We show that the average length $\u27e8L\u27e9$ of threadlike micelles in surfactant solutions predicted by fitting results of a mesoscopic simulation, the “pointer algorithm,” to experimental *G′*(ω), *G″*(ω) data, is longer than, and more accurate than, that from a scaling law that equates $\u27e8L\u27e9/le$ to the modulus ratio $G0/Gmin\u2032\u2032$. Here, *G*_{0} is the plateau modulus, $Gmin\u2032\u2032$ is obtained at the local minimum in *G*″, and $le$ is the entanglement length. The accuracy of the pointer algorithm is supported by the agreement of its predictions with results from a recent application of the slip-spring simulation method to threadlike micelles. Improved fits of the pointer algorithm to the slip-spring results are obtained for weakly entangled micelles (with an average number of entanglements of *Z* < 15) if the full spectrum of Rouse modes is included in the description rather than just the high-frequency modes included in an earlier version. For sodium laureth-1 sulfate and cocamidopropyl betaine in NaCl solutions, we find scaling relations for micelle length, the plateau modulus, and the persistence length that are in rough agreement with the predictions of mean field theory and with the modified scaling relation in which $\u27e8L\u27e9/le$ is raised to the 0.82 power, rather than unity, that we recommend as an improvement to the original scaling law.

## I. INTRODUCTION

Surfactant molecules are used in a variety of industrial and personal care products due to their drag-reduction properties and ability to encapsulate and solubilize hydrophobic molecules in water [1,2]. In solution, they self-assemble, at high enough concentrations forming long wormlike micelles that impart a distinct viscoelastic behavior to the solutions. Understanding the relationship between the microscopic and macroscopic features of these micellar solutions, for example, between the average micelle length and the macroscopic rheology, has remained a problem of particular interest, especially since such properties are very hard to measure except through their influence on rheology.

The first proposed method for determining the micelle length from rheology is the *Cates method* [3–5], which is based on the *Cates theory* [6] for the dynamics and linear rheology of entangled threadlike micelles. The Cates theory, adapted from the reptation theory for entangled polymers, takes the micelles to be contained in a “tube” formed by surrounding micelles, and to relax by diffusing (or “reptating”) along, and eventually out of, the tube. However, unlike polymers, micelles can also reversibly break and rejoin, and the Cates theory adds this mechanism to reptation to obtain its predictions of rheology.

The pointer algorithm is a numerical implementation of the Cates theory that simulates an ensemble of breaking and rejoining micelles with the exponential length distribution expected from random breakage [7]. It allows predictions to be obtained from the Cates theory over a broader range of conditions than allowed by the earlier analyses of the theory and allows some additional physics to be introduced. In the pointer algorithm, at each simulation time step, a micelle can slide randomly a precalculated distance along its tube, and the fraction of the micelle that remains unrelaxed is then updated. A fraction of these micelles will also randomly break or rejoin at each time step, as determined by the breakage/rejoining rate. The reptation process accounts for relaxation at low frequencies. At higher frequencies, other mechanisms become important, including Rouse and bending modes, which are added analytically to the relaxation modulus determined by the simulation. Given measured storage and loss moduli vs frequency, *G*′(*ω*) and *G*″(*ω*), micelle parameters, including the micelle length, can be determined from the pointer algorithm by iteratively adjusting them to give a best fit to *G*′(*ω*) and *G*″(*ω*). Full details of the pointer algorithm and the method of fitting its predictions to rheological data can be found in previous work [7,8].

In this work, we show that micelle lengths estimated from fits of the pointer algorithm to linear rheology significantly improve upon those obtained from the *Cates method,* which gives highly approximate estimates of lengths from the Cates theory. The Cates method uses a scaling law that relates the micelle average length $\u27e8L\u27e9$ to the ratio $G0/Gmin\u2032\u2032$ where *G*_{0} is the plateau modulus, and $Gmin\u2032\u2032$ is *G*″ at the frequency *ω*_{min} at which *G*″ is a local minimum. We argue here that the micelle length estimates from the pointer algorithm are superior than those from the Cates method, in the following ways: (1) the longer micelle lengths from the pointer algorithm are more consistent with the high viscoelasticity and viscosity exhibited by the surfactant solutions; (2) lengths from the pointer algorithm are more consistent with extrapolations of lengths obtained from dilute solution rheometry of the same surfactant; and (3) predictions of the pointer algorithm agree better with those from a more microscopic slip-spring simulation model. We also give a modification of the Cates scaling law that matches more closely the predictions of the pointer algorithm and that can be used to obtain more accurate estimates of micelle length.

## II. DETAILS OF THE CATES METHOD AND POINTER ALGORITHM

In the Cates theory, the stress relaxation of “living polymers,” such as surfactant micelles, is modeled using equations that account for reptation and reversible breakage and rejoining of micelles [6]. Later work [9] also considered the effect of contour-length fluctuations, or “breathing,” as well as Rouse modes, on the stress relaxation. In addition, Cates developed from his theory a scaling relationship linking rheology to average micelle length, namely, $G0G\u2032\u2032min\u223c\u27e8L\u27e9le$ where *l _{e}* is the “entanglement length,” the length of the micelle per entanglement, and other parameters are defined above. Since

*G*

_{0}and $Gmin\u2032\u2032$ are reasonably easy to obtain experimentally, if

*l*is known, and if we take the prefactor to be unity, this scaling relation provides a way to estimate the average micelle length, a property that can otherwise be difficult to determine. In a good solvent, the entanglement length is approximated by $le\u2245\xi 53/lp23$, where

_{e}*ξ*is the correlation length and

*l*is the persistence length. For “loosely entangled” micelles $(le/lp>1)$,

_{p}*ξ*can be estimated by $\xi \u223c(kBTG0)13$ [3], where, again, the prefactor is typically taken to be unity. Thus, if the persistence length is known from another experimental method, such as small-angle neutron scattering (SANS) [10], diffusing wave spectroscopy (DWS) [11], or rheo-optics [12], and

*G*

_{0}and $Gmin\u2032\u2032$ are determined from rheology, then $\u27e8L\u27e9$ can be calculated using the relationships above. Here, the use of the above scaling formulas for micelle lengths and other parameters will be referred to as the “

*Cates method*,” distinguishing it from the more general “Cates theory,” which gives a comprehensive model of dynamics and linear rheology of wormlike micelles.

