Localized Elasticity Governs the Nonlinear Rheology of Colloidal Supercooled Liquids

We propose a microscopic picture for understanding the nonlinear rheology of supercooled liquids with soft repulsive potentials. Based on Brownian dynamics simulations of supercooled charge-stabilized colloidal suspensions, our analysis shows that the shear thinning of viscosity ($\eta$) at large enough shear rates ($\dot{\gamma }$), expressed as $\eta \sim {\dot{\gamma }}^{-\lambda }$, originates from the evolution of the localized elastic region (LER). An LER is a transient zone composed of the first several coordination shells of a reference particle. In response to the external shear, particles within the LER undergo nearly affine displacement before the yielding of the LER. The characteristic strain ($\gamma$) and size ($\xi$) of the LER, respectively, depend on the shear rate by $\gamma \sim {\dot{\gamma }}^{\epsilon }$ and $\xi \sim {\dot{\gamma }}^{-\nu }$. Three exponents, $\lambda$, $\epsilon$, and $\nu$, are related by $\lambda =1-\epsilon =4\nu$. This simple relation connects the nonlinear rheology to the elastic properties and the microscopic configurational distortion of the system. The relaxation of the LER is promoted by the large-step nonaffine particle displacement along the extensional direction of the shear geometry with the step length of 0.4 particle diameter. The elastic deformation and relaxation of the LER are ubiquitous and successive in the flow, which compose the fundamental process governing the bulk nonlinear viscoelasticity. We apply this model to analyze the rheo-small-angle neutron scattering data of sheared charge-stabilized colloidal suspensions. It is seen that our model well explains the neutron spectra and the rheological data.

Popular Summary

Shampoos and liquid detergents are extremely viscous at rest but can still flow easily with a weak squeeze. This phenomenon is known as shear thinning—the nonlinear behavior of fluids whose viscosity decreases with shear rate. A popular understanding of the microscopic origin of shear thinning is built on the concept of cooperatively rearranging clusters in which liquid molecules undergo large displacements to flow. While there is a known correspondence between shear thinning and cluster properties, such as their size and lifetime, the way the flow behavior directly connects to the microscopic distortion in flowing liquids remains elusive. Here, we tackle this connection.

By combining simulations, small-angle neutron scattering, and rheological measurements, we demonstrate the existence of short-lived localized elastic regions (LERs) in model colloidal supercooled liquids under steady shear flow. In response to the external shear, the LERs undergo solidlike deformation before yielding. We find that LERs provide the mechanism for sustaining the imposed shear stress. Moreover, the shear thinning is quantitatively mirrored by the evolution of LERs. The deformation and yielding of LERs are ubiquitous and persistently successive in the flow, composing the fundamental process governing the nonlinear viscoelasticity of liquids.

In addition to colloidal suspensions, rich rheological behaviors are observed in many soft materials, such as micelles, microemulsions, and various biological systems that are often characterized by strong interactions at the molecular level. Our results offer a new perspective for understanding the flow behavior of soft condensed matter in general.

Figure 1

(a) Shear viscosity contributed by the interparticle Yukawa potential (${\eta }_p$). (b) Partial shear viscosity contributed by the Yukawa interactions only between big particles (${\eta }_{\mathrm{pb}}$). Both ${\eta }_p$ and ${\eta }_{\mathrm{pb}}$ are normalized by the solvent viscosity ${\eta }_{\mathrm{s}\mathrm{o}\mathrm{l}}$. Solid lines denote the fits with the power law ${\mathrm{Pe}}^{-\lambda }$ in the shear-thinning regime. (c) and (d), respectively, display the pair distribution function of big particles, $g(\mathit{r})$, at the flow-gradient ($\mathit{v}-\nabla \mathit{v}$ or $x-y$) plane and the flow-vorticity ($\mathit{v}-\nabla \times \mathit{v}$ or $x-z$) plane at the condition of $\varphi =45\%$ and $\mathrm{Pe}=0.75$. The thickness is $0.8d_b$ for both slices. No noticeable layering or crystallization is seen here.

