Lennard-Jones models

Wilson M R 1997 Molecular dynamics simulations of flexible liquid crystal molecules using a Gay-Berne/Lennard-Jones model J. Chem. Phys. 107 8654-63... 

The range of systems that have been studied by force field methods is extremely varied. Some force fields liave been developed to study just one atomic or molecular sp>ecies under a wider range of conditions. For example, the chlorine model of Rodger, Stone and TUdesley [Rodger et al 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules dep>ends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with resp>ect to the bond vectors of the two molecules. The model includes an electrostatic component which contciins dipwle-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

Adsorption of hard sphere fluid mixtures in disordered hard sphere matrices has not been studied profoundly and the accuracy of the ROZ-type theory in the description of the structure and thermodynamics of simple mixtures is difficult to discuss. Adsorption of mixtures consisting of argon with ethane and methane in a matrix mimicking silica xerogel has been simulated by Kaminsky and Monson [42,43] in the framework of the Lennard-Jones model. A comparison with experimentally measured properties has also been performed. However, we are not aware of similar studies for simpler hard sphere mixtures, but the work from our laboratory has focused on a two-dimensional partly quenched model of hard discs [44]. That makes it impossible to judge the accuracy of theoretical approaches even for simple binary mixtures in disordered microporous media. 

TABLE 1 Critical Parameters for the Lennard-Jones Model are Given in Reduced Units such that cr = 1 and e = 1... 

In their work [58], GY demonstrated that a standard Lennard-Jones model grossly over-predicted the well-depth of rare gas-halide ion dimer potential energy curves when they were parametrized to reproduce the neutral rare gas-halide dimer curves. They further showed that the OPNQ model performed just as badly when the charge dependence of the expressions were ignored, but the potential energy curves for both the neutral and ionic dimers could be simultaneously be reproduced if the charge dependence is considered. 

Vrabec, J. Loth, A. Fischer, J., Vapour liquid equilibria of Lennard-Jones model mixtures from the NPT plus test particle method, Fluid Phase Equil. 1995,112, 173-197... 

Bulk phase fluid structure was obtained by solution of the Percus-Yevick equation (W) which is highly accurate for the Lennard-Jones model and is not expected to introduce significant error. This allows the pressure tensors to return bulk phase pressures, computed from the virial route to the equation of state, at the center of a drop of sufficiently large size. Further numerical details are provided in reference 4. 

In this lab, the students determine the compression factor, (9) Z = PV/nRT, for Argon using the hard sphere model, the soft sphere model, and the Lennard-Jones model and compare those results to the compression factor calculated using the van der Waals equation of state and experimental data obtained from the NIST (70) web site. Figure 3 shows representative results from these experiments. The numerical accuracy of the Virtual Substance program is reflected by the mapping of the Lennard-Jones simulation data exactly onto the NIST data as seen in Figure 3. 

From the inception of quantum mechanics, from about 1930 to the late sixties, most research on intermolecular forces was based on two assumptions, namely 1. that the pair potentials could be represented by simple functions, such as the two-parameter Lennard-Jones model,... 

It is now clear that the repulsive energy branch of rare gas pairs is of an exponential form, unlike the R 12 term of the Lennard-Jones model. A few examples of measured repulsive branches of interatomic potentials... 

Approximate Calculation of the Thermal Expansion in the Lennard-Jones-Model (Temperature Dependence of the Activation Energy)... 

In method B, which seems to be devoid of theoretical support, the two functions are fitted in magnitude at the most probable velocity and also at r0, the point of zero potential on the Lennard-Jones model. A comparison with experiment is given in Table 1. Z is the reciprocal probability or number of gas kinetic collisions required for deactivation. The calculated values are derived from equations (17) and (18). 

The relative efficiency per collision for deactivation of excited molecules in thermal reactions increases with the number of atoms in the collider, but reaches a constant limit when this number exceeds about 12. This has been demonstrated for many thermal reactions by studying the low pressure fall-off. It may be noted from eqn. (10) that plots of k /k against pressure for different inert gases should comprise a set of curves dispersed along the pressure axis according to the various efficiencies of deactivation per unit pressure. The relative efficiency per collision can be derived by calculating the collision frequency, Z, with a hard sphere or Lennard—Jones model. 

Liu, Silva, and Macedo [Chem. Eng. Sci. 53, 2403 (1998)] present a theoretical approach incorporating hard-sphere, square-well, and Lennard-Jones models. They compared their resulting estimates to estimates generated via the Lee-Thodos equation. For 2047 data points with nonpolar species, the Lee-Thodos equation was slightly superior to the Lennard-Jones fluid-based model, that is, 5.2 percent average deviation versus 5.5 percent, and much better than the square-well fluid-based model (10.6 percent deviation). For over 467 data points with polar species, the Lee-Thodos equation yielded 36 percent average deviation, compared with 25 percent for the Lennard-Jones fluid-based model, and 19 percent for the square-well fluid-based model. 

Figure 6 The radial distribution function for a Lennard-Jones model of liquid argon at a temperature T = 300 K. A simulation cell of 35 A containing 864 atoms with periodic boundary conditions was used. The simulation was carried out by coupling each degree of freedom to an MTK thermostat, and the equation of motion was integrated using the methods discussed in Ref. 28.

To provide a more quantitative explanation of the magnitudes of the properties of different materials, we must consider several types of intermolecular forces in greater detail than we gave to the Lennard-Jones model potential in Chapter 9. The Lennard-Jones potential describes net repulsive and attractive forces between molecules, but it does not show the origins of these forces. We discuss other intermolecular forces in the following paragraphs and show how they arise from molecular structure. Intermolecular forces are distinguished from intramolecular forces, which lead to the covalent chemical bonds discussed in Chapters 3 and 6. Intramolecular forces between atoms in the covalent bond establish and maintain... 

One important point we should stress, in conjunction with our current interest, is that similar slow relaxation as liquid water is observed in much simpler model systems The binary mixture of Lennard-Jones liquids, which consist of two species of particles, is now studied extensively as a toy model of glass-forming liquids. It is simulated after careful preparation of simulation conditions to avoid crystallization. Also, the modified Lennard-Jones model glass, in which a many-body interaction potential is added to the standard pairwise Lennard-Jones potential, is also studied as a model system satisfying desired features. 

We illustrate the behavior for a first order transition between a vapor and a dense liquid in the framework of a simple Lennard-Jones model. The condensation of a vapor into a dense liquid upon cooling is a prototype of a phase transition that is characterized by a single scalar order parameter - the density, p. The thermodynamically conjugated field is the chemical potential, p. The qualitative features, however, are general and carry over to other types of phase coexistence, e.g., Sect. 3.4. 

Estimate the thermal conductivity of methane at 500°C. Assume the Lennard-Jones model to apply. 

Table 11-3 Values ofQj for diffusivity calculations (Lennard-Jones model)...

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