Universal interrelation between measures of particle and polymer sizeFernando Vargas–Lara, Marc L. Mansfield, and Jack F. Douglas

Citation: The Journal of Chemical Physics 147, 014903 (2017); doi: 10.1063/1.4991011View online: http://dx.doi.org/10.1063/1.4991011View Table of Contents: http://aip.scitation.org/toc/jcp/147/1Published by the American Institute of Physics

THE JOURNAL OF CHEMICAL PHYSICS 147, 014903 (2017)

Universal interrelation between measures of particle and polymer sizeFernando Vargas–Lara,1,a) Marc L. Mansfield,2 and Jack F. Douglas1,b)1Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg,Maryland 20899, USA2Bingham Research Center, Utah State University, Vernal, Utah 84078, USA

(Received 12 April 2017; accepted 19 June 2017; published online 6 July 2017)

The characterization of many objects involves the determination of a basic set of particle size mea-sures derived mainly from scattering and transport property measurements. For polymers, these basicproperties include the radius of gyration Rg, hydrodynamic radius Rh, intrinsic viscosity [η], andsedimentation coefficient S, and for conductive particles, the electric polarizability tensor αE andself-capacity C. It is often found that hydrodynamic measurements of size deviate from each otherand from geometric estimates of particle size when the particle or polymer shape is complex, a phe-nomenon that greatly complicates both nanoparticle and polymer characterizations. The present workexplores a general quantitative relation between αE, C, and Rg for nanoparticles and polymers ofgeneral shape and the corresponding properties

[η], Rh, and Rg using a hydrodynamic-electrostatic

property interrelation. [http://dx.doi.org/10.1063/1.4991011]

I. INTRODUCTION

The shape discrimination of objects on length scales rang-ing from nanoparticles and polymers to people and aircraftinvolves a primitive set of object properties derived from scat-tering or transport property measurements.1 These propertiesinclude the electric polarizability tensor αE, important for thefar-field/low frequency scattering of particles by electromag-netic radiation in the Rayleigh limit where the conductive parti-cles are small in comparison with the radiation wavelength, theradius of gyration Rg also derived from light scattering or imag-ing, the hydrodynamic radius Rh from dynamic light scatteringor sedimentation coefficient S from centrifugation measure-ments, and the intrinsic viscosity [η] particle suspensions andpolymer solutions at low concentrations. The self-capacitanceC of particles is also important in relation to understand-ing the potential energy of conducting objects and the rateof diffusion-limited reaction because the Smoluchowski rateconstant is proportional to C.2 The self-capacitance also arisesin connection to the scattering length in acoustics,3 quantummechanics,4 and other diverse applications.5 Finally, we men-tion that αE emerges in continuum property calculations ofchanges in many material properties such as electric conduc-tivity, thermal conductivity, dielectric constant, and magneticpermeability, among others, when a small concentration ofparticles is added to another medium6–8 so that αE, as inthe case of [η], has a significant potential for particle shapecharacterization.

Although the value of these “shape functionals” for shapecharacterization has long been appreciated in the mathemat-ical community9 and the calculation of these properties hasbeen the preoccupation of many great scientists such as Pois-son, Laplace, Kelvin, Maxwell, and Einstein, this field has

a)Electronic mail: fernando.vargaslara@nist.govb)Electronic mail: jack.douglas@nist.gov

been slow to develop because of the intrinsic difficulty of thecalculation of these measures of particle shape beyond a fewsimple geometries such as the ellipsoid, torus, spindle, andlens shapes. The challenge of calculating these properties hasgenerated a large body of mathematical methods such as theKelvin inversion method, but even great mathematicians of thetwentieth century, such as Polya, had to conclude that the ana-lytic calculation of C of even such “simple” objects as a cubeis basically impossible.9

The computational situation has changed in recent yearsbecause of the confluence of two developments, large scalecomputational facilities and probabilistic potential theory, thatallow for a formulation of problems with complex-shapedboundaries in terms of averaging over random walk paths.This path-integral method allows for the calculation of theproperties of polymers and particles having essentially anarbitrary shape to unprecedented accuracy and with highcomputational efficiency.10,11 We are then in the position toexplore shape-property interrelations that were not previouslyfeasible.

In this work, we focus on the relation between 〈αE〉, C,and Rg of polymer and particles having complex morphologies.Electrostatic-hydrodynamic analogies extend these relationsto those between Rh, [η], and Rg. We validate this relationagainst data for a large variety of particle shapes, where theseproperty measurements lead to rather distinct size estimates.These property interrelations should be useful in particle andpolymer characterization and in the design of new materialsbecause of the significance of 〈αE〉 for estimating propertychanges in materials with additives.

