The Saint-Venant principle for columnar discotic liquid crystals

2014 We compute the high frequency elastic distortion of a columnar discotic single crystal submitted to a longitudinal dilation. The ends of the columns are assumed to be clamped and anchored perpendicularly to two infinitely rigid glass plates, on a width L. The local strain close to the plates relaxes to the homogeneous distortion of an unclamped sample on a characteristic length £ ~ (Lm)1/2, where m is a molecular length. £ is much shorter than L, the usual damping length in standard solids. This results from the mixed (displacement and curvature) elasticity of columnar discotic materials, analogous to the one of smectic materials. The apparent Young modulus of an ideal short discotic single crystal should be length dependent. The free relative glide of columns allows also the conservation, along the columns, of any transverse non-uniformity of the longitudinal strain or stresses. We reformulate the Saint-Venant principle for these anisotropic materials. In practice, permeation and plasticity should prevent these non-uniform distortions from existing permanently. J. Physique Lett. 45 (1984) L-335 L-341 ler AVRIL 1984, Classification Physics Abstracts 61. 30C 62. 20D The structure of discotic columnar liquid crystals [1] is now reasonably well understood [2]. Disc-like molecules pile up in columns. At rest the columns axes are arranged as a two-dimensional solid. The columns are free to glide along each other, and can be bent easily. In practice, discotic samples grow spontaneously between two parallel glass plates in the « homeotropic » texture, i.e. with the columns normal to the plates. In this geometry, by changing the plates spacing, one can determine the stress-strain relationship of discotic single crystals, as is done for other smectic materials [3]. In the present note, we show that the elastic behaviour of columnar discotic single crystals must depend critically on their thickness. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:01984004507033500 L-336 JOURNAL DE PHYSIQUE LETTRES The sample geometry is sketched in figure 1. At rest, the columns are normal to the two plates, at z = ± d/2. A strain is imposed by the normal (z) displacements ± 5/2 of the plates. The free energy of a distorted columnar phase was first written [4] in term of the tube displacement variable u (normal to the columns) and of the volume dilation 0. It was later rewritten [5] using the u and v variables, where v is the local displacement parallel to the columns. For a solid state physicist, this last description suffers from a serious drawback. Since there cannot be any one-dimensional long-range ordering along the columns, the v variable has no meaning on a molecular scale. In fact, we can always define a v variable by proceeding as in fluid mechanics. v will be the displacement along the columns of the centre of mass of a group of molecules in the same column. Let us assume that we deal with mechanical properties on a short enough time scale, so that molecular diffusion and permeation are negligible. The dilation 0 can be written as : 0 = div u Ov In z ° reference [4], the free energy was a quadratic function of e and div u. It transforms obviously into a quadratic function of Ov and div u, so that the two descriptions from [4] and [5] are equivalent. az 1~ ] ] The point is that, because of the one-dimensional liquid-like character of the columns, the knowledge of av/az allows the knowledge of v, i.e. the columns behave as an elastic continuum. This is to be contrasted to the behaviour of a two(or three-) dimensional liquid, where obviously the local dilation does not control the exact shape of the body. In our problem, where the boundary conditions impose a displacement ± 5/2 to the column ends, the use of the v variable is more convenient. To simplify, we deal with a two-dimensional problem (x, z). The glass plates are assumed much more rigid than the 2-D crystal of columns, so that we use clamping conditions on the plates (u(x, ± d/2) =0). We also assume that the columns are strongly anchored on the plates au (x, + d/2) = 0 ). The sides of the sample at rest (at x = + L/2) are assumed to be free, az(~_ p _ and parallel to the columns. For the time being, we neglect surface tension. To lowest order, the elastic free energy density :F can be written as : a and b are elastic constants associated with stretching the columns and the 2-D (here 1-D) crystal. Fig. 1. Undistorted sample geometry. The vertical lines represent the columns, clamped perpendicularly to the horizontal plates. L-337 SAINT-VENANT PRINCIPLE FOR DISCOTICS c is a coupling coefficient. The last term is the curvature energy associated to the bend of columns. 02 u We have neglected the splay term, involving ~ ~ , by assuming L > d, which is the case of Y ~ g ax az’ ’ practical samples (L/d ~ 100). m is of the order of a molecular length ; a, b, c, d are expected of the same order of magnitude ( ~ 108 cgs units). Stability requires a, b, d > 0 and ab C2 > 0. Minimizing ~ with respect to u and v gives the Euler equations : (1) expresses the continuity of the normal stress y~(x) through the sample. (2) can be written as : The solution of (3) is the superposition of a particular solution uo(x) and of the general solution of 2 equation (3) without the term on right hand side. We call (b 2013 2013)~~ = ¡x2, (¡x ’" 1) and ( a u = f(x) g(z). We must have : The boundary condition on the free edges is written as : We must then look for sine-like solutions for J; by taking the constant negative in (4). The normal modes for u are of the shape f" = sin m 2 with the quantization condition (Xn from (5) (n is an integer > 0). To solve the problem, we must Fourier expand the constant dilation ± 5/2 of the plates on the f,,, calculate the corresponding strains and stresses, and sum up again. Let us now for simplicity drop the index n and calculate the stress and strain for one mode. The solution of (3) can be written as : because u is even in z and zero on the plates. A is a complex amplitude to be determined. We now integrate equation (1), and find for v, which must be odd in z, and equal to ± (5/2) cos MXIOCI 2 L-338 JOURNAL DE PHYSIQUE LETTRES on the plates, the expression : To determine A, we use first the strong anchoring condition (3M/~z = 0 on the plate), and the condition for uo(x) from (3) : we then write these two conditions as :