Dynamics and rheology of wormlike micelles emerging from particulate computer simulations

Dynamics and rheology of wormlike micelles emerging from

particulate computer simulations

Citation for published version (APA):

Padding, J. T., Boek, E. S., & Briels, W. J. (2008). Dynamics and rheology of wormlike micelles emerging from particulate computer simulations. Journal of Chemical Physics, 129(7), 074903-1/12. [074903].

https://doi.org/10.1063/1.2970934

DOI:

10.1063/1.2970934

Document status and date: Published: 01/01/2008

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Dynamics and rheology of wormlike micelles emerging from

particulate computer simulations

J. T. Padding, E. S. Boek, and W. J. Briels

Citation: J. Chem. Phys. 129, 074903 (2008); doi: 10.1063/1.2970934

View online: http://dx.doi.org/10.1063/1.2970934

View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v129/i7

Published by the American Institute of Physics.

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Dynamics and rheology of wormlike micelles emerging from particulate

computer simulations

J. T. Padding,1,2,a兲E. S. Boek,2and W. J. Briels1,2

1

Computational Biophysics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands 2

Schlumberger Cambridge Research, High Cross, Madingley Road, Cambridge CB3 0EL, United Kingdom

共Received 19 June 2008; accepted 23 July 2008; published online 18 August 2008兲

We perform coarse-grained computer simulations of solutions of semidilute wormlike micelles and study their dynamic and rheological properties, both in equilibrium and under shear flow. The simulation model is tailored to the study of relatively large time and length scales共micrometers and several milliseconds兲, while it still retains the specific mechanical properties of the individual wormlike micelles. The majority of the mechanical properties 共persistence length, diameter, and elastic modulus of a single worm兲 is determined from more detailed atomistic molecular dynamics simulations, providing the link with the chemistry of the surfactants. The method is applied to the case of a solution containing 8%共by weight兲 erucyl bis共hydroxymethyl兲methylammonium chloride 共EHAC兲. Different scission energies ranging from 15.5kbT to 19.1kBT are studied, leading to both

unentangled and entangled wormlike micelles. We find a decrease in the average contour length and an increase in the average breaking rate with increasing shear rate. In equilibrium, the decay of the shear relaxation modulus of the unentangled samples agrees with predictions based on a theory of breakable Rouse chains. Under shear flow, transient over- and undershoots are measured in the stress tensor components. At high shear rates we observe a steady-state shear stress proportional to

␥˙1/3_{, where␥˙ is the shear rate. This is confirmed by our high shear rate experiments of real EHAC}

in a parallel-plate geometry. © 2008 American Institute of Physics.关DOI:10.1063/1.2970934兴

I. INTRODUCTION

In this paper we will focus on the dynamics and rheol-ogy of amphiphilic molecules forming elongated structures called wormlike micelles. Rheological experiments produce a wealth of data and theories that can explain some of the experimental findings.1–10 Most macroscopic experiments, however, do not provide us with a detailed fundamental un-derstanding of the underlying processes that lead to the emergent rheology. Often their interpretation is based on theories which contain uncontrolled approximations, and their range of applicability is limited to certain flow condi-tions. Simulations may contribute to our understanding of the peculiar rheology of wormlike micelles. With simulations we have the possibility to “zoom in” on the detailed processes, study their influence on the rheology, and test the approxi-mations made in theories. This will contribute to a rational design of new viscoelastic materials based on wormlike mi-celles.

Particle based simulations of wormlike micelles may be performed on many different lengths and time scales, from the atomistic to the mesoscopic. An overview is given in Fig.

1. To realistically simulate the rheology, one would ideally use an atomistic model11–13 or a model in which each am-phiphilic molecule is individually represented by a properly coarse-grained version of it.14–16 However, using such mod-els and current-day computing power, it is impossible to

de-termine the macroscopic rheology of a solution of entangled wormlike micelles. Prohibitively long computation times are needed because of the 共1兲 very large length scales and 共2兲 very long time scales involved.

Focusing first on the length scales, it must be realized that the stress tensor is a collective property of a sufficiently large portion of fluid. In a molecular dynamics simulation, the instantaneous stress components are generally given by

a兲_{Author to whom correspondence should be addressed. Electronic mail:} j.t.padding@utwente.nl.

FIG. 1. Particle based simulations may be performed on many different lengths and time scales. With atomistic force fields 共lower left兲 we can capture the influence of the specific surfactant chemistry on the properties of a piece of wormlike micelle. By coarse graining the surfactant molecules these calculations are greatly accelerated. More macroscopic properties, such as the chain length distribution and rheology, can only be calculated with higher-level models, such as theFENE-CandMESOWORMmodels共upper right兲.

THE JOURNAL OF CHEMICAL PHYSICS 129, 074903共2008兲

␴␣␤= − 1 V

冉

兺

_i=1 N mivi␣vj␤+

兺

i=1 N−1

兺

j=i+1 N rij␣Fij␤

冊

. 共1兲

Here miis the mass andvi␣the␣component of the velocity

of particle i, rij␣ the ␣ component of the vector from the

position of particle j to particle i, Fij␤ the ␤ component of

the force exerted by particle j on particle i 共here we have assumed a pairwise interacting system兲, and V the volume of the simulation box. In our case the volume V is sufficiently large if the simulation box contains at least O共100兲 wormlike micelles, providing a minimal representation of the very polydisperse共bulk兲 length distribution c共L兲. Because the in-teresting rheological behavior occurs in entangled wormlike micellar solutions, the average length of a wormlike micelle must by definition exceed the entanglement length. The en-tanglement length depends on many factors, including the concentration of the solution and the bending stiffness of the wormlike micelle. It is typically O共10–100兲 persistence lengths.17,18This means that the simulation box must contain at least O共104_{兲 persistence lengths. One persistence length of}

wormlike micelle contains O共103_{兲 surfactant molecules,}12

leading to O共107_{兲 surfactant molecules in total. In principle}

the solvent molecules need to be included as well, leading again to a more than tenfold increase in the number of mol-ecules. Even when the surfactant molecules are coarse grained to a few beads, and the solvent effect can be treated in an implicit manner, a simulation of this magnitude will take an appreciable amount of computational time.

Next let us focus on the time scales. Typical rheological experiments are performed at oscillation frequencies or shear rates between 10−2 and 102 s−1 共Ref.19兲 due to mechanical

limitations, with corresponding inverse time scales 共shorter time scales may be accessed through specialized mechanical setups and, notably, by diffusing wave spectroscopy, see, e.g., Ref. 20兲. Most entangled wormlike micelles have a

longest characteristic relaxation time␶ranging from 10−4 to 103 s 共although unentangled wormlike micelles can have smaller relaxation times兲. In order to simulate the rheological properties of entangled wormlike micelles, one must be able to reach time scales of the order of at least milliseconds. For example, for models with coarse-grained amphiphiles a typi-cal time step is 10−14 _{s, leading to a total of 10}11_{time steps.}

Usually the maximum attainable number of time steps in molecular dynamics is of the order of 108_{. So, even when the}

simulation is sped up by coarse graining surfactant mol-ecules to a few beads, this presents us with an unsurmount-able amount of computational work.

