ABSTRACT - The cell size distribution in a thermoplastic foam to a large extent determines its mechanical and thermal properties. It is difficult to predict because of the many physical processes involved, each affected in turn by an array of factors and parameters. The two most important processes are bubble nucleation and diffusion-driven bubble growth. Neither has been thoroughly understood despite intensive and long-standing research efforts. In this work, we consider foaming by a physical blowing agent dissolved in a polymer melt that contains particulate nucleating agents. We propose a nucleation model based on the concept that heterogeneous nucleation originates from pre-existing microvoids on the solid particles. The nucleation rate is determined by a bubble detachment time. Once nucleated, the bubbles grow as the dissolved gas diffuses through the polymer melt into the bubbles, a process that couples mass and momentum transport. By using the Oldroyd-B constitutive equation, we explore the role of melt viscoelasticity in this process. Finally, we integrate the nucleation and growth models to predict the evolution of the bubble size distribution. A cell model is employed to simulate the effects of neighboring bubbles and the depletion of blowing agents. The latter also causes the nucleation rate to decline once growth of older bubbles is underway. Using the physical and operating parameters of a recent foam extrusion experiment, we are able to predict a cell size distribution in reasonable agreement with measurements.