Stretch and Breakage of Wormlike Micelles Under Uniaxial Strain: A

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Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers

Stretch and Breakage of Wormlike Micelles Under Uniaxial Strain: A Simulation Study and Comparison With Experimental Results Taraknath Mandal, and Ronald G. Larson Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02421 • Publication Date (Web): 25 Sep 2018 Downloaded from http://pubs.acs.org on October 1, 2018

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Stretch and Breakage of Wormlike Micelles Under Uniaxial Strain: A Simulation Study and Comparison With Experimental Results Taraknath Mandal and Ronald G. Larson* Department of Chemical Engineering, University of Michigan, 2300 Hayward St, Ann Arbor, MI-48109, USA.

We use coarse-grained (CG) molecular dynamics (MD) simulations to determine the effect of uniaxial strain on the stress, scission stress, and scission energy of solutions of wormlike

micelles

of

cetyltrimethylammonium

chloride/sodium

salicylate

(CTAC/NaSal). We find that the breaking stress, stretch modulus and scission energy of the charged micelles are nonmonotonic functions of oppositely charged hydrotrope (NaSal) concentration. While the stretch modulus shows a peak at a value of surfactantto-hydrotrope concentration ratio (𝑅𝑅) close to unity as expected due to neutralization of head-group charge at 𝑅𝑅 = 1, the breaking stress and scission energy produce a peak at

𝑅𝑅 < 1.0 because of thinning of the micelle diameter with increased R. The breaking

stress from the simulations depends on the rate of deformation, and roughly agrees with the experimental values of Rothstein (J. Rheol. 2003, 47, 1227) after extrapolation to the much lower experimental rates. The method and results can be used to predict the effects of flow and mechanical stress on rates of micellar breakage, which is important in the rheology of wormlike micellar solutions.

*

Email: [email protected]

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Introduction: Surfactant solutions are of immense scientific and technological interest. They display a fascinating variety of self-assembled structures, including spherical, cylindrical micelles, wormlike micelles (WLM), lamellar sheets, and exotic ordered bicontinuous phases, which greatly influence the rheological properties of the micellar solutions.1–8 Such solutions have wide applications in daily life as shampoos, body washes and as well as in industry as oil-production fluids, foaming agents, and dragreduction agents for fluid transportation, among others.9–11 Many of these applications involve fast flow and deformation of the micellar solution which causes interesting and unconventional phenomena such as chain-scission, structure formation and elastic instabilities.12–15 Recently, many experimental studies have been performed to understand the mechanical behavior of micellar solution under uniaxial strain.12,16–22 These studies are useful to understand and predict the behavior of micellar solutions in flows with strong extensional components such as in flow through constrictions during pumping or processing and flow through porous media.13,16 In particular, Rothstein used a filament stretching rheometer to measure the time-dependent tensile force of a wormlike cetyltrimethylammonium bromide (CTAB) micellar solution in uniaxial straining.16 He showed that the breaking stress of the filament is a nonmonotonic function of organic salt (hydrotrope) concentration, and estimated the average scission free energy of an individual CTAB micelle under these flow conditions to be ~4 𝑘𝑘𝐵𝐵 𝑇𝑇, almost independent

of the surfactant and hydrotrope concentration. However, this calculated scission free energy was found to be much less than that reported by other experimental studies,23 and in conflict with the common observation that the micelle scission energy is strongly dependent on the hydrotrope concentration.24,25 The oppositely charged organic hydrotropes have strong adsorption affinity towards the micelle surface which can strongly screen the electrostatic repulsion amongst the charged surfactant head groups. So, the hydrotrope concentration is expected to affect the scission energy of the micelle. Moreover, the adsorbed hydrotropes may also change the micelle’s mechanical properties, for instance its persistence length, which in turn changes its elastic modulus. Thus, to better understand the origin of the nonmonotonicity of the micelle breaking stress observed in experiment and to determine if there is nonmonotonicity in the stretching modulus and in the scission energy as well, we employed molecular dynamics

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(MD) simulations. Although MD simulations have been used extensively to study the self-assembly and rheological properties of micelles,6,25–36 computational studies of mechanical properties of linear micelles have not been performed in detail so far. Recently, Dhakal and Sureshkumar used coarse-grained (i.e., MARTINI) MD simulation to study the configurational dynamics of rod-like and slightly bent “U-shaped” micelles in uniaxial elongational flow.37 They found that above a critical strain rate, hydrodynamic forces overcome the conformational entropy of the micelle and the micelle configuration changes from a folded U-shape to a linear state. They also calculated the breaking strain and the stretch modulus as a function of hydrotrope concentration. However, they did not observe any nonmonotonic behavior in these mechanical properties as a function of hydrotrope concentration.

