By Robert Crispin Evans

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If n is always parallel to the (x, z) plane, φ = 0 and Eq. 74) where K(θ) = K11 sin2 θ + K33 cos2 θ. , no twist is considered. , θ = π/2, Eq. 75) K22 φ , 2 which is relevant to a pure twist deformation. Finally, the one-constant approximation (K11 = K22 = K33 = K) is commonly employed and, in this case, Eq. 73) is written as f (φ, φ ) = f (θ, θ , φ, φ ) = Copyright © 2006 Taylor & Francis Group, LLC 1 2 2 K θ + sin2 θ φ . 77) where fS (θ, φ) takes into account the surface contribution to the total energy and will be considered in detail in the next chapter.

F0 (m) is the surface energy density of a uniformly oriented ﬁlm (m position independent), whereas L, N , and M play the role of elastic constants. Tensors L, N , and M have to be decomposed in terms of the elements of symmetry of the ﬁlm. In the present case, in which the ﬁlm is assumed to be ﬂat, the elements of symmetry are the geometrical normal k (parallel to the z−axis) and the vector m (see details in [29]). The term f0 (m) can be expanded in power series of m · k, or in terms of Legendre polynomials.

8) This simpliﬁed expression permits a simple interpretation for the surface free energy. In this case WP is the anchoring energy which, as deﬁned before, corresponds to the work that has to be done to rotate the director from the stable equilibrium position to the unstable one. In general, it is possible to start with the Rapini-Papoular expression by operating in the following manner. 9) fs = − W (n · n0 )2 , 2 in such a manner to underline the fact that the easy direction is the one which, in the absence of external torques, minimizes the surface free energy.