Alternatively to the Cates *method*, the pointer algorithm (which is itself based on the Cates *theory*) can give the average micelle length through an iterative fitting procedure, as mentioned above. In this method, reptation, breakage, and rejoining are simulated for an ensemble of micelles, and then high-frequency relaxation mechanisms—Rouse and bending modes—are added analytically. Iterating to minimize the error between the simulated and experimental *G*′ and *G*″ curves, micelle parameter values are determined that best fit experimental data. The pointer algorithm offers the following advantages over the Cates method:

The average micelle length $\u27e8L\u27e9$ is not calculated directly from a single rheological feature (i.e., $G0/Gmin\u2032\u2032$) at a single frequency

*ω*but is obtained by fitting the entire frequency dependence of the rheology, making the result less sensitive to errors in either experimental data or the Cates theory at this frequency. Since the micelle length should have its greatest influence in the terminal region well below the frequency_{min}*ω*, fitting the rheology over the entire frequency range should be a more robust method of extracting $\u27e8L\u27e9$._{min}The plateau modulus

*G*_{0}is also a fitting parameter whose value is determined simultaneously with $\u27e8L\u27e9$ from the same fitting of the pointer-algorithm predictions to data. This differs from work by Cates and Turner [13], where*G*_{0}was estimated by extrapolating the linear region of a “Cole–Cole” plot to high frequencies. Other methods used in the literature to obtain*G*_{0}include fitting the semicircular region of a Cole–Cole plot [14], fitting*G*′ and*G*″ to one or two Maxwell elements [5], or using the value of*G*′ at the frequency where*G*″ has its minimum [11]. However, for moderately to lightly entangled micelles where*G*′ never clearly flattens but continuously increases with frequency, these methods likely underestimate*G*_{0}.In the pointer algorithm, the plateau modulus is related to the correlation length by a “crossover formula” [8] that allows for consideration of both tightly and loosely entangled micelles and the intermediate region between these limits. Since most solutions of entangled micelles are in this crossover region while the Cates method assumes that the micelles are in the loosely entangled region, the pointer algorithm treats entanglements more realistically than does the Cates method. Even in the loosely entangled region, the pointer algorithm introduces a prefactor A (discussed later) into the scaling law $\xi \u223cA(kBT/G0)13$, whose value is derived from well-established correlations for polymer solutions and melts [8].

If data are available at a high enough frequency to encompass a second crossover of

*G*′ and*G*″, the persistence length can also be determined as a fitting parameter in the pointer algorithm simulations; otherwise, the persistence length is an input parameter. The frequency required to reach the second crossover is typically up to 10^{5}rad/s, usually obtained from diffusing-wave spectroscopy (DWS).The accuracy of the pointer algorithm can be validated by showing (in what follows) that its predictions of

*G*′ and*G*″ match, using the same parameter values, the results from a more microscopic slip-spring simulation, details of which can be found elsewhere [15].

## III. EXPERIMENTAL MATERIALS AND METHODS

The experimental data are for mixed surfactant solutions containing sodium laureth-1 sulfate (SLE1S) and cocamidopropyl betaine (CAPB) of varying concentration with added NaCl such that the total sodium ion concentration, including the counterions of SLE1S, is held constant at [Na^{+}] = 0.70M. The weight ratio of SLE1S to CAPB is 8.65, and the total volume fraction of surfactant lies in the range of *φ* = 0.015–0.09. These solutions were prepared in D_{2}O with 0.5 wt. % of 630 nm polystyrene latex beads for DWS analysis for the high frequency portion of the rheological spectrum. The added beads do not measurably affect the zero shear viscosity. The high frequency data measured using DWS are shifted vertically to merge smoothly with the mechanical rheology [7].

A second set of data contains only SLE1S (*φ* = 0.067–0.14) and NaCl (3.1 wt. %) in D_{2}O. The rheology for these solutions is obtained from mechanical rheometry and no high frequency data are available. All data for both sets of solutions are taken at 25 °C.

Experimental values for the specific viscosity (defined as $\eta sp=\eta 0\u2212\eta s\eta s$, where *η*_{0} is the zero-shear solution viscosity and *η _{s}* is the viscosity of the pure solvent) are also obtained along with the modulus curves. At low concentrations, when the viscosity is low, the zero shear viscosity

*η*

_{0}is found by steady-state shear using a single-wall, Couette geometry and shear rates from 20 to 0.05 s

^{−1}. The zero shear viscosity is extracted in the limit of the low-shear viscosity plateau. At high enough concentrations,

*η*

_{0}is extracted from the complex viscosity plateau in the low frequency limit of a frequency sweep experiment. The solvent viscosity is determined using a rolling ball rheometer.

## IV. RESULTS AND DISCUSSION

### A. Anomalously small micelle lengths from rheology using the Cates method

Although the simplicity of the Cates method makes it a relatively fast way to determine the micelle length, in some cases, as discussed below, the calculated length is much shorter than that might be expected based on the viscosity of the solution. Even more suspiciously, the micelle lengths predicted by the Cates method can sometimes actually decrease as the surfactant concentration increases, the opposite of the expected behavior predicted by simple laws of mass action (unless somehow concentration-dependent electrostatic interactions reverse this dependence). This anomalous inverted relationship between micelle length and concentration can be seen for our series of solutions containing mixed SLE1S and CAPB surfactants. Figure 1(a) shows the rheological data for a few of these solutions. As the surfactant concentration increases, *G*′ and *G*″ increase and the minimum in *G*″ (which is used in the Cates method) appears and becomes deeper; the terminal relaxation time, calculated as the inverse of the first crossover frequency, increases. All of these features indicate that wormlike micelles are present that, at 5% and 9% concentrations, have become long and entangled. This conclusion is reinforced by a plot of the specific viscosity *η _{sp}* vs volume fraction

*φ*in Fig. 2. Figure 2 shows data for SLE1S solutions, without CAPB, and with constant added NaCl concentration (3.1 wt. %), but the rheology for these solutions, shown in Fig. 1(b), is similar to that of the mixed surfactant solutions [Fig. 1(a)], for which wormlike micelles are also expected to be present. Note in Fig. 2 that

*η*rises slowly with

_{sp}*φ*at low concentrations below about 1%, followed by a much steeper rise at higher concentrations. Figure 2 has a very similar appearance to corresponding plots of viscosity vs concentration for polymer solutions, such as that shown in Figs. 8.11 and 9.10 of Rubinstein and Colby [16]. For polymers, the region of steep increase in viscosity corresponds to a regime of well entangled micelles, suggesting that the micelles are long enough to be entangled at concentrations above 1%, and become longer still at concentrations approaching 10%.

