On the Isotropic–Nematic Phase Transition for the Liquid Crystal

In this paper, we study the isotropic–nematic phase transition for the nematic liquid crystal based on the Landau–de Gennes 𝐐 -tensor theory. We justify the limit from the Landau–de Gennes flow to a sharp interface model: in the isotropic region, 𝐐≡ 0 ; in the nematic region, the 𝐐 -tensor is constrained on the manifolds 𝒩={s_+(𝐧⊗𝐧-1/3𝐈), 𝐧∈𝕊^2} with s_+ a positive constant, and the evolution of alignment vector field 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 ).