In this study we also employ MARTINI coarse-grained MD simulations to investigate the mechanical properties and scission energy of a model charged linear micelle (cetyltrimethylammonium chloride (CTAC)) under uniaxial strain. We calculate the breaking stress and stretching modulus as well as the scission free energy of the micelle as a function of hydrotrope concentration. Interestingly, each of these properties: the micelle breaking stress, the stretch modulus and the scission free energy, are nonmonotonic functions of hydrotrope concentration. While the scission energy and breaking stress show maxima at a hydrotrope-to-surfactant concentration ratio (𝑅𝑅) less than unity, the micelle stretch modulus shows a maximum at 𝑅𝑅~1. We then compare our simulation results with the available experimental results.

Methodology: We used cetyltrimethylammonium and salicylate molecules as the representatives of a model charged surfactant and hydrotrope, respectively. We chose this surfactant since the rheological properties of this micellar solution have been well studied experimentally.38–42 The MARTINI coarse-grained model43 was used to describe the surfactants and hydrotropes as shown in figure 1(a)&(b), since this force field has been used extensively to study this surfactant system.6,25,37,44–48 A linear micelle containing 310 surfactants was built which spans the simulation box along the Z direction as shown in figure 1(c). This corresponds to a surfactant concentration of ~0.15M in the simulation

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box of dimensions 14.3 × 14.3 × 16.6 𝑛𝑛𝑚𝑚3 . Different numbers of salicylate ions were then dispersed randomly in the box to achieve various 𝑅𝑅 values. An appropriate number

of chlorine ions was added to neutralize the charge of the surfactants and, similarly, sodium ions were added to neutralize the charge of the hydrotropes. MARTINI water with 10% anti-freezing beads was used to solvate the surfactant. Energy minimization was then performed using the conjugate gradient method, followed by 1-microsecondlong equilibration runs under constant pressure and constant temperature (NPT) before the micelle was subjected to external strain. The temperature and pressure were controlled using a velocity rescale thermostat49 with a time constant of 1 ps and a Parrinello-Rahman barostat50 with a time constant of 3 ps, respectively. A typical equilibrated micelle is shown in figure 1(c). All simulations were performed using the GROMACS package.51

Figure 1: Coarse grained representation of the (a) cetyltrimethylammonium surfactant and (b) salicylate hydrotrope. The characters on the figures represent the types of MARTINI beads. (c) A typical simulation box setup showing the linear micelle under extension. The applied pressure anisotropy elongates the micelle along the axial direction (black arrows) and compresses the simulation box along the X and Y directions (blue arrows). Red, green, blue, yellow, and gray colored beads represent surfactant heads, surfactant tail beads, and salicylate, sodium and chlorine ions, respectively.

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An external tension along the micelle axis was created by applying a pressure difference between the axial and radial direction of the micelle. As a result, the total force acting on 1

the micelle axis is 𝐹𝐹 = � < (𝑃𝑃𝑋𝑋 + 𝑃𝑃𝑌𝑌 ) > − < 𝑃𝑃𝑍𝑍 >� 𝐴𝐴 , where 𝑃𝑃𝑋𝑋 , 𝑃𝑃𝑌𝑌 and 𝑃𝑃𝑍𝑍 are the 2

pressures along the X, Y and Z directions of the simulation box, respectively, and 𝐴𝐴 is the

cross-sectional area of the simulation box in the XY plane. The pressures along the X and Y directions were set to 1 bar and the pressure along the Z direction (the micelle axis) was gradually decreased to generate an increasing external force acting on the micelle, up to the point at which the micelle broke. Note that the external force generated by the pressure anisotropy acts only on the micelle as the solvent cannot withstand any external force. For different values of 𝑅𝑅, we found the breaking stress acting on the micelle by

dividing the total force acting on the micelle by the micelle cross-section. Note that the micelle cross-section might be different for different values of 𝑅𝑅 for a fixed external force

as we discussed in our previous study.25 We used an umbrella sampling method to calculate the micelle cross-sectional area in the absence of any external force when the micelle may adapt a slightly curved configuration. The details of the calculations are given in the Supporting Information (SI). However, in the presence of an external stress, even at the lowest stress we imposed (~9 bar), the micelle adapts a straight cylindrical configuration so the micelle length (𝐿𝐿) is the same as the box length along the Z direction (see SI). Since the micelle adapts an approximate cylindrical shape under an external force and the surfactant density (𝜌𝜌) is almost uniform throughout the micelle, we have 𝑁𝑁 = 𝜌𝜌𝜌𝜌𝑟𝑟 2 𝐿𝐿, where 𝑁𝑁 is the total number of surfactants in the micelle of length 𝐿𝐿 and

