Bubble nucleation in polymer–CO 2 mixtures Xiaofei Xu, a Diego E. Cristancho, b St´ ephane Costeux b and Zhen-Gang Wang * a We combine density-functional theory with the string method to calculate the minimum free energy path of bubble nucleation in two polymer–CO 2 mixture systems, poly(methyl methacrylate) (PMMA)–CO 2 and polystyrene (PS)–CO 2 . Nucleation is initiated by saturating the polymer liquid with high pressure CO 2 and subsequently reducing the pressure to ambient condition. Below a critical temperature (T c ), we find that there is a discontinuous drop in the nucleation barrier as a function of increased initial CO 2 pressure (P 0 ), as a result of an underlying metastable transition from a CO 2 -rich-vapor phase to a CO 2 - rich-liquid phase. The nucleation barrier is generally higher for PS–CO 2 than for PMMA–CO 2 under the same temperature and pressure conditions, and both higher temperature and higher initial pressure are required to lower the nucleation barrier for PS–CO 2 to experimentally relevant ranges. Classical nucleation theory completely fails to capture the structural features of the bubble nucleus and severely underestimates the nucleation barrier. 1. Introduction Bubble nucleation in polymer–carbon dioxide (CO 2 ) mixtures is a problem of great interest in the manufacturing of polymer foams since it plays a crucial role in determining the cell size and pore density of the foam materials. 1 As a fundamental problem, bubble nucleation in polymer–CO 2 is a rich and complex problem because of the nite compressibility of the mixture and possible interplay between liquid–vapor transition and liquid–liquid phase separation. 2–5 For example, M¨ uller et al. 6 showed that the nucleation behavior and the structure of the critical nucleus are strongly affected by the proximity to the (polymer-rich liquid, CO 2 -rich vapor and CO 2 -rich liquid) triple point. A quantitative theory that predicts the bubble nucleation behavior in polymer–CO 2 mixtures as a function of the pressure, temperature and composition can be a valuable tool for ratio- nally controlling the foaming conditions and can also contribute to the fundamental understanding of nucleation in compressible binary mixtures in general. 3,5,6 However, such a theory has not been available owing to the lack of accurate description of the thermodynamic and interfacial properties of the polymer–CO 2 mixtures at the molecular level. 2,3 Direct molecular simulation of the nucleation behavior is currently impossible due to the molecular complexity of the system and the activated nature of the phenomenon. On the other hand, the classical nucleation theory (CNT) is not only unsatisfactory at the quantitative level, but can even fail to capture some quali- tative features, as will be demonstrated in this work. Density- functional theory (DFT) is an attractive alternative as it can successfully capture the necessary microscopic details of nucleation. 7 DFT treats the nuclei as inhomogeneous density proles, and the free energy of the system is given as a func- tional of the proles. Effects of interfacial curvature and compressibility are incorporated automatically in the DFT description. Under a given nucleation condition, the functional has a saddle point corresponding to the critical nucleus. Nucleation is considered to proceed along the minimum free energy path (MFEP) that connects the metastable uniform bulk state to a well-developed bubble by passing through the critical nucleus. In this work, we use our recently developed DFT 8 based on the perturbed chain-statistical associating eld theory equation of state (PC-SAFT EOS) 9 to study bubble nucleation in polymer– CO 2 mixtures, using poly(methyl methacrylate) (PMMA) and polystyrene (PS) as examples for the polymer. The DFT we use for the polymer–CO 2 is built on a coarse-grained molecular model in which the CO 2 molecule is modeled as a sphere and the polymer is modeled as a freely jointed chain of tangentially connected spheres. The excluded volume of the species is rep- resented by hard-core interactions. Energetic interactions are described by the attractive part of the Lennard-Jones potential. In addition, a weak association interaction is included between the CO 2 molecules. The DFT gives a quantitatively satisfactory description of the thermodynamic bulk and interfacial proper- ties of polymer–CO 2 mixtures in a wide temperature and pres- sure range. The evolution of the bubble along the MFEP is obtained by the string method, which is a modied steepest descent algo- rithm. 10 The free energy barrier and the structure of the critical nucleus are calculated by systematically examining the effect of temperature, pressure and the CO 2 content. The temperature a Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA. E-mail: zgw@caltech.edu b The Dow Chemical Company, Midland, MI 48674, USA Cite this: Soft Matter, 2013, 9, 9675 Received 26th May 2013 Accepted 3rd August 2013 DOI: 10.1039/c3sm51477c www.rsc.org/softmatter This journal is ª The Royal Society of Chemistry 2013 Soft Matter , 2013, 9, 9675–9683 | 9675 Soft Matter PAPER Published on 14 August 2013. Downloaded by California Institute of Technology on 24/10/2013 16:08:14. View Article Online View Journal | View Issue