Metric theory of nematoelastic shells

We consider three-dimensional reshaping of a thin nematoelastic film upon nematic-isotropic transition in the field of a charge one topological defect, leading to either cone or anticone ($d$-cone) shells. The analysis is based on the relation between the shell metric and the tensor order parameter under the assumption of no elastic deformation and volume change. The shape of the shell can be modified by doping, creating cones with curved generatrices. Anticones necessarily have an even number of radial creases. The curvature singularity at the apex is resolved due to decay of the nematic order parameter at the defect core.

Figure 1

(a) Bending the unit circle with $a\left(r\right)=kr$. Numbers at the curves show the values of $k$. (b) A singular shape obtained for $\rho \left(r\right)=n^{-1/2}{r}^n,n=3/2$ and the corresponding function $a\left(r\right)$.

Figure 2

(a) The polar angle at creases as a function of the extension coefficient $\ell$. Numbers at the curves denote the number of creases $2n$. Inset: the maximum values of $\ell$ sustaining $2n$ creases. (b) Relative elevation from the equator $z/\rho =cos\vartheta$ as a function of the azimuthal angle $\phi$. Solid line: $\ell =1.15,n=1$; dashed line: $\ell =1.75,n=2$. (c) 3D view of the anticones at the same values of $\ell$, showing the images of circles and radii in the original disk; one petal of the anticone with four creases is omitted for graphical clarity.

Figure 3

The function $\psi \left(r\right)$. Inset: the shape of the anticone near the apex; $a_0=1/2,\rho <\sqrt{\kappa }$.