Diffusion controlled growth and dissolution of gas bubbles.

The material transfer between an isolated stationary bubble and the surrounding liquid has been formulated mathematically and methods were developed to solve the relevant differential equations. Solutions were made dimensionless to generalize their application. Exact solutions have long been available to describe growth of one- component spheres from zero size. These solutions were used to show that the corresponding finite difference solutions for growth from finite size are accurate. Exact solutions were also derived for diffusion controlled growth from zero size of one-component spheres with concentration dependent diffusivity and for growth of multi-component bubbles. This type of solution was also used to demonstrate the accuracy of the finite difference solutions of corresponding problems for growth from finite size. These finite difference methods were also used for dissolving bubbles where analytical solutions are not possible. Several approximate solutions are discussed and quasi steady-state solutions were obtained for growth or dissolution with concentration dependent diffusivity and inclusion of surface tension. For small solubilities and concentrations, (low solubility parameters), the correct diffusion controlled solutions always converge to the corresponding quasi steady-state. A large number of solutions was obtained to cover the significant ranges of values of the relevant parameters for gases in glass melts. These solutions include the study of limiting regimes for very low and very large solubility parameters as well as the intermediate range. Solutions for multicomponent stationary bubbles were used to discuss some experimental observations of bubbles in glass melts. The effect of surface kinetics on growth or dissolution of spheres was investigated and solved to illustrate the transition between control by diffusion and control by surface kinetics. An extensive study of the effect of surface tension was carried out. The roles of viscosity and inertia were considered and are usually negligible.