Director profile of a nematic between two concentric cylinders with inhomogeneous boundary conditions

Systems that involve liquid crystal alignment between two concentric cylinders have been investigated in connection with the flexoelectric instability [1], with the stability analysis of the orientational profile [2], and the Fréedericksz transition occurring in the absence of external electric field [3]. The starting point of these analysis was the original problem proposed by Meyer, solved in a special case by Parodi, and then discussed in the book by de Gennes [4]. Subsequently, the same problem was reexamined by Williams [5] by considering that the elastic constants of splay and bend are different, in the strong anchoring approximation. Very recently, the equilibrium problem for a nematic liquid crystal confined within two parallel eccentric cylinders with homeotropic anchoring on the lateral boundaries has been rigorously analyzed by Rosso et al. [6] in the framework of a purely director approach. In general, problems dealing with the equilibrium orientational states of nematic liquid crystals are faced in the framework of the elastic continuum theory for liquid crystalline materials. The basic principle of this theory is that the distorted state can be described by the director n that is of unit length but can be of variable orientation, and represents the average molecular orientation. The equilibrium configuration is the one minimizing the total free energy of the sample subjected to appropriate boundary conditions [7, 8]. The determination of the equilibrium director profile can be formulated in terms of boundary value problems [9]. The boundary value problem concerning the situation of strong anchoring on the boundaries corresponds to the Dirichlet’s problem, whereas the weak anchoring situation leads to the mixed Dirichlet Neumann problem. On the other hand, the situation in which the easy axes characterizing the preferred surface alignment change direction continuously with time is relevant to investigate systems whose surfaces are covered with photopolymeric films [10]. In these systems, the orientational changes of the photochromic molecules promoted by incident light lead to remarkable changes in the molecular orientation. In this direction, problems dealing with orientational dynamics and surface viscosity have also been investigated by Mertely and Copic [11]. For this reason, in order to account for these characteristics of a nematic sam-