II. SCALING RELATIONSHIP BETWEEN 〈αE〉,C, AND Rg

The calculation of measures of particle size fromLaplace’s equation, such as the electric polarizability 〈αE〉,

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014903-2 Vargas–Lara, Mansfield, and Douglas J. Chem. Phys. 147, 014903 (2017)

the magnetic polarizability 〈αM〉, the self-capacitance C, andthe hydrodynamic virtual mass [M], is a problem of long stand-ing mathematical and physical interest. Correspondingly, theStokes friction coefficient f t and the intrinsic viscosity

[η]

provide basic measures of particle size associated with thesolution of the Stokes equation.6,9 Hydrodynamic-electrostaticanalogies interrelating these properties are a source of physicalinsight and highly useful approximations.1

The origin of these correspondences can be appreciatedfrom the observation that the angular averaging of the Oseentensor, the free space Green’s function of the Stokes equa-tion, is the free-space Green’s function of Laplace’s equa-tion.12 We do not discuss the details of these relationshipshere but simply note that the knowledge of 〈αE〉 and C forobjects of general shape allows for an estimation of

[η]

andthe Stokes friction coefficient to an accuracy of about 1%,which is usually far less uncertain than experimental estimatesof these quantities.13 The problem of estimating 〈αE〉 and Cthen pertains to the practically important problem of deter-mining

[η]

and Rh of complex shape particles and to theproblem of understanding how these size measurements areinterrelated.

As noted above, the number of object shapes allowing forthe analytic calculation of these measures of particle shapeis rather limited, and for mathematical expediency, it is oftenassumed that complex particles or polymers are effective ellip-soids or some other mathematically tractable form. The press-ing practical need for polymer and particle characterizationhas led established mathematical frameworks for treating morecomplicated particle shapes and we may draw on these frame-works to deduce our general relationship between 〈αE〉, C, andRg given the hydrodynamic-electrostatic analogy mentionedabove.

There are two approximate frameworks for calculatingpolymer hydrodynamic properties, the Kirkwood–Riseman(KR)14 “bead” or “slender-body” model and the Debye–Bueche (DB)15 “porous sphere” model. Each of these modelshave their shortcomings,15 but both models remain of enduringinterest because they make predictions for the hydrodynamicproperties of polymers that have at least qualitative valid-ity. The main approximation of the KR and DB theories isa configurational preaveraging approximation that can leadto an appreciable error as large as O (20%) in the estima-tion of the hydrodynamic properties of flexible polymers.16,17

We are not interested in these shortcomings in the presentwork, but rather on what these approximate theories haveto say qualitatively about how shape influences the variablehydrodynamic interaction in the determination of

[η]

andRh. The self-interaction strength is also at the heart of cal-culating 〈αE〉 and C, and we use these hydrodynamic modelsfor qualitative guidance in developing a scaling relation con-necting these fundamental measures of particle shape. Thepath-integration program ZENO7,13 can then be used to numer-ically calculate 〈αE〉 and C to low uncertainty to validate theproposed relationship for diverse classes of objects listed inSec. III.

Our scaling relation is derived from a consideration ofthe limiting case, when the particle is imagined to be brokenapart into segmental fragments separated at a great distance

from each other and the other extreme limit where the particlecomponents form a compact particle. We are looking here for arelation that subsumes these limits, and everything in between,without any assumption of particle sphericity, configurationalpreaveraging, etc.

Both the KR and the DB framework indicate that theStokes friction coefficient ζ of a collection of well-separatedparticles scales as the number of particles n times the Stokesfriction coefficient of each particle, nζ , and the self-capacityC also exhibits the same simple additive relation in the limitwhere the particles have no interactions. If instead, the particlesare connected into a linear array in three dimensions, f t scalesas nζ/ ln (n), which is similar to the ideal “free-draining” limit,ft ∼ nζ .18 On the other hand, if we compress all our particlesinto a sphere of radius R, then the number of subparticles isclearly irrelevant since the hydrodynamic field does not pen-etrate into the particle cluster.19 The same situation is true, asphere where C equals the sphere radius R, so that C = Rh.The equality between C and Rh also holds exactly for triaxialellipsoids6 and is approximately true (≈1% uncertainty) forthe average friction coefficient of particles having essentiallyan arbitrary shape.7,13 C and Rh, respectively, describe howthe particle shape alters the flux of mass and momentum intothe surrounding medium, so at equilibrium, these measures ofshape are naturally deeply interrelated.

