Effects of liquid fraction and contact angle on structure and coarsening in two-dimensional foams

Aqueous foams coarsen with time due to gas diffusion through the liquid between the bubbles. The mean bubble size grows, and small bubbles vanish. However, coarsening is little understood for foams with an intermediate liquid content, particularly in the presence of surfactant-induced attractive forces between the bubbles, measured by the interface contact angle where thin films meet the bulk liquid. Rigorous bubble growth laws have yet to be developed, and the evolution of bulk foam properties is unclear. We present a quasistatic numerical model for coarsening in two-dimensional wet foams, focusing on growth laws and related bubble properties. The deformation of bubble interfaces is modelled using a finite-element approach, and the gas flow through both films and Plateau borders is approximated. We give results for disordered two-dimensional wet foams with $256$ to $1024$ bubbles, at liquid fractions from $2\,\%$ to $25\,\%$ , beyond the zero-contact-angle unjamming transition, and with contact angles up to $10^\circ$ . Simple analytical models for the bubble pressures, film lengths and coarsening growth rates are developed to aid interpretation. If the contact angle is non-zero, we find that a prediction of the coarsening rate approaches a non-zero value as the liquid fraction is increased. We also find that an individual bubble's effective number of neighbours determines whether it grows or shrinks to a good approximation.