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.
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 (Equation4.2(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 [Citation28].
The graph of the largest eigenvalue has the same qualitative form as that in [Citation64, ] but the values are less. This is because the values in [Citation64, 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 [Citation64] (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.](/cms/asset/5a5d8fa5-6891-407f-96bf-10f3818d8d89/gmcl_a_1030571_f0002_oc.jpg)
![Figure 4. Smectic layer structure in 8CB thin film as identified in [Citation58] (courtesy Emmanuelle Lacaze).](/cms/asset/658c509d-04da-454a-8044-9f6256bd6965/gmcl_a_1030571_f0004_c.jpg)