8

Small Angle Neutron Scatteringas a Tool to Study Kinetics of Block

Copolymer Micelles

Reidar Lund

Summary. Small-angle scattering has long proven capability to extract verydetailed unparalleled structural information on the self-assembly of amphiphilicmolecules into micelles, membranes, etc. With the advent of ever more powerfulneutron/synchrotron sources and better instrumentation, the acquisition time nec-essary for good structural characterizations is now decreasing towards the secondand even sub-second range. This allows for faster time resolved experiment whereone can observe kinetics, transport processes such as diffusion as well as structuraltransitions occurring in the material in real time. Here we will focus on problemsrelated to the kinetics of micelles constituted by amphiphilic molecules and blockcopolymers. We will discuss how SAS and especially neutron scattering in combi-nation with hydrogen/deuterium isotopic substitution can provide very useful anddetailed insight into the equilibrium and non-equilibrium behaviour of micellar sys-tems. Whenever relevant we will highlight the complementarity between X-ray andneutron scattering to resolve such problems.

8.1 Introduction

Micellization is a common self-assembly process where heterogeneous molecules,i.e., molecules having distinct chemical parts, spontaneously aggregate into variousnano-structures that can be spherical, cylindrical or lamellar in shape. This processis usually provoked by a selective solvent, i.e., a solvent that is good for one part butpoor for the other. Often these molecules are amphiphilic i.e., contain a hydrophobic(water insoluble) part and another hydrophilic (water soluble) part and as a resultmicelles are formed in aqueous systems. Such self-assembly of amphiphilic moleculesin water are of immense importance in nature where examples include living cellformation, skin membranes and other biological membranes that are essential forlife. From daily life we usually recognize this phenomenon in detergents or surfactantsystems where micellar formation are responsible for solubilization of fat moleculesand thus used for cleaning purposes.

Another example from physical chemistry is micellization of block copolymerswhich are macromolecular analogues to ordinary surfactant micelles. Such systemsgenerally consist of two or more distinct types of polymeric blocks covalently linked

214 R. Lund

together. Because of the wealth of possible combinations of chemistry and composi-tions of such polymers, the possibilities for tailoring self-assembly and the resultingstructures are virtually end-less. In this contribution we will focus on the simplerblock copolymers having two distinct blocks, combined to form an A–B diblockcopolymers and A–B–A triblock copolymers. In such systems, micellization in aselective solvent (i.e., solvent that is bad for one, B, and good for the other, A),micellization is controlled by three main contributions to the free energy in additionto the translational entropy [1]. The first term is the main driving force for micelliza-tion and is the interfacial free energy attempting to minimize the exposed contactbetween the core forming B block and the solvent. This reduction of free energyis mainly counteracted by the (1) repulsions between the chains constituting thecorona and (2) the loss of entropy associated with the stretching and confinementof the polymer chains. As a result, the equilibrium size and shape of the result-ing polymeric micelle is well defined and given by the minimum in the free energytaking into account all the above mentioned contributions. The question, however,is how or whether such equilibrium structures are really formed and what are thekinetic factors governing the behavior in and out of equilibrium of such micelles.This associated kinetics of polymeric micelles will be the main focus of this work, inparticular we will be concerned with the kinetics associated with the equilibrium.Before going into this subject in more detail, we will briefly outline the importantrole of small angle scattering techniques in this game, especially small angle neutronscattering (SANS).

Small Angle Neutron Scattering has long proven its potential as an extremelypowerful tool to study detailed structure of micelles and other complex nano-structures [2–6]. Here neutron scattering has the advantage over X-ray scatteringas being completely non-invasive and the possibility to vary and selectively enhancethe scattering contrast by simple hydrogen/deuterium substitution chemistry. Fortypical soft matter systems where the electron density difference among the com-ponents is generally small, the corresponding X-ray contrast is also small and highbrilliance beams are often needed. On the other hand, with neutrons the contrast isgenerally much larger and can be enhanced drastically and selectively by simple H/Dsubstitution. This particularly concerns water-systems such as amphiphilic micelleswhere simple H2O/D2O mixtures provides an easy and relatively inexpensive way tostrongly alter the contrast. In such systems neutrons are also particularly preferableover synchrotron SAXS as high energy X-rays can create severe damage on samplesdue to bond breakage, radical formation which thus represents real challenges thatneed to be tackled. For these reasons, in particular isotopic substitution, neutronsoffers a powerful and versatile but “softer” tool that is complementary to X-ray, tostudy kinetic processes. As we soon shall see, isotope substitution in combinationprovide some unique possibilities which open up unique ways to study diffusionalproperties in micelles and other soft matter systems under equilibrium.

The kinetics of micelles can very coarsely be divided into two main subareas: non-equilibrium kinetics and equilibrium kinetics. The first type of processes concernsthe kinetics of a transition involving some sort of thermodynamical gradient. A veryimportant process here is the kinetics of micelle formation, i.e., the self-assemblyprocess from single chains (unimers) to the final equilibrium micelle, induced forexample by sudden change in solvent quality, temperature or pressure. This processis pictured in Fig. 8.1a. Another example is transition kinetics from one micellarstate to another including transitions from spherical to cylindrical micelles etc. Here

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 215

Fig. 8.1. Schematic illustration of two important kinetic processes in block copoly-mer micelles: (a) Non-equilibrium self-assembly kinetics illustrating the formationof a polymeric micelle. and (b) equilibrium kinetics showing the unimer exchangeprocess

scattering techniques are obviously inherently very important as the structure isdirectly observable. Examples of such studies of phenomena in surfactant systemscan be found in a review by Gradzielski et al. in [7].

In this contribution we are mainly concerned with another kinetic processthat is a criterium for equilibrium and that is important for the stability andproperties of micelles; chain exchange kinetics between micellar entities occurringas a result of random stochastic forces. As any dynamical equilibriums wouldimply, the chains constituting the micelles are required to continuously redis-tribute among the micelles. An important process is thus unimer exchange wheresingle chains are continuously emitted and reinserted between micelles as illus-trated in Fig. 8.1a. A central question for block copolymer micelles is not onlyhow and what factors controls this dynamical equilibrium, i.e., in the sense ofphysical pathways and mechanisms; but also if an equilibration mechanism reallyalways exist. That is, under what circumstances polymeric micelles are in equi-librium.

In the following we will concentrate on the equilibrium kinetics of block copoly-mer micelles and how these processes can be elegantly resolved using a novel timeresolved SANS technique that takes advantage of a simple H/D labelling scheme [8–11]. As an attempt to elucidate the mechanism and kinetic pathway characterizingthe kinetics of equilibrium the TR-SANS is applied to various very different experi-mental systems. Before coming to these systems in particular, we will briefly presentthe relevant background for block copolymer and surfactant micelles in general andsome basics of small angle neutron scattering. Because of the space limitation, wewill not go in depth of this important technique but merely emphasize some possi-bilities and views of how time resolved small angle neutron scattering can be usedto study slow kinetic/dynamical features in micellar systems.

216 R. Lund

8.2 Theoretical Background

In the following we will briefly summarize the theoretical results relevant to thestructure and exchange kinetics of block copolymer micelles in equilibrium.

8.2.1 Brief Introduction of Thermodynamics and Scaling Laws

Scaling theory is a quite simple approach pioneered mainly by de Gennes [12]to calculate rather complicated structural and thermodynamical features of poly-meric systems via relatively easy geometrical and physical arguments. For polymericmicelles there are many applications of such theories [13–19] generally using thepseudo-phase approximation, i.e. that a micelles can be regarded as a distinct ther-modynamical phase [20]. In practice, this is satisfied in the limit of high aggregationnumbers, P , and at surfactant concentrations well above the critical micelle con-centration, cmc. In the following we will concentrate on the thermodynamics ofspherical micelles which usually is formed, at least for block copolymer having andasymmetric compositions. In this case it is possible to make several simplificationsand and limiting cases or classes of micelles.