Now, we extend some basic considerations to the lessobvious case of the intrinsic viscosity

[η]

where previoushydrodynamic modeling provides a reference for our scal-ing argument. The application of a shear field to dilute sus-pension creates a dipolar stress response from the particlesin a fashion similar to an electric field on a suspensionof conducting particles in an insulating medium.20 Since,〈αE〉 for rod-like particles exhibits the well-known asymptoticscaling,6

〈αE〉 ∼ n3/ ln (n) , (1)

both the KR and DB theories indicate that[η]

for rod-likepolymers follows the scaling relationship,[

η]∼ 〈αE〉 /Vp, (2)

reflecting the hydrodynamic-electrostatic analogy mentionedabove. Here, Vp is the polymer volume. In terms of the lengthn of the rod,

[η]

then scales as21[η]∼ n2/ ln (n) , (3)

and the electric polarizability of a sphere scales proportion-ally to the sphere volume.6 These limiting scaling relationsare recovered by both the KR and DB theories, which bothindicated a general scaling expression for polymers,[

η]∼ RhR2

g/Vp, (4)

where Rg is the polymer radius of gyration which suggests anapproximation for 〈αE〉. As mentioned before, both the KRand DB theories involve severe approximations.

In the polymer science field, Eq. (4) is known as theWeill–des Cloizeaux approximation,22 and this scaling rela-tionship has often been applied to rationalize the mass-scalingof

[η]

for flexible polymers with and without excluded volume

014903-3 Vargas–Lara, Mansfield, and Douglas J. Chem. Phys. 147, 014903 (2017)

interactions. Unfortunately, an extensive test of Eq. (4) usingZENO has revealed that it is simply inadequate for flexiblepolymers.10 This relation only holds for slender thread-likeand compact bodies where hydrodynamic interactions withinthe particle and conformational fluctuation effects are weak.There has been a long-standing need for a better approxima-tion for the intermediate case between thread-like polymersand compact objects since most real polymers, and manynanoparticles requiring characterization, are objects of thiskind.

Since C arises in so many physical contexts beyond elec-trostatics, and because electromagnetic units can be confusingdue to the multiple systems of units in use, we have fol-lowed standard usage in the mathematical/physics literatureof introducing “reduced units” that emphasize the dependenceof capacity and polarizability on the particle size. In particular,we have used reduced units in which C of a sphere equals itsradius R, corresponding to taking 4πε0 = 1, where ε0 is thepermittivity of the medium in which the conductive particle isplaced, so that the standard definition of C in terms of faradsreduces to C = R for a sphere, and more generally C has units oflength. In the same units, 〈αE〉 has units of volume, and specif-ically 〈αE〉 for a highly conductive sphere is equal 3 times itsvolume.

In the absence of any tractable analytic approach to inter-relate 〈αE〉 to other measures of polymer/particle size, weconsider a scaling approach that respects all limiting knownresults as special cases and reflects our knowledge of 〈αE〉

and C gained by recent computational studies. We estimatethe average electric polarizability tensor, 〈αE〉 = trace (αE) /3,with the requirement that the expression interpolates exactlybetween the slender body and compact particle scaling lim-its, but we do not make the particular choice indicated in thescaling expression given in Eq. (4). We instead take 〈αE〉 toequal,

〈αE〉 = [σ]∞����sphere

(4πC3/3

)Ψ

(df − 3)c , (5)

where the field “penetration function,” Ψc = C/Rg, describesthe extent to which the exterior field penetrates into the poly-mer and df is the fractal dimension of the particle. In particular,df = 1 for a rod, df = 2 for a random coil, df = 3 for a sphere,etc. Here, Ψc plays a central role in both the KR and DB mod-els,14,15 but the power (df − 3) in Eq. (5) is new to modeling thishydrodynamic self-interaction crossover. It is apparent that theslender body scaling limit of Onsager21 is recovered, the casefrom Eq. (5) when taking the slender rod limit (df = 1) andfor a spherical particle where df = 3, and we recover the exactrelation of Maxwell,23

〈αE〉(4πC3/3

) = [σ]∞����sphere

= 3. (6)

Note that Eq. (5) specifies the prefactor so this relation has nofree parameters. Basic scaling consistency then suggests thatwe can approximate 〈αE〉 by a generalized geometrical meanof C and Rg where the exponent involves the fractal dimensionof the object.

