Numerical methods for solvent Stokes flow and solute-solvent interfacial dynamics of charged molecules

Abstract

We develop a computational model for the solvation of charged molecules in an aqueous solvent. Such solvent is modeled as an incompressible fluid, and its dynamics is described by the Stokes equations. All the surface tension force, van der Waals dispersive force, electrostatic force, hydrostatic pressure, and viscous force are balanced on the solute-solvent interface, giving rise to the traction boundary conditions on such an interface. We use the level-set method for the solute-solvent interface motion. To describe the hydrophobic interaction that is characterized by the volume change of the solute region, we design special numerical boundary conditions for the Stokes equations. We then reformulate and discretize the Stokes equations with a second-order finite difference scheme using a MAC grid. At virtual nodes near the solute-solvent interface, we use interpolation to discretize the boundary conditions and formulate the difference equations at such nodes. The resulting linear system is reduced by the Schur complement and then solved by the least-squares method with various techniques of acceleration. Numerical tests show that our method has a second-order convergence in the maximum norm for both the velocity and pressure up to the boundary. We apply our methods to some model molecular systems. With the comparison of our approach with a well established, static solvation theory and method, our dynamic model and numerical techniques accurately reproduce the solvation free energies, and capture the dry and wet molecular states. Our work provides a possibility of describing solvent fluctuations through the solvent fluid flow.