The above discussion shows clearly that the only way forward is to coarse grain even further, to the point where each unit represents several surfactant molecules. This will, of course, be at the cost of losing detailed information about the surfactant molecules. One important development in this direction is the generic FENE-Cmodel. InFENE-Cthe worm-like micelles are represented by flexible chains of relatively hard spheres. Chains can grow by the addition of monomers at the chain ends or by recombination with other chain ends. Conversely, chains can break if any of the bonds are stretched because of thermal fluctuations or tension. The

FENE-C model was studied extensively by Kröger et al.21–23

and Padding and Boek.24,25In the original model21,22solvent beads were included to account for solvent effects, but the solvent effect may be mimicked through Brownian dynamics as well.23The resulting worm length distribution is found to agree well with the theoretical 共mean-field兲 prediction c共L兲 ⬀exp共−L/L¯兲, and shear thinning of the viscosity is observed, although not as strong as in the experiment.22,25

In the FENE-C model, being a generic model, no refer-ence is made to any specific real wormlike micellar system. This may be an advantage, since this allows the simulator of the freedom to scale the simulation results onto experimental results. However, in our opinion great care must be taken if realistic and quantitative results for the dynamics and rheol-ogy of wormlike micelles are required. First, there are al-ways multiple relevant length scales. Scaling one of them onto experimental values does not guarantee that the other length scales will be described correctly as well. For ex-ample, the persistence length of a typical wormlike micelle is several times its diameter,26 whereas in the flexible FENE-C

model these are more or less the same. Second, the kinetics of breakup and fusion of chain ends may not be as fast as predicted by theFENE-Cmodel. In the originalFENE-Cmodel, recombination is relatively easy because chain ends can fuse instantaneously if their separation is smaller than some criti-cal distance. In reality, before two chain ends can fuse, there may be specific demands on the conformations of the surfac-tants in the end caps, giving rise to a considerable free en-ergy barrier. Chain recombination, like scission, may there-fore be an activated process.24,25

These disadvantages may be alleviated by introducing a 共strong兲 bending potential between the spherical beads and an additional radial interaction mimicking the activation bar-rier 共as in the FENE-CB model23兲 or some Monte Carlo equivalent.27Unfortunately, this remains very CPU-intensive because many beads will be required to represent one en-tanglement length of a realistic wormlike micelle, while the integration time step will still be limited by the relatively hard interactions at the scale of the beads.

In this paper we will use and study a coarse-grained model called MESOWORM 共mesoscale wormlike micelles兲.

Preliminary descriptions of this model have been published elsewhere.19,26,28This model is coarse grained to the level of a persistence length, but is still able to capture共most of兲 the specific mechanical properties of a real wormlike micelle.

II. THE MODEL

The philosophy behind theMESOWORM model26 is that the material properties of individual wormlike micelles can in principle be measured from more detailed simulations or targeted experiments. Relevant material properties include the worm diameter, persistence length, elastic modulus, and scission and activation free energies for fusion and breaking. Allowing the model to have these parameters as input, the rheology can be predicted from realistic input, with as few assumptions as possible. This enables us to create a hierarchy of simulation models, ultimately linking the chemical details of the surfactants and other components of the solution to its macroscopic rheology.

Of course, some assumptions still have to be made. First, it is assumed that the stress is dominated by the network of wormlike micelles and that hydrodynamic interactions are relatively unimportant. This allows for an implicit treatment of solvent effects by means of Brownian dynamics 共of rigid rods兲. The model therefore does not apply to dilute solutions, where the micelles are relatively short and do not overlap. Second, it is assumed that excluded volume interactions are relatively unimportant. This allows us to treat the interac-tions between wormlike micelles as chains of infinitely thin lines共but note that the friction with the solvent will be based on the true aspect ratio, vide infra兲. Treating the wormlike micelles as thin lines is allowed if the ratio of persistence length to diameter is much larger than 1 and if the concen-tration is low enough to have an isotropic equilibrium distri-bution. The model should therefore not be used for very high concentrations either, where excluded volume effects may lead to spontaneous nematic ordering of the wormlike mi-celles. There is a large range of concentrations between di-lute and concentrated where the model does apply. Indeed, the semidilute regime is of most practical interest, since the viscosity can be greatly enhanced by adding a relatively low amount of surfactant.

In the MESOWORM model, each persistence length of

wormlike micelle is represented by one unit, see Fig.2. This degree of coarse graining is as large as possible to allow for a large integration time step and few particles, while it is still small enough to allow an accurate description of the overall conformation of the wormlike micelle. The midpoints of bonded units interact with each other through the following radial potential: ␸bond共r兲 =1 2共Ea− Esc兲 − 1 2共Ea+ Esc兲cos

冉

r − lp w

冊

for r苸 关lp−␲w,lp+␲w兴, 共2兲 w =

冑

共Ea+ Esc兲lp 2KL , 共3兲

where Escis the scission energy, Ea the 共additional兲

activa-tion energy associated with the fusion-recombinaactiva-tion pro-cess, lpthe persistence length, and KLthe elastic modulus of

the wormlike micelle. The width parameter w of the potential well has been chosen such that the second derivative in the vicinity of the minimum of the well equals KL/lp, as required

by the elastic response to compression of a wormlike micelle of length lp: ␸bond共r兲⬇共1/2兲共KL/lp兲共r−lp兲2− Esc for small

extensions. The correct stiffness of the wormlike micelle is implemented by means of an angular potential between each bonded triplet共see Fig.2兲,

␸␪共␪兲 = kBT␪tan共␪/2兲. 共4兲

In the limit for small angles this reduces to the often-used potential kBT␪2/2, while the potential diverges for ␪→␲,

which prevents problems with the entanglement algorithm 共vide infra兲.

As soon as a bond is stretched or compressed beyond its limits共lp+␲w and lp−␲w, respectively兲 the bond is broken.