However, Table I(a) shows that the micelle lengths calculated from the mixed surfactant data in Fig. 1(a) using the Cates method are very short, ≤200 nm, and decrease to a length of ≈100 nm, with increasing concentration. We use in these calculations a previously determined persistence length of 70 nm [17] and approximate the plateau modulus as $Gmin\u2032$, the value of *G*′ at the minimum in *G*″. The resulting micelle lengths are shorter than expected, having, on average, only $Z=L/le=3\u22126$ entanglements, which would make them lightly entangled, and in entangled polymers, the rapid 3.4 power-law increase in viscosity with *Z* typically occurs only for *Z* > 2–4 [16]. Given that the zero-shear viscosities are thousands of times higher than the viscosity of water, and that the inverse of the low-frequency crossover frequency indicates a relaxation time near 1 s, it seems unlikely that micelles would be that short and still show such high values of the solution viscosity and relaxation time.

Φ . | $G0\u2248Gmin\u2032(Pa)$ . | $Gmin\u2032\u2032(Pa)$ . | l (nm)
. _{e} | 〈L〉 (μm)
. | η_{0} (Pa s)
. |
---|---|---|---|---|---|

(a) Calculated from the Cates method for SLE1S/CAPB at [Na^{+}] = 0.7M
. | |||||

0.015^{a} | 0.526 | ||||

0.020^{a} | 1.14 | ||||

0.025 | 8.48 | 3.55 | 84.9 | 0.202 | 2.82 |

0.030 | 13.0 | 5.07 | 66.9 | 0.172 | 4.43 |

0.040 | 28.0 | 7.60 | 43.7 | 0.161 | 10.5 |

0.050 | 47.3 | 10.7 | 32.7 | 0.145 | 17.0 |

0.050 | 49.9 | 9.72 | 31.7 | 0.163 | 21.7 |

0.060 | 69.9 | 13.8 | 26.3 | 0.133 | 26.1 |

0.070 | 95.9 | 16.6 | 22.1 | 0.127 | 38.5 |

0.075 | 106 | 18.9 | 20.8 | 0.117 | 41.2 |

0.079 | 125 | 22.7 | 19.0 | 0.105 | 43.3 |

0.089 | 150 | 26.2 | 17.2 | 0.098 | 52.9 |

(b) Calculated from the Cates method for SLE1S at NaCl = 3.1 wt. % | |||||

0.067 | 45.6 | 23.2 | 33.3 | 0.066 | 5.45 |

0.080 | 105.2 | 25.0 | 20.9 | 0.088 | 17.5 |

0.11 | 248.1 | 27.3 | 13.0 | 0.118 | 67.1 |

0.14 | 370.3 | 30.4 | 10.4 | 0.127 | 113 |

Φ . | $G0\u2248Gmin\u2032(Pa)$ . | $Gmin\u2032\u2032(Pa)$ . | l (nm)
. _{e} | 〈L〉 (μm)
. | η_{0} (Pa s)
. |
---|---|---|---|---|---|

(a) Calculated from the Cates method for SLE1S/CAPB at [Na^{+}] = 0.7M
. | |||||

0.015^{a} | 0.526 | ||||

0.020^{a} | 1.14 | ||||

0.025 | 8.48 | 3.55 | 84.9 | 0.202 | 2.82 |

0.030 | 13.0 | 5.07 | 66.9 | 0.172 | 4.43 |

0.040 | 28.0 | 7.60 | 43.7 | 0.161 | 10.5 |

0.050 | 47.3 | 10.7 | 32.7 | 0.145 | 17.0 |

0.050 | 49.9 | 9.72 | 31.7 | 0.163 | 21.7 |

0.060 | 69.9 | 13.8 | 26.3 | 0.133 | 26.1 |

0.070 | 95.9 | 16.6 | 22.1 | 0.127 | 38.5 |

0.075 | 106 | 18.9 | 20.8 | 0.117 | 41.2 |

0.079 | 125 | 22.7 | 19.0 | 0.105 | 43.3 |

0.089 | 150 | 26.2 | 17.2 | 0.098 | 52.9 |

(b) Calculated from the Cates method for SLE1S at NaCl = 3.1 wt. % | |||||

0.067 | 45.6 | 23.2 | 33.3 | 0.066 | 5.45 |

0.080 | 105.2 | 25.0 | 20.9 | 0.088 | 17.5 |

0.11 | 248.1 | 27.3 | 13.0 | 0.118 | 67.1 |

0.14 | 370.3 | 30.4 | 10.4 | 0.127 | 113 |

^{a}

These datasets have no minimum in *G*″ (see 1.5% solution in Fig. 1).

To make these arguments more quantitative, we note first, that, the reptation times for the SLE1S/CAPB micelles predicted by the Cates method are approximately 0.001 s, much shorter than the relaxation times (∼1 s) indicated by the terminal crossover frequency. The reptation time is calculated from $\tau rep=2\u27e8L\u27e93\pi 2\alpha D0$, where $\alpha \u2261le/lp$ is the ratio of entanglement length to persistence length, the diffusivity is given by $D0=kBT\u03c2$ with the drag coefficient for a cylinder given by $\u03c2=2\pi \eta sln\u2061(\xi /d)$, and $\eta s$ is the solvent viscosity (around 1 cP), *d* is the micelle diameter (4 nm), and $\xi $ is the correlation length (30–80 nm). Additionally, the experimental zero-shear viscosities at high concentrations in Table I(a) are orders of magnitude higher than the viscosities we infer for solutions with micelles less than a micrometer in length from the slip-spring model, as shown later. The slip-spring model has been well validated by Likhtman for slightly and densely entangled polymers, and so estimates of viscosity from the slip-spring model should be reasonably accurate, and yet are much lower than the experimental values if micelles are less than a micrometer in length. For example, in one of our comparisons with the slip-spring model, a solution with 0.84 μm-long micelles at a surfactant volume fraction of 0.01, has a predicted zero-shear viscosity of 0.075 Pa s, about an order of magnitude smaller than that of even the most dilute solution in Table I(a). While perhaps not definitive, these arguments strongly suggest that the micelle lengths estimated by the Cates method are too short.

In fact, Granek and Cates [9] first proposed the relationship $G0G\u2032\u2032min\u223c\u27e8L\u27e9le$ as a scaling relationship and simply chose a prefactor of unity to estimate specific values of the micelle length from the literature data. They also remarked that micelle lengths will be underestimated if the timescale for micelle breakage and rejoining is on the same order of magnitude as the entanglement time, as seems to be the case for the experimental data discussed above.

Exploiting the similarity between Fig. 2 and the corresponding data for polymer solutions, we can use the known scaling laws relating polymer length to viscosity, along with neutron scattering data in the dilute regime, to help estimate the lengths of threadlike micelles. Of course, threadlike micelles dynamically break and fuse, while polymers do not. Since for fixed average micelle lengths, breakage/scission *decreases* the viscosity of the micellar solution, by neglecting breakage/scission, the micelle length estimated from the measured viscosity without accounting for breakage/scission will be shorter than the actual length, and thus will be a *lower bound* on micelle length. The actual micelle length will be significantly longer especially at higher concentrations.