Of course, fractal objects are idealizations of realobjects that never conform exactly to self-similar geometrical

structures, just as spheres, cubes, tetrahedra, and octahedraexist as ideal Platonic forms used to model real structures.While true fractal objects exhibit a characteristic mass scalingrelationship between their radius of gyration and mass M,

Rg ∼ M1/df , (7)

many non-fractal objects exhibit this type of mass scaling rela-tionship to a good approximation. This situation arises formany polymeric structures, so we define an effective “massscaling exponent” νeff by the mass scaling relationship,

Rg ∼ Mνeff . (8)

The effective exponent νeff depends on the object geometri-cal parameters. For instance, νeff is a function of the aspectratio r for oblate and prolate ellipsoids, where r is theratio between the largest to the smallest ellipsoid axis.24

The same situation arises for dendrimers,25,26 duplex DNA,27

carbon nanotube domains,8 nanoparticle with grafted lay-ers of DNA chains,28,29 etc. In each case, the molecularparameters describing the object must be prescribed overwhich the mass scaling relation is exhibited. Numericalor analytically estimated νeff values corresponding to theobjects explored in this study are tabulated in the supple-mentary material and values are also summarized in Tables Iand II.

We thus arrive at a general scaling relation between 〈αE〉,C, and Rg,

〈αE〉 = [σ]∞����sphere

(4πC3/3

)Ψ

(1/νeff − 3)c , (9)

that is not restricted to fractal objects. We note that a similarrelation holds for objects in two dimensions,9

〈αE〉 = [σ]∞����disc

(πC2

L

), (10)

where [σ]∞����disc= 2 and CL is the logarithmic capacity (an

extension of the capacity to two dimensions where randomwalks are recurrent). Reciprocity30 implies 〈αE〉 = − 〈αM〉

in two dimensions, and we note that αM in three dimensionsis exactly equal to a hydrodynamic virtual mass tensor M,6

up to a constant of proportionality defined by mathematicalconvention.

The estimation of 〈αE〉 and C can be immediately trans-lated into estimating

[η]

and Rh for polymer and nanoparticles.

TABLE I. The percentage average deviation dev(%) for dendrimers withdifferent generation number (gi).

gi Rg C νeff 〈αE〉 〈αE〉c devj(%)

3 13.3 13.91 0.220 36 256 36 248.5 0.024 16.81 18.60 0.292 84 393 84 413.2 0.025 20.92 24.27 0.315 184 500 184 366.7 0.076 25.82 31.07 0.325 382 600 382 311.3 0.087 31.55 39.07 0.328 757 200 757 302.0 0.018 38.28 48.55 0.331 1 444 000 1 445 308.0 0.099 45.76 59.13 0.332 2 609 000 2 609 038.4 0.0110 53.95 70.81 0.332 4 474 000 4 476 273.1 0.0511 62.37 82.71 0.333 7 120 000 7 116 263.8 0.05

Percentage average deviation dev(%) 0.05

014903-4 Vargas–Lara, Mansfield, and Douglas J. Chem. Phys. 147, 014903 (2017)

TABLE II. List of objects, their effective mass scaling exponents νeff, andpercentage deviation of predictions and simulation estimates, dev(%).

Object νeff dev(%)

Swollen random coils 0.559 1.0Flexible capped cylinders 0.516 0.7Worm-like chains 0.535 0.7

Rods (8 < L < 1900)(

6.1L

)4+ 0.49

L−0.033 0.7

Oblate (1 < r < 100) 1√8+r

0.9

Prolate (6 < r < 100)(

4.6r

)4+ 0.6

r−0.0323 0.3

Flat sheets 0.565 1.2Flexible sheets 0.553 1.3Eden clusters 0.330 0.7Diffusion-limited aggregates 0.32 0.8Lattice animals 0.510 3.6Percolation clusters 0.250 0.8

Tori (5 < r < 1000) 0.7r0.2

1+r0.2 0.6

Star-like polymers (1/3)f 3

400+f 3 0.8

Dendrimers Table I 0.05Knots νeff(m) 0.60Platonic solids 1/3 1.0Polymer-grafted particles 0.333 0.8CNT tumbleweed structures 1/3 0.6Globular proteins 1/3 1.2

In particular, there are hydrodynamic-electrostatic approx-imations linking these properties,13,26 C = qhRh, and

[η]

= qη 〈αE〉 /Vp. In general, qh ≈ 1 to a high approximation[O(1%)] and qη ≈ 0.79 with uncertainty O(5%) having a gen-eral shape,26 so that we also estimate hydrodynamic propertiesto within low uncertainty. This uncertainty can be reduced toO(1%) by using the components of the polarizability tensor tofurther specify qη .26 Below, we focus on interrelating 〈αE〉 toC and Rg since the this interrelation is “equivalent” to the onebetween

[η]

to Rh and Rg.

