.1, plot the classical D L V O potential as a function of the reduced distance χ = r/2ap. Assume a value of AH = 1 0 ~

2 0 J for

the Hamaker constant and calculate the ionic strength on the basis of the released counterions as determined from the reported surface charge density. Assume that y = 1 in the Sogami model for the inter-action potential, and determine Rmin from Eq. (8.11.13).

8.8. The Perçus-Yevick model for the static structure factor for Κ 0 is given by Eq. (8.10.1) (van Helden and Vrij, 1980). Show that this expression gives the correct osmotic susceptibility for hard spheres for Φ Ρ « 1:

^ ^ = 1 + ^ + 3 0 ^ (φρ«1).

8.9. The Percus-Yevick expression for S(K) in the Κ = 0 limit is S(K = 0) = (1 — </>p)

4/(i + 2φρ)

2. This expression has been used on

micellar systems to interpret data taken at a scattering angle θ = 90°. Consider a series of experiments on micellar systems with volume fractions of φρ = 0.01, 0.10, and 0.40. Determine the maximum par-ticle diameter that can be studied at θ = 90° at these volume fractions in which the use of S(K = 0) introduces an error of less than 3 % . Assume λ0 = 488 nm. [Use Eq. (8.10.1) for the comparison.] Discuss any possible difficulties in the analysis for each case if particles of the maxumum size were present.

8.10. A Sierpinski gasket is a fractal object generated by connecting vertices of equilateral triangles (cf. Fig. 8.12). Determine the fractal dimension of the Sierpinski triangle for the increase in mass with the length of a side.

8.11. According to Eq. (8.15.5), the dominant character of an aggregation reaction can change from AL to DL kinetics if the ratio gDkl/gckO can be changed from a value less than unity to a value greater than unity. For simplicity, let us assume that the reaction occurs between two spheres of equal radius, that U(r) is the potential of interaction between the two spheres, and that RT = 2RA. Let us further assume that kx = 1 and that the potential of interaction, U(r\ is not a "piecewise" potential. The former assumption means that there is no

Problems 315

intrinsic activation energy for the reaction (the long-range Coulombic interaction provides the "barrier"), and the latter assumption means that the same functional form is used at the surface as well as into the solution (there is no short-range attractive potential).

(A) Evaluate the ratio gDkl/gc/kO for: (a) the Coulombic potential, U(r) = -Z

2

Ae2/rskT; and (b) the Debye-Hiickel potential, U(r) =

(y/x)exp( — bx), in the weak interaction limit, exp[— U(r)/kT] ~ 1 — [£/(r)//cT]. The parameters in the Debye-Hiickel potential are y = (Z

2JsRA2kT)exp(bl b = 2RA/ÀDH, and χ = r/2RA.

(B) From your results in Part A for the Coulombic interaction, eval-uate the charge Z A for which guî/dc/Ô = 1

m aqueous solution at

25°C. Is there a dependence on the radius of the aggregate? If so, does the reaction move toward AL or D L kinetics as the particle becomes larger?

(C) From your results in Part A for the Debye-Hiickel potential in the weak interaction limit, does the ratio gDki/gc/kO increase or decrease as Is is increased?

8.12. Freltoft and Kjems (1986) reported fractal dimensions for the aggre-gation kinetics of colloidal silica clusters as a function of ionic strength. Representative results are shown in the accompanying tabulation.

The fractal dimension tends to indicate a change from AL to DL kinetics. Is this trend consistent with your results in Problem 8.11 for the Debye-Hiickel potential? Under what conditions might the tabu-lated data be consistent with the assumption that electrostatic interac-tions are responsible for the fractal dimension via AL or D L processes?

8.13. Schaefer and Ackerson (1975) considered a linear array of colloidal particles that interacted through a harmonic potential, φ(χ) = ΑΣ (

xi

— χΐ + ι )

2' where α is a coupling constant that determines the

strength of the interaction. The following expression for G H(K,0) was derived:

[ N a C l ] 0.40

1.55

0.50

1.96

0.60

2.06

0.75

2.49

316 8. C O L L O I D S

where a is the lattice spacing and β = 1/kT. Plot G H(K,0) as a function of Κ for a = 100 A and 1000 Â.

8.14. If a particle rotates through a small angle Ö, then the distance moved by a site on a sphere through the solution is approximated by χ = Rssm(6) ~ Rs6. Rotational motion therefore appears as translational motion with an effective translational diffusion coefficient DeR

2

[cf. Eq. (8.20.1).] For this motion to be "observed" as translational motion in QELS experiment, however, the distance χ must be com-parable to the probe distance 1/X. Estimate the minimum radius of the particle required for such an interpretation of rotational motion if QELS data are collected at a scattering angle of 90° using light of wavelength 488 nm, given that the linear approximation of sin(ö) must be within 10% of the true value. How are these results and interpre-tation of the data affected as the scattering angle is continuously lowered to a value of θ = 20°?

8.15. Discuss how one might use QELS techniques to obtain the distribu-tion of particle sizes Ns(t) as a function of time. Be sure to include possible difficulties in the interpretation of these results.

8.16. Determine the power law RH = (ns)a from the Kirkwood friction factor

dependence on the number of subunits given in Fig. 8.21. Are these structures, which range from a solenoid to a rod, fractal objects?

I C H

9

8 H

7

6 -

5 -

4 -

3 -

2

— I —

3 0 I —

4 0 ~ Ί —

5 0 I

7 0 I —

8 0 9 0 " Ί -

ΙΟ - Γ

-

2 0

Fig. 8.21. Kirkwood friction factor for regular rodlike arrays. Beads of diameter 110 Â were used to calculate the friction factor in accordance with the Kirkwood model for hydrodynamic interaction [cf. Eq. (3.7.11)]. Shown are the linear array (rod), solenoid (six beads per turn that touch the beads of adjacent turns), and open helices with a pitch of 250 Â and radii of 150 Â and 400 Â.

Problems 317

Additional Reading

Everett, D. H. (1988). Basic Principles of Colloid Science. Royal Society of Chemistry Paperbacks.

Burlington House, London.

Hess, W. and Klein, R. (1983). "Generalized Hydrodynamics of Systems of Brownian Particles".

Advances in Physics. 32, 173-283.

Mandelbrot , Β. B. (1982). The Fractal Geometry of Nature. Freeman Press, New York,

van de Ven, T. G. M. (1989). Colloidal Hydrodynamics. Academic Press, Boston.

CHAPTER 9

External Perturbations

"It drew now to evening by the hour, and the light was so dim that even far-sighted men upon the Citadel could discern little clearly upon the fields, save only the burnings that ever multiplied, and the lines of fire that grew in length and speed. At last, less than a mile from the City, a more ordered mass of men came into view, marching not running, still

holding together."

From The Lord of the Rings, Part III: The Return of the King

by J. R. R. Tolkien (1892-1973)

9.0. Introduction

Parallels can be drawn from the above passage and the effects of an external perturbation on a solution of interacting particles. One effect is to alter the magnitude of the interactions between the particles ("... grew in length..."), which changes the random velocity of the particles. Application of a direc-tional external field superimposes a directional velocity on the motion of the particles. The external field may therefore induce long range correlations between the dynamics of the particles ("... still holding together.. ."). The first effect changes the decay rate of C(K, i), or the spectral linewidth of S(K, ω). The second effect introduces oscillations in C(K, f), or a shift in the peak loca-tion in S(K, ω).

9.1. General Mathematical Framework

The full Langevin expression given by Eq. (6.1.1) is used in describing the motion of a test particle, where NT spans polyions and small ions. Hence,

319

320 9. E X T E R N A L P E R T U R B A T I O N S

P a p p ( ^ , ί) <v(0)r * \(t'))exp{icot')dt\ (9.1.1)

where v(f') is the velocity of the particle as modified by </>[{r(f')}], which is also a function of 0(r, t'). The flux of particles of type j is Jy(r,f) = — D a p p( c o , ί) · \ ACj(r,t) + v}(r,i) · AC(r, t\ where v}(r,i) is the superimposed velocity of the j th particle due to Φ ( Γ , t'). Fourier transform of Fick's Second Law, dACp(r,t)/dt = - V · J p , gives

dAC p(K,r)

dt - K

T- p a p p( K , o ) , i ) . K A C p ( K , i )

iv p(K, i )r- K A C p ( K , i), (9.1.2)

which has the general solution

AC p(K,i) = AC p(K,0)exp[- - ΚΓ. Π ( Κ , ω , ί ) · Κ ί ]

χ exp i ν ' ρ(Κ , ί ' )Γ - Κ Λ ' (9.1.3)

Consider the case v p(K,i) = v p , and D a pp

of time). Hence, Eq. (9.1.4) becomes (isotropic and independent

AC p(K,i) = A C p ( K , 0 ) e x p [ - D ^ X2i ] e x p ( - / v p

r. K i ) . (9.1.4)

<ACp(K, 0) ACp(K, i)> is therefore a damped oscillating function in time with a frequency ω ρ

8 = v p · Κ and damping constant DfK

2. In order for this

shift in frequency to be detected the scattered electric field must be "mixed" with a "local oscillator", or a diverted portion of the incident light. This mixing is generally effected by directing a portion of the incident light onto the phototube surface. Denoting the electric field of the diverted beam by £ d(R), the intensity of scattered light is \Ed(R) + £ S(R, t ) \

2. By choosing

|£ d(R,0) | » |£ S(R,0) | , one has for the heterodyne intensity correlation func-tion </(0)/( t)> h et ~ 2<£î (R)£ d(R)><£*(R,0)£ s(R, i )> ,

</(0)/(0>HET - B'VsCpexp(-DfK2t)exp(-iœ0t)

χ exp - f v pX c o s | - ]t (9.1.5)

where B' = A\Id(R)) = A\E%(R)£d(R)>, and A' is a constant. Fourier trans-formation of </(0)/( i)> h et gives 5(K, ω — ω 0) :

S (Κ, ω — ω 0) = • B'VsCpDfK

2

(Dp

sK

2)

2 + ω — ω0 + ν ' Κ cos

(9.1.6)

9.3. Electrophoretic Mobility and the Zeta Potential 321

where ± indicates that the particle motion can be in either direction along the line v p · K. According to Eq. (9.1.6), the peak is Doppler-shifted by an amount v pKcos(0 /2) = œ

ds with a linewidth at half-height of Δ ω 1 /2 = DfK

2.

9.2. The Zeta Potential

Consider the motion of a charged particle that carries along with it counterions and solvent. Because the solvent is incompressible, movement of the particle must also cause a backflow of solvent to take up the void created by the particle displacement. There must exist a region about the particle where the solvent is stationary. This imaginary surface is referred to as the surface of shear. All of the counterions, coions, and solvent particles that move along with the macroion are referred to as the kinetic unit. The potential difference between the moving and stationary solvent layers is referred to as the zeta potential and is denoted by £ p.

9.3. Electrophoretic Mobility and the Zeta Potential

A charged particle is accelerated by applied static electric field E°. Because of solvent resistance that generates a viscous force that balances the applied force, the particle reaches a terminal (constant) velocity v p (oo). Smoluchowski showed that vp(oo) = εζρΕ°/4πη0 for charged particles moving through a porous material (1903) and for particles in the thin-layer limit (1921). Hiickel (1924) examined the thick-layer limit and obtained the result that the electrophoretic mobility (μρ) should be proportional to \/6πη0. Henry (1931) traced the discrepancy between the Smoluchowski and Hiickel models to the way in which the electric field about the particle was treated, whether it is undistorted by the presence of the particle (Hiickel) or distorted in such a way that it is parallel to the surface of the particle (Smoluchowski). The result of Henry's analysis for spherical particles is (Tanford, 1961)

where fi(x\ with χ = αρ/λΌΗ, is Henry's function (Henry, 1931), given by the asymptotic expressions (Hunter, 1981)

E° (9.3.1)

6πη0

fi(x) = 1 +

X 6

exp(x) f e x p ( - f )

dt (x « 1), (9.3.2) 8 96

322

1.5.

χ Q Q.

σ

ι.ο-0.75

9. E X T E R N A L P E R T U R B A T I O N S

cylinder (parallel to field)

sphere / yf cylinder

(perpendicular

to field)

\ conducting

^ sphere

\

\ \

\

-2 0 I 2 l o g ( a p A D H )

Fig. 9.1. Dependence of F ( a p/ A D H, K') on the ratio αρ/λΌΗ for spheres and cylinders. K' is the

particle-to-solvent ratio of conductivity. [Reproduced with permission from Overbeek (1952). In Colloid Science. Vol. I, pp. 194-244. (H. R. Kruyt, ed.). Elsevier Publishing Co., Amsterdam. Copyright 1952 by Elsevier Publishing Co.]

Henry's function for spherical conducting particles is (Henry, 1931; Hunter, 1981)

F(x ,K ' ) = 1 + 2A'[/i(x) - 1], (9.3.4)

where λ' = (1 — K')/(2 + K'\ K' = λ'ρ/λ'0, λ'ρ is the particle conductivity, and λ'0 is the solvent conductivity. The effect of a conducting particle surface is to alter the electric field in the vicinity of the particle. Plots of F(x, K') vs. log(x) are shown in Fig. 9.1 for spheres and cylinders.

Note that for the conducting sphere F(x,K') goes to zero for 1 « αρ/λΌΗ but is unaffected for 1 » αρ/λΌΗ. This is because the polarized medium near the surface of a particle has little effect on a thick ion atmosphere but can virtually nullify the mobility for thin ion clouds. For metallic particles, however, the polarization of the particle may neutralize the conductivity at the surface, hence metallic colloids may behave as if they had a nonconducting surface (Overbeek, 1952).

9.3. Electrophoretic Mobility and the Zeta Potential 323

Overbeek and coworkers (Overbeek, 1943; Wiersema, Loeb, and Overbeek, 1966) examined the motion of an isolated nonconducting sphere of uniform charge density with a mobile G o u y - C h a p m a n electric double layer, which is suspended in a medium whose dielectric constant is independent of position. vp(oo) is reached when the following four forces are balanced: (1) the external force Fp = ZpeE°; (2) the retardation force arising from the separation of positive and negative charge centers, F

e

r; (3) the Stokes viscous force, Fn = — 67D70<3pVp(oo); and (4) the force exerted on the ion cloud, which is then transferred to the solvent molecules and acts as a retarding force, F

s

r.

The total electrical force is obtained by integration of σ°( — V0 e I) over the surface of shear, where σ° is the surface charge density of the sphere and </>el

is the total electrical potential (internal plus external) that is governed by Poisson's equation [cf. Eq. (7.3.1)]. The total hydrodynamic force is obtained by integration of the stress over the surface of shear. Overbeek (1943, 1950) obtained a series expansion solution as a power series in the reduced zeta potential (ζ

τ

ρ = eCp/kT),

6πί/0νρ(οο) <fi(x)-f2(x)(Za-Ze)Cv

Zcmc + Z a m a ^ r ) 2

z c + z a

for asymmetric electrolytes, and

67Γ7/Ονρ(θθ) ( C p ) ' Zîh(x) +

1 imc

+ m a) / 4(x )

(9.3.5)

(9.3.6)

for symmetric electrolytes, where mt = ΝΑεΙίΤΖ^/6πη0λ[, where λ\ is the equivalent conductance of the ith ion. The functions / x(x ) (Henry's function), fi(

x\ h(x\

a nd (x) are tabulated in Table 9.1 for selected values of χ = αρ/λΌΗ (Overbeek, 1950). A plot of 6π^ 0ν ρ(οο ) /ε^ ρ£° vs. \η(αρ/λΌΗ) is com-pared with Henry's function in Fig. 9.2.

A minimum occurs for asymmetric salts in which | Z C | > |Z a | . For the 4 - 1 salt there is an excess of cations that moves in a direction opposite that of the particle in the presence of an applied electric field, thereby retarding the motion of the kinetic unit. For the 1-4 salt, however, there is a deficit of anions that causes the cloud to move along with the particle, thereby accelerating the kinetic unit in the presence of E°, and no minimum occurs.

A comparison of 6πη0\ρ(οο)/εζρΕ° vs. \η(αρ/λΌΗ) for the analytic solutions of Overbeek (1943) and Booth (1950a, 1950b) and the computer-generated solutions of Wiersema et al (1966) is given in Hunter (1981). The models of Overbeek and Booth are virtually identical. The computer-generated solution


Table 9.1

Overbeek Correction Terms to the Electrophoretic Mobi l i ty0

f M Mx)

0.01 1.00000 0.0006 0.00009 0.0006 0.1 1.000545 0.0125 0.00090 0.0107 0.3 1.00398 0.0279 0.0044 0.0218

1.0 1.0267 0.0411 0.0116 0.0387 3.0 1.1009 0.053 0.020 0.051 5.0 1.160 0.057 0.022 0.054

10.0 1.239 0.05 0.021 0.05 100.0 1.458 0.0102 0.00444 0.00992

a F rom Overbeek, (1950). In Advances in Colloid Science. Vol. I l l ,

p. 97. (H. Mark and E. J. W. Verwey, eds.) Interscience Publishers, Inc. New York.

I 1 1 1 1 Γ

- 2 - 1 0 1 2 3

l o g ( x j p / X DH )

Fig. 9.2. The effect of electrolyte charge on the electrophoretic relaxation term as a function of apl A D H. The zeta potential used in these calculations was — 50 m V. The curve for Henry's equation with no relaxation is given by the dashed line. The numbers X- Y associated with each curve give the charge of the ca t i on -an ion , respectively. [Reproduced with permission from Overbeek (1952). In Colloid Science. Vol. I, pp. 194-244. (H. R. Kruyt, ed.). Elsevier Publishing Co., Amsterdam. Copyright 1952 by Elsevier Publishing Co.]

9.3. Electrophorectic Mobility and the Zeta Potential 325

sedimentation

Fig. 9.3. Velocity vector maps. The spheres in the figure have a radius of 23.4 Â; the

temperature is 25 C. The magnitude and direction of the fluid velocity at each grid point and the

direction of motion of the sphere are indicated by the arrows. The shaded half-spheres are neutral.

(A) The total velocity of the fluid flowing around a sphere with zeta potential of 98.4 mV in 0.03 M

NaCl. (B) The total velocity of the fluid flowing a round a neutral sedimenting sphere. (C) The

relaxation component to the fluid velocity is defined as the total fluid velocity for the charged

sphere minus the Stokes fluid velocity for the uncharged sphere. The vectors have been multiplied

by a constant factor for enhancement of visualization. (D) Stokes fluid velocity about an

uncharged sphere. [Reproduced with permission from Stigter (1980). J. Phys. Chem. 84, 2 7 5 8 -

2762. Copyright 1980 by the American Chemical Society.]

of Wiersema et al. has a shallower minimum than those of Overbeek and Booth. The minima for these three models appear at the same location as given in Fig. 9.2.

Stigter (1980) pointed out that the description of a charged particle in a sedimentation field and an electric field is virtually identical. Their difference is the boundary condition at infinity. It is of pedagogical interest, therefore, to present the computer-drawn vector maps reported by Stigter and shown in Fig. 9.3. Note that in the case of electrophoresis, the flow pattern near the surface of the sphere tends to rotate. This is a direct consequence of the relaxa-tion part of the fluid velocity, which is defined as the total fluid velocity of the charged sphere in a sedimentation field minus the Stokes fluid velocity component. Although the scales differ, it is noted that the radial decay of the relaxation velocity is more rapid than the Stokes velocity profile. The decrease


Ο ο

Ο *4

2 3 4 5 ί'ρ

Fig. 9.4. Fluid velocity correction factor b for sedimentation of charged spheres as a function of the reduced zeta potential. The calculations were carried out for charged spheres as described in

Fig. 9.3, where the ionic mobilities were /,· = 70 Ω "1 c m

2 equ iv"

1 (KCl at 25 C). The values of

αρ/λΌΙι are indicated in the figure. [Reproduced with permission from Stigter (1980). J. Phys. Chem.

84, 2758-2762. Copyright 1980 by the American Chemical Society.]

in the fluid velocity by the charge effect is given by

where b depends on the ratio αρ/λΌΗ, ζρ, and the small ion mobilities Xt

(cf. Fig. 9.4). The relaxation velocity set up by the charged sphere acts in a direction opposite to that of the hydrodynamic wake, thereby reducing the magnitude of the fluid velocity and causing the fluid to rotate. Hydrodynamic interaction between two charged spheres should be less than that between two uncharged spheres. A similar conclusion was reached by Ibuki and Nakahara (1986) in their theoretical study of the Stokes radius of L i + using the Hubba rd -Onsage r dielectric friction theory (Hubbard and Onsager, 1977; Hubbard, 1978).

Stigter observed that Henry's function f^x) was related to the function S%(x) of Booth (1954) by ^ ( x ) = 18S?(x). [The proportionality constant of Stigter was 12, but because of the difference in the définition of / x(x ) in this text, the corrected proportionality constant is 18.] The asymptotic limits for S*(x) are (Booth, 1954)

Vp = v J [ l - 6 ( C y 2 ] , (9.3.7)

S?(x) ~ 0.05556 + 0.00347x 2 - 0.00579x 3

S?(x) ~ 0.08333 - 0 . 25x _ 1 + 2.083x" 2 (χ » 1).

(x « 1), (9.3.8)

(9.3.9)

9.4. Nonuniform Distribution and Electric Field S t r eng th -Dependen t Zeta Potentials 327

These expressions can be compared with those of Eqs. (9.3.2) and (9.3.3). The apparent reason that the equivalence f^x) = 18Sf(x) escaped attention was due to the different algebraic expressions for the two functions.

The charge, Z£, of the kinetic unit of radius a'p = ap + SRS is related to the zeta potential by

7Μ =

CP'P ΑΌΗ_

(9.3.10)

where öRs accounts for particles moving with the polyion.

9.4. The Effect of Nonuniform Distribution and Electric Field Strength-Dependent Zeta Potentials

Implicit thus far are the assumptions that the zeta potential is (1) uniformly distributed over the surface of the sphere, and (2) independent of £ ° , and the explicit assumption that (3) the effects of the perturbation are first-order in E°.

The first assumption is intimately related to the question of whether or not an average zeta potential can be substituted for a nonuniform distribu-tion, which was addressed by Anderson (1985). The model employed was a nonconducting sphere in the presence of a uniform electric field in the limit αρ/λΌΗ » 1 and "no slip" boundary conditions. The translational velocity was found to be

- (E°) + a'pPl - (VE°), (9.4.1)

where <( p> is the average zeta potential (monopole moment), P x = — <Cpn> is the dipole moment where η is the unit vector normal to the surface, and P 2 = <Cp(3nn — I)> is the quadrupole moment. Note that the even subscripts are associated with E°, whereas the odd subscripts are associated with VE°, hence the notation "even" and Odd" distributions in the zeta potential.

In the case of the even distribution, where the northern and southern caps are equal in size with a zeta potential ( c and a midregion having a zeta potential CM, Anderson obtained for the translational velocity

Ύ'ρ = < C p >I ~ ~ \ i Cc ~ C J [ c o s ( ö o) - c o s 3< Ö o ) ]

where <ΖΡ> = £ mcos (0 o) + Cc[l — cos(0 o)] , e is the orientational vector, and θ0 is the angle subtended by e with the line connecting the center of the sphere

4πη0

<CP>I

(3ee - I) }· E° (9.4.2)


with the cap region. The rotational velocity is

Ω = ^ ( C c - U [ c o s ( 0 o ) - cos3(0 o) ]ee χ (VE°). (9.4.3)

\6πη0

In the presence of a constant electric field (VE° = 0), the sphere does not rotate. On the other hand, v'p is not in the direction of E° and is less than the Smoluchowski value (1903, 1921) if e is not in the direction of E° .

In the case of the odd distribution, Anderson obtained

v p = ^ E ° + ^ L ( C f - C b) [ I e + 2eee]:(VE°), (9.4.4) 4πη0 96πη0

where <£ p> = ( C f + C b ) / 2 and Ω = (9ε/64πη0α'ρ){ζ{ - C b)e χ E°. The direc-tion of movement of the sphere therefore is not along the vector E° , and the sphere moves more slowly than the Smoluchowski value. The sphere attains its maximum velocity when e is parallel to E°.

For a continuous distribution of odd ( C x ) and even (ζ2) components in the presence of a constant electric field, Anderson obtained the following expressions:

v' - £

<Ζ

Ρ > ί 2 . 3 C 2 I — r ^ ^ e e

5 < C p > " < C p > E°, (9.4.5)

and Ω = (3εζ 1/16π^ 0α ρ)ε χ E°. Hence, application of a constant electric field will result in an additional component to the rotational velocity of a charged sphere with odd distributions in the zeta potential. This additional velocity acts in the direction to align the unit vector e along E° . This, in turn, will affect v p of the particle. A measure of the probability of reorientation due to rotational motion is given by a rotary Péclet number, Pe r = \Q\/De, where De = 1<Τβπη0{α'νγ.

Example 9.1. Orientation of Spheres with Continuous Odd and Even Zeta Potential Distributions.

The questions addressed are: (1) what is the probability of alignment of spheres in an electric field; and (2) what is the effect of orientation on the trans-lational velocity?

it is assumed that two spheres, with radii of 25 Â and 500 Â and identical zeta potentials ζι = 95 mV and ζ2 = 100 mV, are subjected at 25°C to a con-stant electric field of E°= 100 V/cm. Hence Pe r = [ 3 ε ( α ρ )

2Κ 1 | £ ° / 2 ^ Γ ] sin(0).

Upon substitution of the values ε = 78, d = 0.095/300 - 3.167 χ 1 0 "4 stat-

volts, and E° = 100/300 = 0.333 statvolts, the value of the Péclet number


at 25°C for this system is Pe r = \&\/De = 3.0 χ lO1 1

(fl,

p)2sin(0). For the

25 Â particle, one has Pe r = 0.0187 sin(ö), which means that Pe r « 1 for all values of 0. These particles will not, therefore, orient in the presence of the applied field of 100 V/cm. On the other hand, Pe r = 7.5 sin(0) for the 500 Â particle and has the value of unity when 0 = 7.6°. This means that the larger particle will orient in the electric field. For the random orientation the moments in Eq. (9.4.1) average to zero; thus for the 15 Â particle, v p = ε(ζρ}Ε°/4πη0 = 0.0752 cm/s. This is also the electrophoretic velocity of randomly oriented larger particles. However, it is assumed for simplicity that the larger particle attains total alignment (those particles for which 0 > 7.6° become oriented in the field). In this case, v p = (ε(ζρ)Ε°/4πη0) χ [1 - ( 2 C 2 / 5 < C P » ] = 0.0752(1 - 0.41) = 0.44 cm/s. The oriented particle therefore migrates at a slower rate than the random distribution of particles.

The assumption that the potential is independent of E° depends upon the nature of the "local" electric field and what is defined as the kinetic unit. Consider a system of a negatively charged nonconducting sphere with a mobile Stern layer immersed in a polarizable medium. The Stern layer is assumed to be uniform in the absence of an applied field. At low E°, the Stern layer remains unchanged because of the tight binding, but the diffuse ion cloud becomes distorted. As E° increases, the ions within the Stern layer begin to migrate to more favorable chemical potential regions. A nonuniform zeta potential now develops, and pp changes in a manner suggested by Anderson (1985) for an oriented sphere with an "odd" zeta potential distribution. The dipole component that results from the separation of charge centers in the Stern layer also redistributes the ions in the G o u y - C h a p m a n layer. How this redistribution of ions between the Stern and G o u y - C h a p m a n layers affects the zeta potential depends on the location of the "shear boundary". If the shear boundary is at the Stern layer, as assumed by Zukoski and Saville (1986), then the change in the number of ions in the Stern layer may result in a change in the net surface charge, and thus in the value of the zeta potential. On the other hand, if the shear boundary is some distance from the Stern layer, there may not be a significant change in Ζ

μ

ρ of the kinetic unit and <ζ ρ> does not change. Cations flow through the kinetic unit in the direction opposite to that of the negatively charged sphere, whereas anions flow in the same direction as the sphere. Because of coupling of the hydrodynamic, chemical, and electric potentials, a portion of the ion cloud may be "stripped" from the charged sphere as E° is further increased. One must now be concerned with the relative rates of redistribution of small ions between volume regions about the charged sphere. Large variations in these relative conduction rates result in steady-state accumulation of charge. These possible effects are illustrated in Fig. 9.5.

Models for ion movement within the layer of "bound" ions (O'Konski, 1960; Schwarz, 1962) or ions in equilibrium with the bulk medium without


η d o r y ° f S h

D

COUNTERION

Fig. 9.5. Possible effects on an isolated charged sphere with ion atmosphere upon application of a constant electric field. The kinetic unit is defined by the boundary of shear, as indicated in the

figure. The sphere is shown with a Stern layer, al though this is not present for small zeta potentials.

