Polymer foams are usually produced by injecting gas at high pressure into a molten polymer, and then reducing the pressure suddenly to nucleate gas bubbles. The nucleated bubbles grow and form a low density cellular structure that has many applications. This thesis focuses on the development of numerical models for the simulation of foaming phenomena. Different simulation techniques, including the lattice Boltzmann method, molecular dynamics, and the immerse boundary method, are explored to develop a an improved 1D Cell Model, as well as a comprehensive 2D/3D numerical model that accounts for all physical aspects of foaming, and can be used for modeling of the foaming process and polymer foam composites.

An improved 1D Cell Model is presented where the effect of the local variation of viscosity around growing bubbles is investigated. Due to the diffusion of gas from the polymer melt into growing bubbles, the viscosity at the bubble-melt interface can be significantly larger than further from the bubble. This effect of the viscosity profile on bubble growth and deformation is numerically investigated. We show that the viscosity variation slows the bubble growth rate at the early stages, and increases the resistance of bubbles to deformation.

Next, a 2D/3D free surface lattice Boltzmann method simulation package called LBfoam for the simulation of foaming processes is presented. The model incorporates the essential physics of foaming phenomena:​ gas diffusion into nucleated bubbles, bubble dynamics and coalescence, surface tension, the stabilizing disjoining pressure between bubbles, and Newtonian and non-Newtonian rheological models.

Finally, a hybrid lattice Boltzmann method-molecular dynamics-immersed boundary method model is presented for simulating the foaming process of polymer composites. The LBfoam solver is used to resolve the foaming process, and the MD model accounts for filler dynamics. These two solvers are coupled by a direct forcing IBM. This solver can simulate composite foaming processes involving many bubbles and filler particles, including rigid, deformable, and fragile fillers. The solver relaxes most simplifying assumptions of earlier polymer composite models, allowing for a better understanding of filler motion and interaction with growing bubbles, to produce composite foams with improved mechanical, electrical, and electromagnetic properties.