First, note that at low concentrations below around 1%, the observed scaling of viscosity with concentration in Fig. 2 is $\eta sp\u221d\phi 1.7$, while at concentrations above around 2%, the scaling is $\eta sp\u221d\varphi 5.0$. From the dilute polymer theory, we expect $\eta sp\u221d\nu Rg3$, where *ν* is the micelle number density and *R _{g}* is the average radius of gyration of the micelle. For dilute polymers, the chain length is constant and so $\eta sp\u221d\nu \u221d\phi $. However, the micelle length is expected to increase with surfactant concentration. Assuming a mean-field scaling of $\u27e8L\u27e9\u221d\phi 0.5$, and a good solvent scaling of $Rg\u221d\u27e8L\u27e90.6$, we obtain $Rg\u221d\phi 0.3$. Since $\nu \u221d\phi /\u27e8L\u27e9\u221d\phi 0.5$, $\eta sp\u221d\nu Rg3$ implies that $\eta sp\u221d\phi 1.4$, somewhat weaker than the scaling observed in Fig. 2, $\eta sp\u221d\varphi 1.7$. A scaling closer to the observed scaling $\eta sp\u221d\varphi 1.7$ would be obtained if the nonmean-field, excluded-volume, scaling [18] $\u27e8L\u27e9\u221d\phi 0.6$ holds, which leads to $\eta sp\u221d\phi 1.6$.

At high concentrations, the predicted viscosity scaling from the Cates model in the fast-breakage limit, with $\u27e8L\u27e9\u221d\phi 0.5$, is $\eta sp\u221d\varphi 3.5$ [3], which is lower than that seen in Fig. 2, $\eta sp\u221d\varphi 5.0$. The higher power law of the scaling in Fig. 2 is likely to be partly due to micelles lying outside the fast-breakage limit, and perhaps partly due to more rapid growth of micelle length than $\u27e8L\u27e9\u221d\phi 0.5$. The average micelle length for the SLE1S solutions, at the same salt concentration (3.1 wt. %), and at the lowest surfactant concentrations of 0.1%–0.25%, was determined from small angle neutron scattering (SANS) to be around 200 nm [10]. Assuming a constant scaling law $\u27e8L\u27e9\u221d\varphi 0.5$ across the range of surfactant concentrations, at the concentration 1% where the viscosity shows a transition from dilute to entangled, the micelle length should be 400–600 nm, and at 10% it should be 1–2 μm. For surfactant concentrations usually studied rheologically (around 10%), this again indicates that the micelles should be on the order of micrometers, not hundreds of nanometers or less. We also note from Table I(b) that while the Cates method correctly predicts increasing micelle lengths as surfactant concentration increases, the lengths are much less than a micrometer and are shorter than the lengths from SANS (∼200 nm), even at concentrations 100 times higher than those at which SANS measurements were made.

### B. Review of selected literature measurements of micelle length

In other works, micelle lengths were estimated over a range of surfactant concentrations through the use of SANS, cryo-TEM, and rheology. Afifi *et al.* [19] measured the linear rheology and a zero shear viscosity of 10–100 Pa s, or $\eta sp\u2248104\u2212105$ for a wormlike micellar solution of 10 wt. % poly(oxyethylene) cholesteryl ether (ChEO_{10}) and varying amounts of lipophilic monoglycerides. They then imaged micelles in a 5-fold dilution of this solution using cryo-TEM and performed SANS experiments on a 10-fold dilution. From SANS, the micelles were inferred to be ellipsoids or cylinders with elliptical cross section and “lengths” between 16 and 80 nm, which, according to the authors, may be either contour lengths or persistence lengths, indicating either small micellar aggregates or semiflexible wormlike micelles of unknown lengths. For the 5-fold dilution from this concentration, cryo-TEM revealed long wormlike micelles greater than 1 μm in length, although it must be acknowledged that the disruptive preparation methods for cryo-TEM could distort micelle length distributions. As we argued above for SLE1S/CAPB micelles, the high viscosity of the 10% solution (i.e., or $\eta sp\u2248104\u2212105$) suggests that micelle lengths for this solution are likely significantly longer than the SANS estimates for the diluted sample. Given the combined findings from multiple experimental methods, it seems possible that small ellipsoidal or short wormlike micelles exist at low surfactant concentrations, then grow into long wormlike micelles with lengths around 1 μm or more at higher surfactant concentrations.

In an unusually thorough study, Helgeson *et al.* [5] found good agreement between micelle contour lengths from rheology and those from SANS for solutions of CTAB from 40 to 100 mM (i.e., around 1%–3% by volume) in NaNO_{3} at three mole ratios (1, 2, and 3) of salt to surfactant and temperatures ranging from 25 to 45 °C. Other micelle parameters were also obtained. Importantly, the persistence length *l _{p}* for each solution was obtained from the stress-optic coefficient in flow birefringence. The micelle breakage time $\tau br=1/\omega min$ was taken as the inverse of the frequency at which

*G*″ reached a local minimum (which we find from the pointer algorithm not to be a very accurate estimate). The modulus

*G*

_{0}was obtained from Maxwell fits to the rheology, and this combined with

*l*was used to calculate the entanglement length

_{p}*l*. To do so, the formula $\xi =(kBT/G0)1/3$ for the mesh size was combined with the expression $\xi =le3/5lp2/5$, allowing

_{e}*l*to be obtained. The Granek and Cates expression $G0G\u2032\u2032min\u223cLle$ was then used to obtain $\u27e8L\u27e9$. The micelle lengths were obtained from SANS for many solutions and also from rheology for a half dozen of them; agreement between the two measurement was within 10%, and included lengths ranging from 200 to 900 nm, with the highest values at high surfactant and salt concentrations and low temperatures, as expected. Based on the rheology and viscosities of these solutions, and our experience inferring lengths from the pointer algorithm, these lengths seem to be too short. The dependence of specific viscosity on concentration for these solutions showed three scaling regimes corresponding to dilute, semidilute, and concentrated (fully entangled) solutions, very similar to those for polymers, again implying that the micelles are quite long in the entangled regime.