The direction of migration of the kinetic unit is shown by the arrow inside of the (negatively)

charged sphere, and the drift direction of the counterions (cations) is shown by the arrows outside

of the sphere. The steady-state distribution of the cations within the kinetic unit is determined by

the relative conduction rates at the (hypothetical) boundary regions, symbolically indicated by A,

B, C, and D above. Rearrangement of the ions within the Stern layer a n d / o r the applied electric

field may induce a surface polarization in the sphere. Regions of charge accumulation indicated

above are ( / / / / / / / / ) positive charge, and ( \ \ \ \ \ \ \ \ ) negative charge.

radial flow (Dukhin and Deryaguin, 1974) and with radial flow (Chew and Sen, 1982a, 1982b, cf. Section 8.5; Fixman, 1980, 1983; Fixman and Jagannathan, 1983) have been proposed. Since the emphasis of the remaining portions of this section is on the variation of the zeta potential with changes in £ ° , the basic features of the model used by Fixman (1980, 1983) and Fixman and Jagannathan (1983) are reviewed.

Fixman-Jagannathan Model

The assumptions employed by Fixman (1983) and also Fixman and Jagannathan (1983) are virtually the same as those of Overbeek (Wiersema et al. 1966; Overbeek and Wiersema, 1967) outlined above. The macroions are assumed to be sufficiently dilute that they do not interact with each other, and they do not have a Stern layer. The F ixman-Jaganna than model, how-ever, uses "slip" boundary conditions, which was found necessary because of the large velocity gradients near the surface of the sphere.


Electrical potential. The total electrical potential </>el of the system is assumed to be composed of an "internal" part, φ, and an "external" part, Φ. Upon defining the reduced potential φ\χ = e\j\) + 0 ] / /cT = φ

χ + Φ

Γ, the reduced

distance R = r/ÀDH, and the reduced electric field Er = eEÀm/kT = — \φ[},

the Poisson equation is

Λ Ί 2

VR · [ 6 ( r ) V R^ J , ] = - [ρ , + σ°δ(τ - α ρ) ] , (9.4.6)

where the dielectric constant s(r) may be discontinuous across the sphere boundary, ps is the small-ion charge density, and σ° is the surface charge density.

Velocity potential. Ignoring the inertia term, the Langevin equation for the velocity of the small ions is — ξί(\ι; — v 0) — fcTVrln(«J) + qtE = 0, where qt = eZt is the charge of the ith small ion with friction factor and num-ber concentration n\. Rearrangement gives

kT v 0 - — V r A . , (9.4.7)

where the velocity potential is pt = hi + Z t Or, and the concentration of

the small ions is represented as a dimensionless parameter /zt(r) = ln[n;(r)/<ni(r)> e q], where <ni(r)>eq = <n;> uexp(-Z t( />

r) . The reduced charge

density with hydrodynamic interactions is thus given by

Ρΐ = ΣΒι^χΡ(Ρΐ-ΖΐΦ^ where the functional dependence on r is implied and Bt = C^Z-J^ CjZj.

The chemical potential. The function ht is identified as the chemical potential, where n\ obeys the conservation equation dn'Jdt = — Vr · (n-vt). Hence the equation for h{ is

dh-D-'-^ = (DT\- VrPi) - V r ( Z i 0

r - hd + V

2

P i, (9.4.8) dt

where Df = kT/ξι and Vr · v 0 = 0 for an incompressible fluid.

The objective is to solve Eqs. (9.4.6)-(9.4.8) respectively, for the electric, velocity, and chemical potentials. These solutions are determined from the set of boundary conditions at the surface of the sphere ("inner conditions") and at large distances from the sphere ("outer conditions"), and the matching of the solutions at the interface. The approximations of Fixman (1983) and of Fixman and Jagannathan (1983) are now outlined for R » 1.

Because of cylindrical symmetry about the z-axis (direction of flow) in the linear approximation, the functions v t, φ\λ, and ht can be represented as a series


solution in the reduced radial distance R and the angle 0. The velocity potential in the asymptotic limit is represented by the stream function, s [# ,cos(0)] , similar to that derived in Section 6.2. The major difference is the presence of the electrical potential, viz,

d2 sin

2(0) d

2 )

2

^s in2(0 )

dpi δφ\χ dpi δφ\

_dcos(0) dR dR <9cos(0)

Since Ε approaches E° in the limit R » 1, we can write

(9.4.9)

Ε ~ E° + h - q ^ r -1 + ( V ^

1) . μρ + . . . ] , (9.4.10)

where qp is the apparent charge of the sphere and μρ is the dipole moment. Note that these two parameters are nonlinear field effects. The electrical and chemical potentials are related by the Poisson equation, which in reduced units is

V ^r

= - ( P s - e J = - (9.4.11)

where Bt was previously defined and h = £ Bfa. The potentials ht[R, cos(0)], Φ

Γ[Κ, cos(0)], and s[R, cos(0)] can be represented in terms of a multipole

expansion. The functions ht[R, cos(0)] and ΦΓ[Κ, cos(0)] are scalar poten-

tials expressed as a power series in Legendre polynomials, and s[R, cos(0)] is a vector potential expressed as a power series in Gegebauer polynomials. The results of Fixman (1983) for the reduced electrophoretic mobility, μρ = (6^0e/kTDp)\p(cc)/E°, are shown in Fig. 9.6 as a function of the reduced zeta potential, with and without the convection current (v0). The hydrody-namic interactions between particles were calculated at the Oseen level of approximation [cf. Eq. (3.7.8)]. Also shown in this figure for compari-son are the numerical results of O'Brien and White (1978) based on the model of Dukhin and Deryaguin (1974) as reported in Hunter (1981), and the values computed from the analytical expression of Dukhin and Deryaguin (1974) for the added symmetric salt of molality m and charge of magnitude Z s , μρ = (3Cp/2) - 6(F/G). In this expression, F = C p M s i n h

2( 7 ) +

[ ( 2 M / Z s ) s i n h ( 2 y ) - 3 m C p] l n [ c o s h ( y ) ] , and G = (αρ/λΌΗ) + 8M sinh2(Y) -

(24m/Z s

2)ln[cosh(Y )], where Υ = Ζ8ζρ/4 and M = 1 + (3m/Z s

2). The cal-

culations shown in Fig. 9.6 were performed for KCl at m = 0.184 g/kg.

Calculation of qp on the sphere requires more detail about ps in the vicinity of the sphere. The force balance equation at the surface of the particle was the sum of the electrical (pE), buoyancy, F

g = (mt — m0)gVi where Vi is the volume


0 2 4 6 8 10

Fig. 9.6. The reduced electrophoretic mobility as a function of the reduced zeta potential for Û

P M D H = 50. The numerical results of Fixman (1983) with ( · ) and without ( O ) the convection

velocity v 0 [cf. Eq. (9.4.6)] are compared with the numerical results of O'Brien and White (1978,

V ) and the analytical expression of Dukhin and Deryaguin [1974, A ] .

of the particle, and hydrodynamic forces. The force balance equation at the surface of the sphere is

FH + 4 pE + F

g = 0. (9.4.12)

It must be kept in mind that qp in Eq. (9.4.10) is an induced charge, and not the surface charge of the sphere. As pointed out by Fixman and Jagannathan, the


B C D Ε Fig. 9.7. Schematic diagram of the F ixman-Jaganna than model of the electric field strength

dependence of the apparent charge on a sphere. (A) The induced charged on the sphere computed from the first expansion term of Eq. (9.4.10). (B) The positively charged sphere (solid) with associated ion cloud (dashed line) with no applied electric field. (C) The linear regime in which the applied electric field polarizes the ion cloud, which is identical to the relaxation effect discussed previously. (D) The weakly nonlinear regime is characterized by an increase in the positive charge of the kinetic unit that results when there is an unequal conduction rate of ions to the stern and bow of the moving sphere, with a net loss of counterions from the kinetic unit. (E) The strongly nonlinear regime is characterized by a decrease in the value of the apparent charge, eventually changing sign and becoming a large negative value. Detailed numerical evaluation of the small-ion distribution indicates that the classically defined kinetic unit remains positively charged and that the large negative value for the apparent charge results from an accumulation of counterions in a region outside of the kinetic unit.

induced charge is that associated with the sphere and the surrounding ion atmosphere, viz, the kinetic unit. The calculations of Fixman and Jagannathan were for a positively charged sphere. The effects of E° on qp and μ ρ of the sphere were conceptually partitioned into three regions: (1) linear; (2) weakly nonlinear; and (3) strongly nonlinear. The linear region is associated with a competition between the M ax wel l -Wagner effect and polarization of the ion

9.5. Application of a Constant Electric Field: Doppler Shift Spectroscopy (DSS) 335

atmosphere. The Maxwel l -Wagner effect has to do with the difference in the dielectric constants of the sphere and the supporting medium. There is a surface charge buildup to maintain the boundary condition άφ/dr if ε ρ > 0. The ion cloud of the moving charged sphere becomes distorted, thus giving rise to polarization of the ion atmosphere. In this example, the negatively charged counterions accumulate to the stern of the moving sphere, with a deficiency of counterions to the bow. For small αρ/λΌΗ or large £ p, polarization dominates, and the induced dipole moment is positive. For large αρ/λΌΗ or small Cp, the Maxwel l -Wagner effect dominates, and the induced dipole moment is negative. The weakly nonlinear region is manifested by an increase in the induced charge of the kinetic unit. This results when convection and electrical forces remove counterions to the stern of the sphere at a faster rate than they remove counterions to the bow of the sphere. That is, part of the ion cloud is "stripped off" of the kinetic unit. The strongly nonlinear regime is characterized by an accumulation of counterions. According to the charge density distribution calculations, the accumulation of counterions occurs outside of the equilibrium double-layer boundaries. That is, the inner region maintains its positive charge and polarization. These authors suggest that the conductivity of the inner region is strong, whereas the conductivity of the outer region is weaker. The results of these calculations are schematically represented in the cartoons in Fig. 9.7.

9.5. Application of a Constant Electric Field: Doppler Shift Spectroscopy (DSS)

Ware and Flygare (1971, 1972) were the first to perform QELS experiments in the presence of a constant electric field E°. To assure that the particles reached their terminal velocity, data were collected after a brief waiting period after the application of E°. Hence, œp

s in Eq. (9.1.7) is

and linewidth at half-height DfK2, where 2 cos(0/2) sin(0/2) = sin(0) was

used. The experiment in which a constant field is applied throughout the course of obtaining data is referred to as Doppler shift spectroscopy and abbreviated DSS. A schematic of the D D S experiment is given in Fig. 9.8.

Detailed descriptions of electrophoretic cell designs, electrode surfaces, and power supplies are found in review articles by Ware (1974), Smith and Ware (1978), Uzgiris (1981), and Ware and Haas (1983). Four areas must be addressed in the cell design: (1) polarization; (2) Joule heating; (3) electroosmosis; and (4) chemical reaction.

Polarization occurs when ions of opposite charge accumulate near the electrode. This tends to reduce the effective charge on the electrode and

(9.5.1)


Fig. 9.8. Schematic representation of the Doppler shift experiment. The charged particle in the

presence of a constant electric field has a velocity v'p superimposed upon its random Brownian

motion, which results in a frequency shift in the spectral density profile.

therefore the electric field across the sample. To minimize residual polariza-tion effects, a constant-current power supply is recommended over a constant-voltage power supply (Smith and Ware, 1978).

Joule heating occurs upon prolonged application of the electric field, thereby generating convection currents. A measure of turbulent motion is given by the Rayleigh number iVRa = 0(ΑΤ)χατά

3ρο/ηοκτ, where g is the

acceleration due to gravity, (AT)X is the characteristic temperature difference for the region of depth d in the vertical (x) direction, a x is the coefficient of thermal expansion, and κΊ is the thermal conductivity. In general, convection is not a problem when NRa < 1700. The temperature rise at the center of the gap, and applied voltage V° is AT = 0.24V

02 βρ0κτ (Kohlrausch, 1900). To

avoid charge polarization and heating effects that occur for large £ ° , the first experiments performed by Ware and Flygare (1971, 1972) used a "pulsed train" electric field whose polarity was reversed on successive pulses.

Electroosmosis occurs if the surface of the cell is charged relative to the solution, as first examined by Smoluchowski (1921). For example, glass has a negative charge relative to the solution. The "surface-adsorbed positive ions" migrate toward the cathode, dragging solvent along with them. Because of the incompressibility of the solvent, a backflow is created in the center of the chamber. Palberg and Versmold (1989) examined the effect of electroosmosis on S{K,œ). Their calculations indicate that the line shape is distorted from a symmetric peak about the electrophoretic velocity (vp

1), where the direction of

the distortion depends upon the relative signs of vp* and electroosmotic velocity v e 0. The breadth of the distortion depends upon | v e 0| but not D p.


e lectroosmo si s velocity profile

ι

I stick boundary conditions

1 Θ

so/vent >

back flow

cation migration

electrode polarization (cations)

electrode polarization

(h nions)

Fig. 9.9. Schematic representation of electrical effects that may occur in an electrophoretic light scattering cell. Electrode polarization occurs when cations and anions accumulate near the

cathode and anode, respectively, which therefore reduces the effective charge on the electrode.

Electroosmosis occurs if the cell surface is charged relative to the solution, thus adsorbing ions.

These ions migrate in the presence of the applied electric field, thereby giving rise to solvent flow

near the cell surface and solvent backflow in the center of the cell. The solvent velocity profile, with

stick boundary conditions, resulting from these flows is shown. At high field strengths and /o r high

ion concentrations, the cumulative friction resulting from the movement of the ions through the

solvent causes an increase in the temperature (Joule heating).

The fourth effect is undesirable chemical reactions that may occur at the electrodes. Siegel et al. (1978) attempted to minimize electroosmotic effects by coating the cell walls with methylcellulose. This material, however, may interact with the solute particles. Platinum black electrodes affect the pH at the surface; hence one should employ a poisoning agent such as sodium sulfide to minimize this reaction (Uzgiris, 1981). In regard to silver-silver chloride electrodes, Haas and Ware (1976) point out that colloidal silver chloride may result if voltages in excess of 10 V are used. These effects are illustrated in Fig. 9.9.

The original Ware -F lygare cell (1971, 1972) was similar to the free electro-phoresis Tiselius cell design. Uzgiris (1972) used the equivalent of a conduc-tivity cell, where the parallel plate electrodes were submerged directly into the solution. This design eliminates the electroosmosis problem, since the solution surrounds the region where the electric field is present. The third


Ti se liu s cell

GW

Haas - Wore cell

Uzgiris cell

Fig. 9.10. Three standard cell designs used in the QELS experiment with an applied elec-tric field.

design is the H a a s - W a r e (1976) curved electrode cell. In this design, the curved electrodes "focus" the electric field at the sample chamber, thereby "amplifying" the effective potential across the sample. This design allows one to have large electric fields for relatively small applied voltages, thus minimizing electrode polarization effects. The electric field strength E° is related to the applied voltage V° for this cell design by

V° ^ w G H 2GH / C H V

where G W = gap width, G H = gap height, and CH = chamber height. The three cell designs are shown in Fig. 9.10.

Example 9.2. Condensation of Spermidine and Spermine onto λ DNA Wilson and Bloomfield (1979c) reported that when ~ 9 0 % of the charge on DNA was neutralized, there was a substantial structural collapse. This


0 50 100 150 2 0 0

(Hz)

Fig. 9.11. DSS spectra and electrophoretic mobilities of λ DNA in the presence of polyamines. The concentration of the D N A was 10 μ Μ in phosphate , the temperature was 25' C, and the

scattering angle was 20°. (A) Native λ D N A (19.7 V/cm) and λ D N A with 140 μ Μ spermidine

(38.9 V/cm). (Β) Electrophoretic mobilities as a function of polyamine: ( • ) S p3 +

, ( x ) S p4 +

.

[Reproduced with permission from Yen et al. (1983). J. Phys. Chem. 87, 2148-2152. Copyright

1983 by the American Chemical Society.]

observation led Yen et al. (1983) to employ DSS methods to monitor the collapse of λ D N A upon binding spermidine ( S p

3 +) and spermine ( S p

4 +) .

Data were collected over the ranges 10° < θ < 50° and 3.5 < c p (/ig/mL) < 20 in 1 mM sodium cacodylate, 1 m M NaCl, 0.2 m M N a 2 E D T A , pH 7.0 at Τ = 25°C using a H e - N e laser (λ0 = 632.8 nm). The platinized platinum electrodes were not coated with methylcellulose, since this material interacts with the DNA. The electrophoretic mobilities were calculated from data taken at a calibrated distance of 22% of the rectangular length to minimize electroosmosis effects ("zero velocity" position in Fig. 9.9). Representative spectra are given in Fig. 9.11.

It is noted that the broader spectral linewidth for DNA with bound S p3 +

indicates a 2 - 3 times smaller hydrodynamic radius for the complexed DNA. From panel Β it is deduced that S p

4 + binds more efficiently than S p

3 + . It also

appears that the degree of neutralization and extent of condensation are the same for both polyamines. Yen et al. calculate the degree of neutralization for Sp

3+ to be 88.8% and for S p

4 + to be 90.9%.


For a system with more than one electrophoretic mobility species, multiple peaks will appear in the spectral density profile. 5(K, ω — ω0) is now of the form

The resolution R(0) of the Doppler-shifted peaks is defined as (Ware and Flygare, 1971)

Example 9.3. Simulated Resolution for a Mixture of BSA Monomers and Dimers.

It is of pedagogical interest to examine the resolution of a monomer -d imer mixture of BSA as a function of K. The following values were used in this simulation: Dmono = 6.2 χ 10~

7 c m

2/ s ; Ddimer = Dmono/\A; and the electro-

phoretic mobilities (Ware and Flygare, 1972) pmono = 2.5 χ 10~4 c m

2/ V s and

Maimer = 1-7 χ 10~4 cm

2/Vs . The results are given in Fig. 9.12.

Example 9.4. Comparison of DSS and Chromatographic Separation of

Mohan et al. (1976) used DSS methods to obtain the distribution of electro-phoretic species in human plasma. The cell was of the Tiselius design with a 15-mm-long sample chamber of inside diameter 0.83 mm that was separated from the two electrode chambers by membranes to minimize hydrodynamic disturbances and undesirable electrode reactions with the solute particles. The ELS cell was calibrated with BSA in a pH 8.7 buffer with Is = 0.005 M. Data were collected at Θ = 4° and Τ = 25-26°C with 100 V/cm < E° < 130 V/cm. There was no evidence for the dimer even though the supplier indicated 2 - 4 % contamination. (See Problem 9.1.) They reported the values (corrected to 20°C) A*BSA = 3 . 9 X 1 0

- 4 cm

2/Vs , and £>B SA = 6.14 χ 1 0 "

7 c m

2/ s .

These values were in good agreement with the accepted literature values. The human plasma was centrifuged, and the supernatant was dialyzed

against the buffer for several changes of buffer over a two-day period. The supernatant was filtered through two Millipore filters, 0.22 μιη and 0.05 μηι, into the ELS cell. The temperature and buffer was the same as in the BSA

S(K, ω - ω 0) = A'VSK2

(9.5.3)

(9.5.4)

Human Plasma.


CO

100 200 300

AlJ (hertz)

Fig. 9.12. Computer-simulated DSS spectra for a monomer-dimer BSA mixture. S(K, ω - ω 0)

was computed from Eq. (9.5.3) using the parameters £> m o no = 6.2 χ 1 ( Γ7 cm/s ; / z m o no = 2.5 χ

1 0 "4 c m

2/ V s ; D d i m er = £>m o n o/1.4; and / i d i m er = 1.7 χ 10~

4 c m

2/ V s . (The electrophoretic mobil-

ities were taken from Ware and Flygare, 1972).

calibration. The data were collected with a resolution of 0.125 Hz/channel with E° = 66 V/cm. The samples were also characterized by disc gel and free electrophoresis. These data are compared in Fig. 9.13.

Since the DSS results are obtained in far less time than in the disc electrophoresis method, the technique of DSS has potential application as a diagnostic tool in clinical analysis. Besides the well-documented and publi-cized role of lipoproteins in the pathogenesis of coronary heart disease, alterations of lipoproteins and serum lipid levels have been associated with other diseases as well. For example, hypercholesterolemia has long been recognized as a manifestation of hypothyroidism (Peters and Man, 1950; Kutty et al., 1978). Alterations in lipoproteins have also been reported to be associated with diabetes (Nikkila and Hormila, 1978), Tangier disease


I.CL,

0.5 >

σ

Ω-Ο α. ι c

"oit FI : ι •Of

- +

Λ

20 ~ι— 30 (Hz)

A

40 1

50

Fig. 9.13. Electrophoresis studies of human plasma. (A) The 1024-point power spectrum was taken at a scattering angle of 4 with an applied field strength of 66 V/cm at 2 5 - 2 6 C. (B) Disc electrophoresis of human plasma. (C) Free electrophoresis of human plasma. [Reproduced with permission from Mohan et al. (1976). Analytical Biochem. 70, 506-525 . Copyright 1976 by Academic Press.]

(Assmann et al., 1977), obstructive jaundice in infants (Lipsitz et al., 1979), uremic patients on hemodialysis (Minaminosono et al., 1978), and adult respiratory disease syndrome (Kunz et al., 1978). A summary of lipoprotein concentrations of human serum for some diseased states is given in Table 9.2.


Table 9.2

Relative Changes in Lipoprotein Content in H u m a n Serum for Some Diseased States"

Disorder H D L L D L V L D L Chylomicrons

Hyperlipoproteinicimias

Type I

Type II

Type III

Type IV

Type V

Hypolipoproteinemias

abetal ipoproteinemia

familial hypobetal ipoproteinemia

Tangier disease

Liver diseases

fatty acid, acute

fatty acid, chronic

biliary cirrhosis

portal cirrhosis

acute hepatitis

infectious hepatitis

Cancer

child

adult

Circulatory disease

acute leukemia

coronary artery disease

artherosclerosis

Heart disease coronary heart disease cardiovascular disease

Miscellaneous

infectious mononucleosis

malaria

respiratory distress syndrome (abnormal lipoprotein)

diabetes

diabetes (insulin-treated)

+ +

+ + + +

+ + + +

+ + +

+ + +

+ +

+ +

+ +

+ +

+ +

+ +

+ +

a Variations of human serum lipoprotein concentrat ions in the diseased state relative to

the normal state: (-) decrease, (—) marked decrease, ( ) essentially absent, ( + ) increase,

(4- + ) marked increase, (a) abnormal .


9.6. Application of a Periodic Pulsed Electric Field-PPEF

In order to obtain high-resolution data in the DSS experiment, the real-time data collection period must be correspondingly increased. Limitations are therefore placed on the experimental parameters to avoid Joule heating, electrode polarization, and other undesirable effects. Bennett and Uzgiris (1973) attempted to overcome these limitations on the design of the DSS experiment by collecting data during the pulse-reversal period. As in the DSS experiments, a square wave was employed, where the time for one cycle of the pulse-reversed applied field is denoted by τ τ = 2 T w , where T w is the time width of one pulse. We refer to this variation of the square-wave experiment as the periodic pulsed electric field (PPEF) experiment.

The theoretical basis of the P P E F experiment is that the sign of the applied electric field varies in item, but the magnitude remains constant throughout the data collection period. There are three time scales that must be considered: (1) the transient time, or rise time, for the switching process of the applied field, τ Ε; (2) the response time of the charged particles to the change in field, τ ρ ; and (3) the pulse width T w . τ Ε is typically on the order of 1 (T

9 s or less,

which relates to the "sharpness" of the edge of the pulse. Ivory (1984) con-sidered the response of a charged sphere to a step function electric field in the thin-layer limit, where the ion atmosphere served simply as an interface between the particle and the surrounding fluid. Ivory verified the earlier result of Morrison (1969) that transient effects were not important for times that satisfy the inequality t > 9ρρΚζ/{2η0[(ρρ/ρ0) + 0.5]}. It is clear, therefore, that since T w is generally greater than 0.1 s, the inequalities τ Ε < τ ρ « T w hold for all systems that can be examined by the P P E F technique. Hence, the effect of taking data through the pulse-reversal period simply changes the sign of E° without introducing a time-dependent component to the electric field. The particles have therefore reached their terminal velocities, as in the Doppler-shift technique; thus the notation can be shortened to vp(K). One can now extract from the time integral in Eq. (9.1.3) the product | ν ρ( Κ )

Τ· Κ | , with

the result

ν ρ ( Κ )Γ· Κ Λ '

Jo

H v p ( K )r. K |

f t w * 2 r w * 3 T W * 2 N T W

dt' - dt' + dt' - · ·· + dt' _ » 0 2 t w ( 2 N - 1 ) T W

\'p(K)T-Kdt' = μ„Ε°Κχ2Ντν, (9.6.1)

Jo

where Ν is the number of cycles for the applied field and Kx = Κ cos(0/2). The integration over all time is reduced to the integration over the time required

9.6. Application of a Periodic Pulsed Electric F i e l d - P P E F 345

for one cycle, but spaced over periods displaced in time by an integer value of τ τ . S(K, ω — ω0) has peaks located at the harmonic frequencies of the field reversal cycle at either side of the Doppler-shifted peak. Bennett and Uzgiris (1973) obtained

G+(co) 4y 8 [(γ

2 - x

2)Q - 2y(w - a>?)R

B y2 + a

2 ' T w( y

2 + Α

2) Ρ | _ 7

2 + «

2

( Ω2 _ ωα*2 _ y2 ) Q + 2yo)Jq + Τ

2 + (ω + ω*)* J '

( 9·

6·

2 )

where Β is an instrument constant; α = ω — Û)Js; y = DpK

2; Q =

— 0.5 sinh(2yrw) + sinh(yiw) C O S ( < W T w) cos(<x>pSTW); œp

s = μρΕ°Κ cos(0/2); Λ =

cosh(yiw) s in(œi w) co s (œpST W) — 0.5 sin(2orrw); and P = cosh(27iw) — C O S ( 2 C O T W) .

Note that the DSS result obtains for T w oo or ω -> 0, since only the first term survives in these limits. The amplitude from Eq. (9.6.2) is

G+(n) 4y 8 4 Β (y

2 + a«) " y

2 + a

2

"y2 - an

2 ω

2

η - œf2 - y

2 '

.y2 + aB

2 y

2 + K + < )

2

where απ = ωη — œp

s = nœm — ωρ\ ojm = 2π/ττ, and

(9.6.3)

A = [ ( - l ) " s i n h ( Y T w) c o s ( C O PST W) - 0 . 5 s i n h ( 2 Y T w) ] / { i w[ c o s h ( 2 Y T W) - 1]}.

The spectral density profile is interpreted in terms of the peak height ratios in the following manner. The value of Y is obtained by measuring the half-width at half-height of any peak. The constant Β can be eliminated by taking the ratio of the peak heights at the harmonics η and n'. The ratio G

+(n)IG

+{n')

is plotted as a function of E°KX and fitted by varying the parameter œp

s.

Example 9.5. PPEF Study on Poly(styrene). Bennett and Uzgiris (1973) reported a series of P P E F spectra on poly(styrene) spheres of diameter 0.81 pm in distilled water buffered to a pH of 8.5 by addition of N a H C 0 3 to 0.001 M. The data were obtained at θ = 16.4° for E° = 35 V/cm, where the gap width of the electrophoretic cell was 250 pm. A typical spectrum is shown in Fig. 9.14.

The half-width at half-height, Δ ν 1 /2 = y/π, was determined for each peak and averaged. The frequency F = ωρ

8/2π was determined from two ratios

involving the three peak heights Hl, H2, and H3 as labeled in Fig. 9.14, viz, = Hi/H2 and R2 = H3/H2. The agreement between Fl and F2 was good

when the frequency was less than the modulating frequency v m. Their re-sults are summarized in Table 9.3.


l) (Hz)

Fig. 9.14. P P E F spectrum for 0.81 μπι poly(styrene) spheres. The data were taken with a

15 m W H e - N e laser at a scattering angle of 16.4°. The gap between the parallel plate electrodes

was 250 //m and the applied electric field was 35 V/cm. [Reproduced with permission from

Bennett and Uzgiris (1973). Phys. Rev. A. 8, 2662-2669. Copyright 1973 by the American

Physical Society.)

It was reported that the lower the frequency, the poorer the agreement between the experimental and theoretical peak heights. The experimental peaks were higher than the calculated values, which was attributed to an extraneous source of peak strength.

Table 9.3

P P E F Results for Poly(sytrene) Spheres'*

Frequency (Hz)

G+(n) vn Δ ν 1 /2 <Δν 1 / 2> Fi F2

5.75 70.2 3.0 11.66 7.8 78.0 2.6 3.03 77.74 78.33 5.22 26.4 2.0 4.11 67.2 2.1 9.18 9.6 76.8 2.1 2.03 76.73 78.26 4.91 26.4 2.0 5.81 66.0 2.5

12.50 11.0 77.0 2.5 2.63 76.78 77.15 5.00 88.0 2.9 5.68 60.0 2.6

13.81 12.0 72.0 2.8 2.66 72.58 77.81 12.48 84.0 2.6 3.90 52.2 2.4 9.69 17.4 69.6 2.1 2.23 70.86 77.47 8.00 87.0 2.2

9.6. Application of a Periodic Pulsed Electric F i e l d - P P E F 347

Frequency (Hz)

G » vn Δ ν 1 /2 <Δν 1 / 2> Fi

5.90 52 3.1

10.95 26.0 78 2.4 2.86 76.71 74.17

2.80 104 3.1

4.72 31 2.9

9.76 31 62 2.3 2.40 63.64 74.87

6.88 93 2.0

7.80 39 3.1

10.65 39 78 2.6 2.96 72.09 70.07

2.21 117 3.2

a Reproduced with permission from Bennet and Uzgiris (1973). Phys. Rev. A.