radius 𝑟𝑟, and 𝜌𝜌 is the average number of surfactants per unit volume in the micellar state. Thus, the cross-sectional area of the micelle, 𝜋𝜋𝑟𝑟 2 , is proportional to the number of

surfactant molecules per unit length of the micelle, N/L. The surfactant number density (per unit volume) of the micelle, 𝜌𝜌~0.93 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠/𝑛𝑛𝑚𝑚3 , can be computed from the

number of surfactants per unit length N/L and the radius of the micelle under no external force, which is ~2.5 nm. The radius of the micelle obtained from our simulation is very close to the radius of a similar wormlike CTAB/NaSal micelle, namely 2.2 nm, obtained from an experimental study.52 We note that the micelle becomes thinner and longer with increasing 𝑅𝑅 because of salicylate hydrotrope adsorption on the micelle surface as we

will discuss later. The adsorbed hydrotropes slightly change the surfactant headgroup 5


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distribution at the micelle surface but the surfactant number density 𝜌𝜌 remains almost

constant with increasing 𝑅𝑅 . The decrease in the micelle thickness with hydrotrope concentration has a strong impact on the breaking stress and scission energy as we will

discuss later.

Figure 2: (a) Experimental breaking stress of a filament of CTAB micelle solution as a function of surfactant concentration in the solution, from Tables I and II of reference [16]. Inset shows the breaking stress of an individual micelle calculated from the filament breaking stress, as explained in the text. (b) Strain-stress plot obtained from simulations for a micelle at 𝑅𝑅 = 1. The stress is the force per unit cross sectional area of the micelle,

which decreases as the micelle is stretched. See text for the definition of the micelle strain. Inset shows logarithm of micelle breakage time (𝜏𝜏𝑏𝑏 ) as a function of external force

obtained from simulations. The size of the error bars is comparable to the symbol size.

Results and Discussion: First, we discuss the strain-stress relationship for the wormlike micelles under uniaxial strain. In the absence of any external force, the micelles may adapt a slightly curved configuration, particularly at higher 𝑅𝑅 value. However, when the 6


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smallest stress (~9 bar) used in our simulation is imposed, the micelle adapts a straight cylindrical configuration (figure S1, Supporting Information). Thus, the micelle length in the presence of an external stress (>9 bar) is same as the box length (L) along the Z direction. The stress vs. L relationship is linear from a stress of ~9 bar to the breaking strain, around 0.4 (figure 2(b)). So, we define the strain of the micelle as (L-L0)/L0, where L is the box length along the Z direction under stress and L0 is obtained by extrapolating the stress vs. L linear relationship to zero stress. The stress acting on the micelle for 𝑅𝑅 = 1.0 as a function of strain is shown in figure 2(b). Eventually the micelle

breaks into two parts when the stress on the micelle overcomes a critical value which in

this case is ~65 bar. A typical snapshot of the micelle breaking is shown in figure S2 of the Supporting Information. Figure 2(a) gives the experimental breaking stress of the micelle filament from Rothstein,16 which, as might be expected, increases linearly with surfactant concentration, implying that the breakage stress of a micelle is independent of concentration. The average breaking stress for an individual micelle can then be extracted from the filament breaking stress by dividing this stress by the volume fraction of the surfactants in the filament. As shown in the inset of figure 2(a), the average experimental breaking stress of the micelle for 𝑅𝑅 = 1.0 is approximately 3.6 bar, which is ~18 times

lower than the micelle breaking stress obtained from the simulations (~65 bar). This large difference is mainly a consequence of the nine-orders-of-magnitude higher strain rate in the simulations than in the experiments. The typical breakage time of the micelle filament in Rothstein’s experiments is ~1.5 s, whereas in our simulations the strain rate is so high that the micelles break on a time scale of nanoseconds. We expect, based on theory for material fracture, that the average breakage time 𝜏𝜏𝑏𝑏 of the micelles should decrease

0 exponentially with the external force as 𝜏𝜏𝑏𝑏 = 𝜏𝜏0 exp[(𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 − 𝑭𝑭. 𝑑𝑑𝒓𝒓)/𝑘𝑘𝐵𝐵 𝑇𝑇],53 where 𝑭𝑭

and 𝑑𝑑𝒓𝒓 are the external force and the extension of the micelle along the axis from its

0 equilibrium length under the external force, respectively. 𝜏𝜏0 is a constant and 𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 is the

scission energy under no external force. Thus, to estimate the average micelle breaking stress in a typical experimental time scale from the simulation results, we determined the

breakage time from simulations at different external forces as shown in the inset of figure 2(b). Extrapolating this plot to 𝜏𝜏𝑏𝑏 ~1.5 s, a typical experimental time scale, the corresponding breaking force can be estimated as ~108.6 𝑏𝑏𝑏𝑏𝑏𝑏. 𝑛𝑛𝑚𝑚2 . Dividing this force 7