_{e}Additionally, the requirement of multiple Maxwell modes to approximate the data at lower salt and surfactant concentrations indicates that the micelles are not in the fast breakage limit. In that case, *τ _{br}* >

*τ*and

_{rep}*τ*≈

_{R }*τ*, where

_{rep}*τ*is the breakage time,

_{br}*τ*is the reptation time, and

_{rep}*τ*is the terminal relaxation time, all for a micelle of average length. But for a 60 mM CTAB and 120 mM NaNO

_{R}_{3}solution studied by Helgeson

*et al.*, it is found that

*τ*=

_{R}*ω*

_{c}_{1}

^{−1}= 0.04 s and

*τ*= 1.4 × 10

_{rep}^{−4}s, as calculated from the reported parameters ($\u27e8L\u27e9=225nm$,

*l*= 221.5 nm, and

_{e}*l*= 32 nm). The terminal relaxation time

_{p}*τ*is the inverse of the first crossover frequency,

_{R}*ω*

_{c}_{1}, and the reptation time is calculated from $\tau rep=2\u27e8L\u27e93\pi 2\alpha D0$, where the diffusivity is given by $D0=kBT\u03c2$ with the drag coefficient for a cylinder given by $\u03c2=2\pi \eta sln\u2061(\xi /d)$ [8]. This gives a reptation time two orders of magnitude shorter than the relaxation time from rheology, suggesting again that the micelle length used to estimate this reptation time is too short.

For this solution, the micelle length of 225 nm comes from fitting SANS data, but the actual micelle length for this and more concentrated solutions is likely longer than what that can be accurately resolved by SANS. The smallest *q* value for which Helgeson *et al.* report SANS data is about *q*_{min} = 0.03 nm^{−1}, corresponding to $rmax=\pi 0.03nm\u22121=105nm$. Even when micelle radii of gyration fall within this restriction, overlap of micelles leads to the screening of scattering signal, making longer micelle lengths hard to infer by SANS. Others who have performed SANS measurements on nondilute surfactant solutions have concluded that the micelles are too long for their lengths to be accurately determined from SANS. Work by Francisco *et al.* [20] that examined the effect of cosolutes on CTAB/NaSal micelles found a large variation in the rheology and zero-shear viscosity, up to two orders of magnitude in the crossover frequency and viscosity. The SANS results, however, superimpose exactly for all cosolutes, and the authors comment that for micelles greater than a few hundred nanometers in length, a change in length would not be detectable by SANS. A different study of saponin micelles came to a similar conclusion. There, Peixoto *et al.* [21] found relaxation times greater than 100 s from rheology but determined from SANS only that the micelle length was greater than the *q* range of their SANS experiments. Finally, Croce *et al.* [22,23], studying erucyl bis(hydroxyethyl) methylammonium chloride (EHAC) micelles, measured viscosities in the range of what is typically reported for solutions of wormlike micelles (1–100 Pa s). At these concentrations, they note that the micelles are longer than can be measured within their *q* range and that there was no model to usefully fit the whole scattering curve.

As an example of the longer micelle length extracted from our experimental data using the pointer algorithm than is obtained from the Cates scaling law, Fig. 3 shows a fit of rheology data for *ϕ* = 0.05 SLE1S/CAPB, [Na^{+}] = 0.7M, giving a micelle length of 〈*L*〉 = 3.0 μm in contrast to the length of 0.145 μm calculated from the Cates method [Table I(a)]. A length of 3.0 μm seems reasonable considering that (1) the solution viscosity is high, indicating that the micelles are well entangled, that (2) extrapolation of micelle lengths from SANS yields lengths well over 1 μm, and that (3) long ∼1 μm micelle lengths are observed in cryo-TEM results on other solutions with similarly high viscosities discussed above. All of these considerations suggest micelle lengths that are on the order of micrometers for surfactant solutions at our experimental concentrations.

### C. Comparisons with slip-spring simulations

To further validate the micelle length obtained from the pointer algorithm, we next compare rheology generated by the pointer algorithm to results from a more highly resolved slip-spring simulation model. The slip-spring model, originally developed for solutions of entangled polymers as an alternative to the tube model, and which is regarded as quantitatively accurate for polymer solutions and melts, treats each polymer as a bead-spring Rouse chain with entanglements represented by slip-links attached to the chain [24,25]. To adapt this model to micelles, chains were allowed to reversibly break and rejoin [15]. To make the slip-spring simulations numerically tractable, the micelles must be relatively short and, therefore, only weakly entangled with an average number of entanglements *Z* = 3 or 5. Stresses in this limit are often too low to be measurable for experimental solutions, but since the slip-link simulations have proved to be quite accurate for entangled polymers, they can serve to test the accuracy of the rheological predictions of the pointer algorithm, thus justifying their use for estimating micelle length.

The slip-spring model naturally captures reptation, contour length fluctuations, constraint release, and Rouse modes. The pointer algorithm imposes the effects of these phenomena on rheology in a coarser-grained way, but only the short range (fast) Rouse modes were originally included. Slower, “longitudinal” Rouse modes were omitted from the original pointer algorithm because they were assumed to be negligible relative to the stresses produced by entanglements. (Additionally, in the comparisons to the slip-spring model, bending modes usually present in the pointer algorithm are dropped since the slip-spring simulations do not have bending modes.) In previous work [15], we found good agreement between the predictions of the slip-spring and the original pointer algorithm at low frequencies, but pronounced deviation starting at intermediate and high frequencies; this deviation could be alleviated by an inclusion of the full spectrum of Rouse modes rather than just the high-frequency ones. The inclusion of the full spectrum of Rouse modes can be justified by the weakness of the entanglements in the slip-spring simulations, since *Z* = 3 or *Z* = 5 represents only three or five entanglements, which is not enough to create a distinct “tube” able to suppress the slow Rouse modes.

Even for a highly entangled solution, there should be slower, longer-range modes present, but restricted by the tube to one-dimensional relaxation and hence having reduced amplitude. Within the tube model, the one dimensionality of these modes is imposed by reducing their amplitude by a factor of 5 relative to their original amplitude in the absence of the tube. These reduced-modulus slow Rouse modes are called “longitudinal” Rouse modes. Thus, for weakly entangled micelles with *Z* = 3 or 5 (where, we use “*Z*” as a shorthand for the average value 〈*Z*〉), we found in previous work that the full Rouse spectrum of modes should be added to the predictions of the pointer algorithm to be able to match results from the slip-link model. For high degrees of entanglement, the longitudinal Rouse modes, with their reduced magnitude, should theoretically be included, but were assumed to have negligible effect and so were neglected in the original pointer algorithm. However, because we found that these slower modes are clearly important for weakly entangled micelles and must be included, we wish to check whether the longitudinal Rouse modes (i.e., with reduced amplitude) need to be included even for higher levels of entanglement. Thus, there are three options for choice of Rouse modes, shown mathematically below in Eqs. (1) and (2), respectively, where the first terms in Eq. (1) are the longitudinal Rouse modes. In these equations, *ϕ _{i}* and