In all models, the free energy associated with the interfacial tension is consideredto be the driving force for micellization. The interfacial free energy per chain of aspherical micelle is given by:

Fint =4 π R2

cγ

P∼ P−1/3 γ (8.1)

where Rc is the micellar core radius and P is the aggregation number of the micellethat would scale with the volume and thus P ∼ R3

c .This term will favor micellar growth and, in the absence of other effects, lead

to a macroscopic phase separation. However in a real micellar system growth willbe primarily counteracted by a repulsion between the head groups. In polymericsystems entropic forces become dominant, e.g., stretching of chains, which becomesincreasingly unfavorable with increasing aggregation. The major difference betweenthe models is the way the counteracting free energy is calculated, in particular thefree energy of the corona.

Applied to the free energy of single micelles, three main contributions arecommonly considered

Fmicelle = Fint + Fcorona + Fcore (8.2)

where Fint is given in (8.1) above, Fcore is a term related to stretching of segments inthe core and Fcorona originates from repulsions and stretching of the coronal chains.

Within scaling theories, it is possible to distinguish three limiting cases for spher-ical polymeric micelles: crew-cut, intermediate and star-like micelles. For the formercrew-cut micelles, characterized by having a number of repeat units of the B block,NB , much larger than the soluble A block, NA (i.e., NA << NB , the free energy ofthe corona is assumed to be negligible compared of the stretching contribution inthe core. Assuming that all chains are uniformly stretched one obtains [16,21]:

Fcore ∼ R2c

NBl2B∼ P 2/3 (8.3)

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 217

with li, the corresponding characteristic monomer length of the polymer block(i = A,B).

For the other two cases, it is assumed that the balancing free energy is determinedby the free energy of the corona which is calculated by assuming a flat core-coronainterface for intermediate micelles (i.e., suitable when the B block is relatively large);and highly curved when NA >> NB and we have star-like micelle. Using the analogyto grafted polymer chains, the free energy of the corona can be calculated using thephysics of polymer brushes [13–15,18,22]

This gives the following free energies of the corona:

Fcorona/kbT ∼⎧⎨⎩

P 1/2 ln(N3/5A P−2/15 N

−1/3B ) Star-like

P 5/18N−5/9B NA Intermediate

(8.4)

From this formalism the dependence of the various micellar parameters withfor example molecular weight, composition, interfacial tension etc. can be estimatedwhich seems to compare rather well with experimental data- see e.g., [23, 24]. Forexample, for intermediate and star-like micelles, the aggregation number would scalelike:

Pscaling ∼⎧⎨⎩

γ6/5 N4/5B l

12/5B Star-like

γ18/11N2BN

−18/11A l

−30/11A l

30/11B Intermediate

(8.5)

8.2.2 Aniansson and Wall Mechanism

Kinetics: The Aniansson–Wall theory (A–W) [25, 26] was developed for the nearequilibrium relaxation kinetics of neutral low molecular weight micelles. Here it isassumed that all changes in the association/dissociation involve unitary steps inwhich only exchange of one surfactant molecule is allowed at a time. The unitarystep consist of a single exchange of unimers from solvent to micelles and can bewritten as:

MP + Uk+⇀↽

k−MP+1 (8.6)

U is the unimer, MP is a micelle with aggregation number P . k+ and k− are therate constants for the insertion and expulsion. The corresponding rate equation is:

d[U ]

dt= −k+[MP ][U ] + k−[MP+1] (8.7)

Annianson and Wall showed using a system of rate equations constructedfrom the above mechanism, that in equilibrium experiments where only individualunimers/chains are followed, the kinetics is characterized by only one characteris-tic rate constant proportional to the expulsion rate constant, k−. In the case of arelaxation experiments close to equilibrium, i.e., in the linear regime, the kineticsis characterized by two time constants, separated two to three orders of magnitudein time. The first one, τ1 characterizes the unimer consumption/release mentionedabove, and the other, τ2 characterizing the re-equilibration time of the micelles

218 R. Lund

adjusting to their final equilibrium state. This latter process is thus limited by therelease of unimers from the metastable micelles.

It should be mentioned that the situation for low molecular weight surfactantmicelles might be different than for polymeric micelles. In the former case, thekinetics is close to be “diffusion limited” [27], i.e., the diffusion of chains between themicellar droplets might be comparable to the time scale of the expulsion/insertionprocess. For polymeric micelles however, the expulsion time is generally much longer,and, as we will see, can be arbitrarily long for large block copolymers. As an exampleto get a feeling of the time scales, we can estimate the diffusion time of a chain overan typical inter-micellar distance. Taking this distance to be RF = 500 A , and usingτ ≈ R2/6D and a typical diffusion constant of the order of D ≈ 10−11 m2 s−1, wearrive at a very rough estimate of the order of 40 μs.

Halperin and Alexander extended the theory of Aniansson and Wall to calcu-late the detailed rate constants for polymeric micelles and the associated activationenergy. We briefly review the central results in the following section.

8.2.3 Scaling Theory – Halperin and Alexander

The theory proposed by Halperin and Alexander is based on the structural scalingdescription of polymeric micelles outlined above. Using a combination of scalingtheory and the Kramers rate theory for diffusion in an external potential [28], theexpulsion rate for both “crew-cut” and “star-like” spherical micelles was derived.Moreover the different scenarios of chain exchange between micelles were discussed.

In general, the possibility of direct micellar fusion and fission cannot be ruledout.

Mi + Mj ⇀↽ Mi+j (8.8)

The question is, however, how important, i.e., with what probabilities these processesoccurs in comparison with the unimer exchange which anyway will have the lowestactivation energy and hence by far be the fastest equilibration mechanism.

Using the expressions for the free energy terms of a polymeric micelle Halperinand Alexander estimated the activation energies, Ea, for the unimer expulsion andthe fusion/fission mechanisms. Based on these calculations it was concluded thatfusion/fission cannot be important for polymeric micelles as the associating activa-tion energies are very large, especially when the corona is rather dense/extended.This is the case whenever the micelles are well-developed, i.e., at the end of theequilibration process or at equilibrium.

Hence, the most important process for the equilibrium kinetics is the unimerexchange mechanism which, as expected from the Aniansson – Wall scenario, ismainly governed by the expulsion rate constant. In the model of Halperin andAlexander this release of a single unimer from the micelle is pictured to go throughtwo stages:

1. Ejection of the solvophobic part of the block copolymer to form a “bud” on theinterface of the micellar core. Thereby an extra area (≈ l2BN

2/3B ) is exposed to

the solvent2. Diffusion of the whole block copolymer through the micellar corona.

The corresponding free energy profile is shown in Fig. 8.2.For one dimensional stationary flow in a potential well, Kramer’s rate theory

gives the following expression for the outgoing flux, J :

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 219

Fre

e E

nerg

y

Ea

Fig. 8.2. Illustration of the chain expulsion process including the corresponding freeenergy profile, F (y), considered in the Halperin–Alexander model. In the calculationsgiven in the text the reference state is chosen according to F (P + 1) ≡ 0 so thatF ∗ = Ea for the expulsion process

J = −D exp(−F (y)/kb T )∂

∂yφ(y) exp(F (y)/kb T ) (8.9)

where F (y) is the free energy profile along the “reaction coordinate,” y. φ(y) isthe local polymer concentration of the diffusing block copolymer characterized by atypical diffusion coefficient D. Halperin and Alexander demonstrated that the fluxcould rewritten as:

J ∼ exp(−F ∗/kb T ) vdiffusion (8.10)

where vdiffusion is the chain velocity over the barrier and F ∗ = F (y∗) is the the max-imum free energy. This is principally determined by the interfacial energy penaltydue to the expelled B-blocks i.e., F ∗ ∼ r2

budγ ∼ N2/3B l2B γ, where rbud is the radius

of the collapsed B-block.For micelles with a thin corona, NB � NA, vdiffusion is roughly determined by

the time, τB , necessary to diffuse the length of its insoluble block, i.e., τB ∼ N2/3B /D.

