Discontinuous Order Parameters in Liquid Crystal Theories

Abstract

The paper is concerned with various issues surrounding the mathematical description of defects in models of liquid crystals, drawing on experience from solid mechanics. The roles played by a suitable choice of function space and by the growth properties of the free-energy density are highlighted. Models in which the director can jump across surfaces are formulated, and their relevance for nematic elastomers, order reconstruction and smectic A thin films discussed.

Keywords:

• Order parameters, defects, smectic A, order reconstruction, nematic elastomer

Notes

For the validity of the theorems stated in the paper it is enough to suppose that Ω is a Lipschitz domain, examples of which include balls and cubes.

The integrand in (4.2(4.2) ) is polyconvex, that is it is a convex function of the minors (subdeterminants) of ∇y. Polyconvexity is a sufficient condition for ψ to be quasiconvex, a property that is known to be the central convexity condition of the multi-dimensional calculus of variations but which is poorly understood. The free-energy functions usually used for liquid crystals are convex, whereas one would expect on mathematical grounds that quasiconvex free-energy functions would arise. For a survey of these different convexity conditions see [28].

The graph of the largest eigenvalue has the same qualitative form as that in [64, Fig. 4] but the values are less. This is because the values in [64, Fig. 4] were obtained as the average of the largest eigenvalue of Q(z)(jT) over the trajectory j = 1, …, N, where Q(z)(jT) denotes the Q matrix calculated at time jT by averaging over the molecules in the 1nm slice centred at z; since the largest eigenvalue of a symmetric matrix is a convex function of the matrix the maximum eigenvalue of the average of Q(z)(jT) over j is by Jensen’s inequality less than or equal to the average over j of the largest eigenvalue of Q(z)(jT).

Figure 2. Q-tensor values for 12nm thin film between silicon and a vacuum obtained by atomistic molecular dynamics [64] (courtesy C. Zannoni and M. Ricci); the error bars correspond to the standard deviations obtained when averaging over the molecules in each 1nm thick slice and over the trajectory.

Figure 3. Eigenvalues of Q(z) as given in Fig. 2

Figure 4. Smectic layer structure in 8CB thin film as identified in [58] (courtesy Emmanuelle Lacaze).