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by the micelle cross sectional area (~10.8 𝑛𝑛𝑚𝑚2 ), we obtain the micelle breaking stress as

approximately 10.0 bar which compares surprisingly well with the experimental value of

3.6 bar, considering the long extrapolation required. Here we should note that the filament stretching experiments were not performed exactly under constant force. Rather, a constant strain rate was applied so the filaments broke in a time scale of ~1.5 s which is much slower than the simulation strain rate. In our simulations, the average breakage time was determined under constant force. However, our simulation results suggest that if the stretching rate in the experiment is high enough so the micelles break in the timescale of nanoseconds, the micelle breaking stress also would be much higher, comparable to the breaking stress obtained from simulations.

Figure 3: (a) Micelle breaking stress as a function of 𝑅𝑅 obtained from simulations (black

circles). Upper inset (red circles) shows the corresponding results from experiments by Rothstein (Ref. [16]). Lower inset shows a zoomed-in version of the simulated breaking stress in the R~0.5 to R~1.0 range. (b) Stretching modulus of micelle as a function of 𝑅𝑅

obtained from simulations. Blue circles are data points; the black line is a polynomial fit to the simulation data.

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Figure 3(a) shows the micelle breaking stress as a function of 𝑅𝑅 obtained from the simulations and from the experiments of Rothstein, taken from Table II of Ref. [16]. To

calculate the breaking stress, first we ran several simulations to identify a pressure anisotropy (or stress) which is close to, but lower than, the breaking stress. We then gradually increased the pressure difference between the lateral (X and Y axis) and the longitudinal (Z axis) direction of the simulation box in ~0.12 bar steps, which, based on the cross-sectional area of the micelle compared to the box cross section, would imply a ~1 bar increase in the stress acting on the micelle along the axial direction. For each step increase in pressure anisotropy, we ran 10 independent simulations. We note that because of fluctuations in the micelle cross-sectional area, the stress acting on the micelle may vary somewhat (by around ~0.5 bar) during the 10 independent simulations for a given input pressure difference between lateral and longitudinal directions. From these results, we identified two micellar stress thresholds separated by a single step in input stress. At the lower threshold, the micelle is stable (does not break) in more than 50% of the trajectories, each of which lasted at least one microsecond. At the slightly higher stress threshold, the micelle broke in at least 50% of the trajectories again lasting one microsecond long. The breaking stress is then defined as the average of these two stresses and the error is calculated as the difference between the average and any of these two stresses. For example: at a particular given input pressure difference, in 8 of the 10 trajectories the micelle sustained the stress without breaking for at least one microsecond. The corresponding stresses acting on the micelle in the eight simulations were 68.95, 68.35, 68.52, 68.00, 68.47, 68.79, 68.39, 68.50 bar and so the average stress for the micelles that avoiding breaking in 8 of 10 cases was 68.49 bar. When the input pressure difference was increased by 0.12 bar, then in only in 4 of the 10 trajectories did the micelle sustain the stress for one microsecond. The corresponding stresses acting on the unbroken micelles were 69.00, 70.18, 70.47, 70.42 bar and the average of these is 70.0 bar. Thus, the breaking stress is taken as (68.49+70.0)/2 = 69.25 and the error in this estimate can be approximated as 0.76 bar, which is the difference between this average and each of the two threshold stresses.

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The breaking stresses obtained from simulations are in general ~18-20 times higher than the corresponding experimental values due to the higher strain rate in the simulations as discussed above. Interestingly, a nonmonotonic dependence of the micelle breaking stress on hydrotrope concentration is observed in both experiments and simulations. In the simulations, the maximum in breaking stress occurs at 𝑅𝑅~0.6 while in the experimental

results the maximum is at 𝑅𝑅~1.0. The breaking stress does not vary significantly from 𝑅𝑅~0.5 to 𝑅𝑅~1.0 in simulations but it increases significantly in the range from 𝑅𝑅~0.5 to

𝑅𝑅~1.0 in the experimental results. Because of the limited density of data points, the peak in experimental breakage stress could be anywhere between 𝑅𝑅~0.5 to 𝑅𝑅~1.0, but it

seems unlikely that the peak will occur at the value of 𝑅𝑅~0.6 predicted by the simulations. However, the viscosity of the CTAC/NaSal solution has a maximum at

𝑅𝑅~0.6 which corresponds well with the peak of the scission energy at 𝑅𝑅~0.6 , as we

discussed in our previous study.25 The simulated breakage stress is, of course, at much shorter time scales than the experimental one, and therefore it is possible that the peak in the breakage stress might be rate dependent. Thus, more refined experiments on micelle breaking stress as a function of 𝑅𝑅 would probably be worthwhile to determine the experimental peak location conclusively.

