Dynamics of smectic elastomers

We study the low-frequency, long-wavelength dynamics of liquid crystal elastomers, crosslinked in the smectic-$A$ phase, in their smectic-$A$, biaxial smectic and smectic-$C$ phases. Two different yet related formulations are employed. One formulation describes the pure hydrodynamics and does not explicitly involve the Frank director, which relaxes to its local equilibrium value in a nonhydrodynamic time. The other formulation explicitly treats the director and applies beyond the hydrodynamic limit. We compare the low-frequency, long-wavelength dynamics of smectic-$A$ elastomers to that of nematics and show that the two are closely related. For the biaxial smectic and the smectic-$C$ phases, we calculate sound velocities and the mode structure in certain symmetry directions. For the smectic-$C$ elastomers, in addition, we discuss in some detail their possible behavior in rheology experiments.

Figure 1

(Color online) Schematic plots of sound velocities. (a), (b), and (c) Spherical plots of, respectively, mode pairs (i), (ii), and (iii). (d) and (e) Polar plots in the $xy$ and $xz$ planes, respectively, of all three sound-mode pairs.

Figure 2

(Color online) Schematic plots of sound velocities. (a), (b), and (c) Spherical plots of, respectively, mode pairs (i), (ii), and (iii). (d) and (e) Polar plots in the $x^{\prime }{z}^{\prime }$ and $x^{\prime }{z}^{\prime }$ planes, respectively, of all three sound-mode pairs. Plot (a) is magnified relative to the remaining plots by a factor of 2.

Figure 3

Log-log plot of the reduced storage and loss moduli $C_{\xi \chi }^{\prime }\left(\omega \right)∕C_{\xi \chi }$ (solid lines) and $C_{\xi \chi }^{″}\left(\omega \right)∕C_{\xi \chi }$ (dashed lines) versus the reduced frequency $\omega {\tau }_E$ as given, respectively, by the real and imaginary parts of Eq. (6.17) for ${\tau }_n∕{\tau }_E={10}^3$. For the purpose of illustration we have set, by and large arbitrarily, ${\nu }_{\xi \chi }∕\left({\tau }_E{C}_{\xi \chi }\right)=1$ and $\Delta A_{\xi \chi }∕C_{\xi \chi }=3$.