8, 2662-2669. Copyright 1973 by the American Physical Society.

Example 9.6. Ion Exchange Effects on the Electrophoretic Mobility of Strong Acid Latex Particles.

Goff and Luner (1984) used the P P E F method to examine the effect of added NaCl on six different latex particle preparations. The designation and diameters (in μηι) of these particles were: 6M-G (0.2355); LS11 (0.185); Vdh (0.175); PMMA-1 (0.25); PVAc-1 (0.284); and PVAc-2 (0.179). These latexes were cleaned by mixed-bed ion exchange and had strong acid groups on their surfaces. P P E F data were obtained at 25°C using 1/τ τ = 5 Hz. The electrophoretic mobilities were converted to reduced zeta potentials using the tabulated results of Wiersema et al. (1966) or the analytical results of O'Brien and White (1978). A maximum was noted in the plot of ζ

τ

ρ vs. log[NaCl] .

Goff and Luner examined four models to interpret ζτ

ρ vs. [ N a C l ] : (1) a "hairy" surface of flexible coils that extended into the solution as / s was lowered, thereby reducing σ°; (2) counterion adsorption; (3) coion adsorption; and (4) ion exchange. For lack of adequate theoretical models, the hairy surface interpretation was not pursued. Counterion adsorption effectively neutralized Ζ

μ

ρ as / s was increased, but no maximum was obtained. Coion adsorption calculations could be adjusted to fit the data, but this model required the existence of adsorption sites not probed by the titration measurements. The model favored by these authors was the exchange model. Without NaCl, the counterions are H

+ and are assumed to lie within the

boundary of shear (Stern layer, cf. Fig. 9.7). The H+ ions are displaced by N a

+.

However, the N a+ ions are located in a diffuse region outside of the boundary of

shear. The charge density of the diffuse layer (dl) is σ α 1 = — σ°[1 + (β/ηΚ)~]/ [1 + (1/nX)], where the equilibrium distribution of N a

+ between the bulk

solution and the diffuse layer is Κ = [ N a+] d l/ n { ( — o°aJe) — [ N a

+] d l} , with

Table 9.3 (continued)


3 - " " ~ ~ ' Ε 2 ^

Fig. 9.15. Reduced zeta potential vs. NaCl concentration for strong acid latex particles. P P E F

data were obtained at 25°C, and the electrophoretic mobilities were converted to zeta potentials

using the results of Wiersema et al. (1966) or O'Brien and White (1978). Legend for experimen-

tal data: (El) V dh; (E2) P M M A - 1 ; (E3) PV Ac-2. (A) G o u y - C h a p m a n model \_σά =

- ( 8 0 0 0 n ; )1 / 2

s i n h ( ur

p/ 2 ) ] , where (Tl) σ° = 3.56 ßC/cm2 and (T2) σ° = 1.5 / i C / c m

2. (B) Ion

exchange model {σά = - σ ° [ 1 + (jS/wK)]/[l + (l/nK)]} with (T3) β\σ°\ = 0.20, Κ = 480, and

|σ° | = 2.4 and (T4) β\σ°\ = 0.00, Κ = 1600, and = 0.9. [Reproduced with permission from

GofTand Luner (1984). J. Colloid and Interface Sei. 99, 4 6 8 - 4 8 3 . Copyright 1984 by Academic

Press, Inc.]

η = 1000JVA[NaCl]. The parameter β is related to the efficiency of the H+

ions in contributing to σ ά 1, viz, adl/e = [ N a+] d l + j 8 [ H

+] d l. The total charge

density in the double layer is — adl/e = [ N a+] d ] + [ H

+] d l. Continued ad-

dition of NaCl results in a reduction of the charge density outside of the diffuse layer, in agreement with the G o u y - C h a p m a n model, viz, σά = - (8000n;e/cT)

1 /2 s inh(C p/2) . Plots of C p vs. added salt are shown in Fig. 9.15. In

support of the exchange model, the dielectric spectroscopy data of Fitch et al. (1987) on poly(styrene) spheres indicate that N a

+ exchange with the H

+

located in the Stern layer.

9.7. Application of a Sinusoidal Electric F ie ld—QELS-SEF

The theoretical treatment of DSS and P P E F techniques is based on the explicit assumptions that the macroions are independent and are at their terminal velocities. These restrictions are lifted in the present section, where the response of macroions to an electric field E°exp( — iœdt) is also sinusoidal in time. This QELS method is referred to as Q E L S - S E F (quasi-elastic light scattering-sinusoidal electric field). The Q E L S - S E F technique is the optical analogue to dielectric relaxation techniques, except that only the dynamics of the macroions are monitored.

9.7. Application of a Sinusoidal Electric F i e l d — Q E L S - S E F 349

Primary effects are defined as the direct response of the charged particle to E° exp( — iœdt). Because the macroions respond to the internal electric fields of the supporting electrolytes (cf. Chapter 7) and other macroions (cf. Chapter 8), the probe polyion experiences secondary effects of the E ° exp( — iœdt) through the modified dynamics of the other charged particles. The "induced" effects may be stronger than the "natural" interparticle interactions. For example, Kikuchi (1984) reports electric birefringence data on neutral and charged poly (lysine) in a 98% methanol -wate r mixture that indicate that the perma-nent dipole moment is "swamped" by the counterion-induced dipole moment.

The force on the ; t h particle due to the "primary effect" of a sinu-soidal electric field with driving frequency ωά is of the assumed for Zje[ — νΦ(ωαί)] = Z / e £ ° e x p ( — iœdt). The force acting on the ; t h particle due to other charged particles in the solution is assumed to be

f}n t(cod,r) = Zje Σ [i-Wjk(t)l + {A kexp [ - i (û> d + ftk)i]}], (9.7.1) k= ι

where N'T indicates all particles except the j th particle, A k = BkE0/

l(œok —

ωά )

2 + (1 / τ*) 2] where Bk depends upon the strength of interaction of the kth particle with the driving field, x k is the Langevin relaxation time, and ß j k is a phase factor that depends upon the relative location of the 7th and kth particles. The "natural frequency" œ0k is zero for "free particles", such as the kinetic unit composed of the polyion and associated small ion cloud, but is nonzero for the "associated" small ions within the kinetic unit. The equation of motion for the "bound" small ions is

ά^ + νΐί + ω'°Γ>= LυΤ{ωΛ't} + ZjeE°exp("i<0di)]' (9·7·2)

and the Langevin equation for the "free" kinetics units is

^ + -yj(t) = - [ f f ( c o d , i ) + Z , e E 0 e x p ( - i c ö d t ) ] . (9.7.3) dt Tj m}

J

Eqs. (9.7.1)—(9.7.3) therefore couple the motions of all of the charged particles with each other and with the external electric field. These expressions indicate that the external driving field has no effect if cod is much larger than the reciprocal Langevin relaxation time. The effect of a sinusoidal electric field is illustrated in Fig. 9.16.

To obtain S(K, α — ω 0) , additivity is assumed for the various contributions to the velocity of the j t h particle:

ν . ( ω , ί ) = VJ°(0 + v'j(œd9t) + vj(û>? - co d)exp(- ico di ) , (9.7.4)


kinetic unit of charge z£

displacement of kinetic unit

A due to ion cloud distortion

Fig. 9.16. Effect of an external sinusoidal electric field on the interparticle interactions. The

kinetic unit (cf. Fig. 9.5) has an electrophoretic mobility charge of Z£. An applied sinusoidal

electric field (not shown) affects the kinetic unit in two ways. The primary effect is to impart a

periodic displacement of the kinetic unit through space. As a result of this directional periodic

motion, the ion cloud is distorted because of: (1) the viscous drag as the macroion moves through

the solution; (2) the fact that the counterions respond to the electric field in a direction opposite to

that of the polyion; and (3) the finite response time of the ion cloud to the change in direction of

the parent polyion. The displacement of the centers of positive and negative charge within the

kinetic unit is indicated in the figure. The combined effect of these two motions is to generate an

oscillating multipole electric field that affects the motions of the other kinetic units in the solu-

tion. As shown in the figure, the fluctuating multipole fields from kinetic units Β and C arrive at

kinetic unit A with different phase factors because of their different distances from A.

where:\°(t) is the velocity of the jth particle in the absence of an external field that includes the "natural" interparticle interactions; \j(œd9t) is the addi-tional velocity component due to secondary effects; and v

ej(co° — cod) is the

additional velocity component due to primary effects. The matrix D a p p(co, t) is therefore of the form given by Eq. (9.1.1). Since there is at present no com-plete theory that includes all of the possible interparticle interactions implicit in Eq. (9.7.1), we adapt the results of Schmitz (1979,1983) for the case in which vp(r) = Vp(0)exp( — œdt). The expression for ACp(K, t) given by Eq. (9.1.4) thus becomes

ACp(K,i) = A C p( K , 0 ) e x p [ - D p

e fX

2i ] e x p [ z ( > ; - 1)], (9.7.5)

where ζ = μρΕ°Κ cos(0/2)/cüd = cop

s/œd, y = exp( — iœdt\ and D p

e f is the appar-

ent diffusion coefficient obtained from the Q E L S - S E F method. Expansion

9.8. Frequency and Field Strength Dependent D p

ef and μ ρ 351

of the exponential exp[z(y — 1)] as a power series in z(y — 1) and rearrange-ment of these terms to a power series in y

n yields, as the coefficients of y

n,

the Poisson distribution, An = (zn/n!)exp( —z). Thus,

S(K, ω-ω0) = <AC p(K,0)AC p(K,0)>

χ

oo Γ ό ο

Σ An exp[ — D p

e fX

2i ] exp[ — ί(ω — ω0 — nœd)t]dt,

< = 0 Jo S(K, ω-ω0) = <AC p(K,0)AC p(K,0)>

X J o

A" ΙΟ?Κ*ν+(ω-ω0-ηωά)

2-

( 9 J-

6 )

The electrophoretic mobility is obtained from the ratio of adjacent peak heights, HJHn_1 = AJAn_l = z/n9 hence,

H"

V 2y (9.7.7)

9.8. Frequency and Field Strength Dependent D pe f and μρ

Early Q E L S - S E F studies on mononucleosomes (Schmitz, 1982a; Schmitz et al., 1982b; cf. description of these particles in Example 3.6) in 1 m M caco-dylate revealed linewidths that were too narrow to be attributed to D p

expected for RH ~ 55 Â. Likewise, the plot of l n ^ v j vs. l /v d for the 1 m M cacodylate data exhibited curvature, thus indicating a frequency-dependent μ ρ(ω). The asymptotic limits were μρ(ω - • oo) = 4.15 χ 10~

4 c m

2/ V s and

μ ρ(ω 0) = 1.54 χ 10~4 c m

2/ V s . In 10 m M cacodylate, the plot was linear

and yielded j u n u c l e 0s o m e = 0.96 χ 10~4 cm

2/Vs . Q E L S - S E F data for T7

DNA for Is = 0.1 mM, 5 mM, and 10 m M at 20°C also indicated a dispersion μρ (Schmitz, 1983). It was reported that μ Τ 7 ΌΝΑ(ω -+oo) = 4.7 χ 1 0 "

4 cm

2/Vs ,

which is comparable to ^ C T D N A = 5.9 χ 1 0 "4 c m

2/ V s for calf thymus D N A

in 4 m M N a+ at 20°C (Hartford and Flygare, 1975) and / i p l a s m id = 3.5 χ

1 0 "4 c m

2/ V s for plasmid λ DNA in 2 m M N a

+ at 25°C (Rhee and Ware,

1983). The Q E L S - S E F data for mononucleosomes and poly(nucleosomes) are shown in Fig. 9.17. Note that the peak located at œd appears to be "taller" and "more narrow" than might be anticipated from the peak dimensions at the higher harmonics. This observation is consistent with that of Bennett and Uzgiris (1973) on the P P E F study on poly(styrene) (cf. Example 9.5).

The dispersion in Ds

p

{(œ) and μ ρ(ω) is attributed to the terms f}

n t(cod,i)

in Eqs. (9.7.2) and (9.7.3), which were not included in the simple theory out-lined above.

ο. lOmM

Π — I I I I Γ

Ο 2 0 4 0 6 0 \)ύ ( H Z )

V d ( H 2)

Fig. 9.17. Frequency-dependent diffusion coefficient and electrophoretic mobility. (A) The

Q E L S - S E F spectrum of a purified preparat ion of chicken erythrocyte mononucleosomes was taken at an applied field strength of E° = 8.95 V/cm at a scattering angle Θ = 7° and a tem-perature of 20°C. Clearly illustrated are peaks at the driving frequency (vd = 10 Hz) and the overtones.The 60 Hz peak is instrumental and is used as an internal s tandard for the peak height. (B) l /v d dependence of lnfHjVj) for mononucleosomes in two ionic strength buffers. The slope, which is proport ional to the apparent electrophoretic mobility, is frequency-dependent for the 1 m M data. [Reproduced with permission from Schmitz et al. (1982b). Chem. Phys. 66, 187-196. Copyright 1982 by Nor th-Hol land Publishing Co.] C: Dependence of the peak width on the driving frequency for the poly(nucleosome) system. The da ta for the poly(nucleosomes) in 1 m M cacodylate buffer at 21°C were collected at a scattering angle of 8° and E° = 22.4 V/cm. [Reproduced with permission from Schmitz et al. (1982a). In Biomedical Applications of Laser

Light Scattering. Pp. 61 -80 . (D. B. Sattelle, W. I. Lee, and B. R. Ware, eds.) Elsevier Biomedical Press, Amsterdam. Copyright 1982 by Elsevier Biomedical Press] . D: The apparent diffusion coefficient computed from the linewidths as a function of the driving frequency.

9.9. Sinusoidal Enhancement of Correlated S t ruc tures—SECS 353

9.9. Sinusoidal Enhancement of Correlated Structures—SECS

The narrow linewidth in Fig. 9.17 is characteristic of an "induced attractive interaction" between the macroions. The ionic strength dependence of μ ρ(ω) can be interpreted in terms of an increase in λΌΗ, thus giving rise to a significant separation of positive and negative charge centers as the ion cloud oscillates in time (fluctuating dipole). Long-range interactions through an oscillating dipole have been proposed by Fulton (1978a, 1978b) to re-sult in long-range correlations in the interpretation of dielectric relaxation measurements.

It is to be emphasized that the correlated structures in the Q E L S - S E F experiment represent a delicate balance between hydrodynamic and fluctuat-ing dipole forces, in a manner similar to that of small ions distributed about an isolated sphere as in the F ixman-Jaganna than model [cf. Section 9.4 and Eq. (9.4.12)]. It is expected that, at sufficiently large values of E°, these correlated structures will be disrupted by hydrodynamic shear effects. This is realized in the poly(lysine) system to be discussed in Section 10.4. It is also emphasized that these effects attributed to the above mechanism are "damped out" as ωά exceeds the reciprocal relaxation times associated with these motions, l / i k . Homodyne Q E L S - S E F studies on poly(styrene) latex spheres reported by Hubbard et al. (1989), for example, do not show anoma-lous behavior in D p

e f for E° > 30 V/cm and v d > 50 Hz. However, they

reported that μ ρ was smaller than the DSS value and that μ ρ decreased as v d increased over the range 100 Hz to 1000 Hz. These latter observations are consistent with the results of Hurd et al. (1985) on poly(styrene) latex spheres in which V° varied at v d = 1000 Hz. The spheres, of diameter 1.053 χ 1 0

_ 4c m and σ° = 100 mV, were placed on a microscope slide with

conducting stripes separated by a 2 χ 10~2 cm gap. At V° = 5 V, short

"transient strings" were observed to form. The continuous break up and reformation of these strings was attributed to the competition between Brownian motion and the induced dipole moment that results from deforma-tion of the ion cloud and the dielectric material of the particle. As V° 15 V, the strings grew longer ( ~ 1 0 0 particles), but adjacent strings continued to repel each other. At 25 V, the long strings began to attract each other to form side-by-side structures.

The question may arise as to orientation contributions in the above electric field profile, especially in the low-frequency range employed in the Q E L S -SEF experiment. Studies in an a c pulsed electric field on Wyoming sodium bentonite (Jennings and Jerrard, 1965), D N A (Scheludko and Stoylov, 1967; Jennings and Plummer, 1970; Fujikado et al. 1979) and Ε. coli (Jennings, 1981) indicate that orientation effects do make a significant contribution to 7 t i l s. In order to orient these particles significantly, however, fields larger than those used in the Q E L S - S E F experiments reviewed in this must be employed


(E° > 100 V/cm). Furthermore, orientation should contribute to the line-widths at low frequencies of the driving field and diminish as the frequency is increased above 1/τθ, where τθ is the orientational relaxation time. This prediction contradicts the experimental observations in which the linewidth broadens as the frequency is increased.

Folim G. Halaka (1988, personal communication) suggested an interesting combination of high- and low-frequency electric fields on suspensions of neutral dielectric particles. As in the observations of Hurd et al., the high-frequency electric field would induce dipole moments within the dielectric material of the particle but would not cause the particle to migrate. This field is referred to as the "inducing field". Superimposed upon this field is a second electric field, which would interact with the induced fluctuating dipole and cause the particle to migrate. This field is referred to as the "migrating field". We can further speculate upon the effect of the relative strengths of the two fields. Based upon the observations of Hurd et al., low-amplitude inducing fields may result in weakly associated strings that are disrupted by the hydrodynamic shear generated by "dielectrophoresis" of the particles upon application of the migrating field. Large-amplitude inducing fields, however, may result in strongly associated strings that remain intact upon application of the migrating field.

9.10. A Comparison of DSS, P P E F , and Q E L S - S E F Techniques

The principal differences in the DSS, P P E F , and Q E L S - S E F experiments lie in the type of applied electric field and the data collection period, as shown in Fig. 9.18.

It is noted that Eq. (9.7.5) becomes equivalent to the DSS result of Eq. (9.1.5) for V p

7 · Κ = μρΕ°Κ cos(ö/2) in the limit ωά 0. It is clear from comparison of

the spectra, however, that the peak locations and linewidths in the Q E L S -SEF experiment do not asymptotically approach the DSS spectra in the limit ωά -> 0. This is because of the differences in the physical processes between the two experiments. In the DSS and P P E F experiments, the particles attain vp(oo), whereas in the Q E L S - S E F experiment the particles are constantly being accelerated by the external field. It is for this reason that the peak locations in the Q E L S - S E F profile are at ηωά(η = 0,1,2, . . . ) [cf. Eqs. (9.7.6) and (9.7.7)] and are not Doppler-shifted as in the DSS and P P E F methods.

The advantages and shortcomings of these three techniques lie in the type of information desired about the system under examination. Clearly, Q E L S -SEF methods must be used if frequency-dependent properties are to be examined. Polydispersity in the electrophoretic mobilities, either by charge polydispersity of a single species or the presence of two or more species,

9.10. A Compar ison of DSS, P P E F , and Q E L S - S E F Techniques 355

Fig. 9.18. Schematic diagram of the operation and spectra for the D S S , PPEF, and Q E L S -

SEF methods. Time is along the ordinate axis in curves A and B. Curves labeled A represent the

period that da ta are collected in relationship to the application of the applied electric field

indicated by the curves labeled B. The DSS spectrum is drawn with an asymmetric peak to

emphasize that the charge and /o r size polydispersity can be detected by this method. The peaks

are Doppler-shifted by an amount characteristic of the macroions in the DSS and P P E F meth-

ods, and harmonics of the modulat ion frequency appear in the P P E F spectrum. The peaks in

the Q E L S - S E F method are frequency-shifted by an amount equal to the driving frequency

and its harmonics.

is most easily detected by the DSS method. Given a monodisperse size distribution, charge polydispersity is characterized by an artificially broad-ened peak, which may also be asymmetric about the peak maximum. For very large particles, where DpK

2 is much smaller that the instrument resolution, the

spectral linewidth is a direct measure of the polydispersity. Uncertainties arise, however, for small particles in regard to the "true" half-width at half-height. Since the frequency shift and linewidth have different dependences on X, experiments must be carried out as a function of θ to assess the magnitude of the polydispersity (cf. the BSA simulations in Fig. 9.12). Polydispersity cannot be detected in this manner in the Q E L S - S E F experiment, since all of the electrophoretic mobility species lie under the frequency-shifted peaks. As a consequence, however, the peaks are symmetric about the central frequency. One can therefore use the Q E L S - S E F technique to obtain weighted average


values for the diffusion coefficient and the electrophoretic mobility. Using the identity Λ/,·/,· = <AC,-(K, 0) AC,(K, 0)>, where l} is the intensity increment due to an isolated y'th particle, S(K, ω — ω 0) is

Df(œ)K2

00 2π

S(K, ω - ω 0) = Σ I ^ / ^ , j ? Ä ¥- 7 — —ψ- (9.10.1) η = ο j \_vj (ω)Κ J + (ω — ω0 — ηωά)

Assuming that the peak resolution is such that there is no overlap of neighboring harmonic peaks, the relative height of the nth peak at the

A

83 88 93 98 Β

1 :>: ·.

2 0 4 0 6 0 8 0 1 0 0

Ό ( H z )

Fig. 9.19. Comparison of the DSS and Q E L S - S E F line widths for poly(lysine). (A) The

Q E L S - S E F data were obtained for ( l y s ) 3 8 00 at a concentrat ion of 1 m g / m L at 20°C in 0.0005 M K C l at 0 = 6 and E° = 15.6 V/cm. The spectrum was frequency-shifted by 90 Hz, the driving frequency of the applied electric field, in order to examine in more detail the breadth of the peak. [Reproduced with permission from Schmitz and Ramsay (1985c). Biopolymers. 24, 1247-1256. Copyright 1985 by John Wiley and Sons.] (B) The DSS data for ( l y s ) 9 46 at a concentrat ion of 3 m g / m L with no added salt at 22°C and 0 = 14.75° with E° = 15.8 V/cm. [Reproduced with permission from Wilcoxon and Schurr (1983b). J. Chem. Phys. 78, 3354-3364. Copyright 1983 by the American Institute of Physics.] The above data were taken under comparable conditions. Note that the Q E L S - S E F peak is symmetric about the frequency of the peak maximum. One possible source of the asymmetric shape of the DSS spectrum is polydispersity in size and /o r charge. Another possibility is a distortion due to electroosmosis. The theoretical spectra of DSS with electroosmosis of Palberg and Versmold (1989) show a similar shape even though the hypothetical sample is monodisperse.

9.11. Hydrodynamic Fields: Constant and Oscillatory Solvent Flow 357

frequency ω — ω 0 = η ω ά is

_J

Df(ü))K

DSS and Q E L S - S E F spectra for (lys)„ are shown in Fig. 9.19

9.11. Hydrodynamic Fields: Constant and Oscillatory Solvent Flow

Yeh and Cummins (1964) first reported QELS experiments on poly(styrene) particles in a system in which the solvent was in constant flow. These results indicated that the relative velocity of the solvent flow could be determined from the Doppler shift in the power spectrum in accordance with the general formulation given by Eq. (9.1.6), i.e., Δ ω = \'p

T · K. The determination of the

particle velocity relative to an observer is referred to as laser velocimetry. Practical applications of this technique include monitoring blood flow in vivo (Tanaka and Benedek, 1975; Stern, 1975; Powers and Frayer, 1978; Hamilton et al., 1982; Nilsson et al., 1982). The solute particles therefore "go with the flow" of the solvent. Since each friction center tends to travel with the local velocity of the solvent, gradients in the solvent velocity tend to distort long-range solution structures and align rigid and flexible macroparticles.

Ackerson, Clark, and coworkers have examined the temperature- [Schaefer and Ackerson, 1975] and shear-induced [Clark et al., 1979; Clark and Ackerson, 1980; Ackerson and Clark, 1981, 1984; Ackerson et al., 1986] "melting" of colloidal crystals that are assumed to be "charge-stabilized" [Hoffman, R. L., 1972, 1974; Buscall et al., 1982; Lindsay and Chaikin, 1982; Ackerson and Clark, 1983b; Clark et al., 1983; Tomita and van de Ven, 1984; Ackerson et al., 1985; Clark et al., 1985]. Of particular interest in this text are the studies of Ackerson and Clark (1984) on highly charged poly(styrene) spheres of diameter 0.220 μτη at various shear rates (S). The solutions were sufficiently dilute (0.17 wt%) that visible light could be used and video techniques could be employed to monitor the various ordered structures by the changes in the Bragg diffraction spots as a function of shear rate (cf. the photographs of video frames shown for the 2D system in Fig. 8.2). It was reported that at zero applied shear, the Bragg diffraction pattern for the 3D system was that of the bcc crystalline structure. The spots were reported to shift horizontally by an amount dependent upon the shear rate. At sufficiently large shear rates, the difference in the hydrodynamic force exerted on the macroions in two different velocity planes was too great to allow the 3D structure to be maintained. The diffraction spots took on the character of a distorted 2D hcp or fee structure (bottom photograph in Fig. 8.2). Temporal oscillations in the spots were indicative of the dynamic action of the solvent


flow on these particles. This transition was accompanied by a reduction in the solution viscosity ("shear thinning"). An increase in the shear rate disrupted these 2D structures, and "strings" appeared to form along the direction of the solvent velocity vector. At very large shear rates, there was no solution structure, and the liquidlike pattern shown in the top photograph in Fig. 8.2 resulted. This transition to the liquidlike state was accompanied by an increase in the viscosity ("shear thickening"). These structures were reported to be reproducible. Neutron scattering studies at higher macroion concentrations indicate that the equilibrium structure is a closest-packed array rather than the bcc array (Ackerson et al., 1986). Otherwise the more concentrated solutions behave like the more dilute solutions.

The standard interpretation of the S dependence of the crystalline structures is the competition of direct repulsive interactions between the macroions and the hydrodynamic force exerted by the flow field. Ackerson and Clark (1984) expressed the distorted distribution function as g

i2\r) =

0 ( 2 )Κ Γ> ες[1 — (

χζ7Ιγ2)τ

aS]}, where the local relaxation time i a is related to the solvent viscosity and local shear modulus G by τ 3 = η0/ο. Fourier transformation results in an elliptically distorted structure factor.

where ε = zaS. / t i ls for poly(styrene) particles and a plot of ε vs. S are shown in Fig. 9.20. The local relaxation times obtained from the slope were τ 3 = 0.0084 ± 0.001 and 0.013 + 0.003 s, respectively, for particles of diameters 0.109 jum and 0.234 μπι.

An asymmetric particle placed in a shear field is induced to rotate (Flory, 1953). Lomakin and Noskin (1980) examined the effect of particle asymmetry on S(K, ω — ω 0) for a particle subjected to a flow gradient. The velocity gradient was generated by a concentric cylinder arrangement similar to that shown in panel Β of Fig. 9.20. The incident laser beam, which defined the ^-direction, was perpendicular to the cylinder surface. The solvent flow was perpendicular to the laser beam and defined the x-axis. The orientation of the particle was defined by its symmetry axis and represented by the vector n. P(Kd

c) is therefore a function of n T · Κ and changes in time as the orientation

vector η changes in time. The asymmetry of the particle is indicated by the anisotropy parameter a, which can take on the values — 1 < α < 1, where α = 1 for a thin disk, α = — 1 for an infinitely long cylinder, and α = 0 for a sphere.

The angular velocity of the particle was assumed to be proportional to the velocity gradient, which for the stated geometry lies in the x-y plane and is represented by Wxy = W. The time dependence of η found by Lomakin and

(9.11.1)

9.11. Hydrodynamic Fields: Cons tant and Oscillatory Solvent Flow 359

Β

A' B' C

© 9 D' E' F'

C S {Hi)

D Fig. 9.20. Shear-induced melting of Colloidal crystals. (A) The rocking cuvette configuration

for shear gradient formation. The rocking motion and resulting shear gradient profile are shown.

(B) Concentric cylinder configuration for the formation of a shear gradient. (C) Scattered light

intensity distributions observed from a colloidal suspension of charged plastic spheres in a shear

flow (indicated by the arrows). The top row ( A ' - C ) are the results for 0.109 μπι diameter particles

with S = - 1 1 . 9 Hz (1), S = 0(2), and S = 11.9 Hz (3). The bo t tom row ( D ' - F ' ) are the results for

0.234 μπι diameter spheres with S = - 15.4 Hz (4), S = 0 (5), and S = 15.4 Hz (6). (D) The

measured value of ε as a function of shear for 0.109 //m diameter spheres. Both da ta from the

rocking cuvette (O) and concentric cylinder cell ( · ) are presented. The line drawn through

the data represents a relaxation time of 8.4 ms. [Panels C and D are reprinted with permission

from Clark and Ackerson (1980). Phys. Review Lett. 44, 1005-1008. Copyright 1980 by the

American Physical Society.]