Equation共2兲shows that when a bond is broken, its potential energy is not zero but equal to Ea, the activation barrier

associated with the fusion-breaking process. Conversely, be-fore two chain ends are allowed to fuse, they must overcome an energy barrier of height Ea. There are many ways in

which such an energy barrier may be implemented. In the current model it is implemented as a smooth repulsive bar-rier, ␸rep=

冦

1 2Ea

冋

1 + cos

冉

x −␲w w

⬘

冊

册

for x苸 关␲w,␲共w + w

⬘

兲兴 0 for x⬎␲共w + w

⬘

兲,

冧

共5兲 w

⬘

=

冑

Ealp 2KL . 共6兲

Here x is the distance between the endcaps of the wormlike micelle. The position of the endcap of a wormlike micelle is approximated as the extension of the last bond vector, over a length lp/2, i.e., if r is the position of the midpoint of the last

unit and uˆ is the orientation of the last bond vector, then the endcap position is calculated as rend= r + lp/2uˆ. When the

distance x between two endcaps is smaller than␲w, and the distance r between the midpoints is within the range 关lp

−␲w , lp+␲w兴, two chain ends are allowed to fuse. Because

the width w is usually much smaller than the persistence length lp, this means that two wormlike micelles will fuse

only when their extremal bonds are more or less aligned, leading to a smooth transition from being unbonded at en-ergy 0, via the activation barrier at enen-ergy Ea, to being

bonded at energy −Esc, as schematically depicted in Fig. 3.

FIG. 2.共Color online兲 共a兲 The wormlike micelle is represented by a string of thin rods, each of one persistence length.共b兲 The chain can break and fuse. Details of these processes are given in the main text.

0 10 20 30 40 r [nm] -10 -5 0 5 Φb /kB T E_a E_sc l_p scission scission

FIG. 3. 共Color online兲 Interaction between two units in the MESOWORM model. The solid line is the potential well of depth E_sc, which is a function of the distance r between the midpoints of bonded units. The dashed line is the repulsive barrier of height Ea, which is a function of the distance x

between the endcaps of wormlike micellar chains共see main text兲. 074903-3 Wormlike micelles J. Chem. Phys. 129, 074903共2008兲

Note that also in the case of the FENE-CB model,23 chains

tend to recombine in the aligned state.

The fusion energy barrier is purposely not a function of the distance between the midpoints of the units at the ex-tremes of the wormlike micelles because this would lead to far too long wormlike micelles when using realistic values of the scission energy Esc. The average wormlength is actually

greatly overpredicted in the case of both the FENE-C model and mean-field theory. In the latter,1 a wormlike micelle is treated as a random walk on a lattice, with an energy penalty Esc for each pair of chain ends. The random walk lattice

model is usually justified by identifying each occupied lattice space with one persistence length of wormlike micelle. By variationally minimizing the free energy, under the constraint of fixed volume fraction␾, the theory then predicts an aver-age length of L¯ =c␾1/2exp关Esc/共2kBT兲兴 persistence lengths,

where the prefactor c is of the order of 1. Although the scal-ing with Esc/T has been confirmed, the prefactor is

unrealis-tic. For example, experimental estimates of the scission

en-ergy of erucyl bis共hydroxymethyl兲methylammonium

chloride共EHAC兲 wormlike micelles vary between 25kbT and

50kBT.29,30 Mean-field theory then predicts an average

wormlength of O共105_{– 10}10_{兲 persistence lengths,}

correspond-ing to a contour length of O共10−3_{– 10}2_{兲 m. These estimates}

would make wormlike micelles truly giant! It is important to realize that wormlike micelles are relatively thin. By letting the fusion process take place within a small volume at the endcap positions, instead of a large volume associated with a sphere of diameter O共lp兲 around the midpoint of the extremal

units, the average length of the wormlike micelles is brought down to realistic proportions. In Appendix A we will de-scribe the effect of coarse graining the possible breaking points to just one per persistence length. This yields an ad-ditional correction of order 5₂kBT ln共lp/D兲 to the scission

en-ergy.

In the model as described up to this point, there is no excluded volume between the worm segments. In other words, the wormlike micelles would be able to pass through each other like ghost chains, whereas the dominant stress contribution is expected to arise from entanglements between the wormlike micelles. To remedy this, the TWENTANGLE-MENT technique is applied. In this technique, originally de-signed for polymer melt simulations,17,18,31,32 an imminent bond crossing is detected and prevented by the introduction of a new coordinate共an entanglement point兲 at the crossing point. From that time onward, until the entanglement is re-moved again, the interaction between bonded units is a func-tion of the path length measured via the entanglement points, instead of the usual distance between the units, see Fig. 4. Entanglements are allowed to jump over the 共central兲

posi-tion of a unit, and so can slide along the backbone of a wormlike micelle.

We use共overdamped兲 Brownian dynamics to update the positions of the particles. The solvent friction on each par-ticle is anisotropic to reflect its rodlike shape of dimensions lp⫻D. More precisely, we track the centers of mass ri of

individual rodlike particles. The orientation uˆi of rod i

fol-lows from an average of the connectors to the previous and next particle, i.e.,

uˆi= ri− ri−1 兩ri− ri−1兩 + ri+1− ri 兩ri+1− ri兩

冏

ri− ri−1 兩ri− ri−1兩 + ri+1− ri 兩ri+1− ri兩

冏

. 共7兲

The position of rod i is updated according to

ri共t + ⌬t兲 = ri共t兲 + ⌶i−1· Fi共t兲⌬t + vflow共yi兲⌬teˆx+⌬ri R

, 共8兲 where ⌬t is the integration step, Fi the total conservative

force on i, vflow共y兲 the background flow velocity 共in the x

direction兲 at height y, and ⌬ri R

the random displacement of i, which is linked to the inverse friction 共or mobility兲 tensor ⌶i−1, according to具⌬ri

R_⌬r j R_典=2k

BT⌶i−1␦ij⌬t. We note that the

friction tensor⌶iof rod i depends on its orientation uˆi, with

the friction parallel to the rod given by ␰储= 2␲␩slp/ln共lp/D兲

and the friction perpendicular to the rod twice as large. Here

␩s is the solvent viscosity. A derivation can be found in Ref. 33.

In order to study shear flow, Lees–Edwards 共“sliding-brick”兲 boundary conditions are implemented.34

The possi-bility of nonaffine flow is included. This is not trivial for Brownian dynamics simulations where a friction with a static background flowvflow共y兲 is assumed. We solved this by

in-troducing a dynamic background flow, with a velocity coupled to the wormlike micelles through an overdamped feedback mechanism. The feedback is accomplished by mea-suring, at each timestep and height y, the average “velocity” 共excluding random displacements兲 具vworm共y兲典 of the

worm-like micellar material. The background flow velocity field v_flow共y兲 reacts to this by accelerating or decelerating accord-ing to dv_flow共y兲/dt=1/␶f共具v_worm共y兲典−v_flow共y兲兲, where ␶f is the flow reaction time, which must be set sufficiently fast not to interfere with intrinsic timescales of the wormlike mi-celles.