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Figure 4: Representative snapshots of the equilibrated micelle at the following molar ratios of hydrotrope to surfactant 𝑅𝑅: (a) 0.25, (b) 0.62, (c) 1.0, (d) 1.5, (e) 2.0. The color code is as in figure 1. Adsorptions of the salicylate ions (blue) on the micelle surface increases with 𝑅𝑅.

Figure 5: (a) The effective charge per surfactant and (b) the cross-sectional area of the micelle as a function of 𝑅𝑅. The stretching modulus of the micelles, taken as the slope of the linear region of the strain-stress plot in figure 2(b), also shows a nonmonotonic dependence, with a maximum at 𝑅𝑅~1.0 as shown in figure 3(b). The nonmonotonicities in the breaking stress and in the stretching modulus are a consequence of the hydrotrope adsorption on

the micelle surface which changes the effective micelle charge. Figure 4 presents typical micelle configurations at different values of 𝑅𝑅 under no external force. Note in these

figures that the concentration of the hydrotropes (blue) on the micelle surface increases with 𝑅𝑅. The effective micelle charge (surfactant charge + adsorbed hydrotrope charge)

and the cross-section area of the micelle as a function of 𝑅𝑅 are shown in figure 5. The

stretching energy should depend on the surface charge density of the micelle, since the

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positively charged CTAC surfactant head groups repel each other strongly at low 𝑅𝑅

where the electrostatic interaction is not screened much. Thus, the stretching energy and hence the stretching modulus is also low at lower values of 𝑅𝑅 . As the hydrotrope concentration is increased, more and more salicylate ions are adsorbed onto the micelle

surface54–56 (figure 4), and the strength of repulsion between the surfactant head groups decreases because of the electrostatic screening which causes the surfactants to pack more tightly. As a result, more energy is required to stretch the micelle and hence the stretching modulus increases. The maximum of the stretching modulus occurs at 𝑅𝑅~1 where the micelle surface becomes effectively charge neutral (figure 5 (a)), since almost

all of the salicylate ions are adsorbed onto the surface. As the hydrotrope concentration is increased further, salicylate ions continue to be adsorbed on the micelle surface, thus reversing the micelle surface charge (figure 5(a)) so that less energy is required to stretch the micelle. Hence the stretching modulus decreases with the hydrotrope concentration beyond 𝑅𝑅~1, thus explaining the maximum in modulus at 𝑅𝑅~1 in figure 3 (b). We note

that the micelle becomes thinner25 with increasing 𝑅𝑅 as shown in figure 5(b), and can be

observed qualitatively in Figure 4. However, the stretching modulus of the micelle

should not depend much on the micelle thickness, because the stretching modulus is the force per unit area of the micelle divided by strain, and thus the thinning of the micelle is accounted for in the definition of the micellar stretching modulus. So, the maximum of the stretching modulus as a function of 𝑅𝑅 occurs at 𝑅𝑅~1.0, where the effective surface charge density of the micelle is zero, and there is no electrostatic repulsion that might

make the micelle easier to stretch. However, unlike the stretching modulus, the scission energy i.e. the free energy required to break a micelle, depends on the micelle radius as we describe in the following.

Now we discuss the effect of external force or stress on the micelle scission free energy. One might calculate the scission free energy by integrating the area under strain-stress curve. However, as we discussed earlier, the breaking stress depends on the strain rate, and thus the scission “free energy” would also depend on the strain rate if it is calculated from the strain-stress plot. One could, in principle, calculate the scission energy of the micelle by stretching it at an infinitely low strain rate, but this is not practical. In a recent

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study, we developed a method for calculating scission and branching free energies of micelles using a Weighted Histogram Analysis Method (WHAM) with a suitable reaction coordinate. Using this method, discussed in detail in Ref. [25], we here calculate the scission free energy as a function of 𝑅𝑅 for different values of external stress. The scission

free energy (figure 6(a)), like the stretch modulus, shows a nonmonotonic dependence on 𝑅𝑅. However, unlike the stretch modulus, the scission free energy shows a maximum at

