Density-Functional Theory for Mixtures of AB Random Copolymer and CO 2 Xiaofei Xu Center for Soft Condensed Matter Physics and Interdisciplinary Research, Soochow University, Suzhou 215006, China Diego E. Cristancho and Ste ́ phane Costeux The Dow Chemical Company, Midland, Michigan 48674, United States Zhen-Gang Wang* Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, United States ABSTRACT: We propose a density-functional theory (DFT) to describe inhomogeneous mixtures of AB random copolymer and carbon dioxide (CO 2 ). The statistical sequence of monomer in the polymer chain backbone is modeled by a transition matrix in a Markov-step growth process. The parameters of the theory are determined by fitting the bulk experimental data. We apply the DFT to the interfacial properties of binary mixtures of CO 2 with poly(methyl methacrylate co ethyl methacrylate) (P(MMA-co-EMA)), poly(methyl methacrylate co ethyl acrylate) (P(MMA-co-EA) and poly(styrene co ethyl acrylate) (P(S-co- EA)). The dependence of CO 2 solubility and interfacial tension on the copolymer composition and pressure is examined. We find that higher fractions of EA or EMA result in higher solubility of CO 2 at a given pressure, but also results in higher interfacial tension at a fixed CO 2 content in the polymer-rich phase. Using the classical nucleation theory as a rough estimate, we examine the effect of the copolymer composition on the free energy barrier of bubble nucleation in random copolymer-CO 2 mixtures. I. INTRODUCTION Phase separation in polymer and carbon dioxide (CO 2 ) mixtures is a problem of great interest in the manufacturing of polymer foams. 1-3 The fabrication of novel foam materials with nanoscale pores can benefit from systematic and quantitative understanding of the thermodynamics and kinetics of phase separation. 4 In several recent publications, we proposed a density functional theory (DFT) for mixtures of a homopolymer and carbon dioxide, 5 and systematically studied bubble nucleation in these mixtures. 6,7 The theory yields results for the bulk phase behavior and interfacial tension in good agreement with experiments, and predicts the mechanistic pathways for bubble nucleation in these mixtures. In particular, we find that the presence of a metastable transition between a CO 2 -rich vapor and a CO 2 -rich liquid leads to a discontinuous drop in the nucleation barrier as a function of the CO 2 supersaturation. However, experiments have shown that random copolymers made of two different types of monomers are better materials for nanofoaming than homopolymers made of either monomer. 8 We are thus motivated to extend our theory to the case of random copolymers. Random copolymers consist of two or more different monomers with a statistical sequence distribution along the chain backbone. They are of considerable technological importance because of their ability to enhance miscibility or to produce a multiphase region with particular desired morphology, which has led to the development of many special and novel materials. 9,10 There is considerable interest to study their structural and thermodynamic properties. Although a number of theories have been proposed, describing the phase behavior and interfacial properties of random copolymer systems still presents an enormous theoretical challenge. The earliest theoretical descriptions of random copolymer systems are based on an extension of the Flory-Huggins theory (FHT), by assuming an interaction parameter χ ij for different monomer species i and j. 11,12 As the theory did not consider the effect of monomer sequence in the chain, Balazs et al. 13 and later Cantow and Schulz 14 introduced a sequence-dependent χ parameters χ ijk,lmn to describe the interactions between a pair of triads of sequential monomers. Dudowicz and Freed 15,16 developed a lattice cluster theory (LCT) for random copolymers, which is based on the generalized lattice model that endows the monomers with specific structures by allowing them to occupy several lattice sites with specified connectiv- ity. 17 In contrast to the FHT, the LCT treatment requires no ad hoc assumptions concerning the particular form of the sequence-dependent interaction parameters. Fredrickson et al. 18-20 constructed a Landau-type density-functional theory for random copolymer and random block copolymer, in which the free energy is developed as a functional Taylor expansion in a Received: May 25, 2015 Revised: July 15, 2015 Article pubs.acs.org/Macromolecules © XXXX American Chemical Society A DOI: 10.1021/acs.macromol.5b01122 Macromolecules XXXX, XXX, XXX-XXX