Lennard-Jones intermolecular potential function, equation

Within the model represented by equations (1) and (2), the intermolecular potential energy function is fully determined by the set of -1 charges and n n +1) Lennard-Jones parameters, where n is the number of different types of atoms in the system. For example, the water intermolecular potential in this approach requires 5 different parameters. In practice, this is modified in two ways First, one may wish to add additional point charges to provide more flexibility in modeling the molecular charge distribution. In this case, the locations of the point charges are not necessarily identified with the equilibrium positions of the atoms. Second, a major simplification can be achieved if one uses the following approximation[13] ... 

We propose the study of Lennard-Jones (LJ) mixtures that simulate the carbon dioxide-naphthalene system. The LJ fluid is used only as a model, as real CO2 and CioHg are far from LJ particles. The rationale is that supercritical solubility enhancement is common to all fluids exhibiting critical behavior, irrespective of their specific intermolecular forces. Study of simpler models will bring out the salient features without the complications of details. The accurate HMSA integral equation (Ifl) is employed to calculate the pair correlation functions at various conditions characteristic of supercritical solutions. In closely related work reported elsewhere (Pfund, D. M. Lee, L. L. Cochran, H. D. Int. J. Thermophvs. in press and Fluid Phase Equilib. in preparation) we have explored methods of determining chemical potentials in solutions from molecular distribution functions. 

The forces of attraction and repulsion between molecules must be considered for a more accurate and rigorous representation of the gas flow. Chapman and Enskog proposed a well-known theory in which they use a distribution function, the Boltzmann equation, instead of the mean free path. Using this approach, for a pair of non-polar molecules, an intermolecular potential, V (r), is given in the potential function proposed by the Lennard-Jones potential ... 

Extension of more advanced methods, in particular, density functional theory, to non-equilibrium phenomena is the principal aim of this survey. We shall consider a simple one-component fluid with van der Waals interactions as a suitable medium for exploration of basic theoretical problems of interfacial dynamics. In the case when intermolecular interactions are long-range, in particular, in the most important case of Lennard-Jones potential, the transformation from the nonlocal (density functional) to local (van der Waals-Landau-Cahn) equations fails due to divergences appearing in the commonly used expansion of the interaction term in the expression for free energy. Setting bound-... 

In the description of the intermolecular bonding, the Lennard-Jones 6-12 potential function (8) is one of the most common, consisting of an attractive and repulsive contribution to the van der Waals component of the lattice energy (Vydw) as shown in Equation 1. "A" and "B" are the atom-atom parameters for describing a particular atom-atom interaction and "r" is the interatomic distance. This potential function has formed the basis of a variety of different force fields (9-11) that were utilized in this paper. A modified (10-12 version of this potential can also be employed (10,11) to describe hydrogen bonding. The 10-12 potential is very similar in construction to Equation 1 except that the attractive part is dependent on r ° rather than r. ... 

The droplet shape is obtained by solving the augmented Young equation with an appropriate anal5d ical form of derived from density functional theory with intermolecular interactions modeled by pairwise Lennard-Jones potentials (see Fig. 6.10b). Because > scales as for thick films, theory predicts the height of droplets to scale with the width according to a power law with exponent 1 /2 at saturation, as found in the experiments. Moreover, the model describes accurately the droplet shape off-coexistence (solid lines in Fig. 6.10a). 

We have seen above that the 6-12 Lennard-Jones potential closely approximates intermolecular forces for many molecules. Equation (12) can be made dimensionless by dividing F by e. This results in a universal function in which the dimensionless poten-ial is a function of the dimensionless distance of separation between the molecules, r/a. The energy parameter e. and the distance parameter a. are characteristic values for a given molecule. This is a microscopic theory of corresponding states. It is related to the macroscopic theory through the critical properties of a fluid. Because the critical temperature is a measure of the kinetic energy of fluids in a common physical state, there should be a simple proportionality between the energy parameter e. and the critical temperature Tc. Because the critical volume reflects molecular size, there should also be a simple proportionality between a. and the cube root of Vc. For simple non-polar molecules which can be described by the 6-12 Lennard-Jones potential, the proportionalities have been found to be ... 

The second simulation technique is molecular dynamics. In this technique, which was pioneered by Alder, initial positions of theparticles of a system of several hundred particles are assigned in some way. Displacements of the particles are determined by numerically simulating the classical equations of motion. Periodic boundary conditions are applied as in the Monte Carlo method. The first molecular dynamics calculations were done on systems of hard spheres, but the method has been applied to monatomic systems having intermolecular forces represented by the square-well and Lennard-Jones potential energy functions, as well as on model systems representing molecular substances. Commercial software is now available to carry out molecular dynamics simulations on desktop computers. ... 

---