Noskin was

μ /2

(9.11.2)

(9.11.3)

(9.11.4)


where A and τ 0 are constants defined by the initial conditions, and τ = W(l — a

2)

1/2t. If one now employs the homodyne technique, the Doppler

shift due to the translational movement of the particle with the solvent is not observed. One has, however, a periodic orientation of the particle in the flow field that depends only on the velocity gradient and not the orientation of the particle in the gradient. According to Lomakin and Noskin, this periodic motion gives rise to an additional peak in the spectral density profile that is shifted in frequency by Δω = (W/2)(\ — a

2)

112 and has a linewidth that is

proportional to the effective rotational diffusion coefficient.

Lomakin and Noskin chose as a test system E. coli because of it large axial ratio (2 μηι in length and 0.8 μνα in diameter). Spectra for a suspension of E. coli are shown in Fig. 9.21, and contain the frequency-shifted peak. Also shown in

90 θ 60

100 200 300 400 ω - ω0 (Hz) Fig. 9.21. Power spectrum of the scattered light intensity. Spectra 1-3 obtained at the

scattering angles 0 of 90°, 60°, and 45°, respectively, for E. coli suspensions in the flow with velocity gradient; spectrum 4 for the same suspension when the rotor is fixed (scattering angle of 90°); and spectrum 5 for a latex suspension in a flow velocity gradient (scattering angle of 45°). [Reproduced with permission from Lomakin and Noskin (1980). Biopolymers. 19, 231-240. Copyright 1980 by John Wiley and Sons, Inc.]

9.11. Hydrodynamic Fields: Constant and Oscillatory Solvent Flow 361

Fig. 9.21 are the results for poly(styrene) spheres, where no frequency-shifted peak is observed.

Oscillatory hydrodynamic shear flows have been applied in birefringence measurements to obtain orientational relaxation times for rigid and flexible polymers [Thurston and Schräg, 1966, 1968; Thurston and Peterlin, 1967; Thurston and Münk, 1970; Thurston and Wilkinson, 1973; Miller and Schräg, 1975; Wilkinson and Thurston, 1976]. The theoretical basis of the oscillatory flow birefringence is Zimm's beaded string model (Zimm, 1956) for linear flexible polymers. Stasiak and Cohen (1983) examined the effect of an oscil-lating hydrodynamic flow field on Gj(K, t) for Rouse coils and rigid rods. The center-of-mass was treated separately from the internal motions.

The geometry of the experiment is that the solvent flow is along the x-axis and the gradient along the z-axis. The scattering vector is oriented at angles θ and φ, respectively, to the ζ and χ axes. The superimposed velocity is assumed to be along the z-axis, with the gradient in the z-direction, and has a frequency of oscillation ω, \x(t) = yzcos(œi), where y is the maximum shear rate. The correlation functions derived by these authors was

GiiK,*) = e x p [ - ß ( M 0 ) ] exp[-i4(M 0)(i - *o)]. (9.11.5)

For the Gaussian coil of mean-square dimension {R2}, the parameters are

B(t9t0) = [sin(û)i) - 8ΐη(ωί0)]2/4τ

2;

A{t,t0) = -—-[1 + sin(0)cos(0)/o(i , t o)

+ sin2(ö)cos

2((/>)6f0(iô)]

with τ» = l / D p K2 and r s = l/(yKx(R

2}

i/2);

cos(cot) — cos(coi

and

fc

Go(t) =

0(5 *θ) ~~ :

ω _

sin(cot) + œt

ο)".

ω 0.5 + sin

2(<x>0

sin(2coi) — sin(2eoi0)

4o)t

+ œ(t - t0) sm(cot)[cos(cot) — cos(œi 0)]

(9.11.6)

(9.11.7)

(9.11.8)

The correlation function for the translational diffusion mode is obtained by setting t0 = 0, and the time dependence of the total intensity is found by setting t0 = t.


The correlation functions shown in Figs. 2 and 3 of Stasiak and Cohen (1983) indicate that "normal" monotinic decay requires the inequalities ωτΌ » 1 and y/ω « 1. Physically this means that the changes in the hydro-dynamic flow field are too rapid for the particle to follow. At the other ex-treme, the particles can follow the oscillations in the flow field, and GX(K, t) exhibits oscillations with a frequency equal to that of the hydrodynamic flow field.

Ackerson and Pusey (1988) reported light scattering studies on solutions of coated poly(methylmethacrylate) spheres in the presence of shear fields. The intensity pattern indicated a change from "fluidlike" to "solidlike" (the forma-tion of strings similar to those discussed in Section 9.9) solution structures in the presence of a 1 Hz oscillating shear field.

9.12. Mechanical Excitation of Gels

Nossal (1979) has proposed a theory for gels formed in a rectangular cuvette with sides a, fc, and c. The frequencies of the standing waves are given by coaßy = csKaßy, where c s is the speed of sound in the medium, and Kaßy = π[(α/α)

2 + {βlb)

2 + ( y / c )

2]

1 / 2. For slippery gels, c s is the "longitudinal speed

of sound", c l o ng = [ ( / + 2p)/Pmy/2 = { [X c + f G s ] / p m ]

1 / 2, where Kc and Gs

are the compressibility and shear moduli, respectively. For adherent gels, cs

is the "transverse sound speed", c t r an = (p/pm)i/2. In these expressions pm is

the material density, and λ and μ are the Lamé coefficients (Sommerfeld, 1950), which are related to the volume displacement U, and

~dUY. dUy

dXj Γ.Υ,· + λ<$0.ν

Γ·υ (9.12.1)

is the elastic stress tensor σχ. Xj. If g0(t) = g0 cos(codf), then the amplitude of the displacement vector U is the sum of all the "normal mode" contributions for a driven harmonic oscillator. The amplitude of the displacement for ωά near the lowest resonance frequencies (a = 0) is

U A T'

t } π

2 ( Δ ω

2, , )

2 + (ω/τ0βγ)

2 9 ^ '

U'

Z )

odd

where the relaxation time τοβγ = ρηχ/{ξ + ηΚΐβγ), and ΑωΙβγ = ω^βγ — ω2.

The amplitude of the distortion oscillates in time with frequency ωά. Because the oscillations are damped, however, the magnitude of the amplitude is relatively small except when cod approaches a resonance frequency ωαβγ. Hence

9.13. Mechanical Excitation of Gels 363

for ω -» ωαβγ the amplitude is (Gelman and Nossal, 1979)

l J J{ 16flo (ω/ταβγ)2 (9.12.3)

1 1 ~ π

2βγ (Aw

2

ßy)2 + (ω/τ»βγ)

2

where the half-width is ταβγ = ρη/(ξ + ηΚ2

βγ). Brenner et al. (1978) examined the concentration dependence of the reso-

nance frequencies for poly(acrylate) and agarose gels. Gelman and Nossal (1979) reported studies on the temperature and concentration dependence of the shear modulus of poly(acrylamide) gels. Mechanical excitation was effected by means of an audio speaker connected to the light scattering cells. A schematic diagram of the experimental arrangement and representative spectra for poly(acrylamide) gels are shown in Fig. 9.22.

I 1 1 1 1 τ -ο 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0

f r e q u e n c y ( H z )

Fig. 9.22. Light scattered from mechanically stimulated moderately stiff gel. (A) The experi-mental arrangement. The long axis of the cuvette usually is positioned perpendicularly to the Bragg scattering vector. (B) Spectrum of mechanically excited gel. The top record was obtained while the sample was excited by a signal obtained from a sweep generator that was oscillating over the range 161 to 341 Hz; prominent resonance are seen to occur at 287, 307, and 351 Hz. The bot tom tracing was obtained while the generator was sweeping over a range of 48 to 102 Hz; except for the spurious 60 Hz signal, no resonances are discernible. The sample was 7.5% poly(acrylamide) in a 1 χ 1 χ 4 cm cuvette, Τ ~ 4.8°C, and Θ ~~ 22°C. [Reproduced with per-mission from Gelman and Nossal (1979). Macromolecules. 12, 311-316. Copyright 1979 by the American Chemical Society.]

364 9. EXTERNAL PERTURBATIONS

CO

0 10 2 0 30 4 0 50

Τ ( ° C )

Fig. 9.23. Temperature dependence of the diffusion coefficient. The top curve pertains to a 7.5% poly(acrylamide) sample, and the lower curve is data for a 5% gel. [Reproduced with permission from Gelman and Nossal (1979). Macromolecules. 12, 311-316. Copyright 1979 by the American Chemical Society.]

Resonances are clearly shown at ν = 287, 307, and 351 Hz. Gelman and Nossal argued that the narrow linewidths provide evidence that the gel is not moving through the solvent, since typical values of the friction factor ( ~ 1 0

1 0 dyn-s/cm) would lead to very broad linewidths. It was concluded

that the mechanical excitation measured the rigidity of the bulk material, gel plus solvent.

D a pp at Δ ω 1 / 2 is defined by (Gelman and Nossal, 1979)

4

A p p —

ξ ·

It was found that D a pp ~ c p

7 5, which is the de Gennes predicted value for

semidilute solution conditions (de Gennes, 1971, 1976a, 1976b; cf. Sec-tion 5.15). The T-dependence of D a pp for 7.5% and 5% gels is shown in Fig. 9.23.

9.13. Diffusion under High-Pressure Conditions

Patterson et al. (1982) studied the effect of pressure on the glass transition (cf. Section 5.22) of poly(styrene). A block of polymer was introduced into the pressure cell and annealed at 125°C for eight days to remove strains. QELS data were taken over two decades at a time and spliced together. In all, the data were collected over a time range of 10~

6 s to 10 s. The correlation func-

9.13. Diffusion under High-Pressure Condit ions 365

I 1

- 3 H

- 5 4

0 5 10 15

P / 1 0 0 0 ( p s i )

2 0

Fig. 9.24. Plot of log<t) vs. Ρ for poly(styrene) near the glass transition. The ranges represent

the variations from run to run and from time to time observed for the data, including the combined

correlation functions. The temperature was 146°C. [Reproduced with permission from Patterson

et al. (1982). J. Chem. Phys. 77, 622-624 . Copyright 1982 by the American Institute of Physics.]

tions were fitted to the Wil l iam-Wat ts function, and <τ> was determined [cf. Eqs. (5.22.1) and (5.22.2)]. The Wi l l iam-Wat t s fit with β = 0.34 was identical to that shown in Fig. 5.22. <τ> vs. pressure at 146°C is shown in Fig. 9.24.

The "activation volume" was calculated from

The value AV — 318 c m3/ m o l e was noted to be almost twice that of poly-

(methyl acrylate) obtained from dielectric relaxation methods (Williams, 1966). The difference was attributed to the larger side groups in poly(styrene).

Fytas et al. (1984) used high-pressure techniques to separate two relaxa-tion times in the poly(phenylmethyl siloxane) (PPMS) system. The "fast" relaxation process has a Wi l l iam-Wat ts parameter of β ~ 0.4, whereas the "slow" relaxation process has a value of β ~ 0.8, which was attributed to local polydispersity in the sample. An activation energy of A £ a ct ~ 1.4 Kcal/mole was obtained from the T-dependence of the relaxation pro-cess, which is close to A £ a ct for viscous flow.

Nystrom and Roots (1983) studied the dynamics of BSA under high pres-sure. The sulfhydryl-blocked BSA was stated by the manufacturer to be 99% monomer; hence it was used without further purification. The solvent was buffered at pH 4.7 (isoelectric point) with / s = 0.12 M. DLS measurements

(9.13.1)

366 9. EXTERNAL PERTURBATIONS

70H A Β

ο ο Q_ O

CL Q. σ \

20-0 1000 2000 3000 4000 0 1000 2000 3000 4000

Ρ ( A T M )

Fig. 9.25. Pressure dependence of the diffusion coefficient of BSA at 25°C. (A) The con-

centrations ( k g / m3) for the da ta are ( · ) 7.86, ( • ) 33.8, ( χ ) 52.7, and ( Δ ) 79.9. The dashed line

is the expected curve if only the pressure dependence of the viscosity is taken into account.

(B) Illustration of the pressure dependence of the diffusion coefficient when the pressure is first

increased (solid curves) and then decreased (broken curves). Symbols are the same as in panel A.

[Reproduced with permission from Nystrom and Roots (1983). J. Chem. Phys. 78, 2833-2837.

Copyright 1983 by the American Institute of Physics.]

were carried out at 25°C, and the GX(K, t) were analyzed by the cumulant method. D a pp plotted as a function of the pressure for several concentrations is given in panel A of Fig. 9.25. Also shown is the anticipated behavior of D a pp based on changes in the viscosity with pressure, i.e., Dapp(P) = ΟΆρρ(\)η0{\)/η0(Ρ). The curves in panel Β were taken 12 hours after the pressure was released.

D a pp first increases with the pressure, attains a maximum, and then decreases as the pressure is further increased. The behavior of the viscosity with pressure cannot account for the measured decrease in Z) a p p. Nystrom and Roots interpreted the change in D a pp as pressure-induced aggregation, which was assumed to be irreversible since the initial value of D a pp was not recovered upon release of the pressure.

9.14. Reaction Kinetics

The classical method employed to study the kinetics of equilibrium systems is to apply perturbation and then to observe the relaxation to the new equilibrium position. The effect of the perturbation can be assessed in the concentration jump expressions for the bimolecular association reaction. The initial and final concentrations are given as first-order in the pertur-bation concentrations that obey the relationships ACB(t) = ACA(r) = -ACc( i ) . Hence: CA(f) = < C A> e q + ACB(i); CB(i) = < C B> e q + ACB(i); and

9.14. Reaction Kinetics 367

C c(t) = < C c > e q — ACB(t). For generality it is asssumed that k((t) = k° + Akf(t); and kT(t) = k° + Akr(t). The standard form of the forward and reverse reaction in the concentration leads to the rate of decay of ACB(t),

d[_ACB(tJ]

dt = - K 1 ( ( ) A C B ( i ) - M i ) [ A C B ( t ) ] : (9.14.1)

where the effective first-order decay rate R^t) is

R^t) = Mi)C<CA> eq + < C B > e q] + kr(t). (9.14.2)

The solution ACB(f) is found by first solving for the inverse function ACB(t) = l/U(t), whereupon substitution into Eq. (9.14.1) and multiplication by U(t)

2 gives dU(t)/dt = RÛit) + kf(t), which has the general solution

U(t) = exp R^dt'

exp kf(t')dt' + 1/(0) >. (9.14.3)

If k{(t) and k2(t) are independent of time, then

ACB(i) = ACB(0)

1 + y [ l - e x p i - R ^ ) ] e x p i - Ä ^ ) , (9.14.4)

where y = ACB(0)k{/Ri. The amplitude therefore decreases from an initial value of ACB(0) to a final value of ACB(0)/(1 + γ). In order for the decay of the amplitude not to interfere with the interpretation of the data, the quan-tity y must obey the inequality y « 1, or ACB(0) « < C A> e q + < C B > e q + X d i s s. Emphasizing the fact that <C 7> e q is the equilibrium concentration that results at the end of the perturbation relaxation, this inequality places clear restrictions on the systems that one can study by conventional perturbation methods without having to concern oneself with second-order concentration effects.

If the reactants and products differ significantly in their respective polariz-abilities, or scattering powers, then the chemical reaction itself is a source of fluctuations in /(K, t). It is this difference in polarizabilities that was the focus of the early theoretical studies on reaction kinetics (Blum and Salsburg, 1968; Berne et al., 1968; Schurr, 1969; Berne and Giniger, 1973). The funda-mental expression obtained from these theories is that the linewidth is of the form

Δω - DK2 + fcf + kr. (9.14.5)

A study on the ionic association of M g S 0 4 has been reported by Yeh and Keeler (1969), but these results have not been verified. There has not yet been any other experimental studies on association kinetics using DLS methods.


This is probably because of the apparent void in chemical reactions in which the polarizabilities of the reactants and products are significantly different and thus result in a measurable fluctuation in the intensity of scattered light.

Bloomfield and Benbasat examined the unimolecular isomerization and dimerization reactions (Bloomfield and Benbasat, 1971) and generalized one-step association reaction (Benbasat and Bloomfield, 1973) from the point of view of the time dependence of D a p p. The problem is set up as a vector whose elements are the concentrations of the reactants and products that are coupled through a decay rate matrix. The elements of the decay rate matrix are the sum of the diffusion rate matrix (diagonal elements only) and reaction rate matrix (diagonal and off-diagonal elements). The isomerization and dimerization reaction results in 2 χ 2 matrices (Bloomfield and Benbasat, 1971), whereas the one-step association reaction results in 3 χ 3 matrices (Benbasat and Bloomfield, 1973). Numerical evaluation of the eigenvalues of these matrices indicates that favorable conditions for the determination of the reaction rates require one large particle with two smaller particles—that is, two small reactants with a large product, or one large and one small reactant with a small product. In either case, it is necessary to determine by independent measure-ment that Dp of all components involved in the chemical reaction before one can attempt to determine k{ and kT.

Example 9.8. Association Kinetics of Bacteriophage T4D Heads and Tails.

A particularly simple bimolecular kinetics system to analyze is one where all of the scattered light can be attributed to one of the components and D a pp is significantly altered upon attachment of the "invisible" reactant. Such a system is the head- ta i l attachment of T-even bacteriophage, where most of the scattered light is attributed to the head component (Benbasat and Bloomfield, 1975; Wilson and Bloomfield, 1979b). The association reactions for bacteriophage are shown in Fig. 9.26.

As pointed out by Benbasat and Bloomfield (1975), QELS methods are particularly well suited to following the association kinetics of the head- ta i l assembly. Since the resulting fiberless particles are noninfectious, the kinetics cannot be followed by infectivity assays. Also, the association reaction is too fast to be examined by sedimentation methods. The objective of the experi-ment was to follow the molar concentration of head components, C h(i), vs. time. The second-order rate constant k2 can be obtained from the integrated expression for equal concentrations of reactants,

k2t = Ch(t)-l-Ch(0)-

1. (9.14.6)

To achieve this goal through DLS methods it was necessary to determine and for, respectively, the head and the head- ta i l assembly. The heads

9.14. Reaction Kinetics 369

Fig. 9.26. Schematic representation of the head-tail-fiber assembly of T4 bacteriophage. According to Benbasat and Bloomfield (1975) and Wilson and Bloomfield (1979b), virtually

all of the light scattered from the h e a d - t a i l and h e a d - t a i l - f i b e r complexes is due to the head

component because of the high asymmetry in mass distribution. This means that dynamic light

scattering methods are well suited to studying the association kinetics of the h e a d - t a i l reaction,

which is the first step in the reaction diagram shown above. Infectivity assay methods cannot

monitor this step in the process of phage assembly. The assembled phage at the right of the

diagram is shown in its two forms, the fast- and slow-sedimenting structures as examined by

Welch and Bloomfield (1978).

were prepared from an osmotic shock-resis tant multiple T4Dam mutant (10" /18" /19" /34

_/36~/37") lack ing both tails and tail fibers. The tails were

prepared by restrictive growth of the T4Dam mutant (23" /34" /35~/37" /38_)

and purified by sedimentation. The head-ta i l particles were prepared by in vitro mixing of equal portions of head and tail preparations. The diffusion coefficients were determined to be Dh(20°C) = 3.6 χ 1 0

- 8 c m

2/ s (purified

heads), and Dh t(20°C) = 3.14 χ 1(Γ8 cm

2/ s (head-tail fiberless particles).

These values of Dh and Dht were found to be independent of the concentra-tion. Given that the head component accounts for all of the intensity, the


correlation function is

C(t) = [Ah(t)exp(-DhK2t) + A t W e x p ( - £ h tK

2r ) ]

2, (9.14.7)

where Ah(t) and Aht(t) are time-dependent amplitudes for the head and head-tai l components, respectively, and are of the general form Aj(t) = M / j(i) /[M hc h(i) + M h tc h t( i ) ] . Utilizing the normalization condition Ah(t) + AJt) = 1,

[C( i )1'

2 - exp( - DhlK

2t)~] = AMlexpi - DhK h) - exp( - DhlK

2tJ]. (9.14.8)

Given the initial condition that Ch(0) = Ct(0) and the identity ch(i) = M hC h( i ) , one has

ηττ1 ) Ch(0)Ah(t)

r (t) ^M h

' (9.14.9)

1 - AM + Mt)(-r^

Purified heads were filtered into the light scattering cell through 0.22 μηι Millipore filters previously soaked in BSA. Tails were introduced into the cell, and the solution was mixed with a thin glass rod. Correlation functions were taken at a fixed scattering angle at intervals of 20-30 s. Representative values of the < A p P > (in picoficks) as a function of time taken from Fig. 6 of Benbasat and Bloomfield (1975) are given in the accompanying tabulation.

Time(s) 500 1000 3000 4000

<D a p p> 3.7 3.62 3.52 3.5

The reported plots of [C( i )1 /2

- e x p ( - D h tX2i ) ] vs. [ e x p ( - D h X

2i ) -

exp( — DhiK2tJ] were linear in accordance with Eq. (9.14.8) From the

slope of 1/Ch(i) vs. t they obtained k2 = 1.02 χ 107 I v r V

1 at 22°C, or

0.97 χ ^ I v r V1 at 20°C. The Smoluchowski value [cf. Eq. (8.15.4)] was

ks ~ 5 χ 109 M

_ 1s

_ 1. The discrepancy between the experimental and theo-

retical values, k2/ks = 1.02/500 = 0.00204, was attributed to orientation con-straints and/or activation energy requirements on the reaction. Benbasat and Bloomfield suggested that the elongated icosahedral shape of the head may account for a reduction in ks by a factor of 1/24. Assuming that the joint probability for the two reactants to be in the proper orientation is the product of the individual probabilities, then the orientation factor for the tail is 24/500 = 0.048. This means that if the tail were treated as a hydrodynamic spherical surface, approximately l/20th of the effective surface is identified with the reactive site.

Summary 371

Summary

The kinetic unit in a system of highly charged particles is defined as the macroion and the associated ion cloud carried along with the macroion as it moves through the solution. The induced velocity of a charged particle that results from the application of an external field is affected by the distortion of the associated ion cloud, i.e., the relaxation effect. The magnitude of the retardation effect depends upon the charge asymmetry of the supporting electrolyte, the relative values of the particle size and the Debye-Hücke l screening length, the geometric shape and orientation of the macroion, and the charge density distribution within the macroion.

Inclusion of hydrodynamic interaction in the equation of motion results in a solvent flow pattern about a charged particle that is significantly different from that about a neutral particle. The presence of a charge on the sphere results in an apparent rotation of the fluid near the surface of the sphere that can be directly attributed to the relaxation part of the fluid velocity. As a consequence, the hydrodynamic interaction between charged spheres is not necessarily the same as that for neutral spheres. The magnitude of the applied electric field affects the rate at which the ions flow around the charged sphere. Depending upon the strength of the applied electric field, the induced charge first increases in magnitude and then decreases in value, even to the point of changing sign. Physically these situations obtain by a redistribution of the small ions between the kinetic unit and the surrounding medium.

Application of an external field can affect the spectral density profile in four ways: (1) alteration of the linewidth due to changes in the values of the relaxation times; (2) a shift in the peak location due to the superposition of a directed velocity component on the random Brownian motion of the particle; (3) alteration in the peak amplitude; and (4) occurrence of peaks at resonance frequencies of the system. Static applied fields affect only the linewidth, where-as time-varying fields may alter the power spectrum in all four categories.

Three different forms of an applied electric field in the course of a QELS experiment were discussed. Two different techniques employ a constant elec-tric field with periodic polarity. In Doppler shift spectroscopy (DSS), data are collected only during periods in which the field has attained a constant value. The resulting spectra have peaks located at a frequency shift character-istic of the electrophoretic mobility of the species. Particles are therefore "separated" by their instantaneous velocities rather than a physical sep-aration. The linewidth is a measure of the diffusion coefficient. The technique in which data are collected through the field-reversal period is referred to as P P E F , i.e., periodic pulsed electric field. The major difference in this technique is that the spectra contain peaks centered around the Doppler-shifted peak, where the frequency difference is equal to the frequency of the pulse-reversal


period. In the third method, an electric field that varies sinusoidally in time is applied across the sample, with continuous collection of the data. This technique is the optical analogue to dielectric dispersion methods, in which the dynamic response of the macroions is monitored through temporal variations in the scattered light intensity. The spectral density profile in this case has peaks located at the frequency and overtones of the applied electric field. The electrophoretic mobility is calculated from the relative peak heights, and the diffusion coefficient from the half-width of the peak.

Application of constant and sinusoidal hydrodynamic flow fields also results in frequency-shifted peaks in the spectrum of scattered light. For constant-flow fields, the Doppler shift reflects the velocity of the fluid relative to the observer. For oscillatory hydrodynamic movement, the orientational diffusion constant for asymmetric particles can be determined from the fre-quency shift of the peak.

Mechanical excitation of gels by sound waves results in peaks in the spectrum that correspond to the resonance frequencies of the gel. Application of high pressure causes the relaxation processes in polymer melts near the glass transition to slow down. High pressure may also induce aggregation of the solute species.

Although chemical kinetics can be studied by QELS methods in principle, systems appear to be lacking in which reactants and products differ enough in their polarizabilities to be detected. The kinetics of irreversible processes can be followed by QELS methods through changes is D a pp and the relative amplitudes of the decay modes present in the correlation function.

Problems

9.1. You have at your disposal a spectrum analyzer with a resolution of 0.1 Hz. What is the largest radius Rs of a particle that you can study for an accuracy of 10% in Rs at θ = 90°? Assuming that the zeta potential is independent of 7S, what is the minimum value of λΌΗ at which μ ρ is independent of / s? [Hint: Use Eqs. (9.3.1) and (9.3.3).]

9.2. Differentiation of the oldest from the youngest blood cells is one of the many mysteries of life. Lunar et al. (1977) used DSS to test the model that differences in the electrophoretic mobilities enabled the older, denser blood cells to be distinguished from the younger blood cells. Samples were fractionated by density sedimentation, and fractions were taken at the extremes of the distribution. The electrophoretic mo-bilities were reported to be identical at 25°C in 0.15 M NaCl (μ ρ ~ 1.09 χ 10~

4 cm

2/Vs) and 0.03 M NaCl (pp ~ 1.75 χ 10~

4 cm

2/Vs) .

Problems

Assume that RH is the same for the old and new cells. Use up to the first-order correction term in Eqs. (9.3.3) and (9.3.1) to estimate the radius and zeta potential of these red blood cells. What conclusions can be drawn regarding the importance of the correction term?

9.3. Drifford et al. (1981) used DSS methods to study μρ of BSA, DNA, chondroitin sulfate ( C h S 0 4) , poly(methacrylic acid) (PMA), and a copolymer of meleic acid and ethyl vinyl ether. The tabulated data are estimated from their Fig. 5 for BSA at pH 6.8 with c B SA = 50 g/L and Τ = 20°C.

E ° ( V / c m ) 11 12.8 14.8 18

Av d s(Hz) 2 2.4 2.5 3.8

Calculate μ Β 5Α under these conditions. Since the windows of the cell were parallel to each other and perpendicular to the propagation vector of the incident light, the "true" scattering angle differed from the value defined by the photomultiplier tube. The corrected scattering angle was θ = 3.75° and λ0 = 6328 Â.

9.4. In the DSS study of BSA by Drifford et al. (1981, cf. Problem 9.3), the spectral linewidth given in their Fig. 5 is approximately Δ ν 1 /2 ~ 1 Hz for θ ~ 3.75° and λ0 = 6328 A. Calculate RH of the BSA from the spectral linewidth and compare with the QELS value of ap ~ 35 Â under high-salt conditions. If these values are significantly different, how might you account for the difference? If RH > 35 Â, can the difference be attributed to electrolyte dissipation? Use any of the ex-pressions for the electrolyte dissipation term given in Section 7.9 to estimate the value of the charge required for a hard sphere radius of 35 Â. Is this a reasonable value of the charge?

9.5. Mohan et al. (1976) used DSS methods to study human serum plasma, with the results summarized in Fig. 9.13. Although the supplier indicated that the BSA sample used to calibrate their instrument contained 2 - 4 % contamination, these authors were not able to discern monomers and dimers in their DSS spectra. Using the information given in Example 9.3, simulate the BSA monomer -d imer spectrum in a manner similar to that given in Fig. 9.12. Should Mohan et al. expect to resolve the two peaks? (Hint: Assume that the scattering powers of the monomer and dimer are the same, and use the supplier-estimated contamination values.)