Assuming classical diffusion of polymer segments in a homogeneous surrounding ,Stoke-Einstein’s law gives: D ∼ 1/N

1/3B lB and τB ∼ N

2/3B l2B/(N

−1/3B l−1

B ) = NB l3B .The above mentioned expression is valid whenever the core is large compared

to the corona. In the opposite limit, as in star-like micelles, the diffusion throughthe corona has to be considered instead. Using the Langevin equation to describethe stochastic passage of the chain through the corona, Halperin and Alexanderobtained for the characteristic velocity of diffusion through the corona vdiffusion ∼L−2 P 1/2 ∼ N

2/25B N

−6/5A .

Thus the velocity of the block copolymers over the activation barrier is in thetwo limiting cases given by:

vdiffusion ∼

⎧⎪⎨⎪⎩

N1/3B

lBτB

∼ N−2/3B , NB � NA

∼ L−2 P 1/2 ∼ N2/25B N

−6/5A , NA � NB

(8.11)

220 R. Lund

The expulsion rate can be obtained using k− = exp(−F ∗) vdiffusion/Rc and k− =exp(−F ∗) vdiffusion/L for crew-cut and star-like micelles respectively. In this way thefollowing expressions are obtained:

k− ∼⎧⎨⎩

exp(−N2/3B γ l2B/kb T ) N

−4/3B , NB � NA

exp(−N2/3B γ l2B/kb T ) N

−2/25B N

−9/5A , NA � NB

(8.12)

Thus in all cases the activation energy has the form:

Ea = N2/3B γ l2B (8.13)

8.2.4 Other Theories

The theory of Halperin and Alexander was later revised and refined by Dormidontova[29]. In particular the theory was extended to include the non-linear kinetics, e.g.,the kinetics of micelle formation. In this respect all relevant rate constants andthe proper dependence on block copolymer characteristics, concentration etc, wasreconsidered and the corresponding formation kinetics was simulated based on theextracted rate constants and a set of coupled kinetic equations. An important out-come of this work is that micellar fusion/fission is not negligible and plays animportant role for the formation kinetics, especially at short times. The work ofDormidontova also extends the work of Halperin and Alexander by considering spe-cific polymer dynamics and and the particular effect on diffusion times in moredetail. For example, using Rouse or Reptation like dynamics [30], the characteristicdiffusion time in the micellar core, was calculated in somewhat more detail assum-ing that this is the same as in bulk polymers. Thus no particular confinement orinterface related effects because of the large area to volume ratio in nano-segregatedmicellar systems are considered.

A quite different approach has been presented more recently where the micel-lization is viewed as special kind of nucleation and growth process [31–33]. Here thekinetics is treated by using a general micelle potential that is essentially given by thedifference between the micellar free energy and the equivalent amount of unimerstaking into account the translational entropy. In a work for polymeric micelles inparticular by Nyrkova and Semenov [33], the theory of Dormidontova was criticizedclaiming that the role of fusion/fission mechanism was overestimated and the role oftranslational entropy has not been appropriately included. Considering the classicalunimer exchange mechanism only, estimates were performed for the characteristictime of micellization, i.e., the time typically needed to approach a given equilibrium.The results showed that this time scale rapidly approach exceedingly (unpractical)long times as the interfacial tension, or equivalently, the molecular weight of theinsoluble block increases. This caused by the depletion of unimers in the beginningof the micellization process-leading to a broad distribution of micelles of improper(non-equilibrium ) size. For the equilibrium micelles to form, these micelles need todissolve by unimer expulsion- a process which is slow. In the Nyrkova–Semenov the-ory, this is taken into account in a somehow “coarse grained” manner by consideringthe translational entropy and the instantaneous micellization potential. This impliesvery interesting non-equilibrium effects on the structure of polymeric micelles. Firstof all the equilibration time is very sensitive to the unimer concentration and for

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 221

low unimer concentrations the time to reach equilibrium may tend towards infin-ity, practically speaking. Thus the micelles formed might be different depending onthe details of preparations. For this reason Nyrkova and Semenov defined a con-centration, an apparent cmc located above the real cmc, defined where the time ofmicellization is about 1 h which is a typical practical laboratory time.1

8.3 Experimental Background: Small AngleNeutron Scattering

In this section we briefly review the principles of small angle neutron scattering andits applications to micelles with a predominant focus on the unique possibilities toresolve kinetic processes. For a more thorough review on small angle scattering ingeneral and scattering block copolymer micelles in particular, we refer to the generalbook edited by P. Lindner and Th. Zemb [3] and a review article by Pedersen andSvaneborg [4]. For a more fundamental introduction to scattering theory and neutronscattering in general we further refer to the book by S. W. Lovesey [34].

8.3.1 Structure with SANS: Core-Shell Model

The length scale which is probed in a scattering experiment is given roughly by 2π/Q

where the momentum transfer vector is defined by |−→Q | = Q = 4 π sin(θ/2)/λ whereθ is the scattering angle and λ is the wave length of the neutron. A standard SANSset-up usually covers a solid angle of typically 0.001 < θ < 10◦ and a wavelength ofabout 5–10 A which gives a Q range from approximately 10−3 − 0.5 A−1 and thusa structural resolution of about 1,000–1 nm which makes it an ideal tool to studyself-assembled micellar aggregates.

Picturing a typical spherical micellar entity as a central core containing theinsoluble B block and an external corona part consisting of swollen A chains graftedto the micellar core. The total weighed scattering amplitude of such a micelle canbe written as the sum of each part:

A(Q)tot = P VB (ρB − ρ0)A(Q)core + P VA (ρA − ρ0) A(Q)corona (8.14)

where ρi and Vi are the scattering length density and the volume of the solvent(i = 0), or the polymer blocks (i = A, B) respectively.

The absolute scattered intensity, or more precise, macroscopic differential scat-tering cross section, dΣ/dΩ(Q) can, ignoring any structure factor or polydispersity(micellar size distributions) effects, be written as:

dΣ

dΩ(Q) =

φ0 NAvo.

P · VA−B< |A(Q)tot|2 > +Iblob(Q) (8.15)

The term Iblob(Q) is the internal scattering from the chain constituting theswollen corona and can be described by [24,35]:

1 Such a definition is equivalent to what is customary in glass physics where theglass transition temperature, Tg, often is defined as the temperature where thetypical α-relaxation time is about 100 s.

222 R. Lund

Iblob(Q) = (ρA − ρ0)2 B ((erf(Q ξ/

√6))3/Q)d

f (8.16)

where, B is an amplitude related to the volume of the largest polymer “blob”, ξ thecharacteristic blob radius and erf(x) is the error function. Note, as the error functiondecays to zero at Q = 0, this approach ensure proper normalization. df is the fractaldimension which for a chain in a good solvent is given by df = 1.7. A smaller valuewould indicate a more stretched conformation, while a chain in θ-conditions shouldgive a larger exponent, df = 2. A more involved treatment of the scattering due tothese correlations in the radial directions of the internal part of the corona has beengiven by Pedersen et al. (see for example [4]).

