Lennard-Jones equation pair potential

Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but 0 g and were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good an average absolute error of 6.4 percent, considering the complexity of some of... 

This equation acknowledges that real molecules have size. They have an exclusion volume, defined as the region around the molecule from which the centre of any other molecule is excluded. This is allowed for by the constant b, which is usually taken as equal to half the molar exclusion volume. The equation also recognizes the existence of a sphere of influence around each molecule, an interaction volume within which any other molecule will experience a force of attraction. This force is usually represented by a Lennard-Jones 6-12 potential. The derivation below follows a simpler treatment (Flowers Mendoza 1970) in which the potential is taken as a square-well function as deep as the Lennard-Jones minimum (figure 2a). Its width x is chosen to give the same volume-integral, and defines an interaction volume Vx around the molecule, which will contain the centre of any molecule in the square well. This form of molecular pair potential then appears in the Van der Waals equation as the constant a, equal to half the product of the molar interaction volume and the molar interaction energy. 

In Chapter 4 we obtained several equations which relate the density profile of a planar surface, p(z), to the pair potential, u(r), or to functionals of it such as the two-body distribution function or the direct correlation function c(r,2. Zi, z ). None of the equations, however, yields an explicit solution for p(z). in this chapter we describe some of the extra assumptions that have been made to enable them to be solved, and discuss the results for realistic forms of u(r). We consider primarily the Lennard-Jones (12,6) potential, (6.2), since it has been the most widely used, since there are several computer simulations for it, and since it is a reasonably realistic potential for simple fluids. ... 

In the above equation Sq is the contact energy of a pair of segments, which is assumed to obey a Lennard-Jones 6-12 potential... 

In a statistical Monte Carlo simulation the pair potentials are introduced by means of analytical functions. In the election of that analytical form for the pair potential, it must be considered that when a Monte Carlo calculation is performed, the more time consuming step is the evaluation of the energy for the different configurations. Given that this calculation must be done millions of times, the chosen analytic functions must be of enough accuracy and flexibility but also they must be fastly computed. In this way it is wise to avoid exponential terms and to minimize the number of interatomic distances to be calculated at each configuration which depends on the quantity of interaction centers chosen for each molecule. A very commonly used function consists of a sum of rn terms, r being the distance between the different interaction centers, usually, situated at the nuclei. In particular, non-bonded interactions are usually represented by an atom-atom centered monopole expression (Coulomb term) plus a Lennard-Jones 6-12 term, as indicated in equation (51). 

The OPLS model is an example of pair potential where non-bonded interactions are represented through Coulomb and Lennard-Jones terms interacting between sites centred on nuclei (equation (51). Within this model, each atomic nucleus has an interaction site, except CH groups that are treated as united atoms centered on the carbon. It is important to note that no special functions were found to be needed to describe hydrogen bonding and there are no additional interaction sites for lone pairs. Another important point is that standard combining rules are used for the Lennard-Jones interactions such that An = (Ai As )1/2 and Cu = (C Cy)1/2. The A and C parameters may also be expressed in terms of Lennard-Jones o s and e s as A = 4ei Oi and C ... 

When we consider a van der Waals system, we can start with the pair interaction as shown in Figure 2.2. The equation giving the pair potential is the 6-12 or Lennard-Jones-Devonshire equation ... 

A plot of the Lennard-Jones 9-3 form of Equations 7 and 8 for ST2 water interacting with smectite and mica surfaces is shown in Figure 1. Values for the parameters used in Figure 1 are given in Tables II and III, and in reference (23). The water molecule is oriented so that its protons face the surface and its lone pair electrons face away from the surface, and the protons are equidistant from the surface. Note that the depth of the potential well in Figure 1 for interactions with the smectite surface and mica surface are... 

The original work by van de Waals and Platteeuw (1959) used the Lennard-Jones 6-12 pair potential. McKoy and Sinanoglu (1963) suggested that the Kihara (1951) core potential was better for both larger and nonspherical molecules. The Kihara potential is the potential currently used, with parameters fitted to experimental hydrate dissociation data. However, it should be noted that the equations presented below are for a spherical core, and while nonspherical core work is possible, it has not been done for hydrates. 

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... 

Using trajectory calculations with an ab initio pair-wise potential or an assumed Lennard-Jones pair-wise potential, we can calculate the intermoleeular dynamic global potential which can be used to calculate experimentally obtained quantities sueh as a second virial coefficient. From classical statistical mechanics one obtains the following, well known, equation for the second virial coefficient ... 

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 ... 

In the case of a square lattice and for systems with the lateral interaction represented by the truncated Lennard-Jones pair potential with = 2.5cr, the dependence of Us m upon a is represented by the following equation... 

Since the Lennard-Jones potential-energy function is used, the equation is strictly valid only for nonpolar gases. The Lennard-Jones constants for the unlike molecular pair AB can be estimated from the constants for like pairs A A and BB ... 

To compute the detection of steric clashes, we used a Lennard-Jones 6-12 pseudo-potential expression defined in Equation 7. Such a potential, which was successfully applied to the complementarity of several host-guest systems [74], rapidly increases if two elements, e.g. atoms, spheres or peak, approach one another too closely, and interpenetrate. We performed the calculation of for each peaky pair of all complexes ... 

In the case of interacting molecules with distances short enough for the electronic shells to overlap, Mie (1903) suggested a relationship with an empirical constant and a certain decay law of high power against distance. Later, on the basis of Mie s equation and the van der Waals law h" Lennard-Jones (1936) derived an intermolecular pair potential... 

The theory describing diffusion in binary gas mixtures at low to moderate pressures has been well developed. Modem versions of the kinetic theory of gases have attempted to account for the forces of attraction and repulsion between molecules. Hirschfelder et al. (1949), using the Lennard-Jones potential to evaluate the influence of intermolecular forces, presented an equation for the diffusion coefficient for gas pairs of nonpolar, nonreacting molecules ... 

---