9.6. Schlieper and Steiner (1982) used QELS and DSS methods to study the association of tetracaine to liposomes of various charge densities. Phosphatidylcholine (neutral, highly purified) was mixed with 10% (mole percent) of: (1) phosphatidylserine (carboxyl group, negative charge); (2) phosphatidic acid (phosphate group, negative charge); (3) hexadecanoic acid (carboxyl group, negative charge); (4) hexadecylamine (tertiary amine group, positive charge); and (5) hexadecyltrimethylamine (quarternary amine group, positive

charge). The neutral liposome is designated as preparation (6). Liposomes were prepared by ultrasonic irradiation. All DLS experiments were performed in 1 m M KCl, 1 m M glycine, pH 7. QELS methods estab-lished that the particle radii for all of the preparations fell in the range 1700 ± 600 Â. Plots of μ ρ vs. l o g 1 0[ C t e t r a c a i n e] appeared to be linear over the range — 5 < l o g 1 0[ C t e t r a c a i n e] < — 3 for all of the liposome preparations, with the values given in the accompanying tabulation (units of μ ρ are 10~

4 cm

2/Vs) ,

Preparat ion

l°ëlo[^tetracaine] d ) (2) (3) (4) (5) (6)

- 5 - 6 . 1 - 6 . 0 - 1 . 8 + 2.3 + 3.1 0

- 3 + 0.9 + 0.9 + 1.1 + 2.3 + 3.1 + 2.3

What can be concluded from these data in regard to the mechanism of association? The QELS data were collected at θ = 15°, hence it is assumed that the DSS data were also obtained at this angle. They did not report the presence of two or more peaks in the electrophoretic mobility profile. One can conclude either that the binding of the tetracaine did not result in a heterogeneity of electrophoretic mobility species, or that the different peaks could not be resolved. Using the average value of the radius as being the same for all of the preparations and extent of binding to the substrate, calculate the resolution of the Doppler-shifted peaks using Eq. (9.5.4). [Assume λ0 = 632.8 nm and estimate E° needed to obtain R(6) = 1.]

9.7. Ferguson et al. (1981) reported QELS and DSS studies on phos-pholipid liposomes (PL) and brush-border membrane (BBM) vesicles. The tabulated data were obtained (μ in 10~

4 cm

2/Vs and 2ap in Â).

Problems

PL BBM PL + BBM

2640 4350 1600

8.21 3.53 7.68

Use the appropriate limiting expression for Henry's function and calculate the zeta potential from these data, and then the apparent charge of these particles.

9.8. Bovine casein micelles are made up of the a s l, a s 2, β, and κ casein phosphoproteins and calcium phosphate. Holt (1975) described these particles as a central core region with a "hairy" surface of flexible κ casein phosphoproteins. As in the case of the D N A arms and loops in the native adenovirus nucleprotein system discussed in Example 7.9, the extension or collapse of the hairy surface is dependent upon the nature of the solvent. It is the presence of the κ proteins that prevents the casein particles from aggregating, leading to the curdling of milk.

Holt and Dalgleish (1986) used QELS and DSS methods to char-acterize the properties of the hairy surface. The QELS measure-ments were made at 25°C at a scattering angle of 90° with an incident wavelength of 632.8 nm. One of the experiments involved a kinetics study in which the change of the apparent hydrodynamic radius upon reaction with rennet was monitored through the average diffusion coefficient. (Rennet is a dried extract made from the stomach lining of a ruminant.) The concentration of C a

2 + was sufficiently high to prevent

dissociation of the "hairless" casein micelles but sufficiently low to prevent aggregation of these particles. The apparent diameter (nm) as a function of time (minutes) estimated from their Fig. 1 is given in the accompanying tabulation.

Time 0 10 20 30 40 50 80 Diameter 125 120.2 117.6 116 114.5 114.2 114.1

Since the reaction commenced immediately upon addition of rennet, Holt and Dalgleish concluded that the solvent permeated the hairy surface in order to have the reaction site of the κ phosphoproteins readily accessible to the rennet. These authors therefore concluded that the chains in the hairy surface are partial draining, hence the actual diameter of the intact particle was greater than 125 nm.

The experimental conditions actually followed the time dependence


of the diffusion coefficient, given as <D(i)> = Y^X^D^ where X^t) = Ci(t)/CT, Ci(t) is the molar concentration of particles with diffusion coefficient Dh and C x is the total number of particles. Assume a simple model for the reaction in which only "hairy" (h) and "bald" (b) particles are present such that Xh(t) + Xh(t) = 1 and Dh and Db are determined by the asymptotic values of the given diameters. Obtain a first-order kinetics expression for < 1 / R >

_ 1 in terms of the concentration. From

the data, calculate a value for the first-order rate constant.

9.9. In the Q E L S - S E F theory developed in Section 9.7, it was assumed that the response of the particles was in phase with the applied field. This is referred to as the strong interaction limit, which means that per unit time, the distance traveled by the particle as a result of the electric field is much greater than the distance traveled through random Brownian motion. Use the standard one-dimensional relationships < (Ax)

2>

1 /2 = 2 / y

1 / 2 and < x

2>

1 / 2 = μρΕ°ί = (ZpeDp/kT)E°t and cal-

culate the value of the electric field results in the equality of these two values, i.e., < (Ax)

2>

1 /2 = < x

2>

1 / 2. Perform these calculations for 20°C

using the parameters Dp χ 107 = 0.1, 1, and 6, and Z p = 10, 100. How

do these computed values of E° compare with the minimum experi-mental value of ^ 3 V/cm?

9.10. Altenberger, Dahler, and Tirrell (1987) ( A - D - T ) re-examined the response of a charged particle to a constant electric field using the Gibbs canonical ensemble description of the uniform equilibrium mixture. These authors emphasized the importance of fluctuating density fields for inhomogeneous systems. The result of the A - D - T analysis is

DP = 7 7 r D + AZ

2

pCp(K2 + AZ2

cCcn

where A = Απ/ξ^Τ. The electrophoretic velocity is ΚΓ· ν ρ = Κ

Γ·

E°(Zp/kT)Dp

s. Discuss how the Doppler shift and linewidth vary

with Z p , Z c , C p , and C c. The A - D - T result is equivalent to the statement that the mutual rather than the tracer diffusion coefficient should be used in the DSS experiment. Assume that ZpCp = Cp and Z c = 1, estimate the percentage difference in the computed charge Zp

S using the A - D - T and tracer diffusion expressions.

CHAPTER 10

Dynamic Light Scattering from Complex Media

' 'Of late I have been tempted to look into the problems furnished by nature rather than those more superficial ones for which

our artificial state of society is responsible/" (Sherlock Holmes to Dr. Watson)

From The Final Problem by Sir Arthur Conan Doyle (1859-1930)

10.0. Introduction

In the spirit of the above quotation, this chapter focuses on some problems associated with the dynamics of complex systems. While there are many sys-tems that fall under the category of "complex media", the topics discussed in this chapter are restricted to solution properties of charged particles. The mystique of these systems lies in the relative importance of generalized hy-drodynamic, hydrophobic, electrical, excluded-volume, and Brownian forces and how they act in concert in the description of the dynamics of the multi-component system. Because of their intrinsic mathematical complexity, there may not be a general consensus as to the so-called "correct" interpretation of the data reviewed herein.

10.1. Splitting of Relaxation Modes for DNA Fragments as a Function of Ionic Strength

Diffusion of neutral rodlike particles in congested solutions has previously been discussed in Section 5.23. The parent model is that of Doi and Edwards (1978), which is based on the formation of cages and short-range physical contacts that hinder the "escape" of trapped particles. The present section

377

378 10. D Y N A M I C L I G H T S C A T T E R I N G F R O M C O M P L E X M E D I A

extends these systems to charged particles with long-range electrostatic interactions and possibly small-ion coupling between charged rods.

Fulmer et al. (1981) reported QELS studies on D N A fragments obtained by nuclease digestion of chicken erythrocyte chromatin. The mononucleo-somes were isolated by column chromatography, column fractions were pooled, protein was removed by proteinase Κ digestion, and the purity of the preparation was assessed by electrophoresis on 1.4% agarose slab gels. The DNA fragments were reported to be ~ 150 base pairs (L = 510 Â, M r ~ 90,000 Daltons). QELS data were collected at 1-2 (D f a s t) and 10-20 μ 8 (D s l o w) intervals for c p ~ 1.5 g/L. They observed that D f a st ~ D s l ow for C s > 0.05 M. The value <D a p p> = 3.0 χ 1 0 "

7 c m

2/ s in 1.0 M salt is in good agreement with

the T i rado-Garc ia de la Torre [1979, cf. Eqs. (3.8.1)-(3.8.4) and Table 3.2] value of 2.8 χ 10~

7 c m

2/ s for a cylinder of length 510 Â and diameter 27 Â

(Elias and Eden, 1981) (cf. Example 3.6). As C s fell below 0.05 M, D f a st

increased, whereas D s l ow decreased as Is was lowered. The behavior of D f a st as a function of Is is in line with the coupled-mode theories in Chapter 7. The presence of the slow mode at the lower ionic strength was unexpected.

A study similar to that by Fulmer et al. (1981) on dinucleosomal sized DNA was reported by Schmitz and Lu (1984). These DNA fragments were also obtained by nuclease digestion of chicken erythrocyte chromatin, and the dinucleosomes were isolated from the other fractions by column chromatog-graphy. The dimer fraction was deproteinized by a phenol-chloroform pro-cedure after first digesting with protease K. The D N A fragments were sized on a 3.5% Polyacrylamide gel with comigrating φΧ\Ί4 D N A restriction enzyme fragments. The size was reported to be 375 base pairs (L = 1275 À). D f a st and D s I ow for 1.5 mg/mL solutions of dinucleosomal DNA were likewise determined as described by Fulmer et al. The high-salt limiting value Z)p for the dinucleosome was obtained from a plot of <D a p p> vs. [P ] /2 [KC1] and extrapolated to [P] /2[KC1) = 0. As in the case of the mononucleosomal DNA study, the T i rado-Garc ia de la Torre calculations for D p for a rigid cylinder of diameter 27 À were in agreement with the extrapolated high-salt value of <D a p p>. Plots of <D a p p> vs. / s for these studies are shown in Fig. 10.1. The observation that D f a st > D s l ow for the entire / s range for dinucleosomal DNA was attributed to semidilute behavior.

It is noted that an earlier report on sonicated mouse liver DNA (ML) in 1 mM cacodylate buffer at 25°C by Chen and Chu (1977) also indicated two relaxation modes. A biexponential fit of C(K, t) at θ = 30° and 25°C yielded the values D f a s t(ML, 1 mM) - 4.6 χ 1 0 "

8 c m

2/ s and D s l o w( M L , 1 mM) -

1.1 χ 1 0 "8 cm

2/ s . On the basis of a study by Chen et al. (1977) on native

calf thymus DNA (NCT, M w = 12 χ 106 Daltons) in Tris buffer (0.01 M

Tris, 0.14 M NaCl, pH 8), D s l o w( M L , 1 mM) was associated with pure transla-tional diffusion. However, £> s l o w(NCT, 0.14 M) - 1.8 χ 1 0 "

8 c m

2/ s at θ = 30°

10.1. Splitting of Relaxation Modes for D N A Fragments 379

LOG ( I 8 )

Fig. 10.1. Functional dependence of O a pp for mononucleosomal and dinucleosomal DNA as a function of added salt. Da ta were taken at collection intervals of 2 μ$ and 20 μ$, and D a pp was

computed from the first cumulant . ( Ο ) : Mononucleosomal D N A . [Reproduced with permis-

sion from Fulmer et al. (1981). Biopolymers. 20, 1147-1159. Copyright 1981 by John Wiley and

Sons.] ( · ) : Dinucleosomal DNA. [Reproduced with permission from Schmitz and Lu (1984).

Biopolymers. 23, 797-808 . Copyright 1984 by John Wiley and Sons.]

was obtained for the native calf thymus DNA, or D s l o w(NCT, 0.14 M) > D s l o w( M L , 1 mM). It may be concluded that either M p > 12 χ 10

6 Daltons

for the sonicated mouse liver D N A or D s l o w( M L , 1 mM) is not associated with the pure translation mode. Although the mouse liver D N A was not charac-terized in that study, one might infer the approximate length range from other studies using similar preparation procedures. Sonication of mouse liver DNA was carried out for 15 minutes at 0°C using a Branson W140 cell disrupter at a setting of 4. By comparison, Record et al. (1975) reported that sonication of calf thymus D N A in glycerol (η0 = 500 centipoise) for 30 minutes at 2 - 3 minute bursts at 70 W resulted in D N A of molecular weight 3 χ 10

4 < M p < 3 χ 10

5, which brackets M p of mononucleosomal

D N A and, by inference, the sonicated mouse liver DNA.

The increase in D f a st for these systems as Is is decreased can be interpreted in terms of the coupled-mode theories presented in Chapter 7. The monotonie decrease in D s l ow as Is is lowered is consistent with the possibility of attrac-tive interactions that result from fluctuations of small ions along the DNA backbone. Oosawa (1968) has shown that if the small-ion distributions along two neighboring charged rods are correlated, there is an attractive interaction between the two rods [cf. Eq. (8.11.14)]. The range of this type of interaction in the Oosawa calculations is very short, being on the order of the Bjerrum length ( ~ 7 Â). Chance encounters and proper axial alignment may lead to reason-ably large energies of attraction that cause "bundles" of DNA fragments to form. It is also possible that the slow mode might be due to long-range interactions with a solute "lattice", as suggested by Fixman (1985b). It is


conceivable that the repulsive electrostatic interactions between the DNA fragments result in an aligned "lattice" about a "probe" (unaligned) D N A rod. One must also take into consideration the chemical nature of the duplex DNA particles. As the ionic strength is lowered, the double helix structure becomes less stable because of an increase in the mirastrand repulsion interaction energy. It is possible that a fraction of the DNA particles have "frayed ends", or duplex DNA fragments that are partially unwound at the ends. Bloomfield (1987) has suggested the possibility that the slow mode is due to end-to-end mferparticle hydrogen bonding association of the DNA fragments.

10.2 Splitting of Relaxation Modes of Highly Polymerized DNA

QELS studies on highly polymerized D N A were the subject in previous sections of this text (cf. Examples 5.5, 5.6, and 7.12). These studies were carried out under conditions in which Is > 0.05 M. The observation of two (or more) relaxation modes was consistent with the theories of large flexible particles in which internal relaxation processes are observed. However, QELS studies on bacteriophage N l DNA ( M p = 31 χ 10

6 Daltons) in 1 M NaCl (Schmidt,

1973) and 0.012 M NaCl (Schmidt et al., 1977) indicated two relaxation modes, where D s l ow is almost two orders of magnitude smaller than the value anticipated

Two theories have been proposed to explain D s l ow observed for DNA and applied to the N l D N A data of Schmidt (1973). Lee et al. (1977) proposed a "cage" model in which the translational motion of a central D N A molecule was "hindered" by a cage of surrounding D N A molecules. The basis of this model was a polarized and depolarized light scattering study of native calf thymus D N A over the range 25°C <T< 80°C in 0.02 M potassium phos-phate buffer, pH 7.8 (at 25°C) (Schmitz and Schurr, 1973). D s l ow for the polar-ized QELS data was too small to be equated with D p , whereas T s 1 ow in the depolarized light scattering data was found to be in good agreement with the longest orientational relaxation time predicted by the Rouse -Z imm model. These observations were interpreted in terms of a "solute cage" in which the particle was free to rotate but was not able to participate in free translation. To escape the cage the trapped molecule had to have the proper orientation in order to "squeeze" through the obstacle course set up by the surrounding particles. It was assumed that diffusion from the cage was along the long axis of the trapped molecule, and that this motion was coupled with the longest relaxation time of the molecule. The "congested solution" theory of Lee et al. (1977) resulted in two relaxation modes with the decay rates

for D p .

(10.2.1)

10.2. Splitting of Relaxation Modes of Highly Polymerized D N A 381

where Δ - [0.39DjjK4 + (^-)D^DeK

2 + 3 6 D

2]

1 / 2} , and the relative ampli-

tudes are B± = (s± + A)/(s+ — s_) with A = (^j)Dl{K2 + 6De. In the asymp-

totic limits one has for the two relaxation times

— ~ 6De + ^D^K2 (D{lK

2 -> 0), (10.2.2)

τ_ 21

1 D,,K2

1

( D h K2- 0 ) , (10.2.3)

3.913D9 + 0.742L>|,K2 ( D , | K

2^ o o ) , (10.2.4)

— ~ 2.086De + 0.115D,|K2 (DnK

2 ôo). (10.2.5)

τ +

Reihanian and Jamieson (1979) viewed the separation of fast and slow modes as the onset of gellike behavior, in which the molecules first show signs of entanglement, i.e., a pseudogel. The Reihanian-Jamieson model is based on the work of Brochard and de Gennes (1977) for flexible polymers in θ solvents in which cp > c* (cf. Chapter 5). The stress on the system was represented as the sum of a constant part, which was associated with the osmotic stress per unit volume [cp(dn/dcp)], and a time-dependent part, which was related to the longitudinal elastic modulus of the pseudogel (ß g e l) . The model resulted in two decay modes that have the Lorentzian linewidths

Δω, = T, (10.2.6) 1 1 + ( D c o op + D t ) r r K

2'

A œ 2 = - + (Dcoop + Dt)K2, (10.2.7)

(10.2.8)

with relative amplitudes

A, p(l + μ "1) ( 1 +μ + ρ~

ιμ)'

where ρ = {ßge\/cp)(dcp/dn\ τ Γ = η0/β&\ is the characteristic lifetime of inter-chain entanglements, μ = DCOOPTTK

2, and D c o op is the cooperative diffusion

coefficient of the pseudogel network, which is related to the center-of-mass translational diffusion coefficient Dt by

D c o op = DtP. (10.2.9)

The two models predict the same behavior of the two modes in the limit Κ ->0 and are equivalent if one assumes the formal identities 1/τΓ = 6De;


A o o P + A = i i ^ i h a n

d A =

Substitution of D t yields the basic rela-tionship D c o op = 7 Ö | | . The two models therefore cannot be formally distin-guished from data in the low-X region. This is not, however, the situation in the high-K region. According to Eqs. (10.2.4) and (10.2.5), the coupled translation-orientation model predicts a linear dependence of the decay rates on K

2 with nonzero intercepts. While a similar relationship holds for the sec-

ond decay mode in the psuedogel model [Eq. (10.2.7)], the first decay mode becomes independent of Κ and attains the limiting value ( A /

Tr ) / ( A o o P + Α ) ·

Plots of the N l DNA data of Schmidt (1973) and the model fits reported by Lee et al. (1977) and Reihanian and Jamieson (1979) are shown in Fig. 10.2.

It may appear from the curves in Fig. 10.2 that the pseudogel model pro-vides the better fit. One must take into consideration, however, the manner in which the models were fitted to the data. Reihanian and Jamieson have three adjustable parameters in their model, i.e., D t, Dcoop9 and τ Γ. These pa-

0 . 1 0 . 3 0 . 5

s i n2

( 9 / 2 )

Fig. 10.2. Linewidth parameters vs. s in2(0/2) for the fast and slow mode of N l DNA. The

points Τ are from the data of Schmidt (1973). The solid lines are the theoretical calculations of Reihanian and Jamieson ( 1979) using an entangled-chain model. The dashed lines and solid circles are, respectively, the asymptotic limiting slopes and computed points for the coupled t rans la t ion-rotation model for congested solutions (Lee et al., 1977). [Reproduced with permission from Reihanian and Jamieson (1979). Macromolecules. 12, 684-687 . Copyright 1979 by the American Chemical Society. Reproduced with permission from Lee et al., (1977). Biopolymers. 16, 583-599 . Copyright 1977 by John Wiley and Sons.]

10.3. "Order ing" in Polyelectrolyte Solutions 383

rameters may be adjusted independently, and therefore the decay rates for the two modes become independent of each other. On the other hand, in the model of Lee et al. there are only two unknown variables. Dy and De. The two decay rates are therefore interdependent. Lee et al. chose to fit the intercept of the slow mode in accordance with the extrapolation of the high-K expression given by Eq. (10.2.5). The value of DQ obtained in this manner automatically fixed the intercept for the fast mode in accordance with Eq. (10.2.2). The slope of the fast decay mode was fitted by Eq. (10.2.3), which therefore fixed the slope of the slow mode through Eq. (10.2.5).

It may be argued that the data in Fig. 10.2 show a "zero" slope for the slow mode in the high-K region, thereby indicating that the Reihanian-Jamieson model is the appropriate vehicle to interpret these systems. It is noted, how-ever, that the data in Fig. 1 of Schmidt et al. (1977) show a nonzero slope for N l DNA at 34°C in 0.012 M sodium ion buffer. The correct interpretation of these systems therefore remains an open question.

10.3. "Ordering" in Polyelectrolyte Solutions

One of the more controversial topics related to polyelectrolyte solutions is whether or not they form "ordered structures" in solution. Matsuoka et al. (1985) reported a single "Bragg-like" peak in their SAXS data on polyions of a variety of shapes: tRNA (L-shape); lysozyme and BSA (ellipsoidal globular shape); chondroitin sulfate (rod shape). An important feature of the studies on BSA was that the peak disappeared at pH 5.06, the isoelectric point, which argues that the forces responsible for the single peak are electrostatic in nature. The presence of divalent ions tended to eliminate any structure; thus the inference can be made that divalent ions were more efficient in neutraliz-ing the polyion charge in accordance with the condensation theories. Un-expected, however, was the observation that the peak position shifted to larger scattering angles (i.e., small interparticle spacing) as Z B SA was increased. A similar shift was reported by Ise et al. (1984) for poly(styrene sulfonate) (NaPSS). It was concluded that the peak was not intramolecular in origin because mixtures of different molecular weights of NaPSS and mixtures of NaPSS and sodium poly(acrylate) (NaPAA) have scattering profiles that differ from the composite of the parent molecules. Similar results were re-ported on mixtures of ( lys ) 4 06 and (lys) 13 in which a new shoulder interme-diate to the parent solutions was detected (Ise et al., 1983b). The interparticle distance computed from the peak location, denoted as 2 D e x p, decreased with increasing c p and/or added salt concentration. It was concluded from these studies that the peak in the SAXS data resulted from intermolecular ordering.

Kaji et al. (1984) reported SAXS studies on the L i+, N a

+, and K

+ salts

of polyvinyl hydrogen sulfate) (PVHS) in the absence of added salt. The distribution function computed by Fourier transform of the data suggested


two maxima, the first being attributed to intramolecular structure and the second to intermolecular structure. The dependence of the magnitude of the intrasegmental peak on the type of cation present was suggested to constitute proof of counterion condensation.

QELS studies were reported by Ito et al. (1988b) on poly-methylmethacrylate) (PMMA) latex particles, denoted by PM-17 in their paper, with a diameter of 0.17 μιη. Plots of Dp/Dapp(K) vs. Κ exhibited some structure, where Dapp(K) was computed from Ki and Dp was approximated by the most dilute solution examined. It is not clear, however, whether D a p p(K) reflects the dynamics of the particles in the "crystalline state" or the "liquid state". To be consistent with the time-lapse photograph shown in Fig. 8.1, the "crystalline-like" state should be highly immobile. Ito et al. (1989) examined

4 π

Κ/ Κ|,ο

Fig. 10.3. Dynamic structure factor for both liquid and crystalline phases. (A) The liquid phase data (O) exhibit the typical bcc melt characteristics, in particular a second maximum at Κ = 1.8X1 10 rather than 2 K 1 1 0. ( · ) : Crystalline data. [Reproduced with permission from Cotter and Clark (1987). J. Chem. Phys. 86, 6616-6621 . Copyright 1987 by the American Institute of Physics.] (B) The data of Ito et al. for latex PM-17 at a volume percentage of 0 .5%. [Reproduced with permission from Ito et al. (1988). Phys. Rev. B. 38, 10852-10859. Copyright 1988 by the American Physical Society.]


the "lifetime" of a "core particle" using videotapes of poly(styrene) particles. [One operational definition of a core particle was a particle having "six bonds" after a 24-hour period.] They reported that for these core particles, 69% lost one bond after ^ s , the resolution of their video device. It is thus suggested that Dapp(K) in the study of Ito et al. (1988b) is to be associated with the "liquidlike" state shown in Fig. 8.1. To support this conjecture, the QELS study of Cotter and Clark (1987) on both the "liquid" and the "crystalline" phases of a suspension of 0.109-/im-diameter sulfonated poly(styrene) spheres is cited. The coexisting phases were gravity-generated by allowing a suspension of the spheres to stand over a long period of time. The relaxation time τ obtained from the third-order cumulant analysis of correlation functions defined the solution structure factor, S(K AR)/Dp = τΚ

2. These data are compared in

Fig. 10.3.

The Cot t e r -Cla rk study shows that both the liquid and the crystalline phases have a strong first peak. Taking into consideration the large difference in the concentration used in the two studies, one cannot unambiguously conclude that Dapp(K) in the Ito et al. (1988b) study is to be associated with the crystalline or liquid phase.

There are three possible interpretations for the presence of the single peak in the S(K) vs. Κ profile: (1) the correlation hole; (2) the "two-state" structure of polyion solutions; and (3) "temporal aggregates".

The Correlation Hole

The "correlation hole" concept was introduced for polymer melts with chains having labeled regions (de Gennes, 1979; Hayter et al., 1980). The correlation hole refers to a region in the vicinity of one particle that is excluded from the second particle and manifested as the limit S(K AR) 0 as Κ AR 0. Since P(Kd

c) -+ 0 as Κ oo, the total intensity given by I(K) = S(K AR)P(Kd

c)

must have a maximum at some intermediate value of K. The neutron scat-tering profiles of lithium dodecyl sulfate (LDS) (Chao et al. 1985) and BSA at three concentrations in D 2 0 (Bendedouch and Chen, 1983) are shown in Fig. 10.4, along with the pair distribution functions obtained from RMSA calculations.

A single peak in I(K) vs. Κ is clearly in evidence for both of the systems shown in Fig. 10.4. g

(2)(r) for the LDS system indicates some form of "local

structure" as indicated by a maximum in its radial profile. In contrast, g{2\r)

for the BSA system exhibits no such maximum, hence no solution structure. The single peak in I(K) for the BSA system in the absence of local solution structure is therefore due to the presence of the "correlation hole". It is thus concluded that the observation of a single broad peak in the I(K) vs. Κ profile does not constitute evidence of ordered solution structure.


3 _

2 _

I - I

, 8

• χχ Χχ

[ l d s ]

0 . 7 3 4 ·

0 . 3 6 7 0

0 . 1 8 4 X

0 . 0 9 2 V

o ? *xx ο \

0 . 2 5

0 . 2 5

Κ ( A " ' )

Fig. 10.4. Total intensity and solution structure. Neut ron scattering measurements were

performed on solutions of BSA and lithium dodecyl sulfate (LDS). (A) Intensity specta data for

BSA in D 2 0 . The concentrat ions (%) of BSA are given in the figure. These curves were fit with the

expression / t i ns = P{K Ar)S{K AR), where the Hayte r -Penfo ld F O R T R A N program was used to

calculate S{K AR). The apparent charge used in these calculations was Z p = — 8, and the effective

volume fractions were = 0.027, 0.055, and 0.074 for the 4 % , 8%, and 12% solutions,

respectively. Although / (X) exhibits a single maximum, the calculated function S{K AR) shows no

significant structure occurs in the solution. The maximum in I{K) may therefore result from a

"correlation hole," as described in the text. [Reproduced with permission from Bendedouch and

Chen (1983). J. Phys. Chem. 87, 1473-1477. Copyright 1983 by the American Chemical Society.]

(B) The LDS study was carried out in 0.2 M LCI at 37 C at various concentrat ions shown in

the figure. It is deduced from the shape of S(K AR) that there is significant structure in the solu-

tion. [Reproduced with permission from Chao et al. (1985). J. Physical Chem. 89, 4862-4866.


The Two-State Model

Ise and coworkers proposed a "two-state" model for the solution structure of polyions as shown in Fig. 8.1, i.e., regions of highly ordered "crystalline-like" structures and random "gas phase" structures (Ise and Okubo, 1980; Ise et al., 1983a; Sugimura et al., 1984; Ise et al., 1985; Ise, 1986; Ise et al., 1988, 1989). In this model the counterions between the polyions result in an attractive interaction between the polyions of like charge. It is this attractive interaction that is responsible for the observed inequality 2 D e xp ~ AR discussed in the introductory paragraph (Ise et al., 1983,1984; Ise, 1986; Matsuoka et al, 1985). Ise and coworkers (Ito and Ise, 1987; Ito et al., 1988, 1989; Ise et al., 1988,


1989) pointed out that a single broad peak occurs in the /(K) profile if highly ordered structures coexist in solution with disordered structures.

The first attempt to develop a model to explain the crystallinelike stability was that of Sogami and Ise, in which the linearized form of the Po i sson-Boltzmann equation was employed. Sogami (1988, cf. Sec. 8.11) revised the original model (Sogami 1983; Sogami and Ise, 1984) by adjustment of the re-duced potential of interaction by a constant (weak gauge invariance). The pairwise interaction potential obtained by Sogami [cf. Eq. (8.11.11)] from the Helmholtz free energy has a minimum located at a distance given by Eq. (8.11.13). It is emphasized that this minimum occurs only if there is an asymmetric equilibrium distribution of ions between the two polyions.

In contrast to the Sogami model in which an asymmetric (but equilibrium) distribution of small ions between the polyions gives rise to an ordered struc-ture, Imai and Yoshino (1985) and Imai (1988) proposed a model in which charge fluctuations at the boundary region of the ordered phase tend to stabilize the ordered array. The origin of the charge fluctuations is the repeated collisions of particles in the disordered phase with particles in the ordered phase. The Imai model does not address the question as to how the ordered phase is generated.

