Phase separation and folding in swelled nematoelastic films

We explore reshaping of nematoelastic films upon imbibing an isotropic solvent under conditions when isotropic and nematic phases coexist. The structure of the interphase boundary is computed taking into account the optimal nematic orientation governed by interaction of gradients of the nematic order parameter and solvent concentration. This structure determines the effective line tension of the boundary. We further compute equilibrium shapes of deformed thin sheets and cylindrical and spherical shells with the rectilinear or circular shape of the boundary between nematic and isotropic domains. A differential expansion or contraction near this boundary generates a folding pattern spreading out into the bulk of both phases. The hierarchical ordering of this pattern is most pronounced on a cylindrical shell.

Figure 1

(a) Dependence of total energy $\mathcal{F}={\mathcal{F}}_{\mathbf{n}}+{\mathcal{F}}_{\mathbf{e}}+{\mathcal{F}}_{\mathbf{m}}$ and respective nematic, elastic, mixing energies (in the inset) on the solvent fraction $\phi$. (b) The dependence of the scalar NOP $S$ and $\phi$ on the coordinate $x$ normal to the front.

Figure 2

(a) Squared curvature and solvent fraction along the section parallel to the front at $x=10$ shown by a dotted white line in the inset. Inset: the actual shape of a deformed sheet with $h_0=0.8$. (b) Comparison of the profiles along the same section for sheets of different thickness.

Figure 3

The solvent fraction and squared curvature in the isotropic domain of a cylindrical shell with $h_0=0.05$, $R_0=10$, and $L_0=100$. The cylindrical surface is cut along a generatrix for a full view.

Figure 4

The folding patterns (shown by the changes of the local radius) in cylindrical shells with $L_0=100$, $R_0=10$ (a)–(c) and $R_0=15$ (d), and different thicknesses (as indicated). The surface is cut along a generatrix for a full view.

Figure 5

(a) The nematic texture in a spherical shell (in the Mercator projection). (b),(c) The Mercator projection (b) and shape (c) of a spherical shell with the initial radius $R_0=10$ and thickness $h_0=0.01$ after NIT starting from the state with four $+\frac{1}{2}$-charged defects. White dashed lines show the “baseball seam” and the “comet tails” of two pairs of defects in the original nematic texture. (d) The shape of the shell with the same initial radius following isotropic-nematic transition to a biphasic state with an equatorial monodomain nematic belt with fronts located at the polar angle $\mathrm{\Theta }=\frac{1}{2}\pi (1\pm \frac{1}{6})$. Shading in (b)–(d) encodes the local radius.

Figure 6

The shape (a) and distribution (b) of solvent in a spherical shell with the initial radius $R_0=10$ and thickness $h_0=0.01$ following NIT in the southern hemisphere, with the nematic texture in the northern hemisphere remaining frozen (shown in the Mercator projection).