Stresses in curved nematic membranes

Ordering configurations of a director field on a curved membrane induces stress. In this work, we present a theoretical framework to calculate the stress tensor and the torque as a consequence of the nematic ordering; we use the variational principle and invariance of the energy under Euclidean motions. Euler-Lagrange equations of the membrane as well as the corresponding boundary conditions also appear as natural results. The stress tensor found includes attraction-repulsion forces between defects; likewise, defects are attracted to patches with the same sign in Gaussian curvature. These forces are mediated by the Green function of the Laplace-Beltrami operator of the surface. In addition, we find nonisotropic forces that involve derivatives of the Green function and the Gaussian curvature, even in the normal direction to the membrane. We examine the case of axial membranes to analyze the spherical one. For spherical vesicles we find the modified Young-Laplace law as a consequence of the nematic texture. In the case of spherical cap with defect at the north pole, we find that the force is repulsive with respect to the north pole, indicating that it is an unstable equilibrium point.

Figure 1

A $+1$ defect on the top of a mountain. At any point of the curve $\mathcal{C}$, the force per unit length along $\mathbf{l}$ is given by $-(\sigma +{\kappa }_A\chi {\mathcal{R}}_G)$. In addition, there is the anisotropic force ${\kappa }_A{l}^a{l}^b(\chi {\mathbf{\nabla }}_a{\mathbf{\nabla }}_b\chi -{\mathbf{\nabla }}_a\chi {\mathbf{\nabla }}_b\chi )$ along l. Darboux frame adapted to the curve is shown: $\mathbf{T}$ the unit tangent, the unit normal to the surface $\mathbf{n}$, and $\mathbf{l}=\mathbf{T}\times \mathbf{n}$.

Figure 2

The function $\chi$ in Eq. (65) with $+1$ defect at each pole where singularities appear.

Figure 3

A spherical particle with radious $R$ and $+1$ defects on opposite sides; with this nematic texture, the relationship between the parameters is given by the Young-Laplace law, Eq. (66).

Figure 4

Nematic texture on spherical sheets with boundary at $l_0=2Rarcsin(\sqrt{3/2}/2)$ and $l_0=\pi R/2$ respectively. Gauss-Bonnet theorem implies defects with $q=+1/2$ and $q=+1$ on them.

Figure 5

The geometric potencial $\mathcal{U}$ for the spherical caps in Fig. 4. The point $l=0$ being the north pole, where the defect is placed. The force ${\mathcal{R}}_G\mathcal{U}$ before than the root $x_0=l_0/R$ is attractive to the defect point and repulsive after this point.