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)]1744-683X10.1039/c2sm25734c 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.