A systematic model is presented for a free expansion polymer foaming process that includes simultaneous nucleation and bubble growth. An influence volume approach, which couples nucleation and bubble growth, is used to account for the limited supply of dissolved gas. The melt rheology is described using the Larson viscoelastic model. The initial conditions are obtained at the upper bound of critical cluster size under conditions of elastic deformation. The resulting set of equations are solved using a combination of numerical techniques. A parametric study is conducted to examine the effects of key process variables on bubble growth, nucleation, and final bubble size distribution. It shows that the factors influencing nucleation and growth affect the ultimate bubble sizes and their distribution. The Gibbs number, a dimensionless measure of the barrier to overcome for nucleation, has the strongest impact on the cellular structure of the foam.