Solidlike behavior in liquid layers: A theory of the yield stress in smectics

A study of anharmonic effects in the presence of shear flow in the liquid layers of smectic-$A$, hexatic-$B$, and cholesteric liquid crystals is presented. It is shown that the nonlinear coupling of the velocity field to thermally excited undulations of the layers gives rise to a singular $\frac{1}{\left|\dot{\gamma }\right|}$ dependence of the viscosities ${\eta }_i$ on the shear strain rate $\dot{\gamma }$ at small $\dot{\gamma }$. In addition, the coefficient of this anomaly is found to diverge as ${\left(\ln L\right)}^2$ where $L$ is the size of the system. The $\frac{1}{\left|\dot{\gamma }\right|}$ divergence of ${\eta }_2$, which governs shear flow in the layers, is shown to imply that a minimum applied stress is required in order to set up such a flow. It is demonstrated that this effect provides a straightforward explanation of the puzzling non-Newtonian "plastic" behavior widely observed in experiments on capillary flow in the liquid directions of smectics. The magnitude of the observed yield stress is shown to be consistent with the predictions of this work. Further experimental tests of the theory presented here are suggested.