The parameters of the model used in this work, chosen to represent EHAC, are given in TableI. We have determined most parameters from the results of more detailed atomistic molecular dynamics simulations.12,26 Only for the scission and activation energies we do not yet have values from ato-mistic simulations. Very recently, Pool and Bolhuis35 sug-gested a way to estimate such energies by transition path sampling. Note that we have decreased the elastic modulus KL by a factor of 10 relative to the experimental value to

allow a time step of 1 ns. This low value of KLdoes not lead

to a significant change in rheological properties because the rheology is dominated not by the elasticity between succes-sive units but rather by the overall conformations of the

FIG. 4. 共Color online兲 Basic function of theTWENTANGLEMENTalgorithm 共Ref.31兲. 共a兲 A bond crossing is imminent. 共b兲 When two bonds cross, a new

entanglement point is defined at X.共c兲 From then on, the bonded interaction is a function of the path length.

wormlike micelles and the entanglements between them. The large time step of 1 ns is sufficient to reach the time scales of interest共milliseconds or more兲. One of the goals of this work was to investigate the influence of the dimensionless scission energy E_sc/kBT on the behavior of the system. We have

therefore investigated the model at Esc= 5⫻10−20 J 共T

= 333 K兲 and at E_sc= 6⫻10−20 _{J 共T=333 and 300 K兲.}

In-cluding the aforementioned correction of 5₂kBT ln共lp/D兲, this

corresponds to experimental scission energies of 15.5kbT,

17.6kbT, and 19.1kBT, respectively. Experimentally, the

scission energy can be tuned by changing the salt concentra-tion of the soluconcentra-tion, see, e.g., Refs.29,30, and36.

All systems simulated contained 16 384 persistence length segments, corresponding to 10.6⫻106 EHAC am-phiphilic molecules.26These were placed in a periodic box of

dimensions of 1212⫻303⫻303 nm, corresponding to an

experimental wormlike micellar system of 8% 共by weight兲 EHAC. The large system dimensions were chosen purposely to accommodate the radius of gyration of wormlike micelles inside the length of one periodic box, even when they are elongated in the x-direction by shear flow.

III. RESULTS A. Contour length

We first study the contour lengths of the wormlike mi-celles. The contour length distribution in a wormlike micellar solution is very broad—in equilibrium it is exponential—and mainly characterized by its average L. Figure5shows L as a function of shear rate␥˙ for all systems studied. At low shear rates all curves show a constant contour length, the scaling of

which agrees with the theoretical expectation L

⬀exp共Esc/共2kBT兲兲.1 We also observe that beyond a certain

critical shear rate the average contour length gradually de-creases. The critical shear rate decreases with increasing scission energy. This may be expected because wormlike mi-celles with a higher scission energy are longer; therefore they are more easily influenced by the shear flow.

B. Breaking times

Next we measure the average breaking time ␶b of a wormlike chain. In equilibrium, detailed balance requires that the average rate of breaking a bond in a wormlike mi-celle is equal to the average rate of forming a new bond with another wormlike micelle. When a shear flow has just started

up, however, we enter a transient region where these two rates may be very different, causing the average contour length to change. When measuring the average breaking time, we have made sure that steady-state conditions apply. In such a steady state the average breaking time is again equal to the average time for a newly formed chain end to form a bond with another chain end, but its value may be different from the one under equilibrium conditions.

In Fig. 6 we show the average breaking rate per unit length of wormlike micelle as a function of shear rate. We choose to represent the breaking rate per unit length because a longer wormlike micelle, having more bonds to break, will have a higher average breaking rate. Effectively, we are mea-suring the breaking rate constant c1 which appears in the

theory on the rheology of wormlike micelles by Cates et al.37 as␶b= 1/共c1L兲. In the theory of Cates et al., c1is assumed to

be constant. This constant may, of course, differ from one type of wormlike micelle to another. Because an escape over a barrier of height Esc+ Ea is involved, it will certainly

de-crease with increasing scission and activation energies. The theory of Cates et al. applies to the case of linear rheology, where flow rates are infinitesimally small. Indeed, our results for the wormlike micelles with lowest scission energy

共dia-TABLE I. Simulation parameters representing a solution of EHAC worm-like micelles.

Property Symbol Value Persistence length lp 30⫻10−9 m

共Hydrodynamic兲 Diameter D 4.8⫻10−9 _m Elastic modulus KL 2⫻10−10 J/m

Scission energy Esc 共5 or 6兲⫻10−20 J

Fusion activation energy Ea 10−20 J

Solvent viscosity ␩s 10−3 Pa s

Flow reaction time ␶f 10−6 s

Integration time step ⌬t 10−9 _s

101 102 103 104 105 106 107 shear rate [s-1] 10-1 100 101 102 contour lengt hL[ µ m ] T = 300K, E_sc= 19.1 kT, E_a= 2.4 kT T = 333K, E_sc= 17.6 kT, E_a= 2.2 kT T = 333K, E_sc= 15.5 kT, E_a= 2.2 kT

FIG. 5. Average micellar contour length vs shear rate.

101 102 103 104 105 106 107 shear rate [s-1] 109 1010 1011 1012 breaking rate 1/( τ b L) [s -1 m -1 ] T = 300K, E_sc= 19.1 kT, E_a= 2.4 kT T = 333K, E_sc= 17.6 kT, E_a= 2.2 kT T = 333K, E_sc= 15.5 kT, E_a= 2.2 kT

FIG. 6. Average breaking rate vs shear rate.

monds in Fig.6兲 suggest that, at the lowest shear rates

stud-ied, the breaking rate per unit length is constant. At higher shear rates, however, the assumption of constant breaking rate 共per unit length兲 is clearly invalid. We observe an in-crease in the breaking rate over more than two decades. Qualitatively similar but less strong increases in the breaking rate and decreases in the contour length have been observed in theFENE-Cmodel.21–23,25

The observed accelerated breaking under shear is a large effect. From this we learn that one must be careful in apply-ing Monte Carlo–type mechanisms to mimic the activated breaking of bonds27in simulations of wormlike micelles un-der flow. If one breakup probability is used unun-der all condi-tions, the breaking rates will be severely underestimated at high shear rates. This will also lead to an overestimation of the contour lengths at high shear rates.

C. Linear rheology

By analyzing the fluctuations of the microscopic stress tensor in equilibrium simulations, we obtain the zero-shear relaxation modulus as G共t兲=V共kBT兲−1具␴xy共t兲␴xy共0兲典, where ␴xyis given by Eq.共1兲. Other property characteristic of linear rheology can be derived from G共t兲. For example, the storage and loss moduli G

⬘

共␻兲 and G

⬙

共␻兲 are obtained through a Fourier transform, and the zero-shear viscosity is given by the infinite time integral,␩0=兰₀⬁G共t兲dt.33

It is notoriously difficult to measure stress autocorrela-tion funcautocorrela-tions. Indeed, in the system of highest scission en-ergy, the stress correlation turned out to be too noisy to make any definite conclusions. The other two systems gave suffi-ciently converged results. Figure 7 shows the measured re-laxation modulus G共t兲 for these two systems 共squares and diamonds兲.