III. NUMERICAL SUPPORT FOR PROPERTYINTERRELATION

Recent experimental and theoretical studies have indi-cated the failure of scaling arguments to describe measuresof polymer and nanoparticle size. In particular, the mass scal-ing exponents defining size depend on the property, i.e., Rg,Rh,

[η], and S.31,32 This failure of scaling arguments has been

discussed extensively in the case of DNA.32 We next validateour predicted relationship between these measures of polymerand nanoparticle size.

In this section, we test Eq. (9) for linear and branchedpolymers, non-fractal particles and particle clusters, and othercomplex non-fractal structures such as globular proteins andtumbleweed carbon nanotube domains. For each group ofobjects, we plot 〈αE〉 versus 〈αE〉

c in Figs. 1–5, where 〈αE〉

is either determined from ZENO or an analytic result and〈αE〉

c is the value of the electric polarizablity calculated byusing by Eq. (9). Here, the symbols represent the data and thered lines are generated from the linear relation between thesequantities, for comparison purposes. The insets are represen-tative configurations for each object. More information about

FIG. 1. Linear polymeric structures. (a) Flexible polymers with repulsiveexcluded volume interactions (swollen random coils) generated via MonteCarlo simulations,26 (b) flexible capped cylinders generated via MD.8,33,34

(c) Worm-like chains obtained from the fit functions given in Ref. 10. Therods have lengths (L) in the range 10 < L < 1900.10 Calculations for rodshaving a rectangular cross section fit our scaling just as well, but we do notshow these data here.

the objects and their properties is given in the supplementarymaterial.

The uncertainty estimates are calculated from the percentdeviation between the directly computed value, 〈αE〉, and thetheoretical estimate from Eq. (9), 〈αE〉

c,

devj(%) =�����〈αE〉 − 〈αE〉

c

〈αE〉

�����× 100. (11)

For a family of N-objects, we report the percentage averagedeviation value,

dev(%) =

N∑j=1

devj(%)

/N . (12)

FIG. 2. Model oblate and prolate structures. The functional forms for cal-culating 〈αE〉 for oblate and prolate ellipsoids presented in (a) and (b),respectively, are provided in Ref. 7. Here, νeff is a function of the aspectratio r. The flexible and flat sheets were generated to study the electromag-netic properties of graphene, Ref. 8, from which we drew our conformationaldata in (c) and (d).

014903-5 Vargas–Lara, Mansfield, and Douglas J. Chem. Phys. 147, 014903 (2017)

FIG. 3. Equilibrium and non-equilibrium randomly branched polymer struc-tures. In (a) and (b), we show non-equilibrium and equilibrium Eden clustersand diffusion-limited aggregate clusters, respectively, in (c), we show latticeanimals or swollen branched equilibrium polymers, and in (d), we show “per-colation clusters” or equilibrium branched polymers having screened excludedvolume interactions.

Table I shows an example calculation of dev(%) for den-drimers having different generation number. Here, 〈αE〉

c incolumn 6 has been computed using the columns 2, 3, and 4 onEq. (9). The dev(%) values for all the objects analyzed in thisstudy are reported in Table II.

A. Flexible worm-like and rod-like polymersand particles

In Fig. 1, we consider rod-like nanoparticle and semi-flexible and flexible polymer models that have been usedto model a flexible polymer such as polystyrene and rodstructures such as carbon nanotubes and semi-flexible double-stranded DNA molecules (details of these calculations aredescribed in supplementary material). In all cases, Eq. (9)

FIG. 4. Polymers having topological constraints. (a) Tori particles or stiffrings. (b) Worm-like star polymers. (c) Dendrimers involve a cascade ofbranching of fixed units so that this molecular has a hierarchical branchingtopology. (d) We show the influence of knot complexity (average knot crossingnumber) on directly computed and theoretically estimated 〈αE〉 data.