𝑅𝑅~0.62 , whereas the later shows a maximum at 𝑅𝑅~1.0 . The same argument of

electrostatic interaction between the charged surfactant heads and the screening of the electrostatic interactions by the adsorbed oppositely charged salicylate ions that we

invoked to explain the nonmonotonic behavior of the stretch modulus can also be used to understand the nonmonotonic dependence of the scission free energy on 𝑅𝑅. However, in addition to the electrostatic interaction, the radius of the micelle also strongly contributes

to the scission free energy of the micelles. As the hydrotrope concentration is increased, more and more salicylate ions are adsorbed on the micelle surface (figure 4), which makes the micelle longer and thinner as shown in figure 5(b).25 The scission free energy would show a maximum at 𝑅𝑅~1.0, where the micelle is effectively charge neutral, if the

scission energy were independent of the micelle thickness. But, instead, the scission energy shows a maximum at 𝑅𝑅 < 1.0 since the micelle becomes thinner with increasing 𝑅𝑅 , which decreases the scission energy. Thus, while the neutralization of the electrostatic

charge would lead to a peak in scission energy at 𝑅𝑅~1.0, the changing micellar radius shifts this peak to 𝑅𝑅 < 1.0. The average micelle length grows exponentially with the

scission energy. Since the viscosity of the micellar solution increases with the micelle length, our scission energy results predict that the CTAC/NaSal solution would show a large viscosity peak at 𝑅𝑅~0.62, which is consistent with the experimental observation of the viscosity peak at 𝑅𝑅~0.6 for CTAC/NaSal solutions.57

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′ Figure 6: (a) Micelle scission free energy (𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ) determined by WHAM as a function of

𝑅𝑅 for different values of external stress. Symbols are simulation data points and solid

lines are the polynomial fits to these. (b) Scission free energy as a function of external force for 𝑅𝑅 = 1.0 determined by WHAM. Inset shows the scission free energy as a

function of external stress. Dashed lines are linear fits to the simulation data (circles) (c) Slope of decrease in the scission energy with external force as a function of 𝑅𝑅. The red line is a polynomial fit. (d) Persistence length calculated using tangent autocorrelation (black circles) of the micelle and the quantity 𝜋𝜋𝜋𝜋𝑟𝑟 4 /4𝑘𝑘𝐵𝐵 𝑇𝑇 (red triangles), which is the

persistence length predicted from the stretching modulus, as a function of 𝑅𝑅. See text for details. Persistence length simulation data for the same CG model are taken from Ref [25].

The scission free energies of the micelle as a function of 𝑅𝑅 for different external stresses

are shown in figure 6(a). The peak of the scission energy occurs at 𝑅𝑅~0.62 for no

external stress and slightly shifts towards the right with increase of external stress. For a fixed value of 𝑅𝑅, the micelle becomes longer and thinner at a higher external stress. Apparently, since the micelle is already thinner at higher external stress, the thickness effect becomes less important relative to the electrostatic effect at high stresses. As a result, the peak position in the scission energy slightly shifts towards 𝑅𝑅 = 1 at higher external stress. Interestingly, the scission energy decreases linearly with external force or

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stress as shown in figure 6(b) for 𝑅𝑅 = 1. From the slope of the scission energy vs force plot shown in figure 6(c), we can see that the breakage distance decreases with 𝑅𝑅, and is 𝑑𝑑𝑑𝑑 ~ 2.6 nm for 𝑅𝑅 = 1.0 . Figure 6(b) should be useful for estimating the average

breakage time of the micelle at zero external stress, since the average breakage time of the micelle (𝜏𝜏𝑏𝑏 ) increases exponentially with the micelle scission energy as 𝜏𝜏𝑏𝑏 =

′ ′ 0 𝜏𝜏0 exp(𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 /k B T), where 𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 − 𝑭𝑭. 𝑑𝑑𝒓𝒓 is the effective scission energy under

an external force 𝐹𝐹 and 𝜏𝜏0 is a constant. Thus, extrapolating the force vs. scission energy

plot in figure 6(b) determined by WHAM up to the lowest force used in the constant-

force time-dependent breaking simulations discussed earlier, we obtain the scission ′ energy of the micelle to be 𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 6.64 𝑘𝑘𝐵𝐵 𝑇𝑇 at this lowest imposed external force of 729

𝑏𝑏𝑏𝑏𝑏𝑏. 𝑛𝑛𝑚𝑚2 . As shown in the inset to figure 2(b), the average breakage time at 729 𝑏𝑏𝑏𝑏𝑏𝑏. 𝑛𝑛𝑚𝑚2 external force is ~730 𝑛𝑛𝑛𝑛. Using these two values in the expression for 𝜏𝜏𝑏𝑏 =

′ 𝜏𝜏0 exp(𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 /k B T), we obtain 𝜏𝜏0 =

730

exp(6.64)

