An Analysis of Thermal Instabilities in Obliquely Aligned Layers of a Nematic Liquid Crystal

Employing continuum theory, this paper investigates the occurrence of stationary and oscillatory convective instabilities when a 'comparatively thick' horizontal layer of nematic liquid crystal is subjected to a vertical temperature gradient, the director being uniformly obliquely aligned at the boundaries. Two distinct types of roll instability are considered, one being independent of the direction perpendicular to the vertical planes containing the initial director alignment and the other being independent of the horizontal direction in these planes. To obtain accurate numerical values for the threshold gradients, we use Chebyshev polynomial expansions. The variation of the critical temperature gradients for both types of instability with the field . strength of a uniform magnetic field applied parallel to the initial alignment and with the boundary alignment itself is investigated. Introduction The occurrence of thermal instabilities in a layer of nematic liquid crystal contained between two horizontal flat plates has been the subject of several recent investigations [1—9]. The possibility of a stationary convective roll instability in thin layers (typically 1 mm), when the anisotropic axis of these transversely isotropic liquids is initially uniformly aligned either parallel (planar) or perpendicular (homeotropic) to the bounding plates, is considered in references [1—4]. It is found that such an instability is possible when the lower plate is heated in the planar problem, as is the case for a conventional isotropic liquid, but in the homeotropic sample one must heat the upper plate. One notes that typical critical thresholds are much smaller than those appropriate for isotropic fluids with similar average properties. In addition, Velarde and Zuniga [4] obtain a critical threshold at which convection is possible in the planar configuration when heating is from above. However, the threshold value is such that this instability is unlikely to be observed in the laboratory. Extending the analysis to the problem of oblique alignment at the plates, Barratt and Bramley [5] predict two 0340-0204/82/0007-0371 $02.00 © Copyright by Walter de Gruyter & Co. · Berlin New York 372 PJ· Barratt, J. S. Bramley different types of stationary roll, the one expected to appear being dependent upon whether heating is from above or below in agreement with the observations of Guyon, Pieranski and Boix [10]. Investigating the possibility of convection in a homeotropic nematic heated from below, Lekkerkerker [6, 7] and Velarde and Zuniga [4] predict the occurrence of an oscillatory instability. Guyon, Pieranski and Salan [8] observe just such an instability in a comparatively thick layer (typically 5 mm) and they analyse a simple one-dimensional model of the problem. Employing Chebyshev polynomial expansions, Barratt and Sloan [9] obtain accurate numerical results for this problem in the case of the material MBBA and their results agree rather well with the experimental observations. 1. Content In this paper, we extend the analysis of Barratt and Sloan [9] and Barratt and Bramley [5] to investigate the onset of convective instabilities in comparatively thick layers of nematic liquid crystal when the anisotropic axis is obliquely aligned at the plates. Linearising the governing equations about a known equilibrium configuration, the stability of the uniformly aligned configuration with respect to two particular types of perturbation is examined. In one the perturbation variables are considered to be independent of the direction perpendicular to the vertical planes containing the initial director orientation and in the other they are assumed to be independent of the horizontal direction in these planes. To obtain numerical values for the critical temperature gradient at the onset of instability, we employ the Chebyshev polynomial method described by Orszag [11] and Barratt and Sloan [9] and hence reduce the problem to finding the solution of an algebraic eigenvalue problem. The variation of threshold gradients with boundary orientation for both stationary and oscillatory instabilities is examined in the cases for when heating is from above and below. In addition, the effect of a magnetic field applied parallel to the boundary alignment is investigated. Since detailed accounts of the physics of nematic liquid crystals and continuum theory describing their macroscopic behaviour are contained in the books by de Gennes [12] and Chandrasekhar [13] and the reviews by Stephen and Straley [14] and Ericksen [15] and Leslie [16], no further account appears necessary here. Suffice it to say that we use the continuum equations and notation employed in reference [16]. 2. Statement of the problem This paper investigates the stability of a sample of nematic liquid crystal contained between two horizontal plates of large extent when subjected to a vertical temperature gradient. The upper and lower plates are held at temperatures T2 and Tj, respectively, and the director is uniformly aligned at the plates. In addition, we allow for the presence of a uniform magnetic field H applied parallel to'the boundary alignment of the anisotropic axis. Cartesian coordinate axes are chosen so that the J. Non-Equilib. Thermodyn. , Vol. 7, 1982, No. 6 Thermal instabilities in liquid crystals 373 upper and lower plates occupy the planes z = h and z = 0, respectively, the director orientation at the plates has component form n° = (cos ÈÏ , 0, sin È0), 0 < È0 < ð/2 , (2. l) and the applied magnetic field is given by H = (H cos È0, Ï, Ç sin È0) , (2.2) where È0 and H are constants. One simple solution of the continuum equations is the uniformly aligned equilibrium configuration illustrated in Fig. 1. Here í = 0 , n° = (cos È0, 0, sin È0) , Ô = T°(z) = Öæ + Ô, , (2.3) where Ö is the constant temperature gradient (T2 — T^/h. We wish to examine the situation when this uniformly aligned state is subjected to a small amplitude velocity field í associated with which are small perturbations ç and s in the director and temperature fields, respectively. Employing the Boussinesq [17] approximation, the linear equations governing the evolution of the perturbation variables are essentially those given by Barratt and Bramley [5]. The only difference is that now there is a term — %a H i^ on the right hand side of their Eq. (3.8) due to the presence of the magnetic field, %a being the anisotropic part of the magnetic susceptibility. Influenced by the observations of Guyon et al. [10], we consider two particular forms for the perturbed solution. In one the perturbation variables are independent of the spatial variable y and the solution is referred to as a Y-axis roll solution, the axis of the rolls being parallel to the y-axis. In the second, the perturbation variables are independent of the spatial coordinate ÷ and the corresponding roll solution is referred to as an X-axis roll solution. As is common we neglect the director inertia term in the analysis. / / ' vv Ë / ° ••-X = T, Fig. 1: Schematic diagram of equilibrium configuration. J. Non-Equilib. Thermodyn. , Vol. 7, 1982, No. 6 374 Barratt, J. S. Bramley 3. The Y-axis roll solution • f Here we examine the stability of the uniformly aligned equilibrium state with respect to disturbances of the form ft = i(nl5 0,n3) exp (im (÷ — kt)), í = (v4 , 0, v3) exp (im(x — kt)) , s = sexp(im(x — kt)) , (3.1) where nj , n3, Vj , v3 and s are functions of z alone and k is in general complex. With the change of independent variable