Of interest in the present section are the H N C results that predict a "phase separation" into liquid-like and gas-like structures for highly asymmetric charged systems, where under certain conditions attractive interactions occur between colloidal particles (Patey, 1980; Belloni, 1986b). Numerical conver-gence in the calculations of the compressibility of the system becomes more difficult as the solution becomes more dilute, because of the increase in the compressibility of the system. The difficulty of the convergence increases as a spinoidal line is approached. There exists a "temperature" Τ* = αρ/λΒ, however, at which the whole domain is again accessible. The dependence of the "critical temperature" on the charge of the polyion was found by Belloni (1986b) to be Τ* = 0 .123Z*

1 8. Although Belloni is of the opinion that a

"phase separation" into gas-like and liquid-like structures exists, he cautions that T* may not predict the exact location of the separation.

Temporal Aggregates

The "temporal aggregate" (TA) model was proposed to interpret the so-called "ordinary-extraordinary regime" transition (cf. Section 10.4) and Q E L S -SEF data (Schmitz and Parthasarathy, 1981; Schmitz et al., 1982b; Schmitz, 1982a, 1982b). The essential features of the TA model are the delicate balance between the repulsive direct forces between colloidal particles of like sign, the disruptive diffusion forces of Brownian motion, and the attractive interactions that arise from temporal fluctuations in the small-ion distribution. These


fluctuations encompass asymmetric distributions in the ion cloud about the parent polyion, fluctuations in the net charge of the polyion due to adsorption and desorption of small ions, and coupled polyion-polyion dynamics due to the "sharing" of small ions that results from overlap of the ion clouds. The dipole-dipole attractive force between macroparticles j and k, for example, is analogous to the London (or van der Waals) attractive forces at short distances of separation, where the small ions in the temporal aggregate model play the same role as the electrons in the van der Waals force. The fluctuating distribution of ions in the ion cloud is similar to the mechanism proposed by Fulton (1978a, 1987b) for long-range correlations between charged macro-particles. Charge fluctuations at the surface of the particles may also con-tribute to the attractive force between particles j and k in a manner similar to that proposed by Kirkwood and Shumaker (1952a, 1952b). The force equation that stabilizes the TA incorporates the features of Eqs. (9.4.6), (9.4.12), and (9.7.2)-(9.7.4), viz,

Σ [F«(0 - ZjeV<KrJk)] = 0, (10.3.1)

where the double sum is over all of the "free" small ions and the "kinetic units" of the macroions with associated small ions (cf. Fig. 9.5). The internal potential (j)(rjk) between two kinetic units is expressed as a multipole expansion to account for the temporal fluctuations in the small ion distribution about each unit. The time dependence of the forces is to emphasize that the TA is in metastable equilibrium. The TA is therefore disrupted either if the net force becomes repulsive by means of a change in the number of particles within the TA, or if the distribution of particles within the TA gives rise to a net repulsive force.

It is to be emphasized that the Sogami-Ise model (SI) is based on a thermodynamically stable equilibrium description of a highly ordered phase, whereas the TA model is a metastable "aggregate" whose dynamics are correlated by means of temporal fluctuations in the forces between the particles. Possible support of the concept of a TA is the multidetector study of disordered solution phases reported by Clark et al. (1983). They stated that "the scattering becomes characterized by the momentary appearance and disappearance of six-spot patterns reminiscent of the crystalline phase and having the same K10. These fluctuating six-spot patterns appear with random orientation and persist for a few tens of milliseconds." Likewise, Sasaki and Imai (1983) report a slow relaxation process in their QELS studies on a spot in the Laue pattern of an ordered array of poly(styrene) latex particles. Recall also the videotapes of Ito et al. (1989), which indicated "bonds" associated with a "core particle" that are broken within the time resolution of the instrument, O T J Q S = 33 msec.


The Ise-Sogami and Imai -Yoshino mechanisms for stabilizing the crystal-line phase and temporal aggregate mechanism are illustrated in Fig. 10.5.

In view of the above dissection of experimental results and theory, the occurrence of a single peak in the scattered intensity profile does not provide conclusive evidence of any form of order in the solution. While there appears to

E Q U I L I B R I U M DYNAMIC

Fig. 10.5. Comparison of the two-state and temporal aggregate models. (A) Ise and Sogami have proposed a two-state model for explaining the solution structure shown in Fig. 8.1. The "crystalline-like" phase of the solution structure is assumed to be stabilized by an asymmetic

equilibrium distribution of small ions. The model predicts that the attractive part of the interaction increases with an increase in the charge since more counterions and fewer coions should be found in the intervening solution region, as illustrated here by the difference in density of the ion clouds in the ordered and disordered regions. (B) Schmitz and coworkers proposed a "temporal aggregate" model to explain the abnormally slow relaxation mode observed in polyelectrolyte systems under low-ionic-strength conditions and to explain the narrow linewidths obtained in Q E L S - S E F studies on polyions in low-ionic-strength solvents. Because the D e b y e - H ü c k e l screening length is comparable to the dimensions of the particle, asymmetric distortion of the associated ion cloud gives rise to long-range fluctuating dipole interactions between the charged particles. (C) Visualization of the two-state model of Ise and Sogami and the temporal aggregate model. Also shown with the two-state model is the proposed stabilization mechanism of Imai and Yoshino. Note that another stabilizing feature of the temporal aggregate model is the overlap of the ion atmospheres of the participating polyions, thereby providing an additional dynamic coupling mechanism.


be agreement that there is some degree of spatial correlation between the polyions in solution, the extent of polyion "ordering" remains an open question.

10 4 The Ordinary-Extraordinary Transition-/^!/i/w Molécules Somnolentes

Lin et al. (1978) first reported somewhat bizarre behavior of the C s profile of D a pp obtained from QELS data ( D

q e l s) for ( lysine) 9 55 · HBr. As the salt was

first lowered below 0.05 M NaBr, Dq e ls

monotonically increased in magnitude from a value of 1.7 χ 1 0 ~

7c m

2/ s t o ~ 7 - 8 χ 1 0

- 7 c m

2/ s . This dependence of

Dq e ls

on C s is consistent with current polyion-small ion coupled-mode theories (cf. Sections 7.3-7.6) or direct polyion-polyion repulsion inter-actions (cf. Section 8.6) and hence is referred to as the "ordinary regime". At a critical salt concentration, C*, which was on the order of 0.001 M in these studies, the value of D

q e ls catastrophic ally dropped by over an order of

magnitude. Lin et al. pointed out that Dq e ,s

in this lower ionic strength region is not the mutual diffusion coefficient since D m of poly(lysine) obtained from boundary spreading techniques continued to rise as Is was lowered (Daniel and Alexandrowicz, 1963). Furthermore, the drop in Itus(K) was only by a factor of 2 - 3 , thus indicating that the ratio Dm/S(K AR) does not remain constant through this "transition" region. These observations for D a pp have subsequently been verified for (lys)n in KCl (Ware et al., 1983; Zero and Ware, 1984; Wilcoxon and Schurr, 1983b; Schmitz et al., 1984; Schmitz and Ramsay, 1985a, 1985b).

If the catastrophic drop in Dq e ls

obtained by QELS methods reflected a free-rotation-to-hindered-rotation solution condition as effected by a coil-to-rod intramolecular transition, then one would expect the transition region to be highly sensitive to the molecular weight of solute employed in the studies. Current theories of hindered rotation in congested solutions indicate that the orientation relaxation time should be proportional to 1 / L

9 (cf. Section 5.29). A

twentyfold change in length ( M p = 40,000 to M p = 800,000) is accompanied by an 8000-fold change in the volume "swept out" for each molecule. This prediction is not consistent with the experimental observations, where the location of C

c

s is a function of the c p(g /L) rather than the molar concentra-tion of (lys)„ over the range η = 190 (Ware et al., 1983) to η = 3800 (Schmitz and Ramsay, 1985a). The value of C

c

s is not a function of L of the polyion.

Stigter (1979) attempted to interpret the change in Dq e Is

in terms of a coil-to-rod transition. Stigter's analysis predicted an ordered phase at a (lys)„ concentration ~ ^ of the experimental observation, which further indicates that the primary mechanism for the catastrophic drop in D

q e ls is not a result of

physical contacts between molecules of an extended conformation.

10.4. The Ord ina ry -Ex t r ao rd ina ry Transit ion 391

In contrast to the DLS results, the tracer diffusion coefficient obtained by fluorescence recovery after photobleaching (Z) f r a p) showed no unusual behavior. The value of D f r ap in zero-added-salt conditions was only 50% smaller than the high-salt limit (Ware et a l , 1983; Zero and Ware, 1984; Schmitz and Ramsay, 1985a, 1985b; Ramsay and Schmitz, 1985). Zero and Ware (1984) calculated that the ( lys ) 4 29 observations could be interpreted as a polyelectro-lyte expansion of a random coil from ap = 42 Â to ap = 63 Â, where the segment length of 3.61 Â and an average step angle of 60° was used in their computation of { R Q }

1 12 at low salt. Ramsay and Schmitz (1985) attemped to account for the IS dependence of D f r ap of ( l y s ) 3 8 00 in terms of a conformational expansion or electrolyte dissipation. It was estimated that an axial ratio of 18/1 would be required for a sphere-to-prolate-ellipsoid transition. Using the electrostatic dissipation theory of Schurr (1980) [cf. Eqs. (7.8.2) and (7.8.3)], it was found that for a fixed radius of 176 Â, obtained from QELS in the high-salt limit, an apparent charge in the range 194 < Z p < 255 was required to fit the data.

While the conformational change and/or electrolyte dissipation effects may adequately account for the decrease in D f r ap over the salt range examined, these effects cannot account for the catastrophic drop in D q e ls at C!r. We adopt the notation of Drifford and Dalbiez (1985a, 1985b), in which this transition is referred to as the " o - e transition, where " o " denotes the "ordinary" regime and "e" denotes the "extraordinary" regime.

A remarkable characteristic of the " o - e transition" is that no other physical technique exhibits such a dramatic change in molecular parameters as that observed by the QELS methods. There are relatively minor, but noticeable, changes in the values and slopes of the viscosity (Martin et a l , 1979) and conductivity (Shibata and Schurr, 1979). The electrophoretic mobility as determined from DSS exhibited larger changes in slope at CI (Zero and Ware, 1984; Wilcoxon and Schurr, 1983) and in one set of experiments appeared slightly bell-shaped (Wilcoxon and Schurr, 1983). Dp

s computed from the linewidth did exhibit a discontinuity in the vicinity of Q , although the change was not as great as for D a pp obtained by QELS methods. These studies are summarized in Fig. 10.6.

Another system that exhibits an o - e transition which has been exten-sively examined by a variety of techniques is sodium poly(styrene sulfonate) (NaPSS). Drifford and Dalbiez (1985a, 1985b) proposed the following empir-ical ratio for the location of the o - e transition on the basis of the NaPSS and (lys)„ results ("Drifford-Dalbiez ratio"):

m w = 1, (10.4.1) 2 z s v r

where C ^ - e is the total molar concentration of monomer units at the o - e transition point, Z s is the valency of the added salt, 7°~ e is the ionic strength at the the transition point, and the other parameters have been defined.


I

ο

7 -

6 -

5 •

4 .

3-

2-

I -

Qfrap

- 4 I 1 ' -3 -2

log (cs) -I

D - 7

- 6

5 CO

k 4 o >

k2

hi

Fig. 10.6. Comparison of the ionic strength profile of the electrophoretic mobility, apparent diffusion coefficient, and tracer diffusion coefficient for poly(lysine). The ionic strength dependence

of the apparent diffusion coefficient obtained by Q E L S ( Dq e l s

) and electrophoretic mobility

(μ ρ) reported by Wilcoxon and Schurr (1983b), and the tracer diffusion coefficient obtained by

fluorescence recovery after photobleaching ( Df r a p

) reported by Zero and Ware (1984). [Repro-

duced with permission from Wilcoxon Schurr (1983b). J. Chem. Phys. 78, 3354-3364. Copyright

1983 by the American Institute of Physics. Reproduced with permission from Zero and Ware

(1984). J. Chem. Phys. 80, 1610-1616. Copyright 1984 by the American Institute of Physics.]

There is not a large body of information to provide definitive characteris-tics of the "extraordinary regime". For example, if Eq. (10.4.1) is a charac-teristic of this particular transition, then the "splitting" phenomenon exhib-ited by the ionic strength dependence of D a pp on mononucleosomal and di-nucleosomal DNA reviewed in Section 10.1 does not qualify for the designation of an o - e transition. That is, the observations on the D N A fragment system appears to depend upon M p , whereas the o - e transition depends on the mono-meric unit concentration. One possible candidate for another system that ex-hibits the " o - e " characteristics is suggested by the study of Cotts and Berry (1983) on poly(l,4-phenyl benzobisoxazole) (PBO) dissolved in sulfuric acid (SA), chlorosulfonic acid (CSA), and methanesulfonic acid (MSA). Intrinsic viscosity values for PBO in the high-ionic-strength solvents MSA and in CSA with L1CISO3 (0.01 Ν or greater) indicated a rodlike conformation. Since the PBO was in the extended form under high-salt conditions, the increase in [77] and decrease in / t i ls at the lower values of Is was attributed to long-range interparticle electrostatic forces. QELS data obtained for B P O in CSA


Cp[ ] 0.04 0.08 _l L_

20-

ο 2 ιο-

ί 1.0 l04Cp

~i— 2.0

( g / m L )

Fig. 10.7. D a p pi / S as a function of PBO 160/8 in MSA (O), CSA + L1CISO3 ( · ) , and CSA

( · ) . [Reproduced with permission from Cotts and Berry (1983). J. Polym. Sei. Polym. Phys. Ed. 21,

1255-1274. Copyright 1983 by John Wiley and Sons.]

for Δί > 12.8 ps were "pseudo-exponential", and D a pp was associated with the slow relaxation process. For PBO in MSA and CSA + L i C l S 0 3 , D a pp

was found to be independent of θ and was associated with Dm. The product D a p prç s, where rçs is the solvent viscosity, is shown in Fig. 10.7 for BPO in the three solvent systems.

Q E L S - S E F studies on (lys)„ indicate that the slow relaxation process in that system can be disrupted by applying electric fields of larger magni-tude and/or frequency. Shown in Fig. 10.8 are the Q E L S - S E F spectra of poly(lysine) as a function of E° and v d.

Schmitz and Ramsay (1985a) examined the dependence of D p

e f on E° for

( l y s ) 3 8 00 in 0.5 and 10 m M KCl at a fixed driving frequency of 90 Hz. The values of D p

e f were then compared with the values of D

q e ls for (lys)n above

and below I°~e. These data appear to support the identities D p

e f( £ ° 0) =

D a p p(0.5 m M KCl) and Ds

p

e f(£° -> 30 V/cm) = D a p p(500 m M KCl). In other

words, the Q E L S - S E F studies both above and below I°~e exhibited the

same spectrum of relaxation modes obtained by QELS methods as a function


Vd (Hz)

Fig. 10.8. Q E L S - S E F spectral density profile for poly(lysine). (A) The fundamental peak in the Q E L S - S E F signal from ( l y s ) 3 8 00 in 10 m M KCl at Τ = 20°C, p H 8.0. The driving fre-quency is 90 Hz and the scattering angle is θ = 6 ; E° = 3.1 V/cm for the shaded curve and E° = 27 V/cm for the solid line. [Reproduced, with permission, from Schurr and Schmitz (1986). Annual Review of Physical Chemistry. 37, 271-305. Copyright 1986 by Annual Reviews, Inc.] (B) The effect of the magnitude of the applied electric field and the driving frequency on the spectral density profile of poly(lysine) (3 mg/mL) in 1 m M KCl. These spectra indicate that the linewidths are broadened as the field strength and /o r the frequency of the applied electric field is increased. This behavior can be interpreted as induced correlated motion of the polyions that results from fluctuating dipole fields (cf. Fig. 9.16) and is disrupted by increased hydrodynamic flow fields.

of salt. Since the sinusoidal electric field can induce relaxation processes with associated diffusion coefficients comparable in magnitude to D s l ow for 10 m M KCl and disrupt preexisting e-regime decay processes in the 0.5 m M KCl system.

It has been noted that the Drifford-Dalbiez ratio can be written as (Schmitz and Ramsay, 1985a, 1985b)

where Vm = 1000/Ο^"εΝΑ is the volume of a monomer unit and VDH =

π</}>/β Η/Ζ 8 is the Debye-Hückel volume per unit charge of the added electrolyte. This interpretation of the Drifford-Dalbiez ratio indicates that the o - e regime transition occurs when the ion clouds of monomer groups begin to overlap, but it does not distinguish between intra- and interpolyion

D H (10.4.2)

2 Z s / B / s ° -e Vn m


processes. In other words, it may be that the dynamics of all of the flexible coils in the solution become highly correlated in the "extraordinary regime". It is the dissolution of these correlated structures, such as the "temporal aggregate" described in Section 10.3, that is manifested as the "extraordinary" relaxation regime.

The concept of enhanced interpolyion coupling at the " o - e transition" point is supported by the Forster energy transfer study of Bruno and Mattice (1989). A mixture of lightly labeled "donor" molecules [poly(lysine) labeled with fluorescein] and "acceptor" molecules [poly(lysine) labeled with rhoda-mine b] was characterized by their relative emission intensities as a function

low high 1

· charge charge ^ dens i ty d e n s i t y displacement

£5 f l u c t u a t i o n

Fig. 10.9. Model for origin of slow mode in the extraordinary regime. (A) The DrifTord-Dalbiez ratio for the location of the "o rd ina ry -ex t rao rd ina ry" transition for linear polyions can be expressed in terms of the ratio VmKIVm [cf. Eq. (10.4.2)]. The "Debye-Hi i cke l " volume "per unit charge" has a cylindrical radius of λΌη and length </?>/Z s, viz, VDU = (nÀlu)((b}/Zs). The monomeric volume of cylindrical symmetry is Vm = (1000/iVAC^

e) = n{h}r^. Clearly the " o - e

transit ion" occurs when rm = 2XDn. The dashed region is KDH for Z s = 2. (B) The DrifTord-Dalbiez ratio may only describe linear, flexible polyions (Drifford, personal communication). One possible contr ibuting feature to this phenomenon is the charge density fluctuations that result from bending and stretching internal relaxation modes of these polyions. Shown in (B) is a port ion of a linear chain where the equilibrium charge separation distance is <6> = λΒ. Spontaneous longitudinal R o u s e - Z i m m modes may result in attractive interpolyion interactions due to (1) K i r k w o o d - S c h u m a k e r type fluctuations in the net charge as counterions are released or condensed depending upon the local charge density, a n d / o r (2) Oosawa type fluctuations in the condensed counterions along the chain.


of [ N a C l ] a d d e d. The fluorescein was selectively excited with light of wave-length λ0 = 468 nm. The ratio of emission intensity, / 5 8 5/ / 5 1 5( a c c e p t o r / donor), exhibited a discontinuous rate of change [ d ( / 5 8 5/ / 5 1 5) / d [ N a C l ] a d d e d] at [ N a C l ] a d d ed ~ 0.015 M for 0.79 m g / m L < c ; ; s < 0.91 mg/mL. Since the Forster radius is 5.8 nm (Kawski et a l , 1971), the sharp increase in [ i / ( / 5 8 5/ / 5 1 5) / ( i [ N a C l ] a d d e d] in the extraordinary regime indicates that the donor-acceptor separation distance is less in the e-regime than it is in the o-regime. This result is consistent with the point of view that the Drifford-Dalbiez ratio pertains to bulk solution conditions rather than intrachain overlap conditions (cf. Section 7.14).

A possible mechanism for the dynamic coupling between flexible polyions is illustrated in Fig. 10.9.

Although there is compelling evidence to support long-range correlations between the polyions in the "extraordinary" regime, the precise mechanism that generates these interpolyion interactions and the characteristics of the polyions that undergo such transitions, remain to be established. Because of the extremely slow relaxation time associated with the e-regime, this particular phenomenon is referred to as jeu du molécules somnolentes, or "tricks of sleepy molecules", with an apparent diffusion coefficient D j m s.

Additional Reading

Physics of Complex and Supermolecular Fluids. (1987). (S. A. Safran and N. A. Clark, eds.) Wiley Interscience, New York.

Ordering and Organisation in Ionic Solutions. (1988). (N. Ise and I. Sogami, eds.) World Scientific, New Jersey.

APPENDIX A

Mathematical Notation

The mathematical notation for vectors, matrices, and mathematical opera-tions used in this textbook is standard but for a few exceptions.

A column vector is denoted by a boldface character:

a = (Al)

A matrix or second-rank tensor of dimension η χ p, where η is not neces-sarily equal to p, is denoted by an underscored boldface character:

M

M 2 , 1 M 2 , 2

M, η, 2 M„

(A2)

The transpose of the column vector is a row vector, and the transpose of an η χ ρ matrix is effected by interchanging the rows and columns, which results in a matrix of dimensions ρ χ η. The notation for the transpose is respectively, a

T and M

T.

The dot product or inner product between an array A of dimension η χ ρ and an array Β of dimension ρ x m results in an array C of dimension η x m whose elements are given by

(A3)

or, in matrix notation, C = A · B.

397

398 A. M A T H E M A T I C A L N O T A T I O N

The scalar product requires the multiplication involving two vectors that results in a quantity having dimensions l x l ,

ρ

Σ a

kbk

(A4)

or

a r - M - b = Σ Σ akMkJbj. k=l j=l

(A5)

A dyad product between an array A of dimension η x m and an array Β of dimension ρ x q (A and Β can be either vectors or matrices) results in an array C that has the dimensions pn χ qm. The general notation for the dyadic is illustrated by the spatial vector r = (x,y, z):

rr

xx xy xz

yx yy yz

zx zy zz

(A6)

There will be occasion in this text to emphasize that the dyadic is being taken between two arrays. We therefore introduce the notation

rr = r*rr (A7)

The trace of a square matrix of dimension ρ χ ρ is the sum of its diagonal elements, viz,

7>(A) = Σ

It is noted that the following vector identity holds:

ar · a = 7>(a * a

r) .

(A8)

(A9)

APPENDIX Β

Structure Factors for Multicomponent Systems

General Formulation

It is given that there are NL different types of particles present, labeled L = 1, 2, JVL, and the solvent is identified as the L = 1 component, which does not contribute to / t i l s. For generality, each of the NL — 1 solute types may be considered to be polydisperse in its molecular weight distribution and distribution of subunits comprising the macroparticle. The number of distinct weight subgroups is denoted by NLtW, where W = 1, 2, NLtW. The particles within each weight group are assumed to have the same molec-ular composition but may differ in the distribution of subunits. The number of distinct sequence groups within each weight group is denoted by NLtWtS, and the particles are numbered J = 1, 2, 3 , . . . , NL w$j. The subunits within the J t h particle are also numbered, j = 1, 2, 3, n L W S J. To simplify the notation, a single index Ρ is used to include all weight and sequence spe-cific classifications, i.e., Ρ = {W,S}, where Ρ = 1, 2, NL WNL W S = NL P

containing NL P J particles. The total intensity of scattered light from a multicomponent system, </(K)O T>, is given by an expression analogous to Eq. (2.5.5), which, with the identity I0 = | F 0 |

2g e and the definition Β(ω0) =

{œ0lc)\l0/R2\ is

NL>,P',J'

X Σ e x p [ - i K

T. ( R L , W - » L ' . P ' , ; ' ) ] Σ Σ

x <*L,p,jj<*L',p',j'j'expL-iKT 'fa L,P,J,j j —

rL',P',J'J')l . , - ) ] ) . (B l )

399

400 B. S T R U C T U R E F A C T O R S F O R M U L T I C O M P O N E N T SYSTEMS

In the general case of heterogeneous polymers, one can separate neither the polarizabilities from the interference factors nor the center-of-mass co-ordinates from the internal coordinates of the polymers in performing the averages. [This is the basic principle behind DLS and is the basis of an incoherent component to the correlation function (Weissman, 1980; Pusey and Tough, 1985; cf. Section 6.16).] Hence, restrictions are placed on the mathematical approximations that can be used to evaluate the pair contribution to the sum given in Eq. (Bl). Recall that in the single-contact model, exp[ — / K

r · (r7j- — r J T) ] was expressed as exp[ —fK

T · (r 7j — r 7 m) ]

E XP [

— *K

T · (

rJ m '

— rj'yT\ times an excluded volume term involving the

difference r r / — rJm. To rewrite the single-particle interference term as the product of two such terms for a complex system would require the "creation" of a second product of polarizabilities a L PJjOLL. p. J , Y.

Spherical Particles in the Weak Copolymerization Limit

The average segment polarizability for all of the NL P J identical particles in the group Ρ is,

J HL,P,J

< < * L , P , J > = <aL,p,j'> = < X , P > = Σ aL,p,j,r ( B 2)

nL,P,J j = l

We write the double sum over j and / as the product

" L , p j " L ' , p ' , j ' < a L , p > < a L ' , p ' > [ ^ ( ^ A r ) L i P JP ( X A r ) L %p % J01 / 2

, where the particle structure factor P(K Ar)L PJ is

P(KAr)UKJ = P{KAr)LtP = ^—( Σ Σ HL,P,J \ j = l j' = l

x MoiLtPtjj)A((xLtPtjjf)Q\p[iKT - (r L f P t J t J. - r L f P f J J, ) ] ^ (B3)

where A(aLPJj) = a L , P , j , ; / < a L , P J> . Since A(ccLPJJ) may differ greatly from unity for some of the segments in the polymer, one cannot separate the segment polarizability from the associated interference term. (Recall that variation in the scattering power of internal segments is the principle behind contrast matching methods in neutron scattering techniques.) In contrast to the homogeneous sphere case, the center-of-mass coordinates cannot be separated from the internal coordinates in the case of a heterogeneous sphere. This is because any rotation about the center of mass results in a new spatial polarizability configuration. On the other hand, if the chemically distin-guishable segments have almost identical polarizabilities, then the rotation does not result in a different spatial distribution of the polarizabilities. This is

Partial Structure F a c t o r — W e a k Copolymerization Limit 401

referred to as the weak copolymerization limit. Hence, <7(K)m> for spheres is

/ NJL NL> NL,p NL>,P> NL,P,J NL>,P>,J>

</(K)m> ~ Β(ω0)( Σ Σ Σ Σ Σ Σ "L,P,JNL.,P;R

\L = 2 L' = 2 Ρ = 1 Ρ' = 1 J = l J' = l

χ e x p [ - / KT . (R L > P tJ - RL' ,p',j ' )]<aL fp,j><a L, > P, t J,>

x [ ^ ^ ) L , P , ^ ( ^ ^ ) L , P ' , 7 ' ]1 / 2

V (B4)

Partial Structure Factor—Weak Copolymerization Limit

As discussed in Section 2.10, S(K, ω) is given in terms of the pair-to-self ratio of the contribution to 7 t i ] s. A similar expression for a multicomponent system does not readily provide useful molecular information because of the pres-ence of a large number of pair interaction terms. One can, however, express Eq. (B4) in terms of partial structure factors. Let us substitute into Eq. (B4) the expressions

ζ ν "o ( Sns \ cLtPtJVs

2π \dcLtPtjJTtfl. NLPJ

and cL PLVJNL PJ = ML,PJ/NA = ML P/NA. The total intensity of scattered light can now be written as

< / ( κ ' ^ > = β ( ω 4 ( Ι Γ

Χ Σ Σ Σ Σ M)L,PS{K)L,u,P,P;j,r, (Β6)

L = 2 L' = 2 Ρ = 1 P' = 1

where S(K)L P is a self term defined by

S(K)LTP = I ^ - )2 c L , P M L , P P ( X R s ) L , P , (B7)

and S(K)L L, PJ J, is a partial structure factor,

ITCTMTP,

NL,P,J NL,P,J'

Χ Σ Σ < e x p [ - i KT. ( R L . , . J - R t . . p . , J . ) > . (B8)

J = l J' = l

dn*

dnK

x

3CL,P/ Τ,μ'


Mr =

L',P' 1 / 2

L,P

Pr = P(KRS

1/2

(BIO)

(Bl l )

(B12) P(KRS)L,P

From Eqs. (B8)-(B12), S(K)L L P P J3 = 1 since the double summation col-lapses to NL pj terms of unity, which cancels the prefactor l/NL Pj. The indices i and k are introduced to represent the sets of indices {L, P, J) and {L', P ' , J ' } , respectively. The average < > in Eq. (B8) is evaluated through the pair distribution function, or Fourier transform of the total correlation function (cf. Appendix C),

Χ Σ < e x p [ - / K T - (R, - R J > = (N^Kh^KR^l j ~ ι J' = ι

M R f , . ) - l ] e x p ( i K T . R u ) i / R ( ,

(B13)

(B14)

(B15)

The partial structure factor for two different particles is

where n|- = NLPJ/VS and = NL._P,j,/Vs. Hence the combined expression for the partial structure factor is

S(K)Uk = 6Uk + ( π ί π ^ ^ ^ ^ Λ ί , Ρ ^ , , ί ^ ί Χ Α , . k) . (B16)

The factor (lrCrMrPr)ik accounts for differences in the scattering power of the different components. However, it is the product S{K)iS(K)i k that enters into the total intensity calculations. This product is quite small if a weak scattering unit is one of the subunit pairs.

