Bubble Nucleation in Polymer-CO₂ Mixtures

Bubble nucleation is a critical first step in the foaming process, but its theoretical understanding is still rather incomplete. While the classical nucleation theory (CNT) offers a qualitative picture of the process, more advanced theories are needed to better describe molecular details. Here, we combine the string method with the state-of-the-art density functional theory (DFT) to investigate bubble nucleation in CO₂-supersaturated polymers. Nucleation in our study is initiated by saturating the polymer liquid with high pressure CO₂ gas and subsequently reducing the pressure to ambient condition. The string method is used to find the minimum free energy path on the free energy landscape that connects the metastable CO₂-supersaturated state with a well-developed bubble. The nucleation pathway, i.e., the evolution of the density profiles in the formation of the nuclei, as well as the nucleation barrier, are obtained as a function of the temperature and the initial pressure. We present an incipient-phase analysis which allows identification of phases that can form from the metastable CO₂-supersaturated parent phase. As result of an underlying metastable transition from an incipient CO₂-rich-vapor phase to an incipient CO₂-rich-liquid phase, there can be two kinds of nucleating bubbles, a liquid-like bubble and a vapor-like bubble, which can differ substantially in their nucleation barrier. We apply our theory to three common polymers: polystyrene, polymethylmethacrylate, and a commercial polyol. We also examine the effects of adding a third volatile component to the polymer-CO₂ mixture.