An arbitrary Lagrangian-Eulerian method for simulating bubble growth in polymer foaming

We present a sharp-interface algorithm for simulating the diffusion-driven bubble growth in polymer foaming. A moving mesh of unstructured triangular elements tracks the expanding and deforming bubble surface. In the interior of the liquid, the mesh velocity is determined by solving a Laplace equation to ensure spatially smooth mesh movement. When mesh distortion becomes severe, remeshing and interpolation are performed. The governing equations are solved using a Galerkin finite-element formalism, with fully implicit time marching that requires iteration among the bubble and mesh deformation, gas diffusion and the flow and stress fields. Besides numerical stability, the implicit scheme also guarantees a smooth interfacial curvature as numerical disturbances on the interface are automatically relaxed through the iterations. The polymer melt is modeled as a viscoelastic Oldroyd-B fluid. First, we compute three benchmark problems to validate various aspects of the algorithm. Then we use a periodic hexagonal cell to simulate bubble growth in an isothermal two-dimensional foam, fed by a gaseous blowing agent initially dissolved in the melt to supersaturation. Results show two distinct stages: a rapid initial expansion followed by slow drainage of the liquid film between bubbles driven by capillarity. The effect of viscoelastic rheology is to enhance the speed of bubble growth in the first stage, and hinder film drainage in the second. Finally, we use axisymmetric simulations to investigate the thinning film between a bubble and a free surface. Melt viscoelasticity is shown to initially enhance film thinning but later resist it. An important insight from the simulations is that polymer strain-hardening, namely the steep increase of elongational viscosity with strain, helps stabilize the foam structure by suppressing bubble-bubble coalescence and bubble burst at the foam surface. This confirms prior observations in foam extrusion experiments.