FIG. 5. Relatively compact particles and polymers. (a) Ideal platonic solids,which are realized by mineral crystals, viruses, and many other natural forms.(b) Spherical inorganic nanoparticle with grafted chains generated by usingMD.28 (c) Dense cluster of multi-walled carbon nanotubes often observed incommercial nanotube materials.41 (d) A sampling of globular proteins fromthe protein data bank.43

holds to a good approximation, and we provide the effectivescaling exponents νeff in Table II. The flexible and semi-flexible polymer chains were generated by molecular dynamicsimulations, as described in Ref. 8. We also varied the polymercross-sectional shape and the shape of the rod-end to ensurethat these morphological changes do not affect the quality ofthe property interrelationship. In Table I, we report the percentdeviation between the calculated result and the theoreticallyestimated value.

B. Platelets and two-dimensional flexible polymers

Figure 2 considers model oblate and prolate shaped par-ticles. Exact results are available for the ellipsoids, and theyare used to test Eq. (9). We have also shown square plates andcrumpled sheet particles that have been used to model graphenenanoparticles.8 Again, deviations between exact calculationsand the numerical values for 〈αE〉 are ≈1%.

C. Equilibrium and non-equilibriumbranched polymers

In Fig. 3, we investigate randomly branched polymers. In(a) and (b), we show perhaps the most commonly encounterednon-equilibrium branched polymeric structures,5,6,8 Edenclusters,12,13 and diffusion-limited aggregate clusters. Thesecluster types are well known and the conformational data usedin the computations are taken from the study of Mansfieldand Douglas,26 who also provide illustration of representa-tive clusters. Eden clusters are rather compact clusters while(diffusion-limited aggregates) DLA clusters are rather diffusefractal-like clusters because they exhibit a non-trivial massscaling over a large mass range. Both types of non-equilibriumcluster are rather spherical on average. In (c), we show latticeanimals or swollen branched equilibrium polymers from therecent study of Audus et al.35 We contrast these results forswollen branched polymers with “percolation clusters” and

014903-6 Vargas–Lara, Mansfield, and Douglas J. Chem. Phys. 147, 014903 (2017)

equilibrium branched polymers with screened excluded vol-ume interactions. Percolation clusters have been extensivelystudied before36 and we refer to the work of Mansfield andDouglas26 for illustrative images of the percolation clustersutilized in the present analysis. The apparent mass scalingexponents are given in the supplementary material.

D. Polymers and particles having different topology

Polymer structures come in a wide range of topologies andit has traditionally been very difficult to account for topologicaleffects on polymer transport properties. Figure 4 shows sev-eral important topological types. In Fig. 4(a), ring polymericstructures (as observed in some carbon nanotube materials37)are one of the few non-trivial shapes for which exact calcu-lation can be made,6 and direct calculation of νeff from theseresults shows that νeff depends on the ratio r = r1/r2, r1 and r2

being the inner and outer radius. In (b), worm-like star polymerconfigurations having fixed arm length L = 18 beads and persis-tence length of the chains of each arm equals lp = 3 beads in ourcoarse-grained model. A variable number of arms, f, is consid-ered in this plot where f varies from 10 to 35. The star polymermodel uses the same chain model as studied in Refs. 28 and29. Dendrimers involve a cascade of branching of fixed unitsso that this molecular has a hierarchical topology. Here, wedraw upon dendrimer simulations26 ranging from dendrimergeneration gi = 3 to gi = 11, where νeff varies significantly withmolecular mass in this type of polymeric structure. We are notaware of any previous molecular dynamics simulations of thehydrodynamic properties of semi-flexible rings having a largerange of knot complexity, 0 ≤ m ≤ 8, the range encounteredin practice with DNA samples,38,39 and we provide estimatesof 〈αE〉 in Fig. 4(d). We model our polymer by a semi-flexiblechain having a persistence length of 50.2 nm and a diameterof 2.8 nm, appropriate for dsDNA in solution containing 1MNaCl.40

E. Polymers and particles having compact shape

In Fig. 5, we consider model polymers and nanoparti-cles that have a relatively compact structure. In particular, wetest Eq. (9) for (a) ideal platonic solids, which are realizedby mineral crystals, viruses, and many other natural forms.(b) Spherical inorganic nanoparticle with grafted chains. Thepolymer-grafted nanoparticle configurations were generatedby using MD28 and their properties were obtained by usingZENO.7,13 (c) Dense cluster of multi-walled carbon nan-otubes or “tumbleweeds.” These complex-shaped particles areoften observed in commercial nanotube materials,41 and thesimulated tumbleweed structures are derived from molecu-lar dynamics simulations of N worm-like tubes confined toa sphere.8 Here, we vary the number of carbon nanotubes(CNTS) in the range 20 ≤ N ≤ 240. (d) A sampling of globu-lar proteins from the protein database. See the supplementarymaterial for the specification of the 804 proteins involved.The ZENO protein property calculations were taken fromRef. 42.