The individual amplitudes are given by the Fourier transform of the correspond-ing density profile, ni(r). For a centro-symmetric spherical micelle, this can bewritten as:

A(Q)i = 4 π

∫ ∞

0

ni(r)sin(Q r)

Q rr2 dr (8.17)

Thus by performing the integral, we can calculate the scattering from any densityprofile, n(r). For a homogenously packed micellar core, the result is easy and givenby: Acore(Q) = [3 sin(Q Rc) − Q R sin(Q Rc)]/(Q Rc)

3. If we want consider a “dif-fuseness” – a smearing of the core-corona interface, we might assume a Gaussiandistribution of the density which gives a factor exp(−Q2 σ2

c/4) that needs to bemultiplied to the individual amplitudes. This gives for the core:

Acore = exp(−Q2 σ2c/4) · [(3 sin(Q Rc)−Q R sin(Q Rc))/(Q Rc)

3] (8.18)

where σc is a measure of the thickness of the interface. As is evident, the broaderthe interface – the faster the scattering will drop at large Q. Hence, in order to obtaingood information on such details, the signal-to-noise ratios and statistics at large Qneed to be very good which usually demands protonated core and deuterium richsolvents.

For the corona, there are many ways to parameterize the density profile. A versa-tile and robust way is to define a power law profile multiplied by an effective cut-offfunction that takes into account the finite size of the chains.

ncorona(r) = Cr−x

1 + exp((r−Rm)/Rmσm)(8.19)

where C is normalization constant, Rm is the overall micellar radius and σm

is an estimate of the micellar roughness while x is a scaling exponent that canbe varied to yield various segmental distributions of the coronal chains. Such atype of density profile has proved to be a flexible and realistic way to parameterizevarious types of micelles, exemplified by a study of a series of PEP–PEO blockcopolymer where the PEP block was held constant and the PEO block molecularweight gradually increased [23]. In this study a gradual increase in the exponentof x ≈ 0 (homogeneous profile) for symmetric to x ≈ 4/3 for asymmetric diblockcopolymers (asymmetry about factor 20 in molecular weight) was found. Similarexponents close to 4/3 were also found in another study [24] for similarly asymmetricPEP–PEO block copolymers with lower molecular weight as well in a study byForster et al. on poly(styrene)-b-poly(vinyl pyridine) (PS–PVP) in toluene [36].This value is indeed predicted for star-like polymers [15] and micelles with largeasymmetry [17]. Other possibilities to parameterize the density profile and describe

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 223

0.10.0110–2

10–1

100

101

102

103

104

Core Shell Interference Simultaneous Fits

dΣ/d

Ω/φ

(Q

) [c

m–1

]

Q [Å–1]

Fig. 8.3. Structural analysis of block copolymers using SANS and contrast varia-tion. Partial scattering functions of h-PS10-d-PB10 in heptane at 293 K showing thecore (triangles), shell (squares) and an average contrast (stars). The lines displaya simultaneous fit to the experimental data using the core-shell model convolutedwith the experimental resolution [11,35]

the scattering for the corona via the use of for example spline functions have beenpresented in a work by Pedersen et al. [37].

As is evident from the equations above, a change in the contrast will selectivelyeither increase or decrease the scattering contribution from either the core or shellpart. More notably, with isotopic substitution, e.g., by using partly deuterated blockcopolymers and for example H2O/D2O mixtures, we can obtain very clean signalsfrom each part of the micelles. As an examples the principle is shown in Fig. 8.3which display the scattering from a poly(butadiene)-b-poly(styrene) (PS10-PB10)micelles in isotopic heptane mixtures. The different scattering pattern corresponds tovarious isotopic solvent mixtures where the hydrogenated PS core (h-PS) is matchedout – “shell” contrast; d-PB shell matched out – “core” contrast and an “interme-diate” contrast where the d-heptane/h-heptane mixture corresponds to a scatteringlength density in between that of the core and shell. Note that in the latter case,the scattering at low angles (Q −→ 0) is almost disappearing because there is nooverall contrast and only the internal correlations in the micelle are visible wherethere is still local contrast, e.g., between core and shell as well as polymer seg-ments and solvent. The solid lines display fits using the core-shell model above withan almost constant density profile, i.e., x = 0 a micellar smearing of about 10%,σm = 0.1. The core was seen to be uniform but notably considerable swollen withintermixing of about 50% solvent molecules (by volume) in the core. Other exam-ples of such detailed analysis of the structure with aid of contrast variation can befound for poly(styrene)-poly(isoprene) (PS–PI) micelles in decane in [37] and forPEP1–PEO20 micelles in water/dimethylformamide (DMF) mixture in [38].

224 R. Lund

8.3.2 Equilibrium Kinetics and Time Resolved SANS

As should be evident from the previous paragraph, contrast variation in SANS isrelatively easy to perform and gives us access to very delicate features of the struc-ture. It is also shown in Fig. 8.3 that if the scattering length density was adjustedto be close to zero average conditions, i.e., when the scattering length of the sol-vent coincides with the average value of the micelle, we see no scattering at low Q.The same principles of contrast variation can in a similar way be applied to resolvekinetic processes. Imagine two different block copolymers that are identical exceptfor one thing, one is fully deuterated (lets denote this yellow) and the other fullyhydrogenated (blue) (see Fig. 8.5). If these two polymers are mixed randomly in amicelle that are dispersed in a solvent that has exactly the average scattering lengthdensity (green), one would have no overall contrast and no scattering intensity. How-ever, if the polymers are not mixed, and each of the micelles are composed of eitherfully hydrogenated or deuterated polymers, the contrast will be large. This meansthat by mixing fully hydrogenated and deuterated identical micelles in a solvent ful-filling the zero average conditions, we can follow the fraction of chains exchangingin the micelles by simply monitoring the intensity [8–10]. The concept is illustratedin Fig. 8.4.

Fig. 8.4. Illustration of the TR-SANS experiment for the determination of thechain exchange kinetics in block copolymer micelles. Two reservoirs consisting ofdeuterated (yellow) and protonated (blue) micelles are prepared in an isotopic solventmixture (green) which exactly matches the average scattering length density of thetwo constituting block copolymers. As the chains exchange the overall scatteringlength density of the micelles will tend towards that of the solvent (green) andconsequently the contrast decreases

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 225

0.010.02

0.03

1

10

600400

200

Inte

nsity

[cm

–1]

time [min]Q [Å]

Fig. 8.5. Experimental curves showing a realization of the TR-SANS techniqueapplied to PS–PB block copolymers dissolved in DMF which is a selective sol-vent for PS. At time zero a fully deuterated d-PS–d-PB micelles is mixed withfully hydrogenated h-PS–h-PB micelles in a isotopic h-DMF/d-DMF solvent mix-ture exactly matching the average scattering length density. As the block copolymerchains exchange, we see the intensity decrease while the form factor and thence thestructure remains constant [9]

An actual experimental realization of the idea is shown in Fig. 8.5 for poly(styrene)-poly(butadiene) (PS-PB) micelles in dimethylformamide (DMF) obtainedat the D11 instrument at ILL [9].

Mathematically, we might express this more precisely in the following way. Theobserved SANS intensity is determined by: I(t) ∼ (ρm − ρ0)

2 where ρm is theeffective scattering length density of the micelle given by the volume fraction ofhydrogenated and deuterated chains, f and 1−f respectively: ρm = fρh +(1−f)ρd

where ρh and ρd are the scattering length densities of the h and d chains. Since inzero average contrast conditions we have: ρ0 = (ρh + ρd)/2 we see that the squareroot of I(t) is linearly proportional to the excess fraction of the h or d chains, i.e.,√

I(t) ∼ Δρ(t) ∼ (f(t) − 1/2)ρh + (1/2 − f(t))ρd = (f(t) − 1/2)(ρh − ρd). In thisway the analysis of the data is straight forward in contrast to other methods. Henceusing this method the information on the exchange kinetics is unambiguously givenby the relaxation function, R(t):

R(t) =

(I(t)− I∞

I(t = 0)− I∞

)1/2

(8.20)

where I∞ is the scattered intensity of the randomized blend.Following Halperin and Alexander [39], the chain exchange kinetics should be

simply a first order chain insertion/expulsion mediated by the solvent. In our exper-imental setup, we monitor the exchange of d and h chains by effectively measuringthe contrast, which is proportional to (f(t)− 1/2). At the beginning of the experi-ment when hydrogenated and deuterated micelles are mixed together, the fractionof hydrogenated chains is unity (f = 1) for hydrogenated micelles and 0 for deuter-ated micelles. At any time t, the fraction of hydrogenated chains in originally 100%

