We present a dissipative particle dynamics study of surfactant solutions under shear, which allows us to investigate their rheological properties. We consider a variety of concentrations and phase structures, including micellar solutions and liquid crystal phases. It is shown that the viscosity of micellar solutions increases as a function of concentration, in agreement with what is expected from experimental data. We also show that micelles can exhibit shear-thinning behavior when a shear force is applied, which is a result of micelles breaking down into smaller aggregates. Lamellar and hexagonal phases are found to orientate under the application of shear, in agreement with experimental observations. It is normally suggested that lamellar phases under shear can exhibit a transition between orientations as the shear rate is increased, usually as a result of lower viscosity. We calculate the viscosity for different lamellar phase orientations, showing that, although the viscosity of perpendicular orientations is lower than that of parallel orientations, we do not observe a transition to the perpendicular phase at high shear rates. Finally, we show that the choice of Schmidt number has a significant impact on the results, which is important for determining the correct behavior via simulations.
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