Undulation instability under shear: A model to explain the different orientations of a lamellar phase under shear?

Abstract

We study the effects of a dilation on a sheared smectic A phase. Through a linear analysis, we show that undulation may grow in the direction of the flow and of the vorticity as found in previous works. At higher shear rates, we evidence that the undulation along the flow disappears whereas it persists in the vorticity direction. We determine the stable or unstable zone as a function of the shear rate and of the lamellar spacing. This allows us to draw a theoretical shear diagram of the instability. Finally we compare our results with the orientations found experimentally in a lyotropic lamellar phase under shear. Our diagram describes qualitatively and quantitatively the transition observed at high shear rate.

Swelling kinetics of a compressed lamellar phase

“…It is important to note that although quite general [4,5], this behaviour is not universal and remains very system-dependent [6][7][8]. Moreover, the mechanism of onions formation is still under a theoretical debate [9,10].…”

Orientation transition of defective lyotropic triblock copolymer lamellar phase

“…The mechanism of the layer orientation transition in the lyotropic lamellar phase was theoretically argued by some researchers [13][14][15]. Ramaswamy argued that the suppression of undulations triggers the transition [13].…”

“…Oswald et al proposed that defects generated from flowing smectic in misaligned plates also induce the dilation strain [19]. Wunenburger and co-workers extended this idea and demonstrated that the shear induces destabilization in the flow direction [15]. We have also recently presented that the defect formation is essential for the shear-induced onion formation [10].…”

“…Instability of the lamellar phase is explained by the Helfrich-Hurault mechanism [13][14][15][16][17]. The HelfrichHurault effect is undulation instability of the smectic phase subjected to the external electric or magnetic field [17].…”

“…Thus reduced repulsive force generates effective dilation along the layer normal of the membrane in order to retain the equilibrium layer spacing. It has been anticipated that, as the dilation becomes larger than the critical value, coherent buckling of the lamellar membranes appears and either the onion phase formation or the layer orientation is eventually triggered [13][14][15][16]. However, experimental observations present that the onion phase formation and the layer orientation transition occur in different shear rate range [1,4,5].…”