226 R. Lund

hydrogenated micelles is, on average, given by:

f (t) = fo (t) +1

2(1− fo (t)) =

1

2(1 + fo (t)) (8.21)

where fo is the fraction of original chains which remain in the micelle. The secondterm in 8.21 is based on the assumption that chains that have been exchanged withfree unimers in the solution are equally likely to be deuterated or hydrogenated. Thisis correct when the time scale of unimer diffusion is much faster than the time scaleof chain insertion. As a result, the measured intensity of scattering from originallyhydrogenated micelles:

Ih (t) ∼ 1

4f 2o (t) (ρh − ρd)

2, (8.22)

is directly proportional to the amount of original chains remaining in the micellefo. A similar equation can be obtained for the intensity of scattering for deuteratedmicelles. The total scattering intensity would be a sum of intensities of scatteringfrom hydrogenated and deuterated micelles, i.e.,

√I (t) ∼

√1

2fo (t) (ρh − ρd), (8.23)

The intensity will tend to zero only when all chains belonging to the originallyhydrogenated (or deuterated) micelle have escaped at least once. Thus, our time-dependent scattering intensity directly measures the chain exchange kinetics, whichaccording to Halperin and Alexander is expected to be a first order kinetic processwith a single exponential behavior:

R(t) = exp(−k− t) (8.24)

In the following, we will see applications of this technique to various block copoly-mer micelles. But before we will briefly review some results of both low molecularweight surfactant micelles and polymeric micelles using more classical techniques.

8.4 Results – Equilibrium Micellar Kinetics

8.4.1 Low Molecular Weight Surfactant Micelles

Before focussing on polymeric micelles, we briefly summarize the main results forordinary surfactant micelles.

In contrast to polymeric micelles, the kinetics of micelles formed by low-molecular weight surfactants is relatively well understood and have been studiedextensively during the 70–80s using classical relaxation techniques. In particulartheir relaxation behavior close to equilibrium, in the so-called “linear regime,”has been investigated in great detail both theoretically [25–27] and experimentally[27, 40–43] using, temperature/pressure jump, stopped flow combined with fluores-cence and spectrometric techniques using light. The general trend emerging fromthese studies is that the relaxation of micellar systems following a small pertur-bation is characterized by two mechanisms [27, 40–42, 44]. One has been assignedto simple single unimer exchange, with relaxation time τ1 between the bulk phase

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 227

and the micelles, without any net change of the number of micelles. The secondprocess (characterized by τ2) on the other hand, is approximately of the order of102 times slower and has been associated with the dissolution formation equilib-rium of the micelles. This process leads to a change in the aggregation number andconsequently the micellar number density.

The concentration dependence predicted by Aniansson and Wall has been wellcorroborated in many experiments for ionic surfactants where 1/τ1 shows a consis-tent increase with concentration, while the latter process characterized by τ2 seemfirst increase and then decrease again upon higher micelle concentration [27,40,41].This issue was later addressed in a work by Lessner and coworkers [42, 45] andattributed to previously ignored effects of charge screening which can promote micel-lar fusion/fission [46]. Screening of the charges at high ionic strengths in ionicsurfactants lowers the repulsion facilitating fusion as an increasingly importantexchange mechanism. Fusion/fission is in general expected to play more role in micel-lar systems with low repulsion, such as in non-ionic micelles [44]. The relaxation timefor the fusion/fission mechanism is expected to decrease with concentration and atentative description was given by Lessner et al. [45]. However, this does not seemto be strictly necessary to understand the experimental results because also theA–W theory predicts a complicated concentration dependence of τ2 simply becausemetastable micelles restrict micellar growth by consuming and depleting unimers[27]. However, a complication for charged micelles, not considered in the originalA–W theory, is the effect of co-solutes like counter-ions accompanying the main sur-factant chain. These ions will additionally affect the thermodynamics by loweringthe cmc which in turn leads to slower τ2 at higher concentrations. Some correctionsalong this lines, explicitly taking into account issues of charge screening, have beendone by Kahlweit and co-workers [42,45].

8.4.2 Block Copolymer Micelles

In this work we are interested in the equilibrium properties of polymeric micelles.However, since every experimental technique requires some kind of perturbationand/or labelling, this process is much more difficult to assess. Therefore most ofthe experimental studies have been performed under non-equilibrium conditionstypically by a temperature jump/quench from one equilibrium state to another[47–51] or even with jumps from a unimer to a micellar state [52]. However, thesemethods are of very limited use for polymeric micelles as rather large temperaturejumps are generally required for detectable signals which consequently disturb thesystem far away from equilibrium [50]. On the contrary the Aniansson and Walltheory, which is the basis of the Halperin and Alexander model, assumes only asmall linear perturbation. Thus, in this way non-equilibrium properties are mainlyprobed. On the other hand, for equilibrium kinetics, the only experimental techniquethat may give access to the true unperturbed dynamic behavior of the system aretechniques that include some sort labelling scheme.

Experimentally, this can be realized in for example fluorescence spectroscopy[53–57]. In fluorescence quenching, a prerequisite is an introduction of often (bulky)aromatic groups into the backbone of the polymer that can significantly alter thephysio-chemical properties. In addition in fluorescence techniques, energy transfermight occur through unwanted parallel mechanisms which may complicate the dataanalysis [54]. This technique has been applied to various block copolymers with

228 R. Lund

poly(styrene) as the core forming, hydrophobic, block such as PS-poly(ethyleneoxide) (PS-PEO) in methanol [53] and in polyelectrolytes such as PS-poly(sodiummethacrylate) (PS-PSMA) [57]. The general consensus emerging from these inves-tigations is that the kinetics of block copolymer micelles is drastically slower thantheir low molecular weight surfactant counterparts and can reach up to character-istic times from seconds to hours, weeks up to even longer. This means that thekinetics of block copolymer micelles is likely to be “reaction limited” i.e., limited bythe expulsion time2 and not by diffusion time as often surfactant micelles are [58].As stressed above, the kinetics is within the Halperin–Alexander theory expected tobe described using a single rate constant. Interestingly, this is not always found [53]where the kinetics was found to be characterized by a very broad relaxation that wasanalyzed by a double exponential decay. However the fit quality produced by such aparametrization was not very good perhaps suggesting a distribution of relaxationtimes. In addition, in similar cases using fluorescence labelling techniques the datahave been described by a single exponent [57]. Whether these differences are due todifferences in mechanisms between systems or experimental effects is not clear.

In the next section we review systematic study of various block copolymermicelles using time resolved small angle neutron scattering which is much morestraight forward to interpret and that in addition gives us a direct structural andtemporal resolution of micellar kinetics.

8.4.3 Amphiphilic Diblock Copolymer Micellesin Aqueous Solutions

A very good model system for amphiphilic block copolymer micelles is represented bypoly(ethylene-propylene)-b-poly(ethylene oxide) (PEP1-PEO20, the numbers indi-cate the approximate molecular weight in kg/mole) in water/dimethylformamide(DMF). Previous work has shown that in solvent mixtures of water and DMF, the

2 This can rationalized using simple kinetic arguments often encountered in stan-dard textbooks. Consider a diffusion rate constant to, kd, and from k−d a micelle,we denote the intermediate micelle-unimer complex (the fictive activated complexbefore the unimer is inserted) as M − U∗ and write:

MP + Ukd⇀↽

k−d

MP − U∗k+⇀↽

k−MP+1 (8.25)

Assuming that the intermediate configuration is rare (low concentration of fictivereaction intermediate), we can assume d[MP − U∗]/dt ≈ 0. This yields

d[U ]

dt=

(kd k−d

k−d + k+− kd

)[MP ][U ] +

k−d k−k−d + k+

[MP+1] (8.26)

Since in general the diffusion of single polymers is generally relative fast (sometenths of micro seconds), we can thus assume that k−d � k+ leading to the simpleequation:

d[U ]

dt≈ k−[MP+1] (8.27)

and thus the kinetics is intimately controlled by the expulsion rate constant.

