Achieving Convergence in Multiphase Multicomponent Density Gradient Theory Calculations through Regularization

We present a solution strategy for computing the equilibrium density profiles of mixtures consisting of an arbitrary number of components and phases in a closed system at constant temperature. Our approach is based on the density gradient formulation of the Helmholtz energy in the canonical ensemble, which is a functional of the component densities. By extending the corresponding Euler–Lagrange equations with an artificial time, we utilize a time-stepping scheme that converges toward the desired equilibrium state, adopting the approach proposed by Qiao and Sun [ Qiao, Z. ; Sun, S. SIAM J. Sci. Comput. 2014, 36, B708–B728 ]. Numerical methods for this approach are intrinsically incapable of preserving positivity in each time step. This is problematic, since the free energy density, determined by an equation of state, often prohibits nonpositive densities. To overcome this issue, we introduce a regularization of the energy densities that permits nonpositive densities during the time-stepping scheme. At the end of the solution process, the artificial time stabilization and the regularization both vanish, allowing us to obtain the density profile of the mixture, including its bulk compositions and interface profile. In this paper, we utilize the Peng–Robinson equation of state, which is popular in petroleum engineering. However, the proposed solution approach is applicable to arbitrary equations of state.