The role of viscoelastic contrast in orientation selection of block copolymer lamellar phases under oscillatory shear

The mesoscale rheology of a lamellar phase of a block copolymer is modeled as a structured fluid of uniaxial symmetry. The model predicts a viscoelastic response that depends on the angle between the the local lamellar planes and velocity gradients. We focus on the stability under oscillatory shear of a two layer configuration comprising a parallel and a perpendicularly oriented domain, so that the two layers have a different viscoelastic modulus $G^{*}(\omega)$. A long wave, low Reynolds number expansion is introduced to analytically obtain the region of stability. When the response of the two layers is purely viscous, we recover earlier results according to which the interface is unstable for non zero Reynolds number flows when the thinner layer is more viscous. On the other hand, when viscoelasticity is included, we find that the interface can become unstable even for zero Reynolds number. The interfacial instability is argued to dynamically favor perpendicular relative to parallel orientation, and hence we suggest that the perpendicular orientation would be selected in a multi domain configuration in the range of frequency $\omega$ in which viscoelastic contrast among orientations is appreciable.