Early stage spinodal decomposition in viscoelastic fluids

The effect of viscoelasticity on the early stages of spinodal decomposition is examined. In addition to the concentration and momentum equations for the fluid, the effect of viscoelasticity is included using a linear Maxwell equation for the stress tensor. The growth in the amplitude of the fluctuations depends on the transport coefficient, the viscosity of the fluid, and the relaxation time in the Maxwell model. For simplicity, the nonlinearity due to the quartic term in the expression for the Landau–Ginzburg expression for the free energy is neglected, as are the inertial terms in the momentum conservation equation. The momentum and Maxwell equations are solved exactly to obtain the velocity as a function of concentration, which is then inserted into the concentration equation. There are two types of nonlinearities in the conservation equation—one proportional to the cube of the concentration which leads to a four point vertex, and one proportional to the product of the concentration and the random noise in the stress equation which leads to a three point vertex. In the leading approximation, the renormalization of the transport coefficient due to these vertices is determined using the Hartree approximation, and the renormalization of the noise correlation due to the three point vertex is determined using a one-loop expansion. The renormalized transport coefficient and noise correlation are inserted into the concentration equation to determine the effect of the nonlinearities on the growth of the structure factor. It is found that an increase in the relaxation time tends to increase the rate of growth of the structure factor, and tends to decrease the wave number of the peak in the structure factor