Retrograde vitrification equation of state for polymeric foaming*

A key to controlling the cell structure of foams is the interaction between the polymer and the blowing agent. Clearly, the pressure-volume-temperature equation of state relationship as expressed through the solubility of a blowing agent in a polymer melt is at the heart of the design and prediction of polymeric foams, particularly for micro- and nanocellular foams, where the voids are on the order of microns and nanometers, respectively. The Sanchez-Lacombe equation of state is a lattice-based choice that is widely used because it is a first principles rigorous approach while at the same time being coarse-grained with phenomenological parameters. This makes it, in many cases, qualitatively and quantitatively acceptable, provided the parameters are carefully calibrated while still being practical to implement in potentially complicated engineering situations. The Sanchez-Lacombe equation of state suffers from inconsistencies however, and I will discuss what these are and what can be done about them using an off-lattice approach. I will also discuss how finite flexibility of polymers can be incorporated into the off-lattice version in a way that also removes problems with the Gibbs-DiMarzio condition, so that the glass transition and retrograde vitrification in polymer solutions can be characterized. The latter has become particularly important in polymeric foaming. I will give some examples using common commodity polymers and blowing agents, and I'll discuss the range of applicability and limitations of the off-lattice model.