(754b) Calculation of Phase Behavior of Polymer Containing Mixtures - a Review

A polymer is a large molecule (macromolecule) composed of repeating structural units typically connected by covalent chemical bonds. Due to the extraordinary range of properties of polymeric materials, polymers play an essential and ubiquitous role in everyday life from plastics and elastomers to natural biopolymers such as DNA and proteins that are essential for life. Most commercially important polymers today are entirely synthetic and produced in high volume.

The production and processing of polymers are also influenced by the presence of phase separation and segregation, which may be either necessary or highly undesirable. For example, segregation of highly viscous phases during a polymerization process may lead to catastrophic consequences like plugging lines or overheating reactors. In order to manufacture polymers having tailor-made properties, polymer fractionation can be established utilizing the liquid–liquid phase split¹. Knowledge of thermodynamic data of polymer containing systems is a necessity for industrial and laboratory processes. Such data serve as essential tools for understanding the physical behavior of polymer systems, for studying intermolecular interactions, and for gaining insights into the molecular nature of mixtures.

Because of their size, macromolecules are not conveniently described in terms of stoichiometry alone. The structure of simple macromolecules, such as linear homopolymers, may be described in terms of the individual monomer subunit and total molar-mass distribution. Complicated bio-macromolecules, synthetic dendrimers, or hyperbranched polymers require multifaceted structural description like degree of branching. Additionally, polymers can have a certain degree of crystallinity. Macromolecular architecture is receiving increasing interest as the search for new tailor-made polymeric materials with strictly specified properties intensifiers.

In studying polymer systems, it is convenient to imagine all molecules to be divided into segments of equal size. Then, each molecular species possesses a characteristic segment number, and its relative amount may be specified by its segment fraction. The polydispersity can be handled applying continuous thermodynamics², where the mole fraction of every polymer species is replaced by a continuous distribution function of the molecular weight.

For the description of the phase behavior of polymer containing mixtures lattice models or equations of state based on perturbation theories are available. The basic lattice model is the well-known Flory Huggins theory³ allowing the calculation of miscibility gaps in linear polymer in solution or polymer blends composed of linear polymers, however the experimental data can’t be modelled with a high accuracy. Therefore, several extensions, for instance the Sanchez-Lacombe model⁴ and the lattice cluster theory (LCT)⁵, were suggested. The most important feature of these extensions are the introduction of empty lattice places allowing the description of compressibility. Whereas the Sanchez-Lacombe model can only deal with linear polymers, the LCT permits the description of polymers with an arbitrary architecture⁶, without additional adjustable parameters. The semicrystallinity can be included in the thermodynamic equations using the approach developed by Flory⁷. The most important representatives of the equations of state based on perturbation theories, which can be applied for polymers, are the equations of the SAFT-family⁸, especially the PCP-SAFT⁹. These equations of state are able to account for different interaction, such as dispersion, association, chain-forming and polar contributions.

The thermodynamic framework enable us the calculation of different phase equilibria, such as the liquid-liquid equilibria (LLE), the vapor liquid equilibria (VLE, gas solubility) and the solid-liquid equilibria (SLE). Regarding the LLE, which occur in polymer solutions and polymer blends, we will focus on the impact of polydispersity on the demixing behavior as well as the architecture of branched or hyperbranched polymers in solution of a single solvent or a solvent mixture. The solubility of gases in polymers is a key property in foaming by injection molding. The solubility depends strongly on the semicrystallinity of the polymer. We will apply the models above in combination with mechanical arguments in order to calculate the gas solubility as function of pressure and temperature¹⁰. Regarding the SLE, which forms the thermodynamic background of modern fraction methods with respect to the degree of branching of polyolefines, we focus our attention to the impact of the degree of branching and the semicrystallinity on the SLE¹¹.

The thermodynamic model permits also the calculation of other properties, like swelling¹² or glass transition¹³. The swelling of polymers due to a solvent can be modelled by combination of a network model³with the thermodynamic model, like PCP-SAFT. The glass transition can be investigated using a thermodynamic model within the generalized entropy theory, developed by Dudowicz at al. ¹⁴. The generalized entropy theory is a theoretical framework for prediction of the glass transition of pure polymers or polymer mixtures based on the entropy density of the considered system.

We will discuss the performance of the different models, mentioned above, in calculation of phase behavior and additional properties in terms of accuracy in comparison with experimental data and parameter estimation procedure as well as the transferability of the model parameter using examples relevant for polymer production and polymer processing.

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¹² P. Krenn, P. Zimmermann, M. Fischlschweiger, T. Zeiner, Fluid Phase Equilibria 529 (2021) 112881.

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