System size dependence of the structure and rheology in a sheared lamellar liquid crystalline medium

The structural and rheological evolution of an initially disordered lamellar phase system under a shear flow is examined using a mesoscale model based on a free energy functional for the concentration field, which is the scaled difference in the concentration between the hydrophilic and hydrophobic components. The dimensionless numbers which affect the shear evolution are the Reynolds number (γ˙¯L²/ν), the Schmidt number (ν/D), a dimensionless parameter Σ=(Aλ²/ρν²), a parameter μ_r which represents the viscosity contrast between the hydrophilic and hydrophobic components, and (L/λ), the ratio of system size and layer spacing. Here, ρ, ν, and D are the density, kinematic viscosity (ratio of viscosity and density), and the mass diffusivity, and A is the energy density in the free energy functional which is proportional to the compression modulus. Two distinct modes of structural evolution are observed for moderate values of the parameter Σ depending only on the combination ScΣ and independent of system size. For ScΣ less than about 10, the layers tend to form before they are deformed by the mean shear, and layered but misaligned domains are initially formed, and these are deformed and rotated by the flow. In this case, the excess viscosity (difference between the viscosity and that for an aligned state) does not decrease to zero even after 1000 strain units, but appears to plateau to a steady state value. For ScΣ greater than about 10, layers are deformed by the mean shear before they are fully formed, and a well aligned lamellar phase with edge dislocation orders completely due to the cancellation of dislocations. The excess viscosity scales as t^-1 in the long time limit. The maximum macroscopic viscosity (ratio of total stress and average strain rate over the entire sample) during the alignment process increases with the system size proportional to (L/λ)^3/2. For large values of Σ, there is localisation of shear at the walls, and the bulk of the sample moves as a block. The thickness of the shearing region appears to be invariant with the system size, leading to an increase of viscosity proportional to L. The time for structural evolution is found to be the inverse of the strain rate γ˙^-1. In the case of a significant viscosity contrast between the hydrophilic and hydrophobic parts, the average viscosity increases by 1-2 orders of magnitude due to the defect pinning mechanism, where the regions between defects move as a block, and shear localisation at the wall.