Dynamics, dynamic soft elasticity, and rheology of smectic-$C$ elastomers

We present a theory for the low-frequency, long-wavelength dynamics of soft smectic-$C$ elastomers with locked-in smectic layers. Our theory, which goes beyond pure hydrodynamics, predicts a dynamic soft elasticity of these elastomers and allows us to calculate the storage and loss moduli relevant for rheology experiments as well as the mode structure.

Figure 1

Log-log plot of the reduced storage and loss moduli $C_{\xi \chi }^{\prime }\left(\omega \right)∕C_{\xi \chi }$ (solid line) and $C_{\xi \chi }^{″}\left(\omega \right)∕C_{\xi \chi }$ (dashed line) versus the reduced frequency $\omega {\tau }_n$ as given, respectively, by the real and negative imaginary parts of Eq. (9) for ${\tau }_n⪢{\tau }_E$. For the purpose of illustration we have set, by and large arbitrarily, ${\nu }_{\xi \chi }∕\left({\tau }_n{C}_{\xi \chi }\right)=1$ and $\Delta A_{\xi \chi }∕C_{\xi \chi }={10}^2$.

Figure 2

(Color online) Schematic plots of sound velocities: (a) spherical plot of mode pair (i) only; (b) and (c) polar plots in the $y^{\prime }{z}^{\prime }$ and $x^{\prime }{z}^{\prime }$ planes, respectively, of all three sound-mode pairs.