On the Isotropic–Nematic Phase Transition for the Liquid Crystal

Abstract

In this paper, we study the isotropic–nematic phase transition for the nematic liquid crystal based on the Landau–de Gennes $\mathbf {Q}$-tensor theory. We justify the limit from the Landau–de Gennes flow to a sharp interface model: in the isotropic region, $\mathbf {Q}\equiv 0$; in the nematic region, the $\mathbf {Q}$-tensor is constrained on the manifolds $\mathcal {N}=\{s_+(\mathbf {n}\otimes \mathbf {n}-\frac{1}{3}\mathbf {I}), \mathbf {n}\in {\mathbb {S}^2}\}$ with $s_+$ a positive constant, and the evolution of alignment vector field $\mathbf {n}$ obeys the harmonic map heat flow, while the interface separating the isotropic and nematic regions evolves by the mean curvature flow. This problem can be viewed as a concrete but representative case of the Rubinstein–Sternberg–Keller problem introduced in Rubinstein et al. (SIAM J. Appl. Math. 49:116–133, 1989; SIAM J. Appl. Math. 49:1722–1733, 1989).