Undulation instability of lamellar phases under shear: A mechanism for onion formation?

We consider a lamellar phase of bilayer membranes held between two parallel plates and subject to a steady shear. Accounting for the coupling with the shear flow of the short wavelength undulation modes that are responsible for the membrane excess area, we argue that the flow generates an effective force which acts to reduce the excess area. From the viewpoint of the macroscopic lamellar whose geometric dimensions are fixed, this force translates into an effective lateral pressure. At low shear rates (gamma)over dot this pressure is balanced by the elastic restoring forces of the lamellar. Above a critical shear rate (gamma)over dot(c) similar to d(-5/2)D(-1/2), where d is the interlayer distance and D is the gap spacing, the lamellar buckles into a harmonic shape modulation, and we predict its wavelength lambda(c) and amplitude U-o. We show that our model is isomorphic to a dilative strain, which is known to induce a similar buckling (undulation) instability. Indeed, at threshold the wavelength is lambda(c) similar to root Dd and is identical in both cases. Using a non-linear analysis, we discuss how the wavelength and amplitude vary with shear rate away from the threshold. For (gamma)over dot much greater than (gamma)over dot(c) we find lambda(c) similar to (gamma)over dot(-1/3) and U-o similar to (gamma)over dot(2/3). We then focus on the coupling of the buckling modulation itself with the flow, and obtain a criterion for the limit of its stability. Motivated by experiments of D. Roux and coworkers, we assume that at this limit of stability the lamellar breakups into "onion"-like, multilamellar, vesicles. The critical shear rate (gamma)over dot* for the formation of onions is predicted to scale as (gamma)over dot* similar to (gamma)over dot(c) similar to d(-5/2)D(-1/2). The scaling with d is consistent with available experimental data.