Tunable wrinkling of thin nematic liquid crystal elastomer sheets

Instabilities in thin elastic sheets, such as wrinkles, are of broad interest both from a fundamental viewpoint and also because of their potential for engineering applications. Nematic liquid crystal elastomers offer a new form of control of these instabilities through direct coupling between microscopic degrees of freedom, resulting from orientational ordering of rodlike molecules, and macroscopic strain. By a standard method of dimensional reduction, we construct a plate theory for thin sheets of nematic elastomer. We then apply this theory to the study of the formation of wrinkles due to compression of a thin sheet of nematic liquid crystal elastomer atop an elastic or fluid substrate. We find the scaling of the wrinkle wavelength in terms of material parameters and the applied compression. The wavelength of the wrinkles is found to be nonmonotonic in the compressive strain due to the presence of the nematic. Finally, due to soft modes, the critical stress for the appearance of wrinkles can be much higher than in an isotropic elastomer and depends nontrivially on the manner in which the elastomer was prepared.

Figure 1

An illustration is used to represent the geometry and the parameters of the problem. A NLCE plate of thickness $t$, width $W$, and length $L$ is bonded to a substrate, which may be fluid or elastic. The bonded pair is compressed by a percentage $\gamma$ along the longitudinal direction, which we take to be $\widehat{x}$. The nematic director is assumed to lie in the tangent plane of the thin film. The director is parametrized after dimensional reduction by the angle $\varphi$ measured in the tangent plane, and the out-of-plane deformation is $\zeta (x,y)$.

Figure 2

Behavior of the director angle vs compressive strain $\gamma$ for different cross-linking strengths $h/\mu$ and strain-director couplings $\alpha /\mu$. For high $h/\mu$ and low $\alpha /\mu$ (black), the director will not rotate until a very high strain is reached. This critical strain becomes smaller as $h/\mu$ becomes smaller (gray-dashed). For $\alpha >h$, rotating the director becomes easier, and the director can even rotate a total of $\pi /2$ within a reasonable range of compressive strain (red). For increasing values of $\alpha /\mu$ (blue), the director rotates very quickly with any applied compressive strain $\gamma$, then more slowly until rotation is completed. For small $h/\mu$, there can be reentrance as a function of $\alpha /\mu$ (red, blue-dashed, blue-solid).

Figure 3

Areas correspond to parameter values for which the stress in the film is extensile (${\sigma }_{xx}^{\text{(0)}}>0$), which causes the sheet to remain flat regardless of the compression of the substrate. This unusual situation arises from the presence of the soft mode of director rotation. Regions that are not colored either always have ${\sigma }_{xx}^{\text{(0)}}<0$, regardless of the material parameters and compressive strain, or else they require such a large compressive strain to complete the soft mode that they may be well beyond our small-strain theory. For reference, while $h/\mu$ can vary greatly, $\alpha /\mu$ is typically $O\left(0.1\right)$ for side-chain elastomers and $O\left(1\right)$ for main-chain elastomers [62].

Figure 4

Variation of the critical strain as a function of strain-director coupling and cross-linking strength. (a) Position of example values of $(\alpha ,h)$ used, and regions where the example used represents the qualitative behavior of other plates. (b) The internal degree of freedom (nematic director angle) $\varphi$ as a function of the applied compression $\gamma$, which does not depend on an underlying substrate. (c) The dependence of the critical strain for buckling (here shown on the $x$-axis for convenience) on Young's modulus $K$ of the underlying substrate depends drastically on the internal degree of freedom, as shown in (b). For instance, the blue plate never rotates, and it obeys the classical buckling law; the orange plate remains entirely in the soft mode (and in the region of positive stress for $\gamma$ less than a very large value, even larger than those plotted in Fig. 3; therefore, it cannot buckle, and it is not plotted here) for the whole range of $\gamma$ shown; the purple plate remains classical until it begins to rotate at $\gamma \approx 0.11$; the red plate goes through all three phases (unrotated, soft, fully rotated). During the soft mode, the critical strain for buckling (when applicable) is highly insensitive to $K$. For reference, while $h/\mu$ can vary greatly, $\alpha /\mu$ is typically $O\left(0.1\right)$ for side-chain elastomers and $O\left(1\right)$ for main-chain elastomers [62].

Figure 5

Wave number of wrinkles on an elastic foundation, normalized by the natural length ${\ell }_{\mathrm{e}}\equiv t{[\mu /\left(6K\right)]}^{1/3}$, vs $\varphi /\pi$. Different groups for the same value of $h/\mu$ (dashed $h/\mu =0.5$, dotted-dashed $h/\mu =10$, and solid $h/\mu ={10}^3$) refer to the strength of the anchoring term, whereas the color map represents the strength of the elasticity-order coupling, $0\le \alpha /\mu \le 3/2$. The solid black line gives the wavelength when there is no nematic, $\alpha /\mu =0$ and $h/\mu =0$, and the gray line sets the bound for a nematic glass, i.e., $h\to \infty$.

Figure 6

Areas of parameter space in the $\gamma =0.1$ plane for which ${\omega }^2<D^2$ (blue) and therefore $\zeta (x,y)=0$. Here as an example we take $t=1\mathrm{mm}$, $\rho g/\mu =0.05$, representative of a rubber film atop a water substrate. The full finite two-dimensional case has a plane-stress condition that is weakly dependent on the wave number of wrinkling; we choose $q=8\text{mm}$ to mimic the wave number in an elastomer that is not nematic. Because here ${\omega }^2<D^2$ represents one condition rather than the two described in the main text [the stress must be (a) compressive and (b) beyond a certain threshold], we plot both regions (so that the region on the left, which does not appear in Fig. 3, corresponds to a region where the stress is still compressive, but is insufficient to cause wrinkling in the plate with the parameters given).