Lennard-Jones and Devonshire

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

Lennard-Jones and Devonshire summed the interaction (Eq. 30) over z molecules distributed over the surface of a sphere and so obtained a function w(r) describing the resulting field within the sphere, averaged over all orientations. The precise form of w(r) need not be considered here (cf. ref. 17 or 13), suffice it to say that... 

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). 

MSE.9. 1. Prigogine et P. Janssens, Une generalisation de la methode de Lennard-Jones-Devonshire pour le calcul de I integrale de configuration (A generalization of the Lennard-Jones and Devonshire method foir the calculation of the configuration integral), Physica 16, 895-906 (1950). 

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... 

The potential energy of a guest molecule at a distance r from the cavity center is given by the spherically symmetrical potential m (r) proposed by Lennard-Jones and Devonshire (1932, 1938). 

The positional entropy has been calculated (Hirschfelder et al., 1937 Lennard-Jones and Devonshire, 1939) to be approximately 8-12 J / K mol for most organic molecules. This number results from a mathematical expression of the probability of finding the centers of a particular number of liquid molecules arranged as they are in the solid. 

Lennard-Jones and Devonshire, with the potential energy function —ar "+br" ( 41.VIIC) calculated the molar entropy of evaporation ... 

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. 

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. 

A potential function used by many authors, including Mie, Gruneisen, Lennard-Jones and Devonshire, and London is of the form ... 

Lennard-Jones and Devonshire arrive at theoretical critical temperatures which are in good agreement with experimental values for argon, neon, nitrogen, etc. However, the theoretical critical volumes and pressures are off by factors of about 1.6 and 3.9, respectively. 

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. 

The defect crystals of Lennard-Jones and Devonshire possess interstices onto which the motifs of the crystal lattice migrate by leaving a hole. The diffusion, and also the concentration, of the defects and holes are largely independ it of one another. In the transition described by Lennard-Jones and Devonshire defect and hole concentrations play thus both a part. Opposed to this, in molecular crystals with defects, predominantly formed by conformational isomerism in the molecules, the defect and hole concentrations are closdy coupled. In this case the concentrations of conformational isomers alone suffice to describe the transition. 

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... 

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 simplest treatment of the melting of inert gas crystals is that due to Lennard-Jones and Devonshire (LJD) who regarded the mechanism of fusion as a positional order-disorder phenomenon. They postulated that... 

Originally, the L-J potential had a general form [Lennard-Jones, 1929, 1931 Lennard-Jones and Devonshire, 1937]... 

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] ... 

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