For unentangled Rouse chains, in the limit where break-ing of an average chain is faster than the longest relaxation time of an equivalent unbreakable chain, the relaxation modulus arises from a combined effect of breaking and the

usual Rouse stress relaxation of a chain.38In Appendix B we give an explicit derivation, including all prefactors. The re-sulting expression is G共t兲 ⬇ ckT N + 1

冑

␲␶l 6texp兵− t/␶其 共t ⬎␶l/N 2_{兲, 共9兲}

where c is the number of Kuhn segments per unit volume,␶l the longest relaxation time of an unbreakable chain of length equal to the average wormlike micelle, and N the number of Kuhn segments in this average chain. In fact, a precise de-termination of the average length of wormlike micelles is not necessary, as only the combination␶l/共N+1兲2_{matters. For a}

Rouse chain ␶l/共N+1兲2_=␨b2_/共3␲2_{kT兲, with b the Kuhn}

length and␨the共average兲 friction on a segment.33The stress relaxation time ␶in Eq.共9兲is given by

␶⬇ 0.42␶l1/3_␶b2/3_{. 共10兲}

We test Eqs.共9兲 and 共10兲 in Fig.7共solid and dashed lines兲.

In generating the theoretical lines we have used the measured

␶b and calculated ␶l/共N+1兲2 _{by setting b = 2l}

p 关implying N

= L/共2lp兲兴 and by setting␨equal to twice the orientationally

averaged rod friction.

In the case of the lowest scission energy共15.5kT, dashed line兲 we find good agreement between the measured shear relaxation and the theoretical prediction. The agreement is particularly good up to 0.1 ms 共excluding the very shortest times where the t−1/2 scaling diverges and the theoretical derivation is invalid兲. As always when measuring the stress correlation, at larger times the statistical noise starts to in-crease. It is clear, however, that this noise is still centered around the theoretical prediction, as the semilogarithmic plot in Fig.8confirms. This latter plot also emphasizes the near-exponential relaxation at larger relaxation times.

For the intermediate scission energy共17.6kT, solid line兲 we also find perfect agreement up to 0.1 ms, but then the measured stress seems to be biased to somewhat higher val-ues. This is associated with a transition toward entangled wormlike micelles, leading to a higher shear stress than can be expected on the basis of unentangled Rouse theory. This

0.0 0.1 0.2 0.3 t [ms] 0 200 400 600 800 shear modulus G (t) [Pa] E_sc= 17.6 kT, E_a= 2.2 kT E_sc= 15.5 kT, E_a= 2.2 kT T = 333K

FIG. 7. The shear relaxation modulus G共t兲, calculated from the stress-stress autocorrelation in an equilibrium simulation. The smooth lines are not fits, but theoretical predictions for a breakable Rouse chain, Eqs.共9兲and共10兲.

0.0 0.2 0.4 0.6 0.8 1.0 t [ms] 101 102 103 shear modulus G( t) [Pa] E_sc= 17.6 kT, E_a= 2.2 kT E_sc= 15.5 kT, E_a= 2.2 kT T = 333K

FIG. 8. Same as Fig. 7 but on a semilog scale to emphasize the near-exponential relaxation at larger correlation times.

transition from Rouse to entangled behavior is confirmed when we study the stress in sheared systems in the next subsection.

D. Nonlinear rheology

When a shearing motion is suddenly applied to an ini-tially quiescent solution of wormlike micelles, one often ob-serves a characteristic overshoot in the induced shear stress before it attains a stationary value. This is also observed in our simulations. Figure9shows the shear stress␴xy, the first normal stress difference N1=␴xx−␴yy, and the second normal

stress difference N2=␴yy−␴zz as a function of time after a

sudden application of shear flow at a rate of ␥˙ = 104 _s−1 _in

the Esc= 19.1kT system. A clear overshoot occurs in the shear

stress. At the given shear rate, the first normal stress differ-ence is larger than the shear stress and also displays a maxi-mum, but occurring at a later time. Interestingly, the second normal stress difference, although much smaller in magni-tude than the first normal stress difference, shows an initial positive overshoot, after which we find a negative under-shoot and finally a relaxation to a negative value. Such nor-mal stress differences give rise to quite peculiar viscoelastic effects, collectively known as Weissenberg effects.39

Similarly to the case of polymer melts and solutions, it is generally believed that the overshoots in the stress compo-nents occur because the wormlike micelles resist the chain stretch induced by the onset of fast flow, an effect which diminishes once the wormlike micelles become orientated toward the flow direction. An important dimensionless pa-rameter determining the magnitude of this effect is the Debo-rah number, defined as De=␥˙␶, where␶is the longest relax-ation time of the chain in equilibrium. For 共unbreakable兲 polymers, the Doi–Edwards tube model predicts that for De⬎1 a maximum overshoot in the shear stress occurs at a total strain␥=␥˙ t value of 2.33It is well known that at very high shear rates, starting roughly at De⬇102_{, the strain at}

which the maximum in the overshoot occurs actually in-creases with shear rate. This is predicted, at least

qualita-tively, by reptationlike theories which include the convective constraint release mechanism.40,41 At present these theories have not been adapted to the case of nonlinear flow of break-able polymers, but we expect overshoot to occur for shear rates higher than the inverse stress relation time 1/␶. This is tested for the system with lowest scission energy in Fig.10. Here, we plot ␩+_{共t兲, the transient growth of the viscosity}

upon onset of shear, normalized by the steady-state viscosity, against total strain ␥ for three different shear rates. In this system the equilibrium stress relaxation time is approxi-mately ␶⬇4⫻10−4 s. Indeed, we find no overshoot for a shear rate of 103 s−1, which corresponds to a Deborah num-ber of 0.4, but an increasing overshoot is observed for shear rates of 104 _{and 10}5 _s−1 _{共De=4 and 40兲, respectively.}

