Medium amplitude parallel superposition (MAPS) rheology of a wormlike micellar solution

Abstract

The weakly nonlinear rheology of a surfactant solution of wormlike micelles is investigated from both a modeling and experimental perspective using the framework of medium amplitude parallel superposition (MAPS) rheology. MAPS rheology defines material functions, such as the third-order complex compliance, which span the entire weakly nonlinear response space of viscoelastic materials to simple shear deformations. Three-tone oscillatory shear deformations are applied to obtain feature-rich data characterizing the third-order complex compliance with high data throughput. Here, data for a CPyCl solution are compared to the analytical solution for the MAPS response of a reptation-reaction constitutive model, which treats micelles as linear polymers that can break apart and recombine in solution. Regression of the data to the model predictions provides new insight into how these breakage and recombination processes are affected by shear, and demonstrates the importance of using information-rich data to infer precise estimates of model parameters.

Graphical abstract

Appendices

Appendix: A. The reptation-reaction model equations

The stress tensor in the reptation-reaction model is specified by Eqs. 4 and 5, along with the companion Eq. 6. Completing the model specifications requires equations for the tube creation rate ${\mathscr{B}}(v)$, destruction rate $\mathcal {D}(v)$, and the evolution of the $\boldsymbol {Q}(\boldsymbol {E}_{t^{\prime }t})$ tensor describing tube elongation. The creation and destruction rates are specified by two ensemble- averaged integrals over the position of the end of a micelle, X(t):

$$ \mathcal{D} = \frac{2}{\bar{L}\hat{\tau}}\left\langle{\int}_{0}^{\infty}e^{-t/\hat{\tau}}\underset{0<t^{\prime}<t}{\max}[X(t^{\prime})]dt\right\rangle, $$

(A.1)

$$ \mathcal{B} = \frac{2}{\bar{L}\hat{\tau}}\left\langle{\int}_{0}^{\infty}e^{-t/\hat{\tau}}\left( \underset{0<t^{\prime}<t}{\max}[X(t^{\prime})] - X(t)\right)dt\right\rangle. $$

(A.2)

Angular brackets here represent ensemble averages over the stochastic processes of diffusion and reactions at the end of the micelles, such as breakage and recombination. In these expressions, $\bar {L}$ represents the time-averaged micelle length L(t), and $\hat {\tau }$ a time scale related to the recombination process (this time scale is distinct from the time scale τ defined in the main text, which reflects a time scale associated with diffusion of the end of the micelle).

The tensor $\boldsymbol {Q}(\boldsymbol {E}_{t^{\prime }t})$ represents the average orientation over an isotropic distribution of unit vectors u:

$$ \boldsymbol{Q}(\boldsymbol{E}_{t^{\prime}t}) = \frac{1}{4\pi}{\int}_{\mathcal{S}}\frac{[\boldsymbol{E}_{t^{\prime}t}\cdot\boldsymbol{u}][\boldsymbol{E}_{t^{\prime}t}\cdot\boldsymbol{u}]}{\lvert\boldsymbol{E}_{t^{\prime}t}\cdot\boldsymbol{u}\rvert}d^{2}\boldsymbol{u}, $$

(A.3)

where $\mathcal {S}$ represents the surface of the unit sphere, and |x| represents the L²-norm of a vector x.

Appendix: B. MAPS response of the reptation-reaction model

The third-order complex modulus of the reptation-reaction model may be written as the sum of three terms:

$$ G^{*}_{3}(\omega_{1},\omega_{2},\omega_{3}) = (\alpha - 1)G^{*}_{3,\mathcal{B}} + \alpha G^{*}_{3,\mathcal{D}} + G^{*}_{3,\boldsymbol{Q}}. $$

(A.4)

The term $G^{*}_{3,{\mathscr{B}}}$ represents the contribution due to nonlinearity in the tube creation function ${\mathscr{B}}(v)$:

$$ \begin{array}{@{}rcl@{}} && G^{*}_{3,\mathcal{B}}(\omega_{1},\omega_{2},\omega_{3}) = \\ && \quad -\frac{2 G_{0}}{45}{\sum}_{j} {\sum}_{k\neq j}\frac{\omega_{j}\omega_{k}\tau^{2}}{1 + i\tau\omega_{j}}\left[\frac{1}{1 + i\tau(\omega_{j} + \omega_{k})} - \frac{1}{1 + i\tau{\sum}_{l} \omega_{l}}\right]. \end{array} $$

(A.5)

The term $G^{*}_{3,\mathcal {D}}$ represents the contribution due to nonlinearity in the tube destruction function $\mathcal {D}(v)$:

$$ \begin{array}{@{}rcl@{}} && G^{*}_{3,\mathcal{D}}(\omega_{1},\omega_{2},\omega_{3}) = \\ && \frac{2 G_{0}}{45}{\sum}_{j} {\sum}_{k\neq j} \frac{i\tau\omega_{j}\omega_{k}}{\omega_{j} + \omega_{k}}\frac{1}{1 + i\tau\omega_{j}}\left[\frac{1}{1 + i\tau(\omega_{j} + \omega_{k})} \right. \\ && \left. - \frac{1}{1 + i\tau{\sum}_{l} \omega_{l}} + \frac{1}{1 + i\tau\omega_{6-j-k}} - 1\right]. \end{array} $$

(A.6)

The term $G^{*}_{3,\boldsymbol {Q}}$ represents the contribution due to nonlinearity in the $\boldsymbol {Q}(\boldsymbol {E}_{t^{\prime }t})$ tensor. Because, in simple shear, $\boldsymbol {Q}(\boldsymbol {E}_{t^{\prime }t})$ may be written as a quadratic polynomial in the accumulated strain (see the Supporting Information), this factor arises as a time-strain separable contribution to the third-order complex modulus. It may therefore be expressed in terms of the linear modulus (Eq. ??):

$$ \begin{array}{@{}rcl@{}} && G^{*}_{3,\boldsymbol{Q}}(\omega_{1},\omega_{2},\omega_{3}) = \\ && -\frac{1}{7} \left[G^{*}\left( \underset{j}{\sum} \omega_{j}\right) - \underset{j}{\sum} G^{*}\left( \underset{k\neq j}{\sum}\omega_{k}\right) + \underset{j}{\sum} G^{*}(\omega_{j})\right]. \end{array} $$

(A.7)

Finally, the parameter α is defined in terms of the limiting slope of $\mathcal {D}(v)$ (or ${\mathscr{B}}(v)$):

$$ \alpha \equiv \left.\frac{d\mathcal{D}}{dv}\right\rvert_{v = 0} = 1 + \left.\frac{d\mathcal{B}}{dv}\right\rvert_{v = 0}, $$

(A.8)

where v = 0 in simple shear corresponds to the limit of either zero shear-rate or zero shear stress.