Figure 2

(a) and (b) display $g_2^{-2}(r)$ as well as $-(1/\sqrt{15})rd{g}_0^0(r)/dr$ of the $\varphi =45\%$ system at $\mathrm{Pe}=0.002$ (Newtonian regime) and 0.75 (shear-thinning regime), respectively. The magnitude of $-(1/\sqrt{15})rd{g}_0^0(r)/dr$ is rescaled to match that of $g_2^{-2}(r)$ in both (a) and (b). (c) ${\tau }_2(r)$ at the conditions of $\varphi =42.5\%$, $\mathrm{Pe}=0.005$ and $\varphi =45\%$, $\mathrm{Pe}=0.002$ (Newtonian regime). Solid lines denote the linear fits. (d) $\gamma (r)=\dot{\gamma }{\tau }_2(r)$ of the $\varphi =45\%$ system at $\mathrm{Pe}=0.25$, 0.75, 2.5, and 5 (shear-thinning regime). Solid lines denote the fitted curves with Eq. (11).

Figure 3

The shear stress ${\sigma }_M=\dot{\gamma }{\eta }_{\mathrm{pb}}$ (solid symbols) and the microscopic elastic stress ${\sigma }_{\mathrm{el}}=G_{\infty }{\gamma }_{\mathrm{NN}}$ (open symbols) in the shear-thinning regime. Lines denote the fits with the power law $\sigma \propto {\mathrm{Pe}}^{\epsilon }$. The stress is in units of ${\sigma }_0=4k_{B}T/3\pi d_b^3$. With this unit, the stress can be expressed as $\sigma /{\sigma }_0=\mathrm{Pe}·(\eta /{\eta }_{\mathrm{s}\mathrm{o}\mathrm{l}})$.

Figure 4

$\xi (\mathrm{Pe})$ in the shear-thinning regime. Symbols denote the simulated results. Lines denote the fits with the power law $\xi \propto {\mathrm{Pe}}^{-\nu }$.

Figure 5

(a) $C(r_1,r_i)$ as a function of $r_i$ for the $\varphi =45\%$ system at $\mathrm{Pe}=0.25$, 0.75, 2.5, and 5. In this case, $C(r_1,r_i)$ becomes statistically noisy at $C(r_1,r_i)\lesssim 2\%$. Therefore, we set 2% as the effective zero for $C(r_1,r_i)$. (b) Scatter plot of $\xi$ and ${\xi }_c$ for all simulated points in the shear-thinning regime. The solid line denotes the relation of $\xi =1.23{\xi }_c$.

Figure 6

Comparison between $-(\overline{\gamma }/\sqrt{15})rd{g}_0^0(r)/dr$ and $g_2^{-2}(r)$ at $\mathrm{Pe}=0.75$ and $\varphi =45\%$. The phase difference $q(r_1)$ at the first peak is denoted. Inset: $q(r_1)$ as a function of Pe in the shear-thinning regime of the $\varphi =45\%$ system.

Figure 7

(a) The slice of $\mathrm{\Delta }g(\mathit{r})=g_{\mathrm{liq}}(\mathit{r})-g_{\mathrm{aff}}(\mathit{r})$ in the $x–y$ plane. $g_{\mathrm{liq}}(\mathit{r})$ is given by the pair distribution function of the flow at $\mathrm{Pe}=0.75$ and $\varphi =45\%$. $g_{\mathrm{aff}}(\mathit{r})$ is the pair distribution function obtained by affinely shearing the equilibrium structure of the $\varphi =45\%$ system with the strain of $\overline{\gamma }=0.095$. The thickness of the slice is $0.8d_b$. Two arrows denote the basin and the peak on the extensional axis. (b) The profiles of $\mathrm{\Delta }g(\mathit{r})$ along the extensional axis, $\mathrm{\Delta }{g}_{\mathrm{ext}}(r)$, at $\mathrm{Pe}=0.25$, 0.75, 2.5, and 5 for the $\varphi =45\%$ system. The first negative peak and the first positive peak of $\mathrm{\Delta }{g}_{\mathrm{ext}}(r)$, respectively, correspond to the basin and peak denoted by arrows in (a). The distance between these two peaks is about $0.4d_b$.

Figure 8

The step lengths $l_s$ found from the slice of $\mathrm{\Delta }g(\mathit{r})$ in the $x–y$ plane for all simulated conditions in the shear-thinning regime (solid symbols). The cage jump lengths $l_{\mathrm{cj}}$ for the $\varphi =45\%$ system (open symbols) are also shown for comparison.