Be-cause De⬍102_{, the location of the peak always lies at strain}

of 2. For higher scission energies, the stress relaxation time␶ increases, and consequently we reach higher Deborah num-bers. Indeed, Fig. 10also shows that, at fixed shear rate of 104 _s−1_{, the strain at which the maximum in the stress}

over-shoot occurs increases with increasing scission energy. We now turn to the steady-state rheological properties. Figure11 shows the steady-state shear stress and共first兲 nor-mal stress components of the three systems studied here, each versus the applied shear rate on a double-logarithmic scale. At low shear rates the shear stress increases linearly with shear rate, ␴xy⬀␥˙␣, with exponent ␣= 1. This linear regime persists up to a shear rate ␥˙⬇1/␶, after which the slope in the double-logaritmic plot decreases共␣⬍1兲. For the system with lowest scission energy共diamonds in Fig.11兲 the

decrease is gradual, to a value which is close to␣= 1/3. For the system with highest scission energy 共circles兲 the slope first decreases from 1 to a value a bit lower than 1/3, and then up again to␣= 1/3. This indicates that the system with the shortest wormlike micelles still is unentangled, whereas the system with the longest wormlike micelles is entangled. In many experiments on fully entangled wormlike micelles a shear-banding transition is observed.1,8,9,36 Rheologically, shear banding manifests itself as a constant or slowly in-creasing shear stress 共a stress plateau兲 between two critical shear rates ␥˙_c1 and ␥˙_c2, often accompanied by a linear

in-0 1 2 3 4 5 t [ms] 0 1000 2000 3000 4000 stress [Pa ] T = 300K, E_sc= 19.1 kT, E_a= 2.4 kT 104s-1 N₁ σxy N₂

FIG. 9. Time dependence of various stress components measured after start-up of shear flow with shear rate of 104 _s−1_{. Overshoots are observed in the} shear stress␴xyand the first normal stress difference N1. In the case of the second normal stress difference N2we observe an initial overshoot, followed by an undershoot and a relaxation to a negative value.

0 2 4 6 8 10 strain 0 1 2 3 4 5 normalized η + (t) 104s-1, E_sc= 19.1 kT 105s-1, E_sc= 15.5 kT 104s-1, E_sc= 15.5 kT 103s-1, E_sc= 15.5 kT

FIG. 10. Transient growth of viscosity normalized by the steady-state vis-cosity as a function of total strain after startup of steady shear flow. Three curves are for T = 333 K, Esc= 15.5kBT, and Ea= 2.2kBT at shear rates of 103,

104_{, and 10}5 _s−1_{. One additional curve is given for T = 300 K, E} sc = 19.1kBT, and Ea= 2.4kBT at shear rate of 104 s−1.

crease in the 共first兲 normal stress difference. Our wormlike micelles are entangled, but only just so. They are therefore not yet long enough to be in the fully entangled regime where shear banding could be observed. Moreover, the shear rates which we apply are larger共or equal兲 to␥˙_c2共vide infra兲. Remarkably, all flow curves converge at the higher shear rates, irrespective of the scission energy Esc, activation

en-ergy Ea, or temperature T. This is confirmed in Fig. 12,

which shows the steady-state viscosity as a function of shear rate for all systems studied. Experimentally, the steady-state viscosity in a micellar solution at different temperatures is also observed to converge to the same curve.1 To test the predictive power of our simulations we have performed real-life experiments on a solution of 8% 共by weight兲 EHAC solution with 2% KCl,19 which is the system we are sup-posed to simulate. The experimental details are given in Ref.

19. First we measured the shear viscosity in a cone-plate geometry at temperatures T = 308, 318, 328, 338, and 348 K, for shear rates ranging from 3⫻10−4 _{up to 10 s}−1_{. The}

re-sults are shown in Fig.13共open circles兲. Clearly, all

viscosi-ties converge to the same curve at higher shear rates. Ini-tially, this curve has a slope of −1, indicating a stress plateau

共␣= 0兲 with shear banded flow. At higher shear rates the nor-mal forces tend to expel the sample out of the cone-plate geometry. We therefore studied shear rates from 100 to 3 ⫻104 _s−1 _{by using a parallel-plate geometry. With such a}

geometry much higher shear rates can be studied 共stars兲. Note the good overlap of the data where both cone-plate and parallel-plate measurements were done. We confirm that the stress indeed increases like ␥˙1/3 at higher shear rates, yield-ing a slope of −2/3 in the viscosity curve of Fig.13. We find that for 8% EHAC the transition from the −1 scaling to the −2/3 scaling regimes takes place at a shear rate of approxi-mately ␥˙_c2⬇100 s−1_{. All simulations were performed at}

shear rates equal to or higher than this, which explains why shear banding was not observed.

Within the accessible range of shear rates, the simulation results共filled circles兲 tend to systematically overestimate the experimental viscosities 共stars兲, but only by a factor of 1.5. Given the crude friction model used for the rodlike seg-ments, the overall agreement with the experimentally deter-mined results is very satisfactory. In fact, after the simula-tions had completed we realized the agreement would have been even better if we had used a friction model for a finite rod, with ␰储= 2␲␩L/共ln共lp/D兲+␯储兲 instead of the employed

␰储= 2␲␩L/ln共lp/D兲 for the parallel friction, and a similar

cor-rection ␯_⬜ for the perpendicular friction. Estimates for the factors ␯储 and␯_⬜ can be found in Ref.42. The corrections

lead to an essentially unaltered parallel friction for our aspect ratio, but to a perpendicular friction which is 1.5 times lower, leading to a lower viscosity.

IV. CONCLUSION

In this paper we have presented a particle based simula-tion model for solusimula-tions of wormlike micelles. We have

101 102 103 104 105 106 107 shear rate [s-1] 100 101 102 103 104 stress [P a ] T = 300K, E_sc= 19.1 kT, E_a= 2.4 kT T = 333K, E_sc= 17.6 kT, E_a= 2.2 kT T = 333K, E_sc= 15.5 kT, E_a= 2.2 kT normal stress shear stress slope 1/3

FIG. 11. Steady-state shear stress共open symbols兲 and normal stress 共closed symbols兲 vs shear rate for the systems studied.

101 102 103 104 105 106 107 shear rate [s-1] 10-4 10-3 10-2 10-1 100 viscosity [Pa s ] T = 300K, E_sc= 19.1 kT, E_a= 2.4 kT T = 333K, E_sc= 17.6 kT, E_a= 2.2 kT T = 333K, E_sc= 15.5 kT, E_a= 2.2 kT slope -2/3

FIG. 12. Shear viscosity vs shear rate for the systems studied.

10-3 10-2 10-1 100 101 102 103 104 105

shear rate [s

-1

]

10-3 10-2 10-1 100 101 102 103 104

v

iscos

ity

[P

as

]

308 K - 348 K, cone-plate 298 K, plate-plate 300 K, simulation 8% EHAC, 2% KCl slope -2/3 slope -1

FIG. 13. Experimental and simulated shear viscosities vs shear rate of an 8% EHAC, 2%KCl wormlike micellar solution. Cone-plate experiments were performed at 308, 318, 328, 338, and 348 K, respectively共open circles, from top to bottom兲, and parallel-plate experiments were performed at 298 K共stars兲. Simulations data 共solid circles兲 are for T=300 K, Esc= 19.1kBT,

shown that with this model we can study the dynamics and rheology at millisecond time scales, while still retaining the specific mechanical properties of the individual wormlike micelles. The majority of these mechanical properties have been determined from more detailed atomistic molecular dy-namics simulations, providing the link with the chemistry of the surfactants. We have used this model to study the behav-ior of both unentangled and slightly entangled wormlike mi-celles under shear flow. We have made the following obser-vations.

