Interaction energies in nematic liquid crystal suspensions

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 three-dimensional 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_ρ= \sum_j μ_j -4πρ\sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(ρ),, \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 a torque determined by the far-field behavior of the corresponding single-particle minimizer. The above expansion highlights Coulomb-like interactions between the particle centers. This agrees with the $\mathit{\text{electrostatics}}\mathit{\text{analogy}}$ commonly used in the physics literature for colloid interactions in nematic liquid crystal. That analogy was pioneered by Brochard and de Gennes in 1970, based on a formal linearization argument. We obtain here for the first time a precise estimate of the energy error introduced by this linearization procedure.