*Z*are the volume fraction and number of entanglements respectively of micelles of length

_{i}*i*,

*p*is the mode number, and

*τ*is the entanglement time,

_{e}Figure 4 compares results from slip-spring simulations to those from the pointer algorithm, using each of the three options for Rouse modes, namely, (1) fast Rouse modes only [second term only in Eqs. (1a) and (1b)], (2) a “fractionated” Rouse spectrum including fast and longitudinal Rouse modes [both terms in Eqs. 1(a) and 1(b)], or (3) full Rouse modes [Eqs. (2a) and (2b)]. The calculations are carried out for solutions of micelles with a mean length of 0.84 or 1.4 μm, containing, respectively, 3 or 5 entanglements with an entanglement length *l _{e}* of 280 nm; a micelle persistence length of 20 nm, resulting in a semiflexibility parameter

*α*=

*l*/

_{e}*l*of 14; and ratio of breakage to reptation time

_{p}*ζ*= 100. These comparisons show that adding longitudinal Rouse modes improves the comparison marginally, but using a full Rouse spectrum gives us good agreement with the slip-spring simulations across the whole frequency range. It, therefore, seems that at low numbers of entanglements, the entanglement network is not fully formed and at short length scales, the tube has no confining effect, leading to relaxation by unfractionated Rouse modes. Thus full Rouse modes give the best overall agreement to the slip-spring model. Similar results are obtained when

*Z*< 5 for other values of

*ζ*.

We next investigate at what point the tube has enough entanglements to suppress the longitudinal Rouse modes, as originally assumed. To do this, we compare fits of pointer algorithm predictions to SLE1S/CAPB experimental data using our three different options for Rouse modes discussed above. Two sample experimental datasets are fit by the pointer algorithm predictions in Fig. 5, with bending modes included. We find that for all solutions, adding additional Rouse modes results in better fits to the experimental data, particularly beyond the maximum in *G*″, where fits with only fast Rouse motion tend to underestimate the moduli. By modifying the Rouse modes, we can fit the experimental data well across all frequencies. At lower concentrations where *Z* = 7 [Fig. 5(a)], adding full Rouse modes improves the predictions at frequencies above the first crossover. At a higher concentration with *Z* = 30 [Fig. 5(b)], including full Rouse modes no longer results in a good fit to the experimental data, which can be seen particularly near the minimum in *G*″. From the slip-spring simulation data, with *Z* = 3 and 5, we determined that simulations with full Rouse modes best represent the high-frequency relaxation for small *Z*. All these results taken together are consistent with the progressive formation of a confining tube as the number of entanglements per micelle increases. Based on our fits, the transition from full to fractionated (fast and longitudinal) Rouse modes seems to occur at or slightly below around 15 entanglements per micelle. At no point does it appear that the tube entirely suppresses the influence of longitudinal Rouse modes, contrary to the original pointer algorithm model.

### D. Micelle parameters obtained by the pointer algorithm

By fitting a series of experimental datasets using the pointer algorithm, we can determine how the micelle parameters vary with the surfactant concentration. The data we consider here are for the SLE1S and CAPB solutions with NaCl added to maintain a constant sodium ion concentration (while the surfactant concentration varies). Example fits can be found in Figs. 3 and 5. Figure 6 shows selected micelle parameters extracted from the experimental data using our various treatments of the Rouse modes. From Fig. 6, the parameters most sensitive to the treatment of the Rouse modes are *τ _{br}*, which varies by about a factor of three, and $\u27e8L\u27e9$, which varies by a factor of two. However, in all cases, the micelle lengths obtained are on the order of micrometers, not hundreds of nanometers or less. Thus, the conclusion that micelle lengths are on the order of microns is robust to the choice of how the Rouse modes are treated. We note that fits are better when we include the slower Rouse modes. Table II gives our calculated scaling laws along with their mean-field theoretical values in the fast-breakage limit and literature values drawn primarily from experimental correlations. For the case in Table II labeled “full (

*Z*< 15) to fractionated Rouse modes,” (third column), when calculating scaling laws we assume a transition from full Rouse modes at low

*Z*< 15 to a combination of fast and longitudinal Rouse modes for

*Z*≥ 15. Overall, we find that the scaling exponents (except the one for

*τ*) agree fairly well with their theoretical values as well as with literature values. Scaling exponents for the dependence of plateau modulus on surfactant concentration similar to those obtained by our fitting have been reported in the literature—1.85 for CTAB and KBr [3], 2.18 for CPyCl/NaSal and NaCl [14], and 2.12 for CTAC/NaSal and NaCl [26]. For the micelle length, Berret

_{br}*et al.*[14] used a combination of rheology and light scattering to find a scaling relation for micelle length as a function of surfactant concentration with a power of 0.24 or 0.36, though they did not explicitly determine the micelle length. Reanalysis of their data resulted in a power law exponent of 0.60 [27]. These values are similar to the theoretical value as well as to the range of values calculated from fits of our data by the pointer algorithm. The greatest deviation from theory occurs for the breakage time, where the theoretical scaling exponent is simply the negative of the exponent for micelle length, based on the assumption that the product $\tau br\u27e8L\u27e9$ is constant (an assumption not made when using the pointer algorithm). The persistence length from fits by the pointer algorithm is approximately constant at high concentrations, as shown in Fig. 6(d), consistent with theory, but then appears to decrease at lower concentrations. Previous work [7] at lower surfactant concentrations reports persistence lengths of around 20–30 nm, which is consistent with our finding of a lower persistence length at low concentrations.