• With increasing shear rate, beyond a certain critical shear rate, the contour length decreases and the break-ing rate per unit contour length decreases. The critical shear rate decreases with increasing scission and/or ac-tivation energy.

• The linear rheology 共shear relaxation modulus兲 of un-entangled samples is in good agreement with a theory of breakable Rouse chains共see Appendix B兲.

• Upon startup of shear flow, overshoots occur in the shear stress and first normal stress difference at Debo-rah numbers larger than 1, if De is based on the typical stress relaxation time ␶. The maximum overshoot oc-curs at strain of 2 for Deborah numbers lower than 100. • At high shear rates ␥˙ , beyond the stress plateau, the steady-state shear stress increases like ␴xy⬀␥˙1/3. The exponent of 1/3 is confirmed in parallel-plate experi-ments on an 8% solutions of EHAC for shear rates higher than 100 s−1_.

We expect that, given the ever-growing increase in comput-ing power, more highly entangled wormlike micelles can be simulated in the near future. This will open up the possibility to study the shear-banding phenomenon from a micro-/ mesoscopic point of view, complementary to the usual con-stitutive approach共see, e.g., Refs.8 and9兲.

Furthermore, we note that the simulated values of the activation energy Ea, and to a smaller extent also the values

of the scission energy Esc, are lower than in the actual

ex-perimental situation. This is for computational reasons. Fig-ure 13 shows that realistic values would lead to relaxation times which are many orders of magnitude higher. Fortu-nately, the shear stress converges to a universal curve inde-pendent of Esc, Ea, or T, but dependent on other factors such

as the persistence length lp, diameter D, and wormlike

mi-cellar concentration. We found good agreement between the simulated and experimental viscosities at the shear rates that could be studied. We therefore expect that the model can be used, in combination with atomistically detailed simulations to determine the single chain mechanical properties, to de-sign novel wormlike micellar materials.

ACKNOWLEDGMENTS

J.T.P. acknowledges the Netherlands Organisation for Scientific Research共NWO兲 for financial support. Part of this work was made possible by a Schlumberger Visiting Profes-sorship granted to W.J.B. by Schlumberger Cambridge Re-search.

APPENDIX A: CONSEQUENCES OF COARSE GRAINING FOR THE SCISSION ENERGY OF A BREAKABLE GAUSSIAN CHAIN

Suppose we model a solution of linear wormlike mi-celles as a collection of breakable bead-spring chains 共Gauss-ian chains兲. Assuming that all chain lengths are in chemical equilibrium, it can be shown22 that the average number of beads per chain is given by

具N典 =1 2+

共

1 4+ z␳

兲

1/2_{, 共A1兲}

where␳ is the number density of beads that are able to form breakable chains and z is a quantity with dimensions of vol-ume, defined as

z =

共

2₃␲b2

兲

3/2exp共Esc/kT兲, 共A2兲

where b is the 共effective or root-mean-square兲 bond length between consecutive beads.

We will now calculate the scission energy E˜_sc after coarse graining this model. Suppose we represent the mass of ␭ of our beads by one new bead. Distinguishing new quantities with a tilde 共˜ 兲, the number density of beads be-comes˜ =␳ ␳/␭. We demand that the total amount of mass in each chain remains the same, hence a chain which previously consisted of N beads will now be represented by N˜ =N/␭ new beads. Furthermore, we demand that the overall dimen-sions of our wormlike micelles, as expressed for example by the radius of gyration, remain the same. For our random walk chain this means that we require that Nb2_{= N}˜ b˜2_{. Hence}

the effective bond length must increase like b˜ =

冑

␭b. Now, according to Eq.共A1兲the average number of beads per chain in the new situation will be具N˜典=1₂+

共

1₄+ z˜˜␳

兲

1/2. This must be equal to具N典/␭. As can be seen from Eq.共A2兲, the term z␳is approximately equal to the effective volume fraction ␳b3

multiplied by the factor exp共Esc/kT兲 and hence will be very

much larger than 1 in all practical cases where we deal with long wormlike micelles. We can then safely approximate 具N典=共z␳兲1/2 _{and find z˜ = z/␭. Again looking at Eq.共A2兲 we}

therefore draw the following conclusion: in order for the fur-ther coarse-grained solution of chains to have the same dis-tribution of global dimensions 共gyration radii兲 as present in our original solution, we require that b˜3exp共E˜sc/kT兲

= b3_exp共E sc/kT兲/␭ or, using b˜=

冑

␭b, E ˜ sc= Esc− 5 2kT ln␭. 共A3兲

So the scission energy of a chain coarse grained by a factor of␭ must be lowered by an amount of 5₂kT ln␭. Conversely, if one decides to represent each wormlike micelle by more 共but smaller兲 segments, a higher scission energy is needed to get the same distribution of global dimensions.

Finally, we remark that of course there is an upper limit to the number of beads representing a piece of wormlike micelle. In the first place, the random walk approximation used above is no longer valid at scales smaller than the per-sistence length lp. In the second place, one must ask oneself

what is the smallest unit that can break off a real wormlike micelle. This must certainly be at least the thickness of an

amphiphilic molecule. It has recently been shown16 that in order to reasonably explain observations of the kinetics of a stretched and breaking wormlike micelle, the smallest unit of breaking must be set equal to the diameter D of the wormlike micelle. For our coarse-grained simulations this means that, apart from relatively smaller stiffness effects, the experimen-tal scission energy is higher by an amount of 5₂kT ln共lp/D兲

relative to the scission energy used in the simulation.

APPENDIX B: RHEOLOGY OF UNENTANGLED BREAKABLE CHAINS

In this Appendix we calculate the rheological properties of a collection of unentangled breakable chains. The treat-ment is similar to that of Faivre and Gardissat,38but here we give an explicit derivation including the prefactors. We will show that in the fast breaking limit the interplay between relaxation and “breaking” of different normal modes leads to the dominance of a certain length scale. This length scale is independent of the actual length of the chain, so although the results will be derived for a single chain length N, they will apply more generally to a distribution of chain lengths as well.