𝑛𝑛𝑛𝑛 ~ 0.95 𝑛𝑛𝑛𝑛. Now using the values of the

′ constant 𝜏𝜏0 (= 0.95 𝑛𝑛𝑛𝑛) and the scission energy at zero stress (𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ~24.1 𝑘𝑘𝐵𝐵 𝑇𝑇), we

calculate the average breakage time of the micelle under no external stress to be 0.95 ∗

exp(24.1) 𝑛𝑛𝑛𝑛 ~ 28𝑠𝑠 at 𝑅𝑅 = 1.0. The average breakage time for the micelle also can roughly be estimated from the rheological storage modulus (𝐺𝐺′) and loss modulus (𝐺𝐺′′)

data. The intersection of the plots of 𝐺𝐺′ and 𝐺𝐺′′ as a function of the angular frequency (𝜔𝜔) gives the critical angular frequency (𝜔𝜔𝑐𝑐 ). We note that in the experimental results of Rothstein,16 𝜔𝜔𝑐𝑐−1 varies from 1𝑠𝑠 to 35𝑠𝑠 depending on the surfactant concentration at 𝑅𝑅 =

1.0 . The breakage time of the micelle should be much lower than 𝜔𝜔𝑐𝑐−1 . Thus, our

breakage time of ~28𝑠𝑠 at 𝑅𝑅 = 1.0 gives only an approximate upper limit of the micelle

breakage time obtained from experiments, which is not surprising considering the very long extrapolation used to obtain this value. We note that Rothstein16 calculated the scission energy of the micelles using an approximate formula, 𝐸𝐸𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 = 𝐹𝐹𝐹𝐹, where 𝐹𝐹 and 𝐷𝐷 are the micelle breaking force and micelle diameter, respectively. They found in this way that the scission energy of the

micelle is around ~3.9 𝑘𝑘𝐵𝐵 𝑇𝑇, independent of the hydrotrope concentration 𝑅𝑅 in the micelle solution. However, this value is much smaller than the scission energy reported in a

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recent experimental study23 and in our previous study.25 We also showed here that the scission energy of micelles strongly depends on 𝑅𝑅.25 Thus, we emphasize that the scission

energy should be calculated using a direct method like one we used here, rather than from an estimate of the breaking force. The decrease in the slope shown in figure 6(c) can be understood by noting in figure 6(d) that at a higher value of 𝑅𝑅, the persistence length of

the micelle is lower, suggesting that the micelles become more flexible, and hence part of the applied force is required to stretch the micelle into a linear and tight configuration and only the remaining force effectively decreases the scission energy. The persistence length in figure 6(d) was calculated using the tangent autocorrelation of the micelle.25 The

persistence length (𝑃𝑃) and the stretching modulus (𝐸𝐸) should be approximately related to each other58 by 𝑃𝑃 =

𝐸𝐸𝐸𝐸

𝑘𝑘𝐵𝐵 𝑇𝑇

, where 𝐼𝐼 is the moment of inertia about the cylindrical axis of the

micelle. For a uniform rigid rod 𝐼𝐼 =

rigid rod, we obtain 𝑃𝑃 = 𝜋𝜋𝜋𝜋𝑟𝑟 4

4𝑘𝑘𝐵𝐵 𝑇𝑇

𝜋𝜋𝜋𝜋𝑟𝑟 4

4𝑘𝑘𝐵𝐵 𝑇𝑇

𝜋𝜋𝑟𝑟 4 4

. Thus, approximating the micelle as a uniform

. Using the stretching modulus data shown in figure 3(b),

as a function of 𝑅𝑅 is plotted in figure 6(d), and found to be ~3 times higher than the

persistence length calculated using a direct tangent autocorrelation method; see figure 6(d). However, apart from this factor of three, the persistence length calculated from the stretching modulus varies similarly with 𝑅𝑅 as does the persistence length directly

obtained from the tangent autocorrelation function. Note that the formula for the moment of inertia of the micelle, 𝐼𝐼 = 𝜋𝜋𝑟𝑟 4 /4 is valid for an elastic solid rod with uniform modulus.

But the micelle is not uniform (the core and shell regions are different) and the surfactants are mobile (liquid-like), rather than solid. For example, if the micellar headgroup region has a higher local stretching modulus 𝐸𝐸(𝑟𝑟) than the core does, this could

easily increase the bending modulus significantly, since the bending modulus for a

heterogeneous cylinder with r-dependent modulus is given by a radial integral of 𝐸𝐸(𝑟𝑟)

weighted by 𝑟𝑟 3 while the contributions to the average stretching modulus are the same,

per unit area, for any value of 𝑟𝑟. Thus, the bending modulus is much more sensitive to the

modulus of the head-group region than is the average stretching modulus. Hence the formula P = 𝐸𝐸𝐸𝐸𝑟𝑟 4 /4𝑘𝑘𝐵𝐵 𝑇𝑇 may only be qualitatively accurate for the micelles. Thus, while

the formula P = 𝐸𝐸𝐸𝐸𝑟𝑟 4 /4𝑘𝑘𝐵𝐵 𝑇𝑇 correctly implies a stronger dependence on micelle radius 16


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of the persistence length than the stretching modulus, this simple approximation gives only a semi-quantitative estimate of the persistence length.

