Jones and Devonshire Model

Each term has a simple physical interpretation. The excess free energy appears to be due to the change of the lattice energies and the ceil partition functions at mixing. The corresponding formulae for th = excess energy and excess entropy axe 

Bormulae (8.2.7) and (8.2.8) are much simplified if the cell partition functions depend only on the density or more generally if the ratio aI aa (and WbI bb) is temperature independent. This simplifying feature is realized in both the smoothed potential and the harmonic oscillator cell models (cf. Ch. VII, 3, 4). 

We shall now explicitly calculate the thermodynamic excess functions for a solution of molecules of the same size (p — 0) (Prigogine and Mathot [1952], Salsburg and Kirkwoob [1952, 1953]). 

Using the approximation of random mixing, it is quite easy to extend the calculations of the mean potential to mixtures. The probability that a neighbom ig site around a given central molecule is occupied by a molecule of A or B, is then equal to the mole-fractions Xa or Xb, irrespective of the nature of the central molecule. For this reason, we have to replace the parameter s which appears in the cell model for pure liquids by the composition dependent average values 

The same change has to be introduced into o (r) — ca(0) as well as into the definitions of functions g, gi, gm (cf. 7.1.26-7.1.29). A useful quantity which appears in the thermodynamic functions is 

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MSE.6. 1. Prigogine et G. Garikian, Sur le modele de I etat liquide de Eennard-Jones et Devonshire (On the Lennard-Jones and Devonshire model of the liquid state), J. Chim. Phys., 45, 273-289 (1948). 

To gain an initial, qualitative insight into the relationships that exist in crystals formed from chain molecules, we use a simple model with roughly the same rationale as that of the Lennard-Jones and Devonshire model. Let us regard a bundle of N parallel, stretched chain-molecules as the ideal crystal and let its structure be characterized by a mean coordination number q (Fig. 2.4). This model embraces both, extended chain crystals and lamellae formed by folded-chain molecules. Naturally, in the latter ca the chain molecule is merely a segm t of the real chain. It is assumed that all defeats can be built-up from ener ticalfy non-degenerate units with... 

Table 1.2. Values of the critical temperature, found experimentally and calculated by the Lennard-Jones and Devonshire model...

Whilst it does represent real progress in relation to the previous two models, the Lennard-Jones and Devonshire model discussed above has a serious shortcoming-it is incapable of taking account of two dynamic properties of liquids the phenomena of viscosity and self-diffusion. In order to take account of these properties, Ono [ONO 47] introduced the concept of vacancies, comparable to that which takes account of conductivity and diffusion in the solid phase. Ono considers that certain sites in the pseudolattice, or if you prefer, certain cells described in the above model, are not occupied, forming what we call vacancies. Thus, on average, over time, a molecule i will be surrounded by z, first neighbors in accordance with ... 

The critical data as deduced from the Lennard-Jones and Devonshire model are compared with the experimental values in Table 7.2.1... 

CRITICAL PATA ACCORDING TO THE LENNARD-JONES AND DEVONSHIRE MODEL... 

The complexity of the excess functions in mixtures make an analytical discussion highly desirable. For this purpose the Lennard-Jones and Devonshire model is unfortunately not suitable because of the complicated form of the mean potential model valid in a restricted range of density. For hig densities, as in the solid state, we may use a harmonic potential approximation (cf. Fig. 7.1.2). We shall develop this approximation in more detail in the next paragraph. On the other hand, for the range of densities corresponding to the liquid state we may use the smoothed potential model (Prigogine and Mathot [1952]) (cf. Fig. 7.1.2). This is however an oversimplification and the conclusions have to be used with some caution. 

The choice of the upper limit (a — a) is rather arbitrary. In so far as we are interested in excess properties of solutions, we could as well take (a — constant term to the free energy and disappears in the excess functions (cf., however, Ch. XVIII for the quantum case). As in the Lennard-Jones and Devonshire model, we shall use (7.1.23) for the lattice energy" of our system. The cell partition function W is clearly of the same form as for hard spheres (cf. 7.1.11) and depends only on the density. It may be written in the form... 

