Abstract
In this work, we study the nematic–isotropic phase transition based on the dynamics of the Landau–De Gennes theory of liquid crystals. At the critical temperature, the Landau–De Gennes bulk potential favors the isotropic phase and nematic phase equally. When the elastic coefficient is much smaller than that of the bulk potential, a scaling limit can be derived by formal asymptotic expansions: the solution gradient concentrates on a closed surface evolving by mean curvature flow. Moreover, on one side of the surface the solution tends to the nematic phase which is governed by the harmonic map heat flow into the sphere while on the other side, it tends to the isotropic phase. To rigorously justify such a scaling limit, we prove a convergence result by combining weak convergence methods and the modulated energy method. Our proof applies as long as the limiting mean curvature flow remains smooth.
Highlights
Nematic liquid crystals react to shear stress like a conventional liquid while the molecules are oriented in a crystal-like way
One of the successful continuum theories modeling nematic liquid crystals is the Q-tensor theory, referred to as Landau–De Gennes theory, which uses a 3 × 3 traceless and symmetric matrixvalued function Q(x) as order parameter to characterize the orientation of molecules near a material point x
Otherwise there holds ∂t |q dεF (Qε)|2− |Qε|2 0, and the weak maximum principle implies the maximum must be achieved on the parabolic boundary (∂ × (0, T )) ∪ ( × {0}), on which |Qε| is bounded by our assumptions
Summary
Nematic liquid crystals react to shear stress like a conventional liquid while the molecules are oriented in a crystal-like way. It can be shown that, in this case, the two families of minimizers corresponding to (1.8) are the only global minimizers of F(Q): F(Q) 0 and the equality holds if and only if Q ∈ {0} ∪ N , (1.10) At this point we digress to mention that the Landau–De Gennes model (1.5) is closely related to Ericksen’s model, where the energy is eE (s, u):= κ|∇s|2 + s2|∇u|2 + ψ(s) dx. In contrast to (1.5) which uses Q ∈ Q as order parameter, Ericksen’s model uses (s, u) ∈ R × S2 and is very useful to describe liquid crystal defects The analysis of this model is very challenging, mainly due to the reason that the geometry of the uniaxial configuration (1.4) corresponds to a double-cone, and the energy (1.12) is highly degenerate when s = 0. We show that these estimates are guaranteed by our bounds on the modulated energy
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