Let us first consider the case of an unbreakable chain. In the unentangled limit we can treat the dynamics of a selected chain independently of the dynamics of the other chains. In such a case the shear relaxation modulus GN共t兲 can be

ex-pressed as a simple sum over normal modes,33 GN共t兲 =

cNkT

N + 1

兺

k=1 N

C_N,k2 共t兲, 共B1兲

where N + 1 is the number of segments in one chain, cNis the

number of chain segments per unit volume共present in chains of length N兲, and CN,k共t兲=具XN,k共t兲·XN,k共0兲典/具XN,k2 典 the

nor-malized time correlation function of the kth normal mode

XN,k=共1/N+1兲兺n=0 N

Rncos关共k␲/N+1兲共n+1/2兲兴. From the

last expression we see that the kth normal mode represents the dynamics of a wave with a wavelength containing N/k bonds. In the case of an unbreakable Rouse chain, the normal modes decay exponentially, CN,k共t兲=exp兵−t/␶N,k其 with

relax-ation times␶N,k=␶l/k2_.33

Here we have defined␶las the long-est relaxation time 共k=1兲 of an unbreakable chain of N+1 segments, which for a Rouse chain is given by ␶l=␨b2_共N

+ 1兲2_/共3␲2_{kT兲, with b}2_{the mean-square segment bond length}

and␨the friction on a segment.33

Now let us consider a breakable chain. Assuming break-ing is an activated process, the probability that a certain bond will still be in place after a time t decays like exp兵−cbt其,

where cb is a constant characterizing the breaking rate of a

single bond. An entire chain, of course, can break anywhere along its contour and will therefore break much faster than any particular bond. The probability for all N bonds in a chain to survive up to a time t decays like exp兵−Ncbt其,

mean-ing that the average breakmean-ing time of the full chain is ␶b = 1/共Ncb兲. The same reasoning applies also to a subchain.

The stress resulting from normal modes k of all chains of length N/k, or multiples thereof, will continue to decay ex-ponentially as long as they have not yet broken. When they

do break, we expect that the contribution to the stress to disappear quickly. In summary, we expect that the shear re-laxation modulus will decay like

GN共t兲 ⬇ cNkT N + 1

兺

_k=1 N exp兵− 共N/k兲cbt其exp兵− 2t/␶N,k其 = cNkT N + 1

兺

k=1 N exp

再

−

冉

1 k␶b+ 2 k2 ␶l

冊

t

冎

, 共B2兲

where in the second line we have used relaxation times of the Rouse model. This expression shows that large wavelength 共low k兲 modes are “short circuited” by their relatively fast breaking. We can write the effective relaxation rate of mode k as ␯共k兲=1/共k␶b兲+2k2_/␶l.

In the fast breaking limit, which applies to most cases of interest共both experimentally and in our simulations兲, the av-erage breaking time ␶b of a chain is much shorter than the longest relaxation time ␶l of the 共hypothetical兲 unbreakable chain. We still assume that␶bis much longer than the fastest relaxation time of the unbreakable chain, which is the orien-tational relaxation time of a single bond. We therefore as-sume ␶l/N2_{Ⰶ␶bⰆ␶l. In this case ␯共k兲 has a minimum at a}

certain 1Ⰶkⴱ_{ⰆN, where k}ⴱ_{is given by k}ⴱ₌

共

1 4␶l/␶b

兲

1/3_{. For k}

values smaller or larger than kⴱthe effective relaxation rates quickly increase. For long enough times, kⴱwill therefore be the dominant mode. The effective relaxation rate and the second derivative at kⴱare given by␯共kⴱ兲=₂3

冑

34␶l−1/3␶b−2/3and d2_␯共kⴱ_兲/dk2_{= 12/␶l. Defining the effective relaxation time of}

the dominant mode as␶= 1/␯共kⴱ兲, we find

␶⬇ 0.42␶l1/3_␶b2/3_{. 共B3兲}

It is important to note that, at fixed segment density and fixed interaction parameters, the effective relaxation time ␶ does not depend on the actual chain length because ␶l1/3␶b2/3 ⬀N2/3_N−2/3_{= N}0_{. Approximating the sum over k by an}

inte-gral we find GN共t兲 ⬇ cNkT N + 1

冕

₁ N dk exp

再

−

冉

1 ␶+ 6 ␶l共k − kⴱ兲2+¯

冊

t

冎

⬇ cNkT N + 1exp兵− t/␶其

冕

_−⬁ ⬁ dk exp

再

− 6 ␶l共k − kⴱ兲2t

冎

= cNkT N + 1

冑

␲␶l 6t exp兵− t/␶其 共t ⬎␶l/N 2_{兲. 共B4兲}

In the second line we have cut off the Taylor expansion at second order and used the fact that the effective rates ␯共1兲 and ␯共N兲 are much larger than ␯共kⴱ兲 共this follows directly from our assumptions兲. The zero-shear viscosity can now be calculated as ␩0=

冕

0 ⬁ dtGN共t兲 ⬇ cNkT N + 1

冕

0 ⬁ dx

冑

␲␶l 6x2exp兵− x 2_/␶其2x =

_冑

␲ 6 cNkT N + 1

冑

␶l␶. 共B5兲

In the second line we have ignored the fact that for small times tⱕ␶l/N2Eq.共B4兲is not strictly valid. This is allowed

because the contribution of this regime to the integral is rela-tively small.

Since ␶l⬀共N+1兲2 _{and ␶⬀N}0_{, both the stress relaxation}

and the zero-shear viscosity are independent of the actual length N of the chain. The stress in a polydisperse mixture of breakable Rouse chains will therefore also relax as predicted by Eq. 共B4兲, where we have to replace cN by the overall

number density of segments c. Similarly, the viscosity will also be given by Eq.共B5兲.

We warn that the independence of chain length N, re-ferred to above, means independence of N at fixed interaction parameters. If the interaction parameters are changed, the rheology of course will change as well. The most important interaction parameters are the scission energy and activation energy. Keeping all parameters other than Escand Ea fixed,

the average chain length N will increase approximately like N⬀exp兵Esc/2kT其.22 This does not influence the ratio ␶l/N2

appearing in Eqs. 共B4兲 and共B5兲. The breaking rate cb of a

single bond, however, is changed. Assuming breaking is an activated process we expect cb⬀exp兵−共Esc+ Ea兲/kT其 共see

Fig.3兲. The stress relaxation time␶and zero-shear viscosity

␩0 then will increase with increasing Esc and Ea like ␶

⬀exp

兵

2

3共Esc+ Ea兲/kT

其

and␩0⬀exp

兵

1

3共Esc+ Ea兲/kT

其

. We

em-phasize that these scalings hold only as long as the wormlike micelles behave as unentangled breakable Rouse chains. If the average chain length becomes too short, the chains can-not be viewed anymore as Gaussian chains and/or the aver-age breaking time is no longer much shorter than the longest relaxation time共our main assumption兲. If the average chain length becomes too large, the chains start to entangle and the description of the relaxation kinetics will need to be changed accordingly.

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