  In this paper, a thermal-dynamical consistent model for mass transfer across
permeable moving interfaces is proposed by using the energy variation method.
We consider a restricted diffusion problem where the flux across the interface
depends on its conductance and the difference of the concentration on each
side. The diffusive interface phase-field framework used here has several
advantages over the sharp interface method. First of all, explicit tracking of
the interface is no longer necessary. Secondly, the interfacial condition can
be incorporated with a variable diffusion coefficient. A detailed asymptotic
analysis confirms the diffusive interface model converges to the existing sharp
interface model as the interface thickness goes to zero. A decoupled energy
stable numerical scheme is developed to solve this system efficiently.
Numerical simulations first illustrate the consistency of theoretical results
on the sharp interface limit. Then a convergence study and energy decay test
are conducted to ensure the efficiency and stability of the numerical scheme.
To illustrate the effectiveness of our phase-field approach, several examples
are provided, including a study of a two-phase mass transfer problem where
drops with deformable interfaces are suspended in a moving fluid.