Let us now go back to the general case in which we have to take account of the interaction between particles. The interactions of all molecules in the same multiply occupied cell are then treated exactly. The interaction between molecules in different cells are replaced by an average potential similar to that used in the Lennard-Jones and Devonshire model. 

The hole model may serve as a basis in the tmderstanding of the decrease of the first coordination number with decreasing density in the liquid state. However it seems imable to account for the extra-entropy which exists in the liquid state. Ihis is due to the excessively schematic representation of the density fluctuations. Furthermore the problem of correlations in the motions of molecules is not solved, the hole model being a one-particle model exactly as the original Leimard-Jones and Devonshire model. 

This approximation is of the same nature as the use of the cell field in the t nnard-Jones and Devonshire model. It is only reasonable if the number of first neighbours is large and the differences in the inter-molecular forces small. In addition the fluctuations in the inter-molecular field are likely to be smaller in concentrated solutions than in dilute solutions. 

It is now weU understood that the Lennard-Jones and Devonshire model greatly overestimates the local order in liquids, and many improvements have been proposed (cf. Ch. VII). However it gives a consistent picture of some of the main factors which determine the thermodynamic properties of liquids. 

In view of the failure of the rigid sphere model to yield the correct isochoric temperature coefficient of the viscosity, the investigation of other less approximate models of the liquid state becomes desirable. In particular, a study making use of the Lennard-Jones and Devonshire cell theory of liquids28 would be of interest because it makes use of a realistic intermolecular potential function while retaining the essential simplicity of a single particle theory. The main task is to calculate the probability density of the molecule within its cell as perturbed by the steady-state transport process. 

The van der Waals equation of state can be replaced by better models of the liquid state, for example, the gas of hard spheres with intermolecular attractions superimposed (78), or the Lennard-Jones and Devonshire (19) theory of liquids. 

A. The cell model theoiy. The cell model theory was first introduced by Lennard-Jones and Devonshire to calculate the bulk properties of molecular fluids (Barker, 1%3). In this approach, the array of particles is replaced by an array of hypothetical cells inside which the movement of each particle is confined. 

The most successful equation of state for semicrystalline polymers such as PE and PA stems from two unlikely sources (1) calculation of 5 = a of polymeric glasses at T< 80K [Simha et al., 1972] and (2) the Lennard-Jones and Devonshire (L-JD) cell model developed originally for gases and then liquids. Midha and Nanda [1977] (M-N) adopted the L-JD model for their quantum-mechanical version of crystalline polymers, taking into account harmonic and anharmonic contributions to the interaction energy. Simha and Jain (S-J) subsequently refined their model and incorporated the characteristic vibration frequency at T= 0 K from the low-Tglass theory [Simha and Jain, 1978 Jain and Simha, 1979a,b] ... 

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]. 

To obtain the thermodynamic functions of the hard-sphere erystal we use the cell model of Lennard-Jones and Devonshire [17]. The idea of the eeU model is that a given particle moves in a free volume v set by its neighbours whieh are located on their lattice positions (see Fig. 3.2). Then the partition funetion Q takes the form... 

A theoretical treatment of order-disorder phenomena in molecular crystals has been developed by Pople and Karasz. The theory considered disorder in both the positions and orientations of the molecules and it was assumed that each molecule could take up one or two orientations on the normal, a-, and the interstitial, ]8-, sites of the two-lattice model proposed by Lennard-Jones and Devonshire in their treatment of the melting of inert gas crystals. The theory introduced a single non-dimensional parameter, v- related to the relative energy barriers for the... 

Thus, the Lennard-Jones and Devonshire cellular model can be used to calculate thermodynamic functions with only two adjustable parameters. In section 1.7, however, we shall demonstrate that the results obtained are very... 