As before, we find the deviation between the computedand directly estimated properties or the chosen globular pro-teins to be on the order of≈1%. The conformational data for the

spherical nanoparticles with flexible chains is obtained froma previous study of single-stranded DNA grafted to nanopar-ticle surfaces having fixed length L = 10 beads and differentnumber of strands N s, where N s ranges from 5 to 100.

IV. DIRECT ANALYTIC ESTIMATION OF [η] FROM RgAND Rh DATA AND EXPERIMENTAL VALIDATION

We now combine Eq. (9) with the hydrodynamic-electrostatic relationships described in Sec. II to connect

[η],

Rg, and Rh,

[η]=

qηVp

4π(qhRh)1/νeff R(3νeff−1)/νeffg . (13)

For the particular case of a spherical particle, qη = 5/6, qh

= 1, Vp = 4/3πR3h, Rg =

√3/5Rh, and νeff = 1/3, we have[

η]= (5/2), which is consistent with the value determined

by Einstein.44 More generally, qη depends weakly of particleor polymer shape and it is determined from the principle com-ponents of the polarization tensor.13 First of all, if we take thesimple approximation, qη ≈ 0.79 and qh = 1,1 which is suit-able for many complex shaped particles, and if we express

[η]

in terms of polymer mass concentration, then Eq. (13) can bewritten as [

η]=

0.79NA

M4π(Rh)1/νeff R(3νeff − 1)/νeff

g . (14)

Combining Eq. (14) with the mass-scaling relationship forRg = agMνeff which is equivalent for Rh = ahMνheff provides anexplicit and novel expression for the Mark–Houwink exponentrelationship,[

η]=

[0.79NA(4π)a1/νeff

h a(3νeff − 1)g

]Mνheff/νeff + 3νeff − 2, (15)

defined by the scaling relation,[η]=KMa, where K is

a precisely specified constant and the exponent a equalsa= νheff/νeff + 3νeff − 2. For spherical particles, νeff = νheff

= 1/3, so that a = 0, and for the particular case of polymersin a theta-solvent, νeff = νheff = 1/2, so that a = 1/2, asexpected.

We next validate our proposed universal relationshipbetween

[η], Rg, and Rh based on experimental results. We

first consider duplex DNA for this test since there is largeand internally consistent experimental data base32 for

[η], Rg,

and Rh of this polymer because of the fundamental biologicalsignificance of this class of biopolymers. This class of poly-mers is also especially significant since scaling arguments thatassume one scale governs polymer size have been shown to failfor this class polymers.32 Finally, polydispersity of the chainlength does not contribute greatly to experimental uncertaintyin these biological macromolecules.

We may express[η]

in common polymer science concen-tration units, (dl/g), as

[η]

(dl/g) =59.7844

M(g)Rh(nm)1/νeff Rg(nm)(3νeff−1)/νeff . (16)

Figure 6(a) shows a direct comparison between experimen-tal measurements

[η]

and analytic estimations obtained from

014903-7 Vargas–Lara, Mansfield, and Douglas J. Chem. Phys. 147, 014903 (2017)

FIG. 6. Comparison between experimental measurements[η]

and calcula-tions

[η]c for (a) double-stranded DNA and (b) polystyrene in toluene. Inset

shows a direct comparison between our analytic estimates and the measuredvalues of

[η].

Eq. (16) denoted as[η]c as a function of the DNA molecular

mass M. The[η], Rg, and Rh experimental data were taken

from Ref. 32; further information about the calculation can befound in the supplementary material. We find that the aver-age deviation between the computed and directly measuredproperties for DNA is dev(%) = 9.8%. This order of magni-tude deviation for

[η]

between the estimates and experimentalmeasurements is comparable to the deviation between exper-imental estimates from the mass scaling relationships for Rg

and Rh that describes DNA studies from different experimen-tal groups and this higher uncertainty deviations can be tracedto the high uncertainties in the experimental measurements ofthe radius of gyration (see the supplementary material). Theblack circles in Fig. 6 are experimental measurements and thered squares are calculated values by using Eq. (16).