Figure 9

(a) $p$ as a function of Pe in the nonlinear regime. (b) Scatter plot of $D_{\mathrm{naff}}$ and $D_{\mathrm{naff}}^{(\mathrm{LER})}$. The solid line denotes the relation $D_{\mathrm{naff}}=1.26D_{\mathrm{naff}}^{(\mathrm{LER})}$.

Figure 10

(a) Illustration of the rheo-SANS experiment under Couette geometry. $x$, $y$, and $z$ denote the directions of flow, velocity gradient, and vorticity, respectively. (b) 2D SANS patterns obtained from the $x–y$ plane and the $x–z$ plane at $\dot{\gamma }=1$, 10, and $100{\mathrm{s}}^{-1}$ for sample $A$. (c) 2D SANS patterns obtained from the $x–y$ plane and the $x–z$ plane at $\dot{\gamma }=1$, 10, and $100{\mathrm{s}}^{-1}$ for sample $B$. We do not see long-range ordering in all measured 2D patterns. The sample thicknesses along the neutron beam are 5 and 2 mm for the $x–y$ plane and the $x–z$ plane, respectively.

Figure 11

Finding $S_0^0(Q)$ and $S_2^{-2}(Q)$ for sample $A$ from the rheo-SANS spectra. (a) and (b), respectively, show $I_{0,\exp }^0(Q)$ and $I_{2,\exp }^{-2}(Q)$ at $\dot{\gamma }=1$, 10, and $100{\mathrm{s}}^{-1}$ extracted from the 2D SANS patterns through the method given in Appendix pp2. Experimental data are denoted by symbols. Solid lines are calculated by convolving the $I_0^0(Q)$ and $I_2^{-2}(Q)$ given in (c) and (d) with instrumental resolution function. (c) and (d), respectively, show $I_0^0(Q)$ and $I_2^{-2}(Q)$ obtained from $I_{0,\exp }^0(Q)$ and $I_{2,\exp }^{-2}(Q)$ by eliminating the resolution effect. (e) and (f), respectively, show $S_0^0(Q)$ and $S_2^{-2}(Q)$ obtained from $I_0^0(Q)$ and $I_2^{-2}(Q)$ by eliminating the effects of $P(Q)$ and $\beta (Q)$. For clarity, we vertically shift the data shown in (a) and (c).

Figure 12

(a) Experimental ${\gamma }_{\mathrm{NN}}$ found through Eq. (29) for both samples. (b) Comparison between the microscopic shear viscosity ${\eta }_{\mathrm{LER}}$ and the macroscopic shear viscosity ${\eta }_M$ for sample $A$. (c) Comparison between ${\eta }_{\mathrm{LER}}$ and ${\eta }_M$ for sample $B$.

Figure 13

(a) Experimental $g_2^{-2}(r)$ of sample $A$ at $\dot{\gamma }=1$, 10, and $100{\mathrm{s}}^{-1}$. The magnitude of $g_2^{-2}(r)$ is normalized by ${\gamma }_{\mathrm{NN}}$. (b) The range of LER $\xi$ (symbols) of sample $A$ estimated from experimental $g_2^{-2}(r)$. Here, we do not plot the result at $\dot{\gamma }=1{\mathrm{s}}^{-1}$ since its value is unreasonably large. The solid line denotes the fit with the power law $\xi \propto {\dot{\gamma }}^{-\nu }$, where $\nu$ is found to be 0.198.

Figure 14

(a) Trajectory of a reference particle. The color of trajectory changes at every jump. There are four segments shown here. (b) Comparison between $Q_t(a^{*},{\tau }^{*})/⟨Q_t{⟩}_t$ and $P_t({\tau }^{*})/⟨P_t{⟩}_t$. The simulation condition here is $\varphi =45\%$ and $\dot{\gamma }=0$.

Figure 15

(a) Comparison between ${\tilde{g}}_2^{-2}(r)$ and $g_2^{-2}(r)$. (b) Comparison between ${\tilde{g}}_0^0(r)$ and $g_0^0(r)$. The simulation condition is $\varphi =45\%$, $\mathrm{Pe}=0.75$.