Interaction energies in nematic liquid crystal suspensions

Abstract We establish, as $\rho \to 0$ , an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$ -valued maps outside a finite number of particles of size $\rho $ with fixed centers $x_j\in {\mathbb R}^3$ , under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form $$ \begin{align*} E_\rho = \sum_j \mu_j -4\pi\rho \sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(\rho)\,, \end{align*} $$ where $\mu _j$ is the minimal energy after zooming in at scale $\rho $ around each particle, and $v_j\in {\mathbb R}^3$ is determined by the far-field behavior of the corresponding single-particle minimizer. The Coulomb-like interaction in this expansion agrees with the electrostatic analogy : a linearized approximation commonly used in the physics literature for colloid interactions in nematic liquid crystal. We obtain here for the first time a precise estimate of the energy error introduced by that linearization, by developing new tools that address the lack of convergence rate when zooming in at scale $\rho $ .