Equation of state for polymers

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-... 

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

Equation of state for polymer systems based on lattice fluid model... 

A comparison between some empirical equations of state for polymers with regard to their standard deviations was made by Kamal and Levan (1973). 

The procedure is based on the group contribution equation of state by M. S. High and R. P. Danner, "A Group Contribution Equation of State for Polymer Solutions," Fluid Phase Equilibria, 53, 323 (1989) and M. S. High Prediction of Polymer-Solvent Equilibria with a Group Contribution Lattice-Fluid Equation of State, Ph.D. Thesis, The Pennsylvania State University, University Park, PA, 1990. Additional and modified group values are from V. S. Parekh Correlation and Prediction of the PVT Behavior of Pure Polymer Liquids, M.S. Thesis, The Pennsylvania State University, University Park, PA, 1991. 

High, M. S. Danner, R. P., "A Group Contribution Equation of State for Polymer Solutions," Fluid Phase Equilibria, 53, 323 (1989). 

Originally, basically noncubic equations of state, often with some theoretical background on statistical mechanics, have been proposed for polymer systems. To date, it is generally accepted that both cubic and noncubic equations of state can be used for correlating polymer-solvent equlibria. Orbey et al. ° recently reviewed several cubic equations of state for polymers. 

They have been employed in the use of the van der Waals equation of state for polymers. They require combining rules for both the cross energy and cross co-volume parameters. Kontogeorgis et al. have employed the typically used geometric mean for the co-volume parameter, but they used the Berthelot rule for the cross-energy parameter ... 

Einally, in the light of these extensive recent applications, an attempt to justify the use of cubic equations of state for polymer systems seems of interest. Traditionally, cubic equations of state had not been considered applicable to polymer systems because of alleged limitations of their functional form and especially the van der Waals-type repulsive term. We cite below two points that shed some light in why van der Waals-type equations may be applicable to asymmetric systems (e.g., alkane solutions with large size differences, polymer solutions) ... 

Such cubic equations of state as van der Waals correlate very satisfactorily the UCST-type behavior for polymers solutions, as shown by Harismiadis et al. ° A generalized correlation of the interaction parameter of the van der Waals equation of state for polymer blends based exclnsively on polystyrene blends has been presented. By nsing this equation, the van der Waals eqnation of state can be used as a predictive tool for investigating the compatibility of polymer blends. Predictive GC thermodynamic methods such as Entropic-FV, GC-Flory, UNIFAC, and UNIFAC-FV perform rather poorly, at least from a quantitative point of view. Entropic-FV performs best among these models, on a qualitative basis. For semiquantitative predictions in polymer blends, the approach proposed by Coleman et al. is recommended. 

An elementary equation of state for polymer liquids. Polym. Lett. 

In fact, the data, as well as the theoretical equation of state for polymer fluids, exhibit gradual curvature, but this does not have a substantial effect on the determination of Tg. 

Numerous theoretical equations of state for polymer liquids have been developed. These, at the minimum, have to provide accurate fitting functions to experimental data. However, for the purpose of this table, the empirical Tait equation along with a polynomial expression for the zero pressure isobar is used. This equation is able to represent the experimental data for the melt state within the limits of experimental errors, i.e., the maximum deviations between measured and calculated specific volumes are about 0.001-0.002 cm /g. 

From the large number of published equations of state for polymers, only a few are mentioned in this section. Most are based on the Prigogine cell model, others are semi-empirical or based on Ising fluids the smallest group is based on the cell-hole model. One of the reasons for the limited enthusiasm toward the latter models is that by nature they are algebraically more complex. 

Reviews of the Equation of State for Polymer Melts There are many publications comparing different equations of state to each other, and a few more ambitious reviews that discuss theoretical fundamentals, derivation, and application, and provide tabulated characteristic reducing parameters and deviation from the experimental volume [Curro, 1974 Zoller, 1989 Rodgers, 1993 Rudolf et al., 1995, 1996 Lambert et al 2000], For example, Zoller [1989] examined the equations of state by Spencer-Gilmore (S-G) [1949, 1950], Flory-Orwoll-Vrij (FOV), Sanchez-Lacombe (S-L), and Simha-Somcynsky (S-S). Large deviations (<0.01 mL/g) were observed for S-G. While the FOV and S-L expressions were useful at low P, the S-S equations of state consistently provided the best representation of data over extended ranges of T and P, with deviations AV< 0.003 mL/g, eomparable to the experimental error for density. 

Dee, G. T., and Walsh, D. J., A modified cell model equation of state for polymer liquids,... 

Hartmann, B., and Haque, M. A., Equation of state for polymer solids, /. Appl. Phys., 58,... 

Olabisi, O., and Simha, R., A semi-empirical equation of state for polymer melts, J. Appl. Polym. Sci., 21,149-163 (1977). 

Cell and hole models were used to formulate equations of state for polymer liquids or to discuss isothermal expansion and compressibility of the systems [Hory et al., 1964 Simha, 1977 Dee and Walsh, 1988]. In the models, chain segments are placed on lattice sites. All sites are completely occupied in cell models, and volume changes of the system are related solely to changes in cell volume. Hole models as used by Simha and Somcynsky allow for both lattice vacancies and changes in cell volume. 

Today, there are two principal ways to develop an equation of state for polymer solutions first, to start with an expression for the canonical partition function utilizing concepts similar to those used by van der Waals (e.g., Prigogine, Flory et al., Patterson, Simha and Somcynsky, Sanchez and Lacombe, Dee and Walsh,Donohue and Prausnitz, Chien et al. ), and second, which is more sophisticated, to use statistical thermodynamics perturbation theory for freely-jointed tangent-sphere chain-like fluids (e.g., Hall and coworkers,Chapman et al., Song et al. ). A comprehensive review about equations of state for molten polymers and polymer solutions was given by Lambert et al. Here, only some resulting equations will be summarized under the aspect of calculating solvent activities in polymer solutions. 

About ten years after Flory s development of an equation of state for polymer systems, one began to apply methods of thermodynamic perturbation theory to calculate the thermo-... 

There exist several approaches for the development of equations of state for polymer systems. A possibility considered at an early stage was to extend the Flory-Huggins theory by introducing holes into the lattice. Here, the number of holes in the lattice is a measure of the system density. Equations of state based on this idea are, for example, the Lattice-Fluid Theory (often called the Sanchez-... 

Equation (10) might be seen as an equation-of-state for polymer melts. 

Some applications require adhesion between two polymers. Typical examples are systems of polymer blends and structural adhesives. Based on Eqs. (10) and (23), we obtain the equation of state for polymer i in contact with polymer j as... 

S.C. Davis, U.S. Patents 5,372,980, 13 Dec 1994, Appl. 03 June 1993, to Polysar G.T. Dee, D.J. Walsh, Equations of state for polymer liquids. Macromolecules 21, 811-815 (1988) A modified cell model equation of state for polymer liquids. Macromolecules... 

Theories of Homopolymer Surface Tension. More rigorous expressions for predicting polymer surface tension have been derived from equations of state for polymers. Corresponding states principles have been employed to derive expressions for surface tensions in terms of characteristic equation of state parameters and the associated reduced temperature, pressure, and volume. One proposed expression is of the form (14)... 

Numerous equations of state for polymer liquids have been developed. These equations provide valuable thermodynamic information that can be used to predict properties of polymer blends and polymer solutions. The predictions of phase behavior of polymer blends vary considerably from one equation of state to another. EOS theories for polymer liquids can be roughly grouped into three kinds ... 

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