Scattering Power of Molecules

As pointed out by Stockmayer (1950), the equations for the interpretation of 7 t i ls for two-component systems are not adequate to describe data obtained from even slightly polydisperse polymer preparations, polymers that are soluble only in mixed solvents, and heterogeneous preparations of polymers. The specific form of the correction terms depends upon the characteristics of the system. Since the total volume of the solution depends upon the partial specific volumes of the components, vh the volume under consideration is also dependent upon the local chemical composition. The thermodynamic

Pseudo Y-Component System 403

expression for dns at a fixed solvent weight is

( dn \ Nl

dns = (-apdT + KTdP)\^^J + Z2Mft, ( B 1 7) where oc? = (d\nV/dT)Pg is the thermal expansion coefficient and κτ = — (d\nV/dP)Tg is the isothermal compressibility coefficient at fixed weight composition g, and bt = {Sn^/dg^j Pg>, where g' indicates that all components other than the ith component are at a fixed weight composition. The change in the volume is given by

NL

dV = ocPVdT -KTVdP + Σ v.dg^ (B18) i = 2

where vt = (dV/dg^j Pg> is the partial specific volume. To calculate the index of refraction increment, the variation in the chemical potentials of the various components is needed, where the solvent is at fixed chemical potential,

NL

dßk = Σ aikdgh (Β 19)

i = 2

where aik = (dpi/dgk)ZPg> = aki. If all of the components are at constant chemical potential except the ith component, then dpkili = 0. Upon solving Eq. (B19) for dgh one has {dgjdp})TPg> = Λ 0·/ |α 0· | , where |α 0· | is the determi-nant of the matrix containing all of the a 0 , and Ai} is the cofactor of the element aij9 viz, Αυ = 5|α0·|/<9α0·. The general expression for the index of re-fraction fluctuation <(Δη 8)

2> is thus (Stockmayer, 1950)

NL NL U U Λ

<(Δ?Ϊ8)2> = Σ Σ -γ^τ (Β20)

i = 2j=2 \αη\

Pseudo F-Component System

In many situations the exact molecular composition of a sample is unknown, such as in the preparation of complex biological samples. It is therefore desirable to express </(K, Rs)m} in a form similar to that given by Eq. (2.5.15). Without loss of generality, attention is given to a two-component system in which cT denotes the total weight concentration of the NL — 1 solute com-ponents that act as a "pseudo" component. Upon factoring out the prod-uct cT(dnJdcj)Tμ> from the right-hand side of Eq. (B6) and defining Η = [2nn0(dnJdcT)T

2/NAÀQ, one can write

^ = < M m > < S ( K ) m > , (B21)


where the subscript "m" denotes a multicomponent system. The aver-age < M m> is identified from Eq. (B6) in the limit P(KRs)LP-+l and S(K)L,L',P,P\J,J' àL v pp> j j . (small, noninteracting particles), with the result

<MM> = Σ Σ X2

L , P ( " S ) X L A C J ) M L , P , (B22) L = 2 P = 1

where XL,P(ns) = (dns/dcLtP)Ttß./(dns/dcT)Ttß> a n d I L i P( c T ) = c L , P / c x . T h e aver-age <P(K) m> is now obtained from the self term given by Eq. (B7) by factoring out < M m> , viz,

<P(K)m} = Σ Σ ^ Î . p ( « s ) ^ L . p ( c T ) X L , P « M m» F ( X ) t , P , (B23) L = 2 P = 1

where XL,P«Mm» = M L P/ < M m > and both XLjP{ns) and XL,P{cT) have pre-viously been defined. A similar relationship can be obtained for the solution structure factor by factoring out the product < M m> from Eq. (B8).

It is concluded that (I(K)m) for a multicomponent system can be analyzed by expressions similar to those for a two-component system provided that the relative concentrations of the components remains fixed. The resulting average values < M m> , <P(K) m>, and <S(K) m>, however, are weighted by their relative index of refraction ratio XL,P(ns) as well as by their weight ratio.

APPENDIX C

The Ornstein-Zernike Relation and the Pair Distribution Function

Summarized in this appendix are various approximate solutions for the pair distribution function g

i2)

2(r) as derived from the Orns te in-Zernike relation. The starting point is the particle density functions obtained from the grand partition function with no external field, (Ξ),

= o Np\

where λ{ is the activity of the ith particle, and is the pairwise interaction energy between the ith and ; th particles. The method of Ornstein and Zernike (1914) is to calculate the cluster density p

(m) by standard methods of differen-

tiation of Ξ ra-times with respect to λί9 where i runs from 1 to m. The cluster density functions p ^ r )

a nd Ρ ι

22 (

Γ)

a re given by the functional derivatives

P'/V) = § H (C2)

and

^-ΨΜ-, ,c3)

If it is known that particle 1 is at the origin, then, following Perçus (1962), gfli

1) is defined by the "local" single particle density function,

^ = PÏWZKr) = <n'p\g\2Mr), (C4)

405

406 C. T H E O R N S T E I N - Z E R N I K E R E L A T I O N

where <np> u is the uniform distribution concentration. It follows from Eqs. (C2)-(C4) that

W ^ * ^ , C 5 )

where δ12 denotes the functional derivative δλιΙδλ2 and

= g\2M*) - I- (C6)

The function h^l(r) is called the total correlation function because it depends

upon all of the particles as evident in the cluster size expansion of gfyi1)

(Morita and Hiroike, 1960; Hansen and McDonald, 1986):

g[2>

2{r) = exp kT 1 + Σ W U F L C T R ) m — 3

(C7)

g[2

2(r) = e x p ^ - ^ ^ e x p [ ^ . 2 ( R ) ] , (C8)

where ^LI2(R) represents the modification of the pairwise interaction due to the presence of other particles.

Ornstein and Zernike (1914) introduced the direct correlation function

^ > — m ^ - ( C 9)

(Their paper is reproduced in a reprint volume edited by Frisch and Lebowitz, 1964.) Noting that δ12 = δρ^(τ)/δρ^(ι), one has from Eq. (C9)

δ\η(λι)_ _ \ 2 _ _ i 2 ) () ( cm

The left-hand sides of Eqs. (C5) and (CIO) are inversely related; hence a relationship must exist between h

{2)

2(r) and c{2)

2(r). Since h{2)

2(r) depends upon interactions between all of the particles in the cluster, these additional contributions are introduced through the functional derivative representation of δί2. Limiting the cluster size to three particles, δ12 in terms of the functional derivative is

^ = J ^ Ä T A Ö Ü D R - ( C 1 1 )

If we now substitute Eqs. (C5) and (CIO), we arrive at the Ornstein-Zernike relationship (OZ),

hl&(r) = c%(r) + L^hfMr)cf\(r)d

3r3, (C12)

The Orns t e in -Ze rn ike Relation and the Pair Distribution Funct ion 407

where the integration is over the positions of the third particle. The generalized Ornstein-Zernike integral equation to include S types of particles is

s

hf(r) - cf>(r) = Σ (n'k\ fcg>(R - r')c[]\r')dr'. (C13)

Our inquiry into approximate solutions to the pair distribution function focuses on the series expansion in the excess particle density Δ ρ

(

3

Υ( Γ ) , viz,

l n [ f f ? i ( R ) ] ~ +

δ In h ' Αρ«\τ)ά

3ι'3, (C14)

where the integration is over all of the positions of a third particle. Using Eqs. (C4) and (C6), to obtain an expression for Δ ρ

(

3

Υ( Γ ) , and Eq. (C9), one has

l n [ g £ ( r ) ] = - ^ + < n ' p > u h\

2l(t-r')c^(T')d

3r'. (CI 5)

Substitution of Eq. (CI3) for the integral in Eq. (CI5), one has the general expression for the pair correlation function,

Ίψ(τ) = e x p ( - ^ J e x p [ Ä J , 2 ,( R ) - (C16)

Relationships between h\f\r) and c-2 )(r) that give a closed form relationship between the Fourier transforms h\f\K) and c | 2 )(K) are referred to as closure relationships.

In the hypernetted chain approximation (HNCA), the number of particles in the cluster is limited to three. In this case the O Z relation is used to evaluate c

{i\(r') which, when combined with Eqs. (C6) and (CI5), gives the

generalized closure relationship (3 - • i, 1 - • j),

C<jV)HNCA = _ M± + AJ 2 , ( R, } _ 1 „ [ Λ( 2 ) ( Γ' ) + 1], (C17)

where g - 2 )( r ) H NC is given by Eq. (CI6). Linearization of the second exponen-

tial in Eq. (CI6), which limits the H N C approximation to dilute solution conditions, results in the Perçus-Yevick equation (Perçus and Yevick, 1958),

gf (r ) PY ~ e x p ( ^ * ) [ l + A<?>(r) - cf ( R ) ] . (C18)

The mean spherical approximation (MSA) obtains in the limit of weak inter-particle interactions such that ln[/ii?'(r) + 1] ~ Ιιψ(τ), in which case one has from Eq. (CI7)

4 2 ,(r') M S A=--^. (C19)


Citing the works of Hutchinson and Conkie (1971, 1972), in which the PYA and HNCA expressions for purely repulsive potentials bracket the computer simulation (Monte Carlo or molecular dynamic) results, Rogers and Young (1984) proposed a "mixing function" which smoothly approached the PY and H N C approximations in the appropriate limits. The Rogers-Young approxi-mation (RYA) for gfMr) is

g\}WYA = exp

kT

' exp[y, f c(r)/(r)] - Γ

f(r) (C20)

where yu(r) = ηψ(τ) — c^2)(r) is defined in Eq. (CI3), and the mixing function

is f(r) = 1 — exp( — ar), with a to be determined from the thermodynamic consistency requirement. The pair distribution functions for "soft" interaction potentials, i.e., 1/r

11, were said to be in good agreement with the computer

simulations of Hansen and SchifT (1973). Zerah and Hansen (1986) modified the RYA interpolation function approach to include attractive [£/^(r)] as well as repulsive [£/*(r)] interactions. The pair distribution function in the Zerah-Hansen approximation (ZHA) is

^ ( r rA = e x p ( - ^

" , exp[yft(r)/(r)] - 1

fir) (C21)

where yfj(r) = h^\r) — cj2 )(r) — [t/^(r)/ /cT] and f(r) are taken as the same

form as in the RYA. Expanding the exponent containing f(r) and setting f(r) = 0 gives the soft-core mean spherical approximation (SCMSA) distribu-tion function,

— IJK(r)

7(2)/„\SCMSA _ o v rJ u

Ι Α1

* ^S;V) = exp kT

U*(r) (C22)

and the MSA closure for r > r m , c |2 )

( r )M SA

= —UA(r)//cT, where rm is the loca-

tion of the minimum in the potential. The discussion thus far has focused on methods that stem from the HNCA,

which is based on the effect of a third particle on #-2 )(r). Larger cluster size

effects can be examined by repeated use of the O Z relation in Eq. (CI4). The terms that result from the iteration expression for h

(l]2(t) appear as products

of functions involving pairs of particles with a sequential set of running in-dices that start with particle 1, run through the m — 2 other particles in the cluster, and terminate with particle 2. The H N C A is thus limited to a subset of all possible pairwise interactions between the particles in the cluster. Morita and Hiroike (1960) represented these contributions in terms of Mayer cluster diagrams (cf. Mayer and Mayer, 1966). The cluster diagrams consist of two open circles and a varying number of closed circles that are connected by lines, where integration is performed over the coordinates of the closed circles. If a line connects two circles, then the circles are said to be adjacent. The sequence

The Orns t e in -Zern ike Relation and the Pair Distribution Funct ion 409

of connected adjacent circles is called a path. A diagram is called simply connected if it has only one path, and n-tuply connected if there exists at least η independent paths between each pair of circles. Diagrams that contain at least one closed circle such that all paths connecting the two open circles pass through that point are referred to as bridge diagrams. Removal of the bridge point, or node, results into two or more components, where the two open circles appear in différent components. If the removal of one closed circle results in a component with no open circle, then that closed circle is referred to as an articulation circle.

Morita and Hiroike (1960) place four restrictions on the summation of diagrams: (I) each closed circle is independently connected to the open circles; (II) the closed circles are connected among themselves independent of the open circles; (III) the open circles are not directly connected; and (IV) the diagram has no bridge point. Several of these diagrams are illustrated in Fig. C I .

Clearly, H N C A does not include the contribution of highly connected graphs. Work on the density expansion of the radial distribution function (van Leeuwen et al., 1959; Morita and Hiroike, 1960; Green, 1960; Meeron, 1960) resulted in two equations, the Ornste in-Zernike relationship and the closure relationship,

c ( 2 V )H N C A = _ + fc(2y} + B ( r) _ i n [ f c £ y ) + 1], (C23)

where B(r) represents the infinite sum of "bridge" or "elementary" graphs that are highly connected diagrams that cannot be easily evaluated. Rosenfeld and Ashcroft (1979) operationally defined B(r) as that function that accounts

Fig. C.l . Diagrams indicating interparticle interactions.


for the difference between the HNCA computer simulation results and is referred to as the modified hypernetted chain approximation (MHNCA). It was shown that B(r), within the accurracy of computer simulations, was indepen-dent of the form of the potential l^(r). This is due to the fact that B(r) is relatively short-range (Lado et al., 1983).

Belloni (1986a) compared the MSA, RMSA, and H N C calculations of the polyion-polyion partial structure factor for a highly asymmetric polyion ( Z p / Z c = +20/—1). Belloni employed the usual direct correlation functions and also the modified interaction potential of Beresford-Smith et al. (1985, cf. Section 8.11). The results of this comparison are given in Fig. C.2.

• 7~, > 1

Ο Κ ( A- 1

) 0.1 0 .2

Fig. C.2. Polyion-polyion structure factor S(K \R) for a polyelectolyte solution without added salt. The calculations were performed for an asymmetric charge ratio Z p / Z c = + 2 0 / - 1, a polyion radius of 25 Â, and φρ ~~ 5%. The MSA, RMSA, and H C N calculations used point charges for the small ions, whereas the MSA1, RMSA1, and H C N 1 used the value as = 2.5 Â. The effect of the finite size of the small ion is to increase the magni tude of the first peak by a few percent. [Reproduced with permission from Belloni (1986a). J. Chem. Phys. 85, 519-526 . Copyright 1986 by the American Institute of Physics.]

APPENDIX D

MSA and RMSA Solution to the Ornstein-Zernike

Relationship

A general assumption for colloidal systems is that they can be treated as a two-component system (Hayter, 1983), defined in this text as the polymer of interest immersed in a milieu of other particles. For a system of polyions (p), anions (a), and cations (c), the statement for a two-component system is that interparticle interactions between the anions and cations are absorbed in λΌΗ. Hence, 0/pV) = o^V) = 0o

?)(r) = 1, where i, j = a or c but not p.

The mean spherical approximation (MSA) method for solving the Orns te in -Zernike equation for charged spheres of radius ap is characterized by the closure relationships: (1) cp

2)(r) = - Upp(r)/kT, for r/2ap > 1; and (2) ftpp(r) =

- 1 , for r /2a p < 1. Hayter and Penfold (1981) obtained an analytic MSA expression for S(Q)pp, where Q = 2Kap. They assumed that the attractive part of the D L V O potential was not sampled because of strong repulsion interactions between the particles that were much larger than kT. Using the screened repulsive potential U(x)/kT = (y/x)exp( — bx), where χ = Rl2/2ap

and b = 2αρ/λΌΗ, the Orns te in-Zernike equation,

was solved by using the methods of Wertheim (1964) and of Waisman and Lebowitz ( 1972a, 1972b), which were also applied by Palmer and Weeks (1973) for macroions interacting through an unscreened Coulombic potential. The

(Dl)

411

412 D. MSA A N D RMSA S O L U T I O N

first step is to define a new function c0(x) = c(x) + U{x)/kT, which has the property c0(x > 1) = 0. For the integral in Eq. (Dl) one applies the bipolar convolution relationship,

p(\*-y\)q(y)dy = In * o o fu + x

p(u)udu q(v)vdv. ο Jl«-x|

(D2)

The procedure is to substitute these expressions into Eq. (Dl), multiply by χ to clear the denominator, take the Laplace transform, rearrange the resulting expression to a convenient form, and then take the inverse Laplace transform to obtain the desired expressions. The Laplace transform involves the integrals of the type

H(s) = xh(x)çxp( — sx)dx, Jo

G(s) = xg(x) exp( — sx) dx = H(s) + 1

C0(s) = xc0(x) exp( — sx) dx.

(D3)

(D4)

(D5)

After rearrangement, the inverse Laplace transform for χ < 1 can be taken for all terms except G(s)C0( — s). The procedure is then to replace s with — 5 in the rearranged equation and then to multiply through by s 6G(s), thus enabling one to eliminate the term G(s)C0( — s). The resulting analytic expression of Hayter and Penfold (1981) for c(x) and S(Q)pp for the MSA at a volume fraction φρ is

c(x) = -yexp(-bx)

(x > 1).

c(x) = A + Bx + Âx3 +

C sinh(foc) + F cosh(foc) — F

A = Ai+ A2C + A3F,

B = BX + B2C + ß 3 F ,

C = M 1 6F2 + œL5F + ω 1 4

œNF + ω 1 2

and F is the root of the equation

w 4 F 4 + w 3 F 3 + w 2 F 2 + wLF + w 0 = 0.

The parameters Ah w i 5 and œu are given in Table D.I .

(D6)

( x < l ) , (D7)

(D8)

(D9)

(D10)

( D i l )

MSA and RMSA Solution to the Orns te in -Zern ike Relationship 413

δ4

24(/>p{a3[sinh(/?) — frcosh(fr)] + a 2 sinh(fr) — olx cosh(fr)}

δΑ

1 I2 δ

2

- — + a3[cosh(fr) — 1 — bsinh(b)] — olx sinh(fr) + a 2cosh(b )

3 0 P

( 0 P^

2 )2 - 1 2 φ ρ 7e x p ( - f c ) [ f t + ft + (1 +

12(/>p{ft|>cosh(6) - sinh(&)] - ft sinh(b) + ft cosh(fe)}

2" 1 2 ψ ρ ^

2 , + 2) Γ ψ ρ +

ι — 3 Η " τ - ft[cosh(6) - 1 - fcsinh(fc)] + ft sinh(fc) - ftcosh(b)

<54

w 0 = ω2

4 - ω 1 2ω 2 4

w x - 2 ω 1 5ω 1 4 - ω 1 3ω 2 4 - ω 1 2( ω 34 + ω 2 5)

w2 = ω is + 2ω1 6ω1 4 - ω13(ω34 + ω25) - ω12(ω35 + ω26)

νν3 = 2 ω 1 6ω 15 - ω 1 3( ω 35 + ω 2 6) - ω 1 2ω 3 6

w 4 = ω\6 - ο ^ 1 3ω 3 6,

where ωί7· = μίλ} — μ^λ{

δ = 1 - φ ρ

-(2</>ρ+1)<5 1 4 0 ρ

2- 4 0 ρ - 1

«ι- τ «2= π

36</>2

-((/>2 + 7 0 ρ + 1 ) ^ ο 9φρ(φ

2

ρ + 4</>ρ - 2)

6

1 2 φ ρ( 2 02 + 8 ψ ρ - 1 )

4 μ ι - ί2Λ2 - 12φρΚ

2 μ2 = txA2 + ί2Λι - 24ψρΚχΚ2 {continues)

Table D. l

Hayte r -Penfo ld MSA Parameters for S{Q) Calcula t ions0

_ 240 py cxp( - fe ) [g 1 + ol2 + (1 + fr)«3] - (2φρ + l )2

Ax —


Table D. l {Continued)

μ 3 = Μ 3 + Μ 2 - 2 4 0 ρ Κ 2 Κ 3 μ 4 =

μ 5 = ί 1 Λ 3 + ί 3 Λ 1 -24φρν,Υ3 ^ 6 =

λ{ = \2φρρ2

λ2 = 2 4 ψ ρρ , ρ 2 - 2 Β 2

λ3 = 24φρρ2ρ3 λ* = 1 2 ψ ρρ ? - 2 β ,

λ5 = 24φρΡιρ3-2Β3-ό2

κ =

yexp( </>2)2 </>Ρ + 2

Ρ 2 : {φ\ + 0l)sinh(fe) + 2(/>1(/>2cosh(fr)

Ρ 3 : + 0

2) cosh( /? ) + 2φιφ2 s'mh(b) + </>

2

h = Τ

3 + Τ

4 ^ 1 + Τ

5 # 1

ί 2 = τ 4Λ 2 + τ 5 β 2 + 12ψ ρ[τ! cosh(fc) - τ 2 sinh(/>)]

ί 3 - τ 4Λ 3 + τ5Β3 + \2φρ{τχ sinh(6) - r2[cosh(/>) - 1]} - - φ ρ ( 0 ρ + 10) - 1

( 2 ψ ρ+ \)(φ2

ρ-2φρ + 10) yexp(-f r ) ( i ; 4 + υ 5

i?4cosh(6) — ν5 sinh(fr)

- 6φΙ + 5)<5 - 6φ 2φ

3

ρ-3φ2

ρ + 1 8 0 ρ + 10 24φρν3 + U4sinh(fr) — U5cosh(fr)

- (φ ρ

3 + 3 φ

2 + 4 5 0 ρ + 5)<5

5<54

2 03 + 3</>

2 + 4 2 0 ρ - 20

2 03 + 30</>ρ - 5

•• 6ψ ρ(ϋ 2 + 4ν3)

5/?

: -\2φργ&\ρ{-ο)(τι + τ 2)

3 0 Ρ ( 0 Ρ + 8) Γ ( 2 0 ρ+ 1)'

10 L

υ 4 = νγ + 24φρον3

, + 2

Τ 4 = 3 0 Ρ ^2

( Τ2 - Τ

2)

α A copy of the M S A / R M S A program can be obtained upon request from Dr. John Hayter ,

Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, Tennessee, 37831-6031.

MSA and RMSA Solution to the Ornstein-Zernike Relationship 415

The analytic expression for the structure factor is

S ( Q) = [1 - 2 4 0 ρ Δ ( β ) ] *

A(ß) = 4 [ s i n ( ß ) - ß c o s ( ß ) ]

(Dl 2)

Β

2ß3lß3 + 4

12 24"

ß c o s ( ß ) + 2 s i n ( ß ) -

_6

ß sir 2 ι sin(Q)

ß c o s ( ß )

+

+

ß ( ß2 + b

2)

F

Q(Q2 + b

2)

[ftcosh(fc)sin(ß) - β sinh(fo)cos(ß)]

{frsinh(fr)sin(ß) - ß[cosh(b)cos(ß) - 1]}

y exp( — b) + ψ Ccos(ß) - 1] b2 } ] Cfcsin(ß) + ß c o s ( ß ) ] . (Dl3)

The procedure is to calculate S(Q) over a limited range in β for each root F for Eq. (D i l ) . The radial distribution function g(x) is then calculated from the expression

9(x) 1 + 1

\2πφρχ Jo

lS(Q)-l]Q*in{Qx)dQ. (D14)

The root that gives the correct physical picture is the root that gives g(x) = 0 inside the particle.

Hansen and Hayter (1982) pointed out that the MSA appeared to work quite well for highly concentrated micellar solutions in which y » 1 and φρ > 0.2. However, when applied to dilute solutions of highly charged poly(styrene) spheres, the MSA calculations gave unphysical results, namely, negative values for g(x). They argued that the strong Coulombic interaction effectively prevented the particles from experiencing the hard-core potential. To remedy the situation, they proposed a rescaled mean spherical approxi-mation (RMSA), which incorporated a "rescaling" of the lengths used in the computation of the structure factor. There are two lengths to be considered. The first is the average distance of separation between neighboring particles, {AR} = ( 3 / 4 π < Π ρ »

1 / 3. It is at this distance that the Coulomb coupling

constant T b = 2U((AR}/2ap)/kT is calculated. The second length is the


particle diameter, which is used to calculate the value of the MSA distribution function at the point of contact of the two spheres for an "effective diameter" 2<2p, given in the above notation as

gmA(x')=-(p1+p2C + p3F) (D15)

where x' = r/2ap. The rescaling prescription of Hansen and Hayter com-mences with the calculation of the Coulomb coupling constant T b and g

MSA(x') with ap = ap, using the proper root for F. If öf

M S A(x') is zero or positive

for this "natural" set of paramaters, then one proceeds with the usual MSA calculation. If, on the other hand, g

MSA(x

f) < 0, then the parameter a'p is

increased while a fixed value of the Coulomb coupling constant is maintained. The calculations are repeated until a value of 2ap is obtained for which g

MSA(x

f) = 0. This value is denoted as the "effective" particle radius, ap

{. The

data are then fitted with the MSA method using ap

f, resulting in effective

values of ße f f

, freff, and y

e f f. The relevant parameters are related to the effective

values by the ratio s' = ap/af, i.e., y = ye f f

/ s ' , β = s ' ße f f

, and b = s'beff.

APPENDIX Ε

The Medina-Noyola Formalism for the Tracer

Friction Factor

Medina-Noyola examined the velocity self correlation function of a probe particle in the presence of other solute particles (personal communication; Medina-Noyola, 1987; Medina-Noyola and Del Rio-Correâ, 1988).

It is given that there are S different types of solute species present in solution, with Sj denoting the number of particles of the 7th solute component having a local concentration n}(r, t). The evolution of the probe particle velocity is then given by the Langevin equation

dt m p 7- t l m p

[V0(r)] j pn;-(r,i)rf3r + - ^ , (El)

and n'j(r, t) is assumed to obey the diffusion equation

dn'j(r, t)

dt = v p

r(i).[Vn;-(r,i)]

4ί ν Γ ' D j n X f r t) · V /Ur, t) - \T - J,-(r,r), (E2)

where D j m is the diffusion tensor for solute species j in the presence of the other solute species m, μ^τ, t) is the chemical potential for species 7, and Jy(r, t) is the random flux of species j that arises from thermal fluctuations. Eqs. (El) and (E2) are coupled through the terms v p (t) · [Vn}(r, i)] and [V(/>(r)]Jpn}(r, t). The objective is to solve Eq. (E2) for rc}(r, t) and then substitute into Eq. (El) and solve for vp(r).

417

418 Ε . T H E M E D I N A - N O Y O L A F O R M A L I S M

Linearization of the equations. The local concentration and chemical potential are linearized in their fluctuations about the equilibrium quantities: n'j(r,t) = <«}(R)> + ôn'j(r,t); and μ/R.t) = <^(r)> e q + δμ/τ,ί). It is empha-sized that vP(R, f) is itself a first-order fluctuation quantity since its average value must be zero by conservation of momentum arguments. With these ideas in mind, one arrives at the following first-order terms:

[V0(R)],PK(M)]rf3R = [V (R)]JP[5«}(R,i)]d3R, (E3)

(E4)

since }K[V0(R)]Jp[<nj(R)>EQ] d3r = 0 because to spherical symmetry:

v;(r).[V«>,f)]sv;(i).V<«}(R)>E(P

where vp(f) · \ ön'j(r,t) is neglected because it is a second-order term; and ex-pressing oßj(r, t) in terms of <5n}(R, r),

1 T

^rVT-D J M>4(r,t)- V/UM)

= VT · L ,„ S

•VI Ejk(r9T')ôn'k(T'9t)d*r', (E5)

where Ejk(r,r') = (l/ /cT)[<^(r, t)/dn' k(r\f)] e q, with the assumption that ^

T · Ojmdrim(r,t) · \ôpj(r,t) ~ 0 because it is a second-order term, and

V</i_y(r)>eq = 0 by definition of the equilibrium state. One may express Ojm

in terms of the Onsager phenomenological coefficients L j m = D j m<n}(r)> e q. [Note that this formalism is slightly different from that described in Section 3.1 where L p p = C p / / m , cf. Eq. (3.1.13).] Making these substitutions in Eqs. (El) and (E2) gives

d\p{t)

dt

d6n'(t)

dt

= -G 1 1-v p( i ) -G 1 2.EN ' ( i ) + g(i),

= - G 2 1- v p ( i ) - G 2 2- 6 n ' ( i ) + j ( i ) ,

where the stochastic vectors are defined as

δη ' τ(ί)

g T(0

{ [v P(0L , [v p( t ) ] y, [v p( i ) ] 2}

[δη\(ή, ôn'2(t),...,ôn's(t)],

flU) flit) fun

jT ( I ) ={[_VT.J(T)] 1,...,[-VT.J(t)] s},

(E6)

(E7)

(E8)

(E9)

(E10)

(E l l )

The Medina-Noyola Formalism for the Tracer Friction Factor 419

and the matrix elements are (α, β = x, y, ζ),

(E12) m p

[ G I 2 ( R ) ] A E. , = - — {ΐΝΦ(Τ)1Ρ}ΛΒ, (E13) m p

ÎQNIRÏÏUJ = - [V< " }(')>«,]« 4 , , (E14)

[ G 2 2( R , R ' ) ] M = - J Vr · L im . V £ M J( R , R ' ) . (E15)

m = 1

STATISTICAL PROPERTIES OF THE STOCHASTIC VARIABLES. One combines Eqs. (E6) and (E7) to give an expression formally equivalent to Eq. (6.5.4), hence the solu-tion given by Eq. (6.5.5). We adopt these expressions to obtain the velocity correlation function, thus our derivation differs from that of Medina-Noyola and co-workers (Medina-Noyola and Vizcarra-Rendon, 1985; Ruiz-Estrada et al., 1986; Vizcarra-Rendon et al. 1987; Medina-Noyola, 1987; Medina-Noyola and Del Rio-Correa, 1988).