Parameter . | Fast Rouse modes only . | Full (Z < 15) to fractionated Rouse modes
. | Fractionated Rouse modes . | Mean field [3] . | Literature [3,14,26,28, and 29] . |
---|---|---|---|---|---|

τ _{br} | −1.1 | −0.84 | −1.5 | −0.5 | |

〈L〉 | 0.48 | 0.28 | 0.64 | 0.5 | 0.24–0.6 |

G_{0} | 1.77 | 1.83 | 1.69 | 2.2 | 1.5–2.18 |

l _{p} | ∼0 | ∼0 | ∼0 | 0 | |

l _{e} | −0.57 | −0.65 | −0.56 | −0.75 | From −0.65 to −0.72 |

η_{0} | 2.9 | 2.9 | 2.9 | 3.5 | 2.42–5 |

Parameter . | Fast Rouse modes only . | Full (Z < 15) to fractionated Rouse modes
. | Fractionated Rouse modes . | Mean field [3] . | Literature [3,14,26,28, and 29] . |
---|---|---|---|---|---|

τ _{br} | −1.1 | −0.84 | −1.5 | −0.5 | |

〈L〉 | 0.48 | 0.28 | 0.64 | 0.5 | 0.24–0.6 |

G_{0} | 1.77 | 1.83 | 1.69 | 2.2 | 1.5–2.18 |

l _{p} | ∼0 | ∼0 | ∼0 | 0 | |

l _{e} | −0.57 | −0.65 | −0.56 | −0.75 | From −0.65 to −0.72 |

η_{0} | 2.9 | 2.9 | 2.9 | 3.5 | 2.42–5 |

### E. A new Cates-like correlation for micelle length

Finally, we attempt to recover a Cates-like scaling for the micelle length, i.e., resembling $G0G\u2032\u2032min\u223c\u27e8L\u27e9le$, but matching our micelle lengths fitted from the pointer algorithm. Fitting rheological data for a variety of mixed SLE1S/CAPB and pure SLE1S solutions with NaCl using the pointer algorithm with only fast Rouse modes, we find that by plotting the experimental values for $Gmin\u2032/Gmin\u2032\u2032$ rather than $G0/Gmin\u2032\u2032$ against $\u27e8L\u27e9/le$ (Fig. 7), we obtain a best-fit relationship $G\u2032minG\u2032\u2032min=0.225(\u27e8L\u27e9le)0.99\xb10.03$, which is essentially the Cates scaling with a prefactor less than unity and that uses the experimental value of $Gmin\u2032$ instead of an estimated plateau modulus. If we use results from full Rouse modes for *Z *< 15 and from fast and longitudinal Rouse modes for *Z* > 15, we obtain $G\u2032minG\u2032\u2032min=0.317(\u27e8L\u27e9le)0.82\xb10.05$, which more closely resembles a refinement of the Cates scaling law by Granek [27] that includes contour length fluctuations as well as Rouse modes and predicts a 0.8 power-law exponent. The similarity between our exponent of 0.82 and the 0.8 power-law exponent derived by Granek may be significant since the Granek correction includes contour length fluctuations which are also part of the pointer algorithm. However, the pointer algorithm also includes constraint release and bending modes, which are lacking from the Granek calculation, and so the similarity of the exponents may be fortuitous.

When we used the original Cates scaling law to calculate average micelle lengths for our experimental data in Table I(a) and Table I(b), we found that the lengths were too short and for the data in Table I(a), the micelles were predicted to shrink as surfactant concentration increased. Our modified Cates scaling relation derived from the pointer algorithm, however, is consistent with micelles growing as surfactant concentration increases and yields micelles that are at least 10–20 times longer than calculated from the original Cates method. Both the micelle growth and lengths given by the modified scaling better agree with the high solution viscosities. For the solutions with a constant concentration of added salt [Fig. 1(b) and Table I(b)], the micelle lengths are also longer than the lengths determined from SANS at lower concentrations, not shorter as is found from the Cates scaling law.

The increased micelle lengths come partially from the crossover formula in the pointer algorithm which accounts for a transition between loosely and tightly entangled micelles and an additional factor of 9.75 in the formula $G0=9.75kBT/\xi 3$ for loosely entangled micelles, rather than the simple scaling law $G0=kBT/\xi 3$ typically used along with the Cates scaling law to determine micelle length. The factor of 9.75 is based on equations for the packing of polymer chains in good solvent, with the full derivation given in [8]. Inclusion of the prefactor 9.75 increases *ξ* by 9.75^{1/3} = 2.14, which, in turn, increases *l _{e}* and thus $\u27e8L\u27e9$ by 2.14

^{5/3}= 3.5. For more tightly entangled micelles, the effect of the prefactor lessens. The additional prefactor of 0.317 in the modified Cates relationship further increases the micelle length by (0.317

^{−1})

^{1/0.82}= 4.1. Assuming loosely entangled micelles, the cumulative effect is to increase the micelle length by a factor of about 15 compared to the original Cates method.

The Cates model in the “fast breakage limit” also allows the derivation of a relationship for the terminal relaxation time—$\tau R=(\tau br\tau rep)0.5$. If we allow the exponents for *τ _{br}* and

*τ*to vary separately but constrain them to sum to unity so that the expression remains dimensionally correct, we calculate $\tau R=0.48\tau br0.58\tau rep0.42$ and $\tau R=0.48\tau br0.63\tau rep0.37$ for simulations with fast Rouse modes only and modified Rouse modes, respectively. These scaling laws indicate that the pointer algorithm results are in general agreement with the Cates model scaling for the relaxation time but have a stronger dependence on the breakage time. Thus, for both the micelle length $\u27e8L\u27e9$ and time scale $\tau R$, we recover scaling relationships that are similar with those from the Cates theory, but with somewhat different exponents and prefactors.

_{rep}We now use the correlations established above, in particular, the new Cates-like scaling law

to predict the micelle length for the 60 mM CTAB/120 mM NaNO_{3} solution from Helgeson *et al.* [5]. This prediction enables us to further test the pointer algorithm and compare the micelle length obtained from the pointer algorithm with the published value. The input parameters for the pointer algorithm are taken to be *G*_{0} = 22.2 Pa, $\u27e8L\u27e9=1.2\mu m$, *l _{p}* = 32 nm, and ζ = 200. The micelle length is obtained using Eq. (3), and details of how the other parameters are estimated for this solution, from information provided in the paper of Helgeson

*et al.*, can be found in the Appendix. Figure 8 compares experimental data to predictions of the pointer algorithm obtained from the above micelle parameters and from those published in Helgeson

*et al.*The most significant difference is that our correlation in Eq. (3) gives a micelle length of 1.2 μm, while in Helgeson

*et al.*, the reported length is 225 nm, obtained from the Cates correlation.

We can see in Fig. 8 that the published parameter values, when used in the pointer algorithm, result in *G*′ and G″ curves that relax one to two orders of magnitude more quickly than in the experiments, as might have been expected from the calculation of the reptation time above. Furthermore, using the published values, the shapes of the relaxation curves do not match the experimental data, and no crossover between *G*′ and *G*″ is observed, all of which signifies that the micelle length of 225 nm is too short. Alternately, the results from the micelle length obtained from Eq. (3), with other parameters given in the Appendix, match the experimental data fairly well. The micelle length from Eq. (3), 1.2 μm, corresponds to about five entanglements. Since our new parameters for this solution were themselves obtained from the predictions of the pointer algorithm, it is no great surprise that the rheology predicted by the pointer algorithm using these same parameters agrees with the data. However, recall that the predictions of our pointer algorithm also agree well with those from the slip-spring model, where this model has proved to be able to predict well the effects of entanglements in polymers, and so is likely quite accurate. This, plus the fact that the micelle length we estimate from Eq. (3) is consistent with the relatively high solution viscosity, strongly indicates that the micelle lengths we obtain from the pointer algorithm are more accurate than those estimated from the original Cates scaling law.

