Bubble dynamics in viscoelastic fluids with application to reacting and non-reacting polymer foams

Abstract The effects of fluid viscoelasticity on the expansion of gas bubbles in polymer foams for the cases of reactive and non-reactive polymers are investigated. For non-reactive polymers, bubble expansion is controlled by a combination of gas diffusion and fluid rheology. In the diffusion limited case, the initial growth rate is slow due to small surface area, whereas at high diffusivity initial growth is rapid and resisted only by background solvent viscosity. In this high Deborah number (De) limit, we see a two stage expansion in which there is an initial rapid expansion up to the size at which the elastic stresses balance the pressure difference. Beyond this time, the bubble expansion is controlled by the relaxation of the polymer. In the model for reactive polymer systems, the polymer molecules begin as a mono-disperse distribution of a single reacting species. As the reaction progresses molecules bond to form increasingly large, branched, structures each with a spectrum of relaxation modes, which gel to form a viscoelastic solid. Throughout this process gas is produced as a by-product of the reaction. The linear spectrum for this fluid model is calculated from Rubinstein et al. [Dynamic scaling for polymer gelation, in: F. Tanaka, M. Doi, T. Ohta (Eds.), Space–Time Organisation in Macromolecular Fluids, Springer, Berlin, 1989, pp. 66–74], where the relaxation spectrum of a molecule is obtained from percolation theory and Rouse dynamics. We discretise this linear spectrum and, by treating each mode as a mode in a multimode Oldroyd-B fluid obtain a model for the non-linear rheology. Using this model, we describe how the production of gas, diffusion of gas through the liquid, and evolution of the largest molecule are coupled to bubble expansion and stress evolution. Thus, we illustrate how the rate of gas production, coupled to the rate of gas diffusion, affects the bubble size within a foam.