Wrinkling of a thin film on a nematic liquid-crystal elastomer

Wrinkles commonly develop in a thin film deposited on a soft elastomer substrate when the film is subject to compression. Motivated by recent experiments [Agrawal et al., Soft Matter 8, 7138 (2012)] that show how wrinkle morphology can be controlled by using a nematic elastomer substrate, we develop the theory of small-amplitude wrinkles of an isotropic film atop a nematic elastomer. The directors of the nematic elastomer are initially uniform. For uniaxial compression of the film along the direction perpendicular to the elastomer directors, the system behaves as a compressed film on an isotropic substrate. When the uniaxial compression is along the direction of nematic order, we find that the soft elasticity characteristic of liquid-crystal elastomers leads to a critical stress for wrinkling which is very small compared to the case of an isotropic substrate. We also determine the wavelength of the wrinkles at the critical stress and show how the critical stress and wavelength depend on substrate depth and the anisotropy of the polymer chains in the nematic elastomer.

Figure 1

Schematic diagram of a thin film of thickness $h$ deposited on a liquid-crystal elastomer substrate of thickness $d$. The mesogens are not drawn to scale. The film has wrinkles of wavelength $2\pi /q$, with the modulation along the $x$ direction. The rotation of the directors penetrates a distance $l_{\mathrm{p}}$ (discussed in Sec. 3b).

Figure 2

Compressive stress $-{\sigma }_0$ as a function of wrinkle wavelength $2\pi /q$ for an ideal nematic elastomer substrate with $r=1.68$ (black dotted line), a semisoft nematic elastomer substrate with $r=1.01$ (blue dashed line), and an isotropic elastomer substrate with $r=1$ (red solid line), all with a shear modulus $\mu =0.24\mathrm{MPa}$, and a polystyrene film with thickness $h=100$ nm and stiffness $E_s=3.5$ GPa. For each curve, the critical stress ${\sigma }_c$ corresponds to the minimum at $2\pi /q_c$; the critical stress for the nematic elastomer substrate is much smaller than for the isotropic elastomer substrate.