Certain data appear in Table 1.2 for Lennard-Jones and Devonshire s model (see section 1.4). Others are given for the solids of rare gases in Table 1.3, and pertain to Eyring s model (see section 1.5). 

Note that both models yield satisfactory results on this point. However, it is important to apply the comparison to several types of results. For example. Figure 1.11 shows that, for the representation of the distribution function, Lennard-Jones and Devonshire s model, Eyring s model and the calculations performed by numerical simulation are very similar. Meanwhile, Figure 1.12, which gives the variation of the compressibility coefficient as a function of a reduced volume, illustrates the significant behavioral difference between the molecular dynamics simulation and Eyring s model, on the one hand, and Lennard-Jones/Devonshire s, Guggenheim s (see section 1.3.1)... 

The formula for the pressure given in Eq. (28.5-5) approaches the ideal gas value for large molar volume, and diverges as the molar volume approaches the molar volume of the solid at 0 K. This behavior is qualitatively correct, but the cell model does not predict accurate values of the pressure. Lennard-Jones and Devonshire developed an improved version of the cell model, in which they explicitly summed up the potential energy contributions for the nearest neighbors, obtaining better results. 

The above picture of slowly cooled SCLs allows considering the liquid cell model of Lennard-Jones and Devonshire [34] (Figure 10.1 and its various elaborations [35]. In the figure, we show a cell representation of a dense liquid in (a) and of a crystal in (b). Each cell is occupied by a particle in which the particle vibrates. A defect in the cell representation corresponds to some empty cells. The regular lattice in (b) is in accordance with Einstein s model of a crystal. In the liquid state, this regularity is absent. We consider the conjiguratiorud partition JunctionZ T, V) (Appendix lO.A),... 

This more or less regular structure forms the basis of the cell model in its simplest form it is assumed that each molecule is confined to its own cell. The first attempts to give a description of the liquid state with the ceil model are due to Eyeing [1936] and Eyeing and Hirschfel-DER [1937J. However, Lennard-Jones and Devonshire [1937, 1938] were the first to iss the cell model in the interpretation of the thermodynamic properties of a liquid in terms of intermolecular forces. 

The constant factor t v y) may be absorbed in the kinetic part of the partition functions. From this relation for the cell partition function all relevant thermodynamic properties may easily be deduced. The two models we have considered in 3 and 4 are dearly ovnsimplifled. Their intor t is that they retain the most important features of the Lennard-Jones and Devonshire cell model of condensed states with an appreciable gain in simplidty. 

The main progress in the theory of concentrated solutions came from two somewhat complementary directions of approach. A decisive step toward the understanding of the liquid state was made in 1937 by Lennard-Jones and Devonshire using a free volume theory (or cell model). Before Lennard-Jones and Devonshire, the cell model had been used by many authors (mainly by E3rring and his coworkers) to correlate the thermodynamic properties of liquids. However, Lennard-Jones and Devonshire were the first to use it to express the thermodynamic properties in terms of intermolecular forces (as deduced for example, from drial measurements). 

Simple Cell Model of Prigogine et al. The cell model by Frigogine et al. (10-12) is an extension of the cell model for small molecules by Lennard-Jones and Devonshire (68) to polymers. Each monomer in the system is considered to be trapped in the cell created by the surroimdings. The general cell potential, generated by the smroimdings, is simplified to be athermal. This turns the simple cell model into a free volmne theory. The mean potential between the centers of different cells are described by the Lennard-Jones 6-12 potential. The dimensionless equation of state has the following form ... 

An ab initio potential for the methane-water bimolecular system has been developed for use in modeling methane hydrates and in order to evaluate currently used statistical thermodynamic models. In this paper, an introduction to gas hydrates is first given, and the problem with the Lennard-Jones and Devonshire (LJD) approximation, typically used for modeling hydrates, is described. Second, the methodologies for generating the ab initio potential energy surface are described and results discussed. Third,... 

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