Shear-induced instabilities in layered liquids

Abstract

Motivated by the experimentally observed shear-induced destabilization and reorientation of smectic A like systems, we consider an extended formulation of smectic A hydrodynamics. We include both, the smectic layering (via the layer displacement u and the layer normalp) and the director n of the underlying nematic order in our macroscopic hydrodynamic description and allow both directions to differ in non equilibrium situations. In an homeotropically aligned sample the nematic director does couple to an applied simple shear, whereas the smectic layering stays unchanged. This difference leads to a finite (but usually small) angle betweenn andp, which we find to be equivalent to an effective dilatation of the layers. This effective dilatation leads, above a certain threshold, to an undulation instability of the layers. We generalize our earlier approach [Rheol. Acta 39 (3), 15] and include the cross couplings with the velocity field and the order parameters for orientational and positional order and show how the order parameters interact with the undulation instability. We explore the influence of various material parameters on the instability. Comparing our results to recent experiments and molecular dynamic simulations, we find a good qualitative agreement.

Boundary Layers in Pressure-driven Flow in Smectic A Liquid Crystals

“…2 since n and a are unit vectors, which is the form used in [1,2,29]. Although B 1 has been investigated theoretically, measurements for it are scarce in the literature.…”

“…Auernhammer et al [2]): the exact solutions can still be found when this approximation is not made but the results become unwieldy and tend to obscure the key roles of various material parameters. Notice also that, using (3.3), (3.2) can be rewritten as (3.5) ηu − uλ…”

“…Equations (2.2), (2.3), (2.7), (2.12), and (2.13) provide nine equations in the nine unknowns Φ, n i , v i , p, and μ ; the smectic layer normal a is, of course, determined by (2.1) from the solution for Φ. One elementary candidate for an energy density is that used by Stewart [33], which has been based upon the one evidently first introduced by Ribotta and Durand [27] and other variants deployed in references [1,2,29,33,32,30,34,13,37,38]. It is given by [33] (2.14)…”

“…Note that J is a natural nonlinear extension to the versions discussed in [1,2,16,11,12]: when it is suitably linearized for small changes in the layer and director orientations, then it reduces to the explicit expressions found in [1,2] when n and a are allowed to separate. It further reduces to the classical results in [12,16] when n ≡ a.…”

“…The first term on the right-hand side of (2.14) represents the usual elastic splay deformation of the director n, while the second term is a measure of the bending of the smectic layers; both K n 1 and K a 1 are positive elastic constants (measured in newtons). The third term represents smectic layer compression and originates from the classical descriptions of SmA liquid crystals [12,16,2]; B 0 is the positive layer compression constant. The fourth expression accounts for the strength of the coupling between n and a with the positive constant B 1 having the same dimensions as B 0 (energy per unit volume): in an equilibrium state this energy contribution is clearly minimized when n and a are parallel.…”