Shear-Induced Buckling Instability in The Lamellar Phase:a Mechanism for Onions Formation

We present a model of coupling of shear flow to microscopic degrees of freedom in the lyotropic Smectic A (lamellar phase ). This coupling is then shown to produce macroscopic buckling instability analogous to well known undulation instability appearing in thermotropic smectics under dilatation in the direction perpendicular to the layers. The Iyotropic Lamellar phase is experimentally known to become unstable under shear flow. The instability occurs at shear rate γ ∼ 1sec-1and the so called 'onions' (multi lamellar vesicles) are formed. We propose that the instability occurs in two stages. At the first stage, the layers are buckled by shear -induced microscopic tension σ ≅ ηeγd where ηe is an effective viscosity of the system (which can be very large :ηe ≅ 1000ηsolvent) and d is all inter- layer spacing. As the membrane projected area cannot increase (for example, being sandwiched between defects) it has to buckle in the direction perpendicular to itself.There is a critical tension however due to energy cost of compression of layers in the buckled state and of bending and expansion of individual layers. Then one is able to calculate the buckling amplitude Uo and the wavevector k as a function of γ. Denoting K usual smectic bending modulus and B the compressional one, the result is that the transition occurs at σ ≅ T/Dd, where D is the width of a film. For σ - σc >> σc we have k ≅ √(8σ/3dB). At the second stage the buckled structure is deformed by shear (by coupling to Macroscopic degrees of freedom) nad eventually broken, resulting in the formation of onions. We speculate that the buckled state will become unstable when the relaxation time of the buckling structure τc ≅ ηc/Kk2 is shorter than the one needed for shear to deform the structure to a degree that maxima are shifted to the position of neighboring minima. The estimate is γ ≅ Kk2(U0k)-1 which gives γ ∼ d-a with 7/3 < a < 17/7 which is consistent with experimental data a= 2.4 ±: 0.4