The plateau modulus (𝐺𝐺 ) is inversely proportional to the persistence length (𝑃𝑃 ), as 𝐺𝐺~𝑃𝑃−0.4 in the “loose entanglement” limit, where the entanglement spacing along the micelle, 𝑙𝑙𝑒𝑒 , exceeds the persistence length 𝑃𝑃.59 Thus, the rapid fall in the persistence length in the range 𝑅𝑅~0.5 to 𝑅𝑅~1.25 (figure 6(d)) might indicate a rapid increase in the plateau modulus with increasing R in this hydrotrope concentration range. Indeed, Lutz-

Bueno et al.57 recently showed that the plateau modulus (𝐺𝐺) of CTAC/NaSal solution increases rapidly in the range 𝑅𝑅~0.5 to 𝑅𝑅~1.0. However, they found that the plateau modulus remains almost constant beyond 𝑅𝑅~1.0 even though our results show that the

persistence length slightly decreases with 𝑅𝑅 beyond 𝑅𝑅~1.0. This is probably not too

surprising as the plateau modulus also scales with the entanglement length 𝑙𝑙𝑒𝑒 , as 𝐺𝐺~𝑙𝑙𝑒𝑒−0.6 in the “loose entanglement” limit, and as 𝐺𝐺~𝑙𝑙𝑒𝑒−1 in the tight entanglement

limit.59 Thus, it is difficult to predict the peak of the plateau modulus at 𝑅𝑅~1.0 and its

nearly constant value at higher R, without the knowledge of the exact variation of the entanglement length with 𝑅𝑅, which is beyond the scope of our present work. Conclusion: In summary, we employed coarse-grain molecular dynamics simulations to study the behavior of wormlike micelles under a uniaxial external force. We calculated the stiffness, the breaking stress and the scission energy of the wormlike micelles as a function of hydrotrope concentration and compared our simulation results with the available experimental results. The simulation results show a linear, Hookean, response of micelle extension to imposed force up to a very high external force and linear strains of 0.4. The breaking stress of an individual micelle, obtained from the simulations, is around ~65 bar which is much higher than the corresponding experimental value of ~4 bar, because of the very high strain rates in the simulations relative to the experimental strain rates. However, extrapolating the simulated breakage time of the micelles to the experimental timescale, we find that the breaking stress is reduced to ~10 bar which is comparable to the experimental value. Our simulation results capture the nonmonotonic behavior of the breaking stress as a function of hydrotrope-to-surfactant ratio 𝑅𝑅 that is 17


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typically observed in experiments. While both the stretching modulus and scission free energy, the latter calculated using a Weighted Histogram Analysis Method (WHAM), show a nonmonotonic behavior as a function of 𝑅𝑅, the stretching modulus shows a peak

at 𝑅𝑅~1.0, whereas for the scission energy the maximum is at 𝑅𝑅~0.62. We argued that

even though the micelle is effectively charge neutral at 𝑅𝑅~1.0, the radius of the micelle

decreases with 𝑅𝑅, which also decreases the scission energy which therefore shows a peak

at 𝑅𝑅 < 1.0. However, the stretching modulus does not depend on the micelle radius and

hence shows a peak at 𝑅𝑅~1.0, where the micelle is effectively charge neutral. The persistence length of the micelle can be predicted, within a factor of three, using a

combination of the stretching modulus and the micelle radius, using the theory for the bending modulus of cylinders. The scission free energy strongly decreases with external force. Since the micelles are more flexible at higher 𝑅𝑅, the rate of decrease in the scission

energy decreases with increased 𝑅𝑅. The simulations results presented here can be used as

input for theoretical modeling to predict micellar rheology and micellar length under an imposed stress.

Supporting Information: Method for calculation of micelle radius, snapshot of micelle breaking and a typical GROMACS input file are provided in the Supporting Information.

Acknowledgement: We are grateful for financial support from the National Science Foundation under grant CBET-1500377 and to the Procter and Gamble Company. Computational resources and services were provided in part by the Advanced Research Computing at the University of Michigan, Ann Arbor, and in part through the Extreme Science and Engineering Discovery Environment (XSEDE, grant number TGCHE140009).

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Stretch and Breakage of Wormlike Micelles Under Uniaxial Strain: A

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