Prigogine cell model

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. 

Table 6 gives the three characteristic parameters P, V, T for the Prigogine cell model for the polymers in Tables 1 and 2. The average deviation between Eq. (AlO) and regenerated PVT data is 0.0008 cm /g. 

Cell Model by Flory, Orwoll, and Vrij (FOV) The cell model by Flory et al. (14) has the simpler mathematical form than the Prigogine cell model ... 

The factor s/3c in Eq. (A13) is called the flexibility ratio, which is usually set to 1. Other factors are the same as those in the Prigogine cell model. 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

From the quantitative point of view, the success of the cell model of solutions was more limited. For example, a detailed analysis of the excess functions of seven binary mixtures by Prigogine and Bellemans5 only showed a very rough agreement between theory and experiment. One should of course realize here that besides the use of the cell model itself, several supplementary assumptions had to be made in order to obtain numerical estimates of the excess functions. For example, it was assumed that two molecules of species and fi interact following the 6-12 potential of Lennard-Jones ... 

Concerning point (b), a generalized theory was developed independently by Prigogine and his co-workers10 11 and by Scott.18 The main idea was to combine the concept of average potential involved in the cell model with the theorem of corresponding states for pure compounds, in such a way that ... 

SOL. 15.1. Prigogine et V. Mathot, Application of the cell model to the statistical thermodynamics of solutions, J. Chem. Phys. 20, 49-57 (1952). 

MSE.IO. I. Prigogine et L. Saraga, Sur la tension superficielle et le modele ceUulaire de I etat liquide (On the surface tension and the cell model of liquid state), J. Chim. Phys. 49, 399-407 (1952). 

More recently, Arora et al. [2001] derived equation of state starting with the Prigogine et al. cell model and full L-J potential ... 

Equations of state derived from statisticai thermodynamics arise from proper con-figurationai partition functions formuiated in the spirit of moiecuiar modeis. A comprehensive review of equations of state, inciuding the historicai aspects, is provided in Chapter 6. Therefore, we touch briefly in oniy a few points. Lennard-Jones and Devonshire [1937] developed the cell model of simple liquids, Prigogine et al. [1957] generalized it to polymer fluids, and Simha and Somcynsky [1969] modified Pri-gogine s cell model, allowing for more disorder in the system by lattice imperfections or holes. Their equations of state have been compared successfully with PVT data on polymers [Rodgers, 1993]. 

Dee and Walstf developed a modified version of Prigogine s cell model that provides an excellent description of the PVT-behavior of polymer melts ... 

The first statistical mechanical theory describing the interface of liquid-gas far from critical point was formulated by Prigogine and Saraga [19]. They used a cell model with each molecule of the liquid locked in a cell, with u being potential energy at the cage center and z the partition... 

Flory et al. (1964) (FOV), Sanchez-Lacombe (1976,1977,1978) (S-L), Simha and Somcynsky (1969) (S-S), Prigogine et al. (1953, 1957) (P), Dee and Walsh (1988) (D-W), Hartmann and Haque (1985) (H-H), and Sanchez and Cho (1995) (S-C), and tabulated the respective corresponding state values P, V, and T ) for most common polymers. These comparisons span across the different types of EoS models, from cell models (FOV, P, D-W), to lattice-fluid (S-L) and hole (S-S) models, to semiempirical approaches (H-H, S-C), comparing the validity of distinctly different EoS approaches across large numbers of different homopolymers and copolymers. All reviews seem to build a consensus on the comparative accuracy of the various EoS Zoller (1989) reported large deviations (<0.01 mL/g)... 

The Prigogine simple cell model (P) considers each monomer in the system to be trapped in the cell created by its surroundings. The general cell potential, generated by the surroundings, is simplified to be athermal (cf. free volume theory), whereas the mean potential between the centers of different cells are described by the Leimard-Jones 6-12 potential. The P model EoS can be summarized as... 

Equation (3.121) is identical with that obtained by Prigogine [4] in cell theory. Prigogine has further employed the cell model to calculate Eg he evaluted the potential field of a representative molecule of the system as a function of its position in the cell, defined by the geometry of the nearest neighbours whose positions were assumed to be fixed. Flory assumed that Eg depends only on the volume of the system according to Hildebrand and Scott s relation ... 

Redlich-Kwong equation of state and Soave modification Peng-Robinson equation of state Tait equation for polymer liquids Flory, Orwoll, and Vrij models Prigogine square-well cell model Sanchez-Lacombe lattice fluid theory... 

Polymer molecules are modeled as having two distinct sets of modes contributing to the partition function in the cell models. The two modes are internal and external modes. The internal modes are used to represent the internal motions of the molecules, and the intermolecular interactions are accounted for by the external modes. Prigogine and Kondepudi [12] proposed the conceptual separation of the two modes. The PVT properties of the polymer systems will be affected by the external modes. A polymer molecule is divided into r repeat units. Each repeat unit has 3 degrees of... 

Two forms of the cell model (CM) are then developed harmonic oscillator approximation and square-well approximation. Both forms assnme hexagonal closed packing (HCP) lattice structure for the cell geometry. The model developed by Paul and Di Benedetto [13] assumes that the chain segments interact with a cylindrical symmetric square-well potential. The FOV model discnssed in the earlier section uses a hard-sphere type repulsive potential along with a simple cubic (SC) lattice structure. The square-well cell model by Prigogine was modified by Dee and Walsh [14]. They introduced a numerical factor to decouple the potential from the choice of lattice strncture. A universal constant for several polymers was added and the modified cell model (MCM) was a three-parameter model. The Prigogine cell EOS model can be written as follows. 

The Prigogine square-well cell model is based upon conceptual separation of internal and external modes for polymers separately accounted for in the partition... 

What is the significance of the parameters of the Prigogine square-well cell model ... 

Nine different equations-of-state, EOS theories are described including Flory Orwoll Vrij (FOV) Prigogine Square Well cell model, and the Sanchez Lacombe free volume theory. When the mathematical complexity of the EOS theories increases it is prudent to watch for spurious results such as negative pressure and negative volume expansivity. Although mathematically correct these have little physical meaning in polymer science. The large molecule effects are explicitly accounted for by the lattice fluid EOS theories. The current textbooks on thermodynamics discuss... 

The SS theory can be generalized to multicomponent systems [61, 66] by adopting a simplification introduced by Prigogine et al. in their theory of mixtures, which is based on the simple cell model [70]. 

The harmonic oscillator cell model (Prigogine and Garikian [1950], Rowlinson [1952], Prigogine and Mathot [1952], Prigogine, Trappenbbrs and Mathot [1953]) is obtained by retaining only the first term in this expansion. As can be ihferred from Fig. 7.3.1 this modd is only valid at low temperatures and may be of interest in the study of solid solutions. For an example of application to solid solution see Sarolea [1953]. 

The invariant hole size and energy of hole formation in the Lacombe-Sanchez model implies a dependance of the internal energy on the density that is strictly of the van der Waals form, as in Flory s theory. Another, somewhat more complicated, model developed by Simha and his collaborators is similar to that of Sanchez and Lacombe in its use of a liquid lattice with vacant sites, but it also retains features of Prigogine s earlier cell model in the c parameter for external degrees of freedom and a lattice energy with a density dependence based on an effective (6-12) pair potential. Like the other theories, this one has been successful in correlating equation of state data both for neat polymer liquids and for nondilute solutions. 

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