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 229

block copolymer spontaneously self-assembly into well-defined star-like micelles [24].The latter property has been thoroughly tested by small angle neutron scatteringwhere the results have confirmed a microscopic star-like density profile of the coronan(r) ∼ r−4/3 [24], corresponding ultra-soft intermicellar potential [38, 59] and thescaling between the aggregation number and interfacial tension (P ∼ γ6/5) [24].Moreover, while water and DMF both are highly selective for PEO, their interfacialtension towards PEP is very different (γ ≈ 46 mNm−1 and 8 mNm−1 respectively).This allows for highly controlled tunability of the system in terms of the interfacialenergy which we might expect to play a very intimate role on the kinetics.

In Fig. 8.6 the scattering intensity for several different mixtures is plotted forvarious DMF/water mixtures showing the influence of interfacial tension. Thesedata were obtained at the KWS-2 instrument in the old experimental hall at theFRJ-2 reactor at FZ-Julich (now relocated to FRM2 at Garching, Germany) withan acquisition time of some 5 min. Here the time evolution of the scattered intensityafter mixing d-PEP1-PEO20 and h-PEP-h-PEO20 in various DMF/water mixturescorresponding to different interfacial tensions is shown. The isotopic solvent mix-ture is always such that the scattering length density equals that of the arithmeticaverage of d-PEP1-PEO20 and h-PEP-h-PEO20. As reference, we have plotted the“blend” scattering which results from micelles formed by random 50:50% mix of

0.010.01

100

101

100

101

t = 5 min at 20 °C24 h ∞

t = 3 min 9 h at 70 °C∞

XDMF = 0.1XDMF = 0

XDMF = 0.3 XDMF = 0.5

c d

dΣ/d

Ω(Q

) [c

m−1

]dΣ

/dΩ

(Q)

[cm

−1]

t = 3 min at 50 °C9 min2 h 6 h 24 h ∞

Q [Å−1] Q [Å−1]

bt = 0t = 24 h at 70 °Creservoir∞

a

Fig. 8.6. Time evolution of the scattering curves of several kinetic mixtures ofh-PEP1-h-PEO20 and d-PEP1-d-PEO20 in several DMF/water mixtures at a fixedpolymer concentration of φ = 1 vol.%. The data shows the effect of interfacialtension changed by varying the DMF/water composition: from (a) XDMF = 0 (γ= 47 mNm−1); (b) XDMF = 0.1 (γ = 27.4 mNm−1); (c) XDMF = 0.3 (γ = 19.7mNm−1) to (d) XDMF = 0.5 ( γ = 14.5 mNm−1)

230 R. Lund

h-PEP-h-PEO/d-PEP-d-PEO polymers. This was prepared by dissolving the poly-mers in a good solvent (toluene) and freeze drying the sample followed by addingDMF/water to form micelles. The kinetic runs were performed by manually mix-ing equal amounts of the fully deuterated and hydrogenated PEP1–PEO20 micellesusing a pipette. In Fig. 8.6a the reservoir scattering (mean arithmetic scatteringintensity of the two polymers separated) is plotted in addition. Here we observefirst that the intensity at low Q is lower for the reservoir than that for the kineticmixtures. This can simply be understood from a cancellation effect of the struc-ture factor once the deuterated and hydrogenated micelles are mixed randomly in a50:50% mixture under zero average contrast conditions. A similar observation andmore detailed justification of this can be found in [60].

Now concentrating on the kinetics, we see that in pure water the intensityobtained just after mixing (<3 min) and after heating for 10 h at 70◦C are identi-cal. If any exchange had occurred, we would expect the intensity to decrease. For acomplete intermixing, the intensity should approach that intensity of the “blend.”In water we thus do not observe any intensity change and thus these results demon-strate that the exchange kinetics is frozen at high interfacial tension. This is alsothe case even if we lower γ to almost the half by adding DMF to give a 10 mole%DMF solvent mixture (see Fig. 8.6b). Here the intensity only decrease very slightlyafter heating the sample a long time indicating that only a tiny fraction of chainscan exchange on this time scale.

However, now increasing the DMF content to 30 mole%, we observe a monotonousdecrease of the intensity with time. Hence at this composition the kinetics occurs ona time scale optimal for this TR-SANS. Increasing the DMF content even furtherto 50%, the kinetics occurs on a time scale of some minutes which is too fast for theconventional mixing technique. This can be improved by using stopped flow mixing,which for a current typical set-up at ILL [61] can bring the dead time down to about0.5 s or less – instead of some minutes (0.5–2 min) with conventional mixing. In thefollowing we will concentrate on systems with 25 mole% DMF where the kineticscan be most easily followed with TR-SANS and conventional mixing.

In Fig. 8.7 a), the corresponding relaxation functions, R(t), from the TR-SANSexperiment of PEP1-PEO20 in water/DMF with XDMF = 0.25 is plotted as afunction of time for various temperatures.

From the data we observe the general trend that the kinetics is fast in the begin-ning and slows down considerably at long times and is not fully relaxed even at timesup to 12 h. The kinetics is significantly faster at elevated temperatures, however aslow-down at long times is still prominent where the decay almost freezes. Recall-ing the theory of Halperin and Alexander outlined in Sect. 8.2.3 we expect a singleexponential decay (8.24) for this slow kinetics where the diffusion of single chains inthe solvent obviously is very fast compared to this time scale. However, comparingwith the experimental data, we find no agreement (full line in Fig. 8.7a) Only in thevery beginning the single exponential decay coincides with the experimental data.

Now considering the results depicted in Fig. 8.7b, we observe no significant con-centration dependence on the kinetics in the range 2–0.25%. Some small deviationsmight be seen for the lowest concentration, but here some uncertainty in the datamust be tolerated due to low signal-to noise ratio. As expected we might thus con-clude that the kinetics seem to be “reaction limited” i.e., controlled by the expulsiontime and that fusion/fission mechanisms do not seem to play a significant role as inthe opposite case we would expect an increase in rate with concentration. It should

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 231

0 100 200 300 400 500 600 700

0.4

0.6

0.8

1

0 200 400 600 800 10000.5

0.6

0.7

0.8

0.9

1

R(t

)

time [min]

a

0.25 %0.5 %1.0 %2.0 %

R(t

)

time [min]

b

Fig. 8.7. Relaxation kinetics of the PEP1–PEO20/DMF/water system at 25 mole%DMF/water solvent mixture. (a) Temperature dependence. Various temperaturesat a fixed total volume fraction of 1%. From top to bottom T = 47◦C, 55◦C,60◦C, and 65◦C. The solid line displays an example of a single exponential pre-dicted by Halperin and Alexander. Dotted lines display simulated curves taking intoaccount the polydispersity of the insoluble PEP-block (c.f. (8.28) and (8.29)) (b)Concentration dependence. Relaxation kinetics of the PEP1-PEO20/DMF/water,XDMF = 0.25, system at different polymer concentrations at 50◦C. All data arepresented in a semi-logarithmic representation [9,10]

be recalled that although in this kind of experiment we only observe a net reduc-tion of contrast and thus parallel relaxation channels/modes cannot be separated,we do observe the structure (scattering curves) which do not seem to evolve withtime. If fusion and fission would play an important role we might also expect tosee a different evolution of the scattering curves- and not just a simple decrease inintensity.

In order to validate the expression suggested for the expulsion kinetics byHalperin and Alexander, we must consider polydispersity effects as the correspond-ing rate depends on the exponential of an exponential of the molecular weight (see8.12–8.24).

