Inclusion of Viscosity Into Classical Homogeneous Nucleation Theory for Water Bubbles in Silicate Melts: Reexamination of Bubble Number Density in Ascending Magmas

To evaluate the effect of melt viscosity on bubble nucleation, we formulated the homogeneous nucleation rate of water bubbles to explicitly include melt viscosity. The viscosity coefficient appears in the preexponential factor of the nucleation rate in terms of the Péclet number: the ratio of the bubble growth timescale by molecular diffusion and the viscous relaxation timescale. The preexponential factor is almost constant when viscosity is low (or a high Péclet number), whereas it linearly decreases with increasing viscosity (or a decreasing Péclet number) exceeding the crossover value of viscosity, under a given supersaturation. The crossover point depends on whether homogeneous or heterogeneous nucleation takes place. We numerically solved the evolution of bubble nucleation and growth processes in ascending magmas by using the new nucleation rate formula and a precise approximation of moment equations of the bubble size distribution function. The resultant bubble number density has two regimes, similar to the previous study, but the transition point between the diffusion-controlled regime and the viscosity-controlled regime moves to higher viscosity or higher decompression rates by 0.6 log units at the maximum. In the viscosity-controlled regime, the effect of the better approximation of bubble size distribution moment equations reduces bubble number density by a few orders of magnitude compared with the previous study. As a result of compiling the past laboratory experimental data, it turned out that all the experiments are conducted under the conditions equivalent to the diffusion-controlled regime. We propose an experimental condition to confirm the presence of the viscosity-controlled regime.