Polymer equation of state

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

The sorption of supercritical solvent and flic resulting swelling could be very high, for example around 30% and 20% respectively for carbon dioxide in polymethylmefliacrylate (PMMA). In such instances, the experimental information could be summarized using polymer equation of states such as Sanchez-Lacombe where a single mixture fitting parameter is used. ... 

The Gruneisen parameter is widely used in polymer equation-of-state calculations. Bulk modulus is primarily a function of volume. The volume of a pol5uner can be changed by varying either the pressure or the temperature, leading to two equivalent expressions for the Gruneisen parameter y ... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

A number of theoretical models have been proposed to describe the phase behavior of polymer—supercritical fluid systems, eg, the SAET and LEHB equations of state, and mean-field lattice gas models (67—69). Many examples of polymer—supercritical fluid systems are discussed ia the Hterature (1,3). 

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and ac tivity. The Floiy-Huggins theory does not work well with the cross-linked semi-ciystaUine polymers that comprise an important class of pervaporation membranes. Neel (in Noble and Stern, op. cit., pp. 169-176) reviews modifications of the Stefan-Maxwell approach and other equations of state appropriate for the process. 

A. Milchev, K. Binder. Osmotic pressure, atomic pressure and the virial equation of state of polymer solutions Monte Carlo simulations of a bead-spring model. Macromol Theory Simul 5 915-929, 1994. 

It is an arduous task to develop thermodynamic models or empirical equations that accurately predict solvent activities in polymer solutions. Even so, since Flory developed the well-known equation of state for polymer solutions, much work has been conducted in this area [50-52]. Consequently, extensive experimental data have been published in the literature by various researchers on different binary polymer-solvent sys-... 

For application to gas sorption in polymers, we have modified the Prigogine-Flory formalism to apply to low- and high-density fluids and their mixtures (12). The modified equation of state has the form... 

Matsuoka S., Cowman M.K. 2002. Equation of state for polymer solution. Polymer 43, 3447-3453. 

Such weaknesses of the present implementation include the lack of an explicit inclusion of intermolecular forces other than excluded volume, resulting in a qualitatively inaccurate description of the equation of state. Another weakness is that the model shows lattice artefacts when dealing with problems of polymer crystallization or liquid-cristalline order only rather flexible poly-... 

Dodd, L. R. and Theodorou, D. N. Atomistic Monte Carlo Simulation and Continuum Mean Field Theory of the Structure and Equation of State Properties of Alkane and Polymer Melts. Vol 116,pp, 249-282,... 

The quantity of primary interest in the study of nonuniform fluids is the density profile of the fluid at a surface. Dickman and Hall [28] reported the density profiles of freely jointed hard-sphere chains at hard walls. Their focus was on the equation of state of melts of hard-chain polymers, and they performed simulations of polymers at hard walls because the bulk pressure, P, can be calculated from the value of the density profile at the surface using the wall sum rule ... 

Equation of State of Thermal Polymer Solutions and Melts. 

The object of this work was to extend the field of application of the equation-of-state method. The method was applied to aqueous systems in conjunction with a model that treats water as a mixture of a limited number of polymers, an approach similar to that previously adopted for the carboxylic acids (2). Association is calculated by the law of mass action corrections for non-ideal behaviour are made by means the equation of state. A major problem of the method is the large number of parameters needed to describe the properties and concentrations of the polymers together with their interaction with molecules of other substances. The Mecke-Kemptner model (15) (also known as the Kretschmer-Wiebe model (16) and experimental values for hydrogen-bond energies were usecT for guidance in fixing these parameters. 

Using a recent equation of state of the van der Waals type developed to describe non-polar components, a model is presented which considers water as a mixture of monomers and a limited number of polymers formed by association. The parameters of the model are determined so as to describe the pure-component properties (vapour pressure, saturated volumes of both phases) of water and the phase equilibria (vapour-liquid and/or liquid-liquid) for binary systems with water including selected hydrocarbons and inorganic gases. The results obtained are satisfactory for a considerable variety of different types of system over a wide range of pressure and temperature. 

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