Thus even though both polymers considered here have essentially the samemolecular weights and are made by living anionic polymerization (for details con-cerning synthesis see [24]) and thus near mono disperse, any finite molecular weightspread must be taken into account. This can be done by employing the Poisson

232 R. Lund

distribution which is well-known to describe the molecular weight distribution,P (NB), of polymers produced by living anionic polymerization techniques [9,10]:

P (NB) =(〈NB〉 − 1)NB−1 exp(−(〈NB〉 − 1))

Γ(NB), (8.28)

where Γ(x) is the gamma function replacing the normal faculty function for con-tinuous variables. The resulting relaxation function can thus be calculated in thecontinuous limit by:

R(t) =

∫ ∞

1

P (NB) exp(−k t)dNB (8.29)

where k ≡ k− is given by (8.12–8.24).The results from this analysis are already shown in Fig. 8.7 and is represented

by the dotted lines. Here only τ0 has been fitted from the initial decay while allother parameters characterizing the distribution (〈NB〉 = 16) and the parametersfor the activation energy (γ = 21.8 mNm−1 for XDMF = 0.25 and lB ≈ 5.1 A) havebeen independently determined by the polymer characterization as well as pendantdrop interfacial tension and density measurements [24]. As seen the agreement withthe data is slightly improved but is still very poor – the experimental data exhibit amuch broader decay than what would arise from polydispersity effects alone. We canalso exclude any isotope effects as both the interfacial tension and micellar structurewere found to be essentially independent of H/D content [10,24].

As a second step we can consider any broadening effect by assuming a stan-dard Gaussian distribution of activation energies, Ea, which would give a lognormaldistribution of expulsion rate constants, k, and calculate the relaxation kineticsaccording to:

g(ln k) =R T√2πσEa

exp

(− (−R T ln(k τ ′

0)− 〈Ea〉)22σ2

Ea

)(8.30)

where σEa is the width of the distribution in units of energy and 〈Ea〉 the meanactivation energy. Here we have re-expressed the rate constants using a generalArrhenius type expression: k = 1

τ ′0

exp(−Ea/R T ) where τ ′0 is the characteristic

“attempt time” to escape the potential well. As demonstrated in [10], this approachyields a good description of the data- even a simultaneous fit for all temperaturescan be done, but the there is no physical meaning of the resulting fit parameters.First of all, the mean activation energy is found to be 165 ± 1.7 kJmole−1 andthe width σEa = 17.6 ± 0.2 kJmole−1 which are very large numbers. Comparingwith the calculated activation energy from the theory of Halperin and Alexander,Ea = 21.6 kJmole−1, this is almost one order of magnitude higher. For comparisonthe dissociation energy for a carbon–carbon (C–C) covalent bond is approximately347 kJmole−1. Secondly, the characteristic attempt time, τ ′

0, of the order of 10−23

s, is unphysically low and completely out of any soft-matter frame. In contrastfrom the polymer dynamics we would expect the longest Rouse time for PEP tobe of the order of 10−6 s in this temperature range [62]. Thus we might concludethat such a scenario, with parallel relaxation channels as represented by a Gaussiandistribution of activation energies, does not seem to support the physics behind thebroad relaxation kinetics. This is further supported by the fact that on a logarithmictime scale the relaxation functions show straight lines, i.e., the kinetics is describedby a logarithmic law: R(t) ∼ − ln t [9, 10]. An example is shown in Fig. 8.8.

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 233

1 10 100

0.2

0.4

0.6

0.8

1.0

time [min]

R(t

)

Fig. 8.8. Logarithmic relaxation kinetics. Data of Fig. 8.7 on a logarithmic timescale. Solid lines display linear fits. PEP1-PEO20 in XDMF = 0.25 DMF/watersolvent mixture

As a logarithmic relaxation implies that there is no typical time scale – just ahierarchical type time distribution [63], it is tempting to interpret this finding asa signature of cooperative behavior inside the micellar core [9, 10]. This would alsoallow an explanation for the very large value for the strong temperature dependencegiving a large apparent activation energy using the Gaussian distribution ansatzdiscussed above. Such a scenario, discussing the possibility for a strong couplingbetween chain conformation and chain expulsion probabilities, has been presentedin [9] and [10].

The idea here is firstly that only chains having a small compact conformation hasa reasonable probability to escape the micelle; and secondly the transition betweenthese states is slow because it involves a cooperative movement of chains inside thecore that is strongly confined due to a large interfacial tension. Consequently chainshaving a compact conformation near the interface can exchange many times beforeany more deeply buried chains are released. This gives rise to the very fast drop in theintensity followed by the increasingly long release time observed in Fig. 8.7. Hencethis implies a strong influence of the nature of the core (compactness, topologicalinteractions etc) on the chain exchange kinetics. In the next section we will presentresults of other block copolymer systems where we effectively vary the interactionsinside the core.

8.4.4 Diblock Copolymer Micelles in Organic Solvents

A very different block copolymer system is poly(styrene)-poly(butadiene)(PS10–PB10) which forms micelles in selective solvents such as n-alkanes which aremoederately good for PB and poor for PS. An example a detailed scatteringanalysis of PS–PB micelles in heptane using contrast variation and careful datamodelling, was already shown in Sect. 8.3.1. Contrary to PEP–PEO and other typi-cal amphiphilic systems characterized by large interfacial tensions, PS–PB in alkanesforms micelles which are not strongly segregated and a considerable amount of

234 R. Lund

solvent penetrates the core [11, 35]. In this system the interaction parameter orinterfacial tension remains almost constant and small while the amount of solventswelling the core depends on the temperature and the length of the n-alkanes prob-ably due to solvent entropy effects [35]. This makes this system a very interestingmodel system to study the effect of varying the polymer density and hence; theconfining properties of the core.

The results show that exchange kinetics is critically dependent on the alkanesolvent; in heptane and decane the kinetics is too fast to be resolved with manualmixing; while in tetradecane and hexadecane the kinetics occurs on a suitable timescale of minutes-hours. This correlates strongly with the degree of swelling, i.e.,the amount of solvent, in the core which decreases from about 50% in heptane toaround 30% in hexadecane [11, 35] and thus means that the kinetics slows downwith an increasingly dense core and packing of chains. Changing the temperaturehas qualitatively the same effect where the core is gradually dissolved at highertemperatures. This is reflected in the kinetics of the PS–PB/hexadecane micelles asshown in Fig. 8.9.

At the two lowest temperatures the data display a very broad heterogeneousdecay that deviates strongly from a single exponential decay. As discussed above,it is essential to include polydispersity in the analysis of the data and the resultsfollowing the same approach as discussed in the previous section ((8.28)–(8.29)) isincluded as a solid line in the figure. Please note that since the interfacial tension ofthis system is rather low (γ ≈ 1–3 mNm−1) [11, 35], the corresponding activation

0 100 200 300 400 500 600

0.01

0.1

1

1 10 100 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time [min]

R(t

)

40 °C

30 °C

20 °C

R(t

)

time [min]

Fig. 8.9. Equilibrium chain exchange kinetics of the PS10–PB10 diblock micelles inhexadecane at various temperatures. The data are displayed in a semi-logarithmicrepresentation. Solid lines display the predictions using the Halperin and Alexandertheory including the experimental polydispersity. Inset plot: the same data on alogarithmic time scale. The lines represent the best least square linear fits [11]

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 235

energy is low which gives rise to a weak temperature dependence and close to a singleexponential decay law in all cases. As seen, this approach works very bad at the twohighest temperatures (20 and 30◦C), but at the highest temperature, T = 40◦ C) wesee that the theoretical prediction indeed seems to approach the experimental data.This immediately suggests that correlations in the core gradually appear to vanishat higher temperatures when the core is sufficiently swollen and the chain densitylow. This is additionally corroborated by the observation elucidated in the insetplot of Fig. 8.9 where we see a clear logarithmical decay for the highest temperaturewhile for the lowest a much poorer agreement to this type of law is observed. Thisobservation is in excellent agreement with the hypothesis outlined above; at highchain packing and density of the core the exchange exchange kinetics is slow dueto a cooperative release mechanism which involves slow conformational changes.However, decreasing the density, the correlations between the chains disappear andwe see an approach to single chain behavior and thus a Halperin-Alexander likerelease rate.

