Transitions through critical temperatures in nematic liquid crystals

We obtain estimates for critical nematic liquid crystal (LC) temperatures under the action of a slowly varying temperature-dependent control variable. We show that biaxiality has a negligible effect within our model and that these delay estimates are well described by a purely uniaxial model. The static theory predicts two critical temperatures: the supercooling temperature below which the isotropic phase loses stability and the superheating temperature above which the ordered nematic states do not exist. In contrast to the static problem, the isotropic phase exhibits a memory effect below the supercooling temperature in the dynamic framework. This delayed loss of stability is independent of the rate of change of temperature and depends purely on the initial value of the temperature. We also show how our results can be used to improve estimates for LC material constants.

Figure 1

The static equilibria plotted in the $(\tilde{A},\tilde{S})$ plane. Unstable branches are marked with dashed curves and the arrows indicate the sign of $d\tilde{S}/d\tau$. The tildes have been removed from the dimensionless variables for brevity.

Figure 2

Numerical solutions of the differential Equation (10) plotted in the $(\tilde{A},\tilde{S})$ plane with $\tilde{A}\left(\tau \right)=1.5-\tau$, $\tilde{S}\left(0\right)=0.5$, and $\epsilon =0.5$ (yellow, solid), $0.25$ (green, dashed), $0.125$ (red, dotted), and $0.0625$ (pink, dot-dashed). The equilibrium solution is plotted as a thick blue curve.

Figure 3

Numerical solutions of the differential Equation (10) plotted in the $(\tilde{A},\tilde{S})$ plane with $\tilde{A}\left(\tau \right)=-3+\tau$, $\tilde{S}\left(0\right)=3$, and $\epsilon =0.5$ (yellow, solid), $0.25$ (green, dashed), $0.125$ (red, dotted), and $0.0625$ (pink, dot-dashed). The equilibrium solution is plotted as a thick blue curve.

Figure 4

The interval $\delta A$ over which $S\left(\tau \right)$ transitions to an $\epsilon$ neighborhood of $S=0$. The dashed curve shows the asymptotic prediction (15).