Squint-a Nonlinear Feature of Flow Patterns in Nematics

The curvature $ of an electronematodynamic flow pattern obeys, in the steady state, the nonlinear equation $" + qZ $ t,h3 + b$$' = 0. The term $ $ I , called squint, is studied. Squint originates in the distortion torque and the distortion stress. Squint brakes the axial inversion symmetry of a simple flow roll but preserves the reflection symmetry of a roll pair. Flow patterns are predicted to show an observable pairing of domain stripes. Terms unjustifiably discarded in the one-constant approximation are explicitly introduced. The mean angle approximation which replaces the variable coefficients qZ and b by constant mean values is also discussed and justified. Applied electric, magnetic and thermal fields induce in nematic liquid crystals a variety of interesting flow patterns [l-41. The well known Williams domains may be taken as a typical representative. Recently, the importance of nonlinear phenomena in these flow patterns has been increasingly recognized [5-111. Moritz and Franklin [7] presented a concise but reasonably self-contained derivation of the governing equations for the charge density q and the pattern curvature $. A comprehensive and deatiled treatment was given by Moritz [12]. Their result is : This paper focuses on the remarkable nonlinear term with t,b dt,b/dx that has been neglected heretofore. For reasons that will become clear later, let us call it squint. The presentation will be concise. A full report will be published elsewhere. We deal with a nematic in a standard sandwich geometry, independent of the z-coordinate. An electric field is applied across the sample in the y-direction. The angle between the director and the x-axis is cp. The squint terms arise from two sources : the distortion torque density m, and the distortion stress density %? Moritz, in his analysis [7, 121, follows de Gennes [ I ] and chooses the pattern curvature t,b = acplax, rather than the director angle c p , as the relevant variable. Accordingly, we are interested in the derivatives of the distortion torque density : m,x = am,/ax = k(cpxxx + cp,,,) + 2 6 ( C[1/2(vxxx cp,,,) + cpxx 9, + 3 c p , cpx v: + cpx cp;1 + + S[cpXXY 2 cpxx cpx + cpx, cp, + ( P y y cpx 2 cp; cpyl I 3 m,, = am,ldy = k(cp,,, + cpxxy) + 2 6 ( C1112(cp,,, + cpxxy) + cpyy cpx + 3 40xy c p g + 40; 40,. 4021 + (2) + S[cpXYY + 2 cp,, cp , cp,, vx cpxx cp , 2 cp; vxlI . Here we have introduced the notations (*) On sabbatical leave from : Department of Physics BenGurion University of the Negev 84 120, Be'er-Sheba', Israel. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979349 C3-260 SHELAMO I. BEN-ABRAHAM and k = l/2(k33 + k l l ) , 6 = 1/2(k33 k l l ) , where k,, and k, , are the bend and splay modulus, respectively. The subscripts x, y denote partial derivatives, of course. The body force density components fx, f,, due t o 7 a r e given by f x = k(2 cpxx cpx + cpx, cp, + cp,, cpx) + 6[C(2 cpxx cpx cpx, cp, cp,, cpx + 4 cp; cp,). + + S(cpxx (P, + 3 4oxy 4ox 2 cp: + 2 cpx cpy2)I 3 fy = k(2 (Pyy (Py + 4oxy cpx + cpxx (P,) + 6[C(2 cpyy cp, + (Px, cpx + cpxx cpy + 4 cp; 4ox) + (5) + S(cpyy cpx + 3 cpxy cp, + 2 cp; 2 cpy 931 In eqs. (2) and (5) we have conveniently separated the terms kept in the one-constant approximation from the (( correction )) terms which are by no means small. Following Penz and Ford [4, 7, 121, we now eliminate the dependence on y by introducing an arbitrary dispersion parameter p such that All derivatives in eqs. (2) and (5) can now be expressed in terms of Then, eqs. (2) and (5) yield, respectively, the following squint terms :