8.4.5 Triblock Copolymer Micelles in Organic Solvents

In order to further investigate the role of topological interactions on the chainexchange kinetics, the TR-SANS technique has also been applied to A–B–A tri-block copolymer micelles. In a selective solvent for the A block, we expect formationof micelles but in this case but with knots induced by the connectivity of the thetwo terminal A blocks. Thus by comparing the kinetics of a A–B diblock with aA–B–A triblock type block copolymer micelles, we can directly study the effect oftopological interactions and their effect on release rates. This can be experimentallyrealized using the PS10–PB10 diblock copolymer system introduced in the previoussection combined with a triblock copolymer which is a “double diblock” PB10–PS20–PB10. Since the composition and corona block is equal for both polymers,all structural properties including the aggregation behavior and micellar dimensionsare similar [11]. However, a single but important difference between the resultingmicelles will be the connectivity between the two PB blocks present only in the tri-block micelle which may lead to additional topological interactions of chains. Centralresults displaying the comparison of the PS10–PB10 and PB10–PS20–PB10 blockcopolymer micelles in dodecane, where the cores are strongly swollen, are shown inFig. 8.10.

As anticipated, the kinetics is generally much faster in the diblock micelles thanthe corresponding constituted of the triblock copolymers.This can be attributedto primary two different factors; first (1) the increased activation energy becauseof the double length of the PS block in the triblock; and/or secondly, (2) chainconnectivity between the two PB blocks present in the triblock but not in the diblock.However, while the former effect certainly will be present, this will only lead tolarger H–A activation energy which will thus give a small shift in the time scalebut no additional broadening of the relaxation spectra. On the contrary we see inFig. 8.10 a logarithmical like relaxation behavior for the triblock micelles while thediblock copolymer micelle exhibit a more typical relaxation behavior closer to singleexponential decay.

This leads us back to the second point, (2), which indeed will lead to less trivialtopological effects. In order to liberate one chain from the micelle, either one PBchain has to pass through the PS rich core leading to an extra energetical penalty and

236 R. Lund

0 100 200 300 400 500 600 700 800

0.1

0.2

0.3

0.4

100 101 102 103

0.0

0.1

0.2

0.3

0.4

0.5

0.6

time [min]

R(t

)

Fig. 8.10. Chain exchange kinetics of triblock copolymer micelles. Semi-logarithmicrepresentation of the relaxation curves of PS10–PB10 diblock (triangles) and PB10–PS20–PS10 triblock (stars) in dodecane at 20◦C. The dotted lines represent fits usingthe Halperin and Alexander theory taking into account the Poisson distribution forpolydispersity. Inset plots: same data on a logarithmical time scale including a linearfit [11]

thus a slow-down of the time scale; and/or the test chain as well as the neighboringchains have to rearrange in such a way that the chain can pass out to the core-coronainterface and later be expelled. The latter effect is more prominent for the micellesin dodecane in Fig. 8.10a). Here the triblock copolymer exhibits a logarithmic likerelaxation indicating some cooperative behavior while the diblock system exhibits aclose to the single exponential decay signaturing single chain behavior. This clearlydemonstrates the importance of topological interaction: while for the PS10–PB10system the micellar core is sufficiently swollen such that the internal correlations havenearly vanished, there are still topological “knots” present for the PB10–PS20–PB10system. Thus, in conclusion it seems that the logarithmic relaxation is caused bycooperative effects (involving several chains).

8.5 Concluding Remarks and Outlook

Hopefully, in this work we have demonstrated the versatility and usefulness of smallangle neutron scattering to study very sensitive and slow kinetic processes in micel-lar systems. Instead of relying on neither physical or chemical perturbations wedemonstrate that simple H/D substitution can be used to label and obtain singlechain dynamical behavior in complex self-assembled structures. We have further

8 Small Angle Neutron Scattering as a Tool to Study Kinetics of BCM 237

demonstrated that logarithmical relaxation kinetics seem to be an inherent phe-nomenon to well-segregated block copolymer micelles but appears to transform toa single exponential decays upon a sufficient reduction in micellar core density.We interpret this as a sign of cooperative behavior in the expulsion process, a phe-nomenon that would be beyond the theory like the one from Halperin and Alexandertheory which seems to be more appropriate for swollen micelles without strong topo-logical interactions in the core. Hence the results would demand a a revision of thecurrent theories and a need for a careful investigation, possibly by means of com-puter simulations, elucidating global polymer dynamics and diffusion processes insuch strongly confined systems. From an experimental point of view, much workalso remain in exploring the equilibrium kinetics of micelles going from polymer like(block copolymers) to more ordinary low-molecular weight surfactant micelles. Thisway we expect a transition from “constrained” exchange kinetics in the case of poly-meric dense cores to more classical Halperin and Alexander/Aniansson and Wall likesingle process mechanism for surfactant micelles. In this way also the expected grad-ual onset of importance of fusion/fission upon reducing the inter-micellar osmoticrepulsion can be explored. However, since reducing molecular weight also meansshorter characteristic times, a greater demand on fast data acquisition and mixing.In order to bring this down to the necessary millisecond range, we thus need faststopped flow mixers and a powerful neutron flux. Such equipment already existsat for example at ILL, but in order to further reduce the deadtime as well as toreduce the number of necessary repetitions (consuming excessive amounts of expen-sive deuterated material) new neutrons sources such as the proposed ESS spallationsource in Europe or the new SNS at the Oak Ridge National Laboratory in USAwill bring new dimensions and possibilities for time resolved neutron scattering ingeneral.

In any case, as has been shown here, that apart from advances in sample environ-ments and beamlines, time resolved small angle neutron scattering in combinationwith H/D substitution tricks, is a very powerful and valuable tool which do giveunique information about single particle diffusional properties which cannot readilybe obtained with any other technique. It is encouraging to see that very recently,the same H/D substitution trick was employed to measure lipid kinetics in biologicalvesicle systems using TR-SANS [64] where for the first time two distinct activationenergies characterizing a trans-bilayer and a intra-bilayer kinetic mode were mea-sured in equilibrium. In another recent study the formation kinetics of micelles wasstudies by using time resolved synchrotron SAXS which is capable to reach timesscales of only some few ms [65]. Here it was shown that the same unimer exchangemechanism limits the growth kinetics of micelles. This is an example of the com-plimentary roles of SANS and SAXS where the former can measure equilibriumkinetics whereas very fast non-equilibrium kinetics can be resolved by SAXS” [65].Undoubtedly, many other examples will be demonstrated in the future.

Acknowledgments

The part of the author’s own work presented here was mainly done in the group ofProf. Richter at the Institute of Solid State Research, Forschungszentrum Julich. Theauthor is indebted to the crucial contribution and support to this work especiallyby Dr. Lutz Willner and Prof. Dieter Richter. Prof. Juan Colmenero is gratefullyacknowledged for support. Dr. Peter Lindner at ILL is gratefully appreciated for

238 R. Lund

kind assistance and support in connection with experiments at the D11 instru-ment where much of the presented data were obtained. In addition we acknowledgehelpful discussions and help with experiments by the following colleagues at FZ-J:Dr. Michael Monkenbusch, Dr. Jorg Stellbrink, Dr. Aurel Radulescu and Dr. WimPyckhout-Hintzen. Finally but not least, we wish to express the gratitude for thekind invitation to this contribution by the organizers, in particular Prof. Gotz Eckoldat the university of Gottingen.

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