We employ some of the approximations of the Medina-Noyola formalism in regard to the properties of the system: (1) The average values of the ele-ments of the stochastic vectors are <[v p(i)] f> = 0 = <öff(i)>

a r>d (ônk(t)} =

0 = 0' f c(i)>; (2) the initial conditions of the vectors vp(0) and £n'(0) are not correlated with the stochastic driving vectors g(i) and j(i), viz, the dyadics are <vp(0) * g

T(i)>o = 0 and < & I ' ( 0 ) * j

T( i ) > m p = 0; and (3) the cross-correlation

terms < & Ι } ( ί ) * <5η^τ(0)> are not in general zero because they are coupled

through the Onsager coefficient and the function E F C 7( R , R ' ) in the matrix G 2 2. The first condition is satisfied by limiting the discussion to linearized fluctuations about the average value. The second condition is satisfied by the large difference in the time scale of solute and solvent relaxation processes. It necessarily follows from (3) that the cross-correlation terms for the initial conditions < & I } ( 0 ) * <5n^

T(0)> are nonzero.

Multiplying Eq. (E7) on the right by <5n'T(0) and taking the ensemble

average gives

* Λ 0 ) ) = Q 2 . < V p ( f ), f c f T ( 0 ) >_ Q i . < f t i , ( t ) . ^ n , T ( 0 ) >

(E16)

The propagator associated with solute relaxation is X s(t) = S(t) · S_ 1

( 0 ) = e x p ( - G 2 2t ) , where S(t) = <Αη'(ί) * fti'

T(0)>. The dyadic of Eq. (E7) with v£(0)

is averaged,

<in'(r) * vj(0)> = S(t) · S - ' ( 0 ) . <ft,'(0) * v£(0)>

S J O

S(t - ί') · S-\0). G 2 1 · <v.(t') * vî(0)>df, (E17)

420 Ε. T H E M E D I N A - N O Y O L A F O R M A L I S M

where <<5n'(i) * vj(0)><<5n'(0) * v ^ O ) ) " 1 = e x p ( - G i ) by Eq. (6.5.6), and therefore is equivalent to S ( i ) ' S _ 1( 0 ) . We now form the dyadic of Eq. (E6) with vj(0), take the ensemble average, and substitute Eq. (El7) to obtain

GU • S« - f ) • §- ' (0) • Gj, • <»„«') « v ; ( 0 » A ' 0

- G 1 2 · S(t) · S- ' (0) · <δη'(0) * vj(0)>. (E18)

The first two terms on the right-hand side are contained in the Langevin expression for the velocity correlation function derived by Medina-Noyola and Vizcarra-Rendon (1985). The third term represents the decay of the initial coupling between the probe particle and the solute particles. Medina-Noyola and Vizcarra-Rendon set the concentration-velocity cross-correlation equal to zero at the onset of their derivation because of time reversal arguments (Medina-Noyola, personal communication). The third term may be of impor-tance only when external fields are applied to the system, and it is excluded in the remaining development herein. Hence,

£ κ ( ί - ί ' ) · < ν ρ ( ί ' ) * * ϊ ( 0 ) > Λ ' , (Ε 19)

where the kernel is Κ(ί - ί') = - G 1 2 · exp[ — G 2 2( i — t')] · G 2 1. Using Laplace transforms, one arrives at the expression

<vp(î) * vj(0)> = X p( t) · <vp(0) * vj(0)>

Χρ(ί - ί') · G 1 2 · X,(t') · <vp(f') * vj(0)> dt', (E20) J c

where X p( i — ί') is the probe particle propagator and is the Laplace transform of X p(z) = [ G n - K(z) ]" 1, and K(z) is the Laplace transform of Κ(ί - ί'). The velocity correlation function, now denoted by c(i), can be expressed as either the scalar or dyadic products, viz,

Φ) = <v p(i) T - vp(0)> = T r « v p ( i ) * vj(0)>}, (E21)

and is related to DTr by the time integration of c(i), which is expressed as a Laplace transform in the variable s and then evaluated at s = 0. Because of spherical symmetry, the tensors <vp(i) * vj(0) and K(t — ί') are diagonal and isotropic matrices. Hence only the Laplace transform of <νρ(ί) * νJ(0)> zz need

The Medina-Noyola Formalism for the Tracer Friction Factor 421

be considered. One therefore has,

D T r = — { ^ - [ ^ . ( G ^ r - G ^ V , (E22)

where use was made of the fluctuation-dissipation theorem, kT/mp = <v p(0)T · v p(0)> z z. It is deduced from Eq. (E22) that

/Tr = ^ h - m p [ G 1 2 . ( G 2 2) - 1 . G 2 1] z z .

This expression was first derived by Medina-Noyola and Vizcarra-Rendon (1985).

Formal evaluation of fTr. Recall that Ekj(r,r') in Eq. (E5) is the concentration derivative of the chemical potential evaluated at the equilibrium concentra-tion. We write μ , ( Γ , ή = μ? + kT ln[n}(r, i)] - kTcf^r, n ' ] + μ]ρ(τ\ where μ° is the standard state value, fcTln[w}(r, i)] is the ideal solution term, μ]ρ(τ) = Z/ [0(r)]J-p is the energy of intersection of the jth solute species at position r with the probe particle ρ at the origin, and ^ [ Γ , Η ' ] = (l//cT)<30[r,tt']/3n} is a functional of η (denoted by [ ]) and is the contribution due to interactions with the remaining solute species such that οο^Ιτ,η'Ι/δη'^τ') = ^ [ Γ , Γ ' , Π ' ] ,

where ^ [ Γ , Γ ' , Η ' ] is the two-particle direct correlation function (Evans, 1979; cf. Appendix C). It follows that the general form for ^ ( Γ , Γ ' ) is

E ' * r ' ) = eq

V<tt(r)> eq = * r £ ί ^ ^ Γ , ) ν < π ; · ( Γ ' ) > Ε ς ί / ν + ν Ζ ^ [ ( / > ( Γ ) ] Ί Ρ. (E24) J=i Jv

The elements of G are evaluated through the condition for equilibrium, V<^j(r)> eq = 0, hence one can solve for V <n}(r)>eq by inversion of Eq. (E24),

V<"}(r)>eq = - 7 ^ Σ ί ^ ( Γ , Γ " ) ν ρ Ζ £ β [ φ ( Γ ' ) ] Ι ρ ί 3 Γ " (E25) Kl i=l Jy

where ^ ( Γ , Γ " ) must satisfy the relationship

X f ajk(r, r")EH(r", r') </3r" = δβδ(τ - r") (E26) *=i Jv

and is identified as the equilibrium cross-correlation function of the concen-tration fluctuations, σβ(τ,τ") = <cn}(r)<5nÎ(r")>eq. The choice of the function-al form for {c^ }[r,r ' ,n ' ]} and VpZ/e[0(r)]_/p fully determines the functional form of the elements in the matrix G. The element G 1 2 is determined directly from Eq. (E12). The element G 2 1 is evaluated through Eqs. (E23)-(E25). The element G 2 2 is generated through Eq. (E23) and additional assumptions about the Onsager transport coefficient L i f c.

Glossary

at Normalized relative scattering amplitude of compo-nent i.

(xs Isotropic polarizability of a segment in a polymer or a particle in the solution.

a(r, t) Polarizability tensor at point r and time t.

c Speed of light.

CP Constant-pressure heat capacity.

Cv Constant-volume heat capacity.

[7] or Cj Concentration of species j in units of moles/liter.

Cp Polymer concentration in units of g/L, mg/mL.

cj, Polymer concentration in units of g/dL.

c p Polymer concentration in g/mL.

C T , c T, Total solute concentration in appropriate c'j,orcj units.

C* or c* Concentration at which volumes of adjacent polymers begin to overlap.

Z) a pp Apparent diffusion coefficient, which is operationally defined as 1/τΚ

2.

Infinité dilution limit of the translational diffusion coefficient of the j t h species, equal to kT/f*}.

D|| Anisotropic diffusion coefficient of a cylindrical particle associated with motion parallel to its major axis.

4 2 3

424 G L O S S A R Y

DL Anisotropic diffusion coefficient of a cylindrical particle associated with motion perpendicular to its major axis.

D c o l l(K, C) Collective diffusion coefficient, which is equal to kT/f(C)S(K,C).

A o o P ( Q Apparent diffusion coefficient associated with long-range diffusion processes involving the cooperative motion of several solute particles.

Dm Mutual diffusion coefficient, defined by (kT/fmNA)(dn/dcp)T .

Dp Diffusion coefficient for a polymer or polyion.

Z) s e l f(T) Time-dependent [ τ > mp/£(/z)] self diffusion coefficient, where D s e l f( i -» oo) = D T r(C) .

D^ e lf t oo limit of a time-dependent self diffusion coefficient.

Z>self t 0 limit of a time-dependent self diffusion coefficient.

D T r(C) Tracer diffusion coefficient, equal to kT/f(C) = D^cl{.

De Rotational diffusion coefficient.

Dt Center-of-mass translational diffusion coefficient.

Ds Diffusion coefficient for a small ion.

Dr

slf{ Self diffusion coefficient associated with polymer rep-tation motion.

Df r ap

Diffusion coefficient obtained from fluorescence recov-ery after photobleaching techniques.

Dq e ls

Diffusion coefficient obtained from QELS.

dc Characteristic distance associated with a particle, such

as the length of a rod or the radius of a sphere.

Arjk or rjk Distance between the subunits j and k within a single particle, or distance between centers of spherical particles.

ArJjKk Distance between the subunit j of polymer J and the subunit k on polymer K.

ARjk or Rjk Distance between centers-of-mass of particles j and k.

AR Generic notation indicating the average distance be-tween any two particles.

Glossary 425

Δί Data collection interval.

η0 Solvent viscosity.

η8 Solution viscosity.

η0/Ρο Kinematic viscosity.

E0 Amplitude of the incident light (electric field).

Es Amplitude of the scattered light (electric field).

E° Amplitude of external applied electric field.

ε Dielectric constant of the solvent.

f° General friction factor associated with a particle in the absence of any interactions.

/ ' Functionality of a star or a branch polymer.

fs(i) Random fluctuating force acting on a polymer due to

solvent collisions (white noise).

fd(r) Random fluctuating force acting on a polymer due to

other solute particles (colored noise).

fi n t

(co d, i ) Induced interparticle force acting on a polymer due to the presence of an external field.

g{l\t) Normalized time-dependent component of the electric

field correlation function.

g{2)(t) Normalized time-dependent component of the scattered

intensity correlation function.

g(rU2) Radial distribution function involving only particles 1 and 2.

g^r) Radial distribution function for the pair of particles ί

and

g(2)(r) Radial distribution function for two particles in the

presence of other particles that affect the pairwise distribution.

g{n'm)(r) Radial distribution function for η particles in a cluster of

m particles.

GX(K, t) Generalized form of the molecular correlation function.

G t Hydrodynamic shielding coefficient for the ith subunit.

426 GLOSSARY

Is Ionic strength, equal to \ £ c^z2.

Κ Scattering vector, with magnitude equal to (4πηΟΜο)δίη(0/2).

λ0 Wavelength of the incident light in a vacuum.

λΒ Bjerrum length, equal to e2/skT.

λΌΗ Debye-Hückel screening length, equal to (1000/8πΛΓΑΑΒ/ 5)

1 / 2.

L p Persistence length of a wormlike coil.

LK Kuhn statistical persistence length.

ns Index of refraction of the solution.

n0 Index of refraction of the solvent.

Nc Number of points in the autocorrelation function.

Ni Number of particles of type i.

nb Number of blobs in a linear polymer.

n'i Number of particles of type i per cm3.

np Number of persistence lengths in a linear polymer.

ns Number of subunits in a polymer.

Ω Generalized frequency matrix for the initial decay of a correlation function.

ω Generalized parameter indicating angular frequency.

ω 0 Angular frequency of incident radiation.

ω 0 8 Natural frequency of a material oscillator.

ωά The driving frequency of an external applied sinusoidal field.

P(r, ή Polarization of matter at location r and time t.

Pn(x) Lengendre polynomial of order n with the argument x.

P(Kdc) or P(K Ar) Particle structure factor.

Q e Quantum efficiency of the photomultiplier tube.

Re Rayleigh ratio.

< # G > Mean-square radius of gyration.

Glossary 427

Rapp Apparent particle radius as computed from D a pp using the Stoke s-Einstein relationship.

RH Effective hydrodynamic radius.

Rs Radius of a hard sphere.

σ, Effective hard sphere radius of the ith scattering center.

σ° Surface charge density.

S(KAR) Static structure factor that includes self and pair inter-action terms.

S'(KAR) Static structure factor that includes only the pairwise terms.

ξΌ Dynamic correlation length.

ξα Static correlation length.

ξρ(ή Time-dependent particle friction factor.

τ A unit of time, generally associated with either a relaxation process or a time window.

τ,· Relaxation time associated with particle i.

p0 Solvent density in g /cm3.

Vp Volume of a solute particle.

Vs Scattering volume.

Vs Volume of a sphere.

Z, Magnitude of the charge on particles of type denoted by i.

Ζμ

ρ Electrophoretic mobility charge of the polyion.

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Nota t ion

[S 10.2] means Section 2 in Chapter 10.

A = Appendix Ε = Example F = Figure

Ρ = Problem S = Section Τ = Table

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Index

Absorption of light, 35

Adenovirus nucleoprotein, 231-232, 375

Autocorrelation function, 31, 77, 88, 92,

97, 168

center-of-mass, 32, 167

electric field, 31 ,99

intensity of scattered light, 31

internal relaxation (general), 99, 100

flexible particle, 116-120

rigid particle, 101-105, 107

semi-flexible particle, 109-112, 123

photo tube current, 31, 42

velocity, 38, 165-168

Bacteriophage, 107-108, 120, 338, 368-369

Bessel function [B,(KL)] , 101-102,

108-109

Blood cells, 372

Bovine casein micelles, 375-376

Bovine mercaptalbumine (BMA), 252

Bovine serum albumin (BSA), 3, 33 -34 , 74,

80, 146, 1 7 5 -176 ,211 ,216 -217 ,

247-250 , 3 4 0 - 3 4 1 , 365-366 , 373, 383,

385-386

Brownian motion, 1, 128

Chondroi t in sulfate, 257, 373 Coherence area, 31 -32 , 39 Collective diffusion coefficient (D C 0 l l) ,

130, 196

poly (adenylic acid), 131

poly(di-n-hexylsilane), 154 poly(styrene), 139-142

Colloidal gold, 222, 261, 291, 303 -305 Colloidal silica, 9 ,315 Cooperat ive diffusion coefficient {Dcoop),

381-382

D e b y e - H ü c k e l potential, 206, 265-266 , 315

D e b y e - H ü c k e l screening length (Λ.Ο Η),

206-208 , 213, 236-237 , 240, 244 -245 ,

2 6 9 - 2 7 1 , 2 7 9 - 2 8 1 , 283, 321-322, 324,

389, 394-395

Deoxyribonucleic acid (DNA) (general)

circular, 39, 73, 124, 152

linear, 39, 57, 73, 120-122, 152-153, 252,

259, 338-339 , 353, 373

plasmid, 103-104, 351

supercoiled, 103-104, 2 5 4 - 2 5 6

Deoxyribonucleic acid (DNA) (specific)

calf thymus, 3 5 1 , 3 7 8 - 3 8 0

chicken erythrocyte, 57, 378-379

lambda (circular), 124

lambda (linear), 338-339

mouse liver, 378-379

N l , 3 8 0 , 382-383

0 2 9 , 120-122

plasmid D N A s

Col E l , 7 3 - 7 4

pBR322, 254 -256

p U C 8 (circular), 241-244, 255

p U C 8 (linear), 2 4 2 - 2 4 4

T7, 2 4 - 2 5 , 351

Dextran, 196-198

Dielectrophoresis, 354

E. coli, 353, 360 Electrolyte dissipation, 2 1 8 - 2 2 3

poly(styrene), 222 -223 Electroosmosis, 336-337 Electrophoretic mobility, 321, 323-324 ,

332 -333 , 335-336 , 372, 376

blood cells, 372 -373

bovine serum albumin, 340, 373

DNA, 3 3 8 - 3 3 9 , 3 5 1 , 3 7 3

445

446 I N D E X

Electrophoretic mobility (contd)

Doppler shift spectroscopy, 226-227 ,

335-336, 339^342, 353-356

human plasma, 340, 342

latex spheres, 346-348

liposomes, 374-375

nucleosome, 351-352

periodic pulsed electric field, 344-348 ,

354-355

poly(lysine), 356, 394

poly(nucleosome), 352

poly(styrene), 347-348 , 353

6,6-ionene, 226-227

sinusoidal electric field, 348-357 , 394

vesicles, 374-375

Fick's first law of diffusion, 46, 174

Fick's second low of diffusion, 32, 320

Flory limit, excluded volume, 62, 129

Flory radius (RF), 62, 140, 244-245

Flux, 4 3 - 4 6 , 173, 292

F o k k e r - P l a n c k equation, 4

Fractal dimension (general), 292, 2 9 8 - 3 0 1 ,

304, 309-312

colloidal gold, 291,303, 305

colloidal silica, 9, 315

Gelatin, 137-138

Glu tamate dehydrogenase ( G D H ) (beef

liver), 41

Guggenheim method of analysis, 97

Hemoglobin, 3 4 - 3 7 , 3 9 - 4 0 Heparin, 257 -258 Holtzer plot, 75

H u m a n serum plasma, 340-343 , 373

Insulin, 21, 202 -203

Internal relaxation modes (general), 101-102, 105-106, 110-113, 115-117, 119-120, 122-124

D N A (circular), 103-104, 124-125,

241-244, 254-256 D N A (linear), 120-122, 241-244 hydrodynamic interaction, 114-115,

123-126

poly(y-benzyl-L-glutamate), 154-155 poly(styrene), 118-119,139-142, 153 T-even bacteriophage, 107-109 tobacco mosaic virus, 102-103

Intrinsic viscosity (M), 62, 64, 115, 140

gelatin, 137-138

poly(acrylate), 151

poly(di-n-hexylsilane), 154

poly(ethyl ether ketone), 152

poly(styrene), 74, 118

Joule heating, 3 5 - 3 7 , 336

Langevin equation, 38, 47, 150, 157-158,

349

Liouville equation, 4

Liposome, 374-375

Lithium dodecyl sulfate (LDS), 29 -30 ,

385-386

Lysozyme, 383

Manning condensation (general), 225-229

adenovirus nucleoprotein, 231-232

bromine ions, 228-229

poly(acrylate), 258-259

poly(styrene sulfonate), 238-239

6,6-ionene, 226-227

sodium ions, 227-228

Memory function, 150, 164-167, 187-190

Micelles, 2 9 - 3 0 , 9 4 - 9 5 , 258, 274-276 , 375-376 , 385 -386

Molecular weight, number average (<M>„), 97

Molecular weight, weight average ( < M > J , 2 0 - 2 1 , 5 4 - 5 5 , 97

E.coli, 7 3 - 7 4

insulin aggregate, 21

phosphofructokinase, 4 9 - 5 0 phycocyanin, 4 0 - 4 1

poly(acrylamide), 4 0 - 4 1 T7 D N A , 2 4 - 2 5

Mutual diffusion coefficient {Dm), 3 2 - 3 3 , 36, 4 6 - 4 7 , 5 0 - 5 1 , 170, 173-174, 176, 178-181, 203, 209-214 , 216-218 , 273

adenovirus nucleoprotein, 231-232 bovine mercaptalbumin, 252 -253

bovine serum albumin (BSA), 33 -34 , 2 1 1 - 2 1 2 , 2 1 6 - 2 1 7

chondroit in sulfate, 257-258 dextran, 196-197

D N A (circular), 103-104, 124-125, 254 -256

D N A (linear), 7 3 - 7 4 , 120-122, 378-380 , 382

I N D E X 447

fractal object, 302 -306

hemoglobin, 3 4 - 3 5 , 3 9 - 4 0

hemoglobin (carbon monoxide), 3 6 - 3 7

hemoglobin (oxygenated), 40

heparin, 257-258

insulin, 2 0 2 - 2 0 3

micelles, 258, 275 -276

nucleosome, 251-252

phosphofructokinase, 49 -52

poly(acrylamide/acrylic acid), 253

poly (lysine), 390 -393

poly(maleic acid/ethyl vinyl ether),

253 -254

poly(methymethacrylate), 384

poly( 1,4-phenyl bensobisoxazole),

392-393

poly(styrene), 74, 76, 118-119, 384-385

silica, 198-200, 286-289

T-even bacteriophage, 107-109

tobacco mosaic virus, 102-103

xanthan, 7 5 - 7 6

Mutual friction factor ( /„) , 33, 4 5 - 5 0 , 224

phosphofructokinase, 4 9 - 5 0

Polyethylene oxide), 4 8 - 4 9

Nav ie r -S tokes equation, 160

Nucleosome, 39, 57, 251-252, 351-352

Octyl t imethylammonium bromide (OTAB), 228-229

Osmotic susceptibility (dn/dc), 3 3 - 3 5 ,

4 5 - 4 6 , 50

bovine serum albumin, 216-217

hemoglobin, 3 4 - 3 5

Péclet rotary number (Pe r) , 328-329 Phosphofructokinase (PFK) , 4 9 - 5 2 Phycocyanin, 5 4 - 5 5 Polarizability, 16-18 Polydispersity, 7 8 - 8 1 , 8 3 - 8 4 , 9 1 - 9 4 ,

9 7 - 9 8 , 355-356 , 4 0 0 - 4 0 4 Poly(-)

(acrylamide) (PAAm), 4 0 - 4 1 , 8 4 - 8 5 , 253, 363 -364

(acrylamide/acrylic acid), 258 (acrylate), 151,363,383 (acrylic acid), 143-145 (adenylic acid), 131 (n-butyl methacrylate) (PBMA), 138

(y-benzyl glutamate) (PBG), 2 7 - 2 8 ,

154-155

(η-butyl isocyanate) (PBIC), 149

(η-butyl methacrylate) (PBMA), 138

(ethylene oxide) (PEO), 4 8 - 4 9 , 135-137,

146

(ether ether ketone) (PEEK) , 151-152

(rc-hexyl isocyanate) (PHIC) , 149,153

(di-n-hexylsilane), 154

(isoprene), 63

(lysine), 356-357 , 383, 390 -396

(maleic acid), 2 5 3 - 2 5 4

(methacrylic acid) (PMA), 365, 373

(methyl methacrylate) (PMMA) ,

168-170, 188-192, 201,384

(nucleosome), 351-352

(n-octyl isocyanate) (POIC) , 149

(1,4-phenyl benzobisoxazole) (PBO),

392 -393

(phenylmethyl siloxane) (PPMS) , 365

(styrene) (latex spheres), 3, 3 9 - 4 0 , 6 4 - 6 5 ,

8 0 - 8 1 , 9 4 , 9 6 - 9 7 , 139, 143-146,

198-199, 222 -225 , 2 6 1 - 2 6 3 ,

276-279 , 312 -313 , 345-348 , 351,

353 ,357-359 , 364 -365 , 384

(styrene) (linear), 6 0 - 6 1 , 63, 74, 76,

118-119, 134-135, 139-144,

147-148, 153

(styrene sulfonate) (PSS), 238-239 , 257,

383, 391

(vinyl hydrogen sulfate) (PVHS), 383

(vinylacetate) (PVA), 188

Porod tail, 9, 300-301

Pullulan, 41

Pyruvate oxidase, 73

Radius, hydrodynamic (general), 5 0 - 5 1 ,

7 5 - 7 6

bovine serum albumin, 74 colloidal gold, 305 glutamate dehydrogenase, 41 hemoglobin, 40 linear, flexible polymer, 6 2 - 6 3 phosphofructokinase, 4 9 - 5 2 phycocyanin, 5 4 - 5 5 poly(acrylamide), 41 poly(di-n-hexylsilane), 154 poly(isoprene), 63 poly(styrene), 6 0 - 6 1 , 63, 76, 153 pullulan, 41 star, flexible, 6 6 - 6 9

448 I N D E X

Radius of gyration (general), 22, 69, 75

linear, flexible polymer, 5 7 - 6 0 , 66

poly(acrylamide), 4 0 - 4 1

poly(acrylate), 151

poly(di-n-hexylsilane), 154

poly(ether ether ketone), 151-152

poly(hexyl isocyanate), 153

poly(isoprene), 63

poly(styrene), 39 -40 , 6 0 - 6 1 , 63, 74, 76,

153

pullulan, 41

star, flexible, 6 6 - 6 9

T7 DNA, 2 4 - 2 5

xanthan, 7 5 - 7 6

R a y l e i g h - G a n s - D e b y e scattering, 16

Rayleigh peak, 2, 7

Rayleigh ratio, 19

Reptation diffusion coefficient (D r e p) , 134

Second virial coefficient {B2 or A2), 26, 40, 173, 248

bovine serum albumin, 247-249 exclude volume, 26 insulin, 202 -203

poly(acrylamide/acrylic acid), 253 poly(di-n-hexylsilane), 154 poly(y-benzyl glutamate), 2 7 - 2 8 weak, long range interactions, 26

Sedimentation coefficient (sT), 46, 73 bacteriophage, 107,109 C o l E ! D N A , 7 3 - 7 4 phosphofructokinase, 4 9 - 5 0 phycocyanin, 55 plasmid DNA, 103

Self diffusion coefficient {DseU), 134-138,

145-146, 166-167, 184-190, 193, 199-203

bovine serum albumin, 146-147 gelatin, 137-138 hemoglobin, 145

poly(styrene), 134-135, 142-145, 222-225

polyethylene oxide), 135-137 poly(methyl methacrylate), 168-170,

188-192, 201 silica particles (coated), 186-187

Silica particles, 9, 186-187, 199, 286-289 , 315

Smoluchowski equation (mutual diffusion), 4

Smoluchowski equation (bimolecular

kinetics), 293, 297

Sodium dodecyl sulfate (SDS), 9 4 - 9 5

Structure factor [ P ( K A r ) ] , 17, 19, 21 -24 ,

400 -404

branched, flexible, 67

bromine ions (around OTAB),

228-229

circular coil, 6 5 - 6 6

circular cylinder, 22

disk, infinitely thin, 22

fractal object, 299-301

Gaussian coil, 2 2 - 2 3

lithium dodecyl sulfate, 30

poly(styrene) latex spheres, 3 9 - 4 0

silica aggregate, 9, 301

sphere, hollow, 22

sphere, solid, 21, 23, 27

star, flexible, 67

thin rod, 2 2 - 2 3

Structure factor [S(K AR)], 2 8 - 2 9 ,

215-217 , 221,273, 279,314, 336,

339-342 , 345-346 , 351, 356, 358,

399-403 , 410

bovine serum albumin, 216-217 , 341,

385-386

E. Coli (oscillatry hydrodynamic field),

360

gel (mechanical stimulation), 362-364

lithium dodecyl sulfate, 2 9 - 3 0 , 385-386

poly(styrene), 276-279

silica, 287-288

single contact flexible coil, 29

spherical particle, 29, 78

Thermal conductivity, 3 5 - 3 6 Theta condition, 26, 58 -60 , 7 0 - 7 2 , 74, 135,

139

poly(styrene), 60, 74, 139-141, 143 Tobacco mosaic virus (TMV), 102-103 Tracer diffusion coefficient (D T r) , 3 4 - 3 5 ,

73, 167

calcium ions, 227

hemoglobin, 3 4 - 3 5

phosphofructokinase, 51 -52 phycocyanin, 55 poly(lysine), 391-392 poly(styrene), 142 pyruvate oxidase, 73 sodium ions, 227 -228

I N D E X 449

Tracer friction factor (fT r), 4 7 - 4 9 , 5 3 - 5 5 ,

7 2 - 7 3 , 75, 421

helix, 316

phosphofructokinase, 5 1 - 5 2

phycocyanin, 55

poly (ethylene oxide), 4 8 - 4 9

rod ,316

Viscoelastic relaxation time, 115

Wyoming sodium benonite, 353

Xanthan, 7 5 - 7 6

Zimm plot, 27

poly(acrylamide), 41

poly(y-benzyl glutamate), 2 7 - 2 8

poly(hexyl isocyanate), 153

Documents

An Introduction to Dynamic Light Scattering by Macromolecules