Surface free energies for nematic shells

We propose a continuum model to describe the molecular alignment in thin nematic shells. By contrast with previous accounts, the two-dimensional free energy, aimed at describing the physics of thin films of nematics deposited on curved substrates, is not postulated, but it is deduced from the conventional three-dimensional theories of nematic liquid crystals. Both the director and the order-tensor theories are taken into account. The so-obtained surface energies exhibit extra terms compared to earlier models. These terms reflect the coupling of the shell extrinsic curvature with the nematic order parameters. As expected, the shape of the shell plays a key role in the equilibrium configurations of nematics coating it.

Figure 1

We have set ${\lambda }_1=L_2/L_1$, ${\lambda }_2=L_{24}/L_1$. $S_1$ is the region in which the elastic energy density in Eq. (16) is nonnegative (see Ref. [24]). $S_1\cup S_2$ represent the domain in which the surface elastic free energy in Ref. [7] is non-negative. $S_1\cup S_2\cup S_3$ is the region where inequalities (28) hold. $A\equiv (-3/2,1/2)$, $B\equiv (-1,1)$, $C\equiv (1,1)$, $D\equiv (6/5,4/5)$.