We also consider the mass scaling of polystyrene intoluene in Fig. 6(b), another polymer for which tabulationsof

[η], Rg, and Rh have been made. The comparison of

our analytic estimate from Eq. (16) to experiments reportedby Roovers and Toporowski45 is also in good agreement,dev(%) = 3.0. We again find that our analytic estimate of[η]

performs rather well without any free parameters.Quite apart from the estimation of

[η], Rh, and Rg, the

calculation of the polarizability tensor 〈αE〉 is interesting inits own right with respect to particle characterize and materialproperty design. The leading coefficient [σ] for the electricor thermal conductivity of a medium containing randomlydistributed and oriented highly conductive particles such as

carbon nanotubes, graphene, or carbon black in a relativelynon-conductive matrix exactly equals [σ] = 〈αE〉 /Vp. Forreference, this result exactly reduces to Maxwell’s estimate[σ] = 3 for spherical conductive particles.23 The virial coef-ficient of the dielectric constant of a heterogeneous mediumcontaining randomly distributed inclusions having relativelylarge dielectric constants in comparison to the medium inwhat the particle are placed is given by the same expression.1

It is evident from our discussion above that [σ] can be esti-mated from Rh and Rg of the particles, or from measurementsof

[η]. Previous work has shown that electric conductivity

percolation threshold, φc, and momentum conductivity thresh-old, φη , of isotropically distributed and oriented particlescan be estimated from φc ≈ 1.6/ [σ]1,8 and φη ≈ 1.6/

[η].46

These concentrations, which are distinct from the percolationthreshold of geometrically overlapping objects, define the crit-ical concentrations at which electric conductivity and shearviscosity become rapidly varying with the particle concentra-tion. The prediction of these critical concentrations, based onresummation of the transport virial expansions, has obvioussignificance for materials design and for the characterizationof nanoparticle shape in fashion that relates to the ultimateproperty of composite materials. Finally, we point out that[η]

arises as a basic correlating parameter in the rate ofseparation of polymers and nanoparticles by size-exclusionchromatography.47,48

V. CONCLUSIONS

We provide a general relationship that connects the prop-erties, C, Rg, and 〈αE〉 for particles having complex shape withno free parameters. This relationship has been tested for a vari-ety of objects showing deviations from simulation estimateson the order O(1%). In contrast, expressions developed in thecontext of the approximate Kirkwood–Riseman and Debye–Bueche models have previously shown much larger deviationsin general.7,32 Comparison of hydrodynamic theory to poly-mers and nanoparticles involves additional uncertainties, suchas hydration in the case of DNA and chain polydipersity inpolystyrene solution, but we find that we can estimate theproperties to an uncertainty close to inherent experimentaluncertainties.

Our universal relationship appears to explain the oftenobserved difference in the mass scaling of polymer andnanoparticle properties. This new relation should allow forself-consistent estimates of polymer and nanoparticle sizeand a better characterization of the structure of polymersand nanoparticles based on precise static and dynamic lightscattering measurements, in conjunction with solution vis-cosity and sedimentation measurements. Equation (9) can beused in conjunction with precise

[η], Rh, and Rg observa-

tions, to estimate νeff for particular objects where no variablemass structures exist. This relation provides a measure ofrelative shape “irregularity” of particles that are not truly frac-tal and that in combination with other measures of polymershape anisotropy, such as Rh/Rg, should have applications inclassifying polymers and nanoparticles. Finally, Eq. (9) pro-vides a useful consistency relationship among

[η], Rh, and Rg

measurements.

014903-8 Vargas–Lara, Mansfield, and Douglas J. Chem. Phys. 147, 014903 (2017)

SUPPLEMENTARY MATERIAL

See supplementary material for more information aboutthe computational models used to generate some of thenanoparticle and polymer configurations utilized to test thegeneral relationship described in the main manuscript. We alsoinclude tables with the property values, their estimations, andtheir corresponding uncertainties.

ACKNOWLEDGMENTS

We thank Dr. Michael Zachariah and Dr. Mingdong Lifor posing the problem and Dr. Kathryn Beers for helpful dis-cussions. We also thank Dr. Debra Audus and Dr. AchilleasTsortos for providing us with the data shown in Figs. 3(c) and6(a), respectively. This work was supported by NIST AwardNos. 70NANB13H202 and 70NANB15H282.

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