A Generalized Rayleigh--Plesset Equation for Ions with Solvent Fluctuations

We introduce a mathematical modeling framework for the conformational dynamics of charged molecules (i.e., solutes) in an aqueous solvent (i.e., water or salted water). The solvent is treated as an incompressible fluid, and its fluctuating motion is described by the Stokes equation with the Landau--Lifschitz stochastic stress. The motion of the solute-solvent interface (i.e., the dielectric boundary) is determined by the fluid velocity together with the balance of the viscous force, hydrostatic pressure, surface tension, solute-solvent van der Waals interaction force, and electrostatic force. The electrostatic interactions are described by the dielectric Poisson--Boltzmann theory. Within such a framework, we derive a generalized Rayleigh--Plesset equation, a nonlinear stochastic ordinary differential equation (SODE), for the radius of a spherical charged molecule, such as an ion. The spherical average of the stochastic stress leads to a multiplicative noise. We design and test numerical methods for solving the SODE and use the equation, together with explicit-solvent molecular dynamics simulations, to study the effective radius of a single ion. Potentially, our general modeling framework can be used to efficiently determine the solute-solvent interfacial structures and predict the free energies of more complex molecular systems.