Lennard-Jones, and Devonshire cell

Careri (Ref 7b) suggested a procedure for computing exactly the thermodynamic functions of a nonequilibrium system. The state of the system was then varied, at fixed volume and temperature, so as to give a minimum Helmholz free energy, consistent with such conditions as are imposed to permit the exact computation. The condition under which this method leads to self-consistent equations is discussed in detail. The method is then applied in a way that is very close to the Lennard-Jones and Devonshire cell method, bur with cells of variable size. The distribution within a cell is assumed to be Gaussian. Mayer Careri claimed that the method is easier to apply than the cell method, but it seems to be rather complicated... 

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 evaluation of K requires multidimensional integration (the position and orientation of the guest relative to individual water forming the cavity). Employing the Lennard-Jones and Devonshire cell modeP° for liquids by... 

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. 

E. Evaluation of the Cell Partition Function by the Method of Lennard-Jones and Devonshire... 

N. G. Parsonage I can see no possibility of obtaining an analytical expression for the partition function for cavities containing several molecules. For cells containing n molecules, we would need to integrate a complicated function over — 3n coordinates. Lennard-Jones and Devonshire considered only the case of one molecule per cell, and for this needed only one coordinate, the radial distance. Even so, they had to evaluate their integrals numerically. 

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... 

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 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. 

Once the mean potential to r) is known, we may use the general relations (7.1.3)-(7.1.6) to compute the thermod3mamic functions. This can imfortunately not be done in closed form because of the complicated analytical form of the mean potential. However, tables of the cell partition function W and its derivatives with respect to volume and temperature now exist (Lennard-Jones and Devonshire [1937, 1938], Hill [1947], Prigogine and Garikian [1948], Wentorf, Buehler, Hirschfelder and Curtiss [1950], Hirschfelder, Curtiss and Bird [1954]), which permit to compare the Lennard-Jones and Devonshire theory with experimental data. 

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. 

Finally we quote in Table 11.5.2 some results obtained by Kirkwood and Salsburg [1953] by the use of the cell method of Lennard-Jones and Devonshire, for some of the mixtures analysed here in detail. 

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 ... 

The Lennard-Jones-Devonshire theory (as summarized by Fowler and Guggenheim, 1952, pp. 336ff) averaged the pair potentials of Equation 5.24a and b between the solute and each water, for Zi molecules in the surface of the spherical cavity to obtain a cell potential r) of... 

There are substantially fewer MC studies of hydrates than there areMD studies. The initial MC study of hydrates was by Tester et al. (1972), followed by Tse and Davidson (1982), who checked the Lennard-Jones-Devonshire spherical cell approximation for interaction of guest with the cavity. Lund (1990) and Kvamme et al. (1993) studied guest-guest interactions within the lattice. More recently Natarajan and Bishnoi (1995) have studied the technique for calculation of the Langmuir coefficients. 

The other cause, the density effect, is especially important at high densities, where molecules are more or less confined to cells formed by their neighbors. In analogy to the well-known quantum mechanical problem of a particle in a box, the translational energies of such molecules are quantized, and this has an effect on the thermodynamic properties. In 1960 Levelt Sengers and Hurst [3] tried to describe the density quantum effect in term of the Lennard-Jones-Devonshire cell model, and in 1980 Hooper and Nordholm proposed a generalized van der Waals theory [4]. The disadvantage of both approaches is that, in the classical limit, they reduce to rather unsatisfactory equations of state. 

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