First-Principles Constitutive Equation for Suspension Rheology

Using mode-coupling theory, we derive a constitutive equation for the nonlinear rheology of dense colloidal suspensions under arbitrary time-dependent homogeneous flow. Generalizing previous results for simple shear, this allows the full tensorial structure of the theory to be identified. Macroscopic deformation measures, such as the Cauchy-Green tensors, thereby emerge. So does a direct relation between the stress and the distorted microstructure, illuminating the interplay of slow structural relaxation and arbitrary imposed flow. We present flow curves for steady planar and uniaxial elongation and compare these to simple shear. The resulting nonlinear Trouton ratios point to a tensorially nontrivial dynamic yield condition for colloidal glasses.

Figure 1

Steady state stress $\sigma ={\sigma }_{xy}$ under shear (full lines, ${\kappa }_{ij}=\dot{\gamma }{\delta }_{xi}{\delta }_{yj}$) and stress difference $\sigma ={\sigma }_{xx}-{\sigma }_{yy}$ under planar elongation [broken lines, ${\kappa }_{ij}=\dot{\gamma }({\delta }_{xi}{\delta }_{xj}-{\delta }_{yi}{\delta }_{yj})$] as a function of ${\mathrm{Pe}}_0=\dot{\gamma }{d}^2/D_0$, where $d$ is the sphere diameter. Each curve is labeled according to the distance from the glass transition $\varphi -{\varphi }_c$, (a) $-{10}^{-4}$, (b) $-{10}^{-3}$, and (c) ${10}^{-4}$. The inset shows a possible realization of planar elongational flow. The lower left panel shows the Trouton ratio for such flow while the lower right panel shows the ratio for uniaxial elongation [${\kappa }_{ij}=\dot{\gamma }({\delta }_{xi}{\delta }_{xj}-{\delta }_{yi}{\delta }_{yj}/2-{\delta }_{zi}{\delta }_{zj}/2)$].