Boundary conditions and reversibility in diffusion controlled reactions. II. A three state model for steady state evaporation of a spherical drop

Abstract

A simple three state model of the reaction complex was introduced into reversible diffusion controlled reaction theory and the flux balance equations were derived for all times. The long time asymptote of this relation was used to describe the evaporation from and condensation on a spherical drop immersed in a dilute gas infinitely dilute in vapor. The molecular velocity distribution is required to satisfy the linearized steady-state Boltzmann equation with the molecular interaction taken to be Maxwell’s potential. The solution to this and the boundary condition imposed by the flux-balance equations of the three state model was approximated by a four mode ansatz. The coefficients of these modes were determined by four moment equations which could be shown to satisfy a variation principle due to Cercignani. Two of the modes corresponded to two of the components of the usual Chapman–Enskog ‘‘normal’’ solution to the problem and were found to be dominant at long distances, falling off as r−1 and r−2. The other two modes were discontinuous in velocity, and, outside a boundary layer several mean free paths thick, were found to decay as r−4 and r−3. The boundary condition at the drop surface was found to be of the usual radiation boundary condition type used in diffusion controlled reaction theory, modified by a factor dependent on mean free path and reaction probability.