While the modification of the Cates method by Granek results in somewhat longer micelle lengths and a scaling exponent (0.8) in closer agreement to what we obtain from the pointer algorithm, the Granek modification does not change predictions of decreasing micelle length with increasing surfactant concentration obtained in some cases from the Cates method, and the micelle lengths remain much smaller than those inferred from the pointer algorithm, unless the prefactor of the scaling law is changed to a value suggested by fits to results from the pointer algorithm. Although most of the fits and our proposed scaling laws are based on the SLE1S + CAPB/NaCl system, the pointer algorithm should be generally applicable to solutions of entangled wormlike micelles. Our prediction of the experimental data from Helgeson *et al.* indicates that this true, but to verify that the pointer algorithm and the derived scaling laws are applicable to any given surfactant solution with entangled micelles, more pointer algorithm fits with different systems would need to be run. Further comparisons of the pointer algorithm with predictions of the slip-spring model for more highly entangled wormlike micelles should also help determine the general accuracy of the pointer algorithm.

## V. CONCLUSIONS

Using a mesoscopic simulation, the pointer algorithm, we have extracted surfactant micelle parameters from linear rheology data for entangled wormlike micellar solutions for a series of solutions of SLE1S and CAPB in NaCl solutions. In particular, the extracted average micelle lengths $\u27e8L\u27e9$ are more consistent with the high solution viscosities, in excess of 10^{2} Pa s, measured for these solutions, than are the values extracted from the Cates method in which $G0G\u2032\u2032min\u223c\u27e8L\u27e9le$, where $le$ is the entanglement length, *G*_{0} is the plateau modulus, and $Gmin\u2032\u2032$ is the local minimum value of *G*″ as a function of frequency. Similar data in the literature also give submicrometer micelle lengths despite high solution viscosities, again suggesting underprediction of micelle lengths by the Cates method.

To further validate the micelle lengths from the pointer algorithm, we compared *G*′ and *G*″ curves generated by the pointer algorithm to results from a slip-spring simulation model adapted to breakable chains and found good agreement between the two methods after modifying the Rouse modes in the pointer algorithm to include low-frequency modes. By modifying the pointer algorithm by switching to an unfractionated Rouse spectrum at low concentrations and adding longitudinal Rouse modes to the high-frequency modes at higher concentrations where the number of entanglements per micelle exceeds *Z* = 15, we also improved our fits to experimental data. From the fits to experimental data by this and the original version of the pointer algorithm, we calculated scaling laws for the micelle parameters that generally agree with theoretical and literature values, and that do not depend severely on the choice of which Rouse modes to include. Additionally, the micelle lengths from fits of the pointer algorithm to experimental data for a variety of surfactant solutions follow a scaling law, $G\u2032minG\u2032\u2032min=0.317(\u27e8L\u27e9le)0.82$, similar to the scaling law $G0G\u2032\u2032min\u223c\u27e8L\u27e9le$, of Cates and even more similar to the scaling law $G0G\u2032\u2032min\u223c(\u27e8L\u27e9le)0.8$ of Granek. But, because of the smaller prefactor (0.317), the new scaling law is more consistent with the longer micelle lengths obtained from the pointer algorithm. Thus, the pointer algorithm allows extraction of micelle parameters, in particular, the average micelle length, that are consistent with the high solution viscosities of these solutions, and gives dependencies of micelle parameters on surfactant concentration that are generally in agreement with theoretical scaling laws. In the future, we recommend use of either the pointer algorithm or the scaling law $G\u2032minG\u2032\u2032min=0.317(\u27e8L\u27e9le)0.82$, rather than the Cates scaling law, to extract average micelle lengths from rheological data.

## ACKNOWLEDGMENTS

Funding was provided by Procter & Gamble, as well as the National Science Foundation (NSF) under Grant No. CBET-1907517. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

### APPENDIX: PARAMETERS FOR 60 mM CTAB/120 mm NaNO_{3} SOLUTION FROM HELGESON *et al*.

The experimental data in Fig. 8 drawn from Helgeson *et al.* show no minimum in *G*″, but other data in Helgeson *et al.* show that an increase in either surfactant concentration or salt concentration produces a minimum in *G*″. Thus, we take the data in Fig. 8 to be at the threshold of having a minimum in *G*″. (We can also see that the data are near this threshold by examining the two-mode fit to these data in Fig. 5 of Helgeson *et al.*) Thus, we take our scaling relation $G\u2032minG\u2032\u2032min=0.317(\u27e8L\u27e9le)0.82$ in the limit $Gmin\u2032/Gmin\u2032\u2032=1$ and thereby obtain $\u27e8L\u27e9=4.06le$. To get *l _{e}*, we assume loosely entangled micelles, which is usually true of less concentrated/entangled solutions, and our crossover formula then reduces to

The plateau modulus is estimated by $G0=4.88Gmin\u2032$, which comes from fitting pointer algorithm results to the equation

(shown in Fig. 9) and again taking $Gmin\u2032/Gmin\u2032\u2032=1$. We can also estimate from Fig. 8 that the storage modulus at the frequency where this incipient minimum will appear, $Gmin\u2032$, is approximately twice its value at the crossover frequency. Then from Fig. 8, we find $Gmin\u2032=4.54Pa$ and, therefore, Eq. (A2) gives *G*_{0} = 22.2 Pa. From the persistence length of 32 nm from Helgeson *et al.*, and Eq. (A1), the entanglement length is 298 nm. Then, from $\u27e8L\u27e9=4.06le$, we find $\u27e8L\u27e9=1.2\mu m$. For a solution well outside the fast breakage regime, we use a large value of ζ = 200 so that results are insensitive to ζ. These calculations have given us all the independent input parameters needed for the pointer algorithm, namely, *G*_{0} = 22.2 Pa, $\u27e8L\u27e9=1.2\mu m$, *l _{p}* = 32 nm, and ζ = 200. (Other calculated parameters used in the pointer algorithm are derived from these, namely,

*α*=

*l*/

_{e}*l*= 9.30,

_{p}*Z*= 5.1, and

*l*= 298 nm.) Using these input parameters, the pointer algorithm predicts the

_{e}*G*′ and

*G*″ curves shown by the dotted lines in Fig. 8.

## REFERENCES

_{3}wormlike micelles