Lennard Jones

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

The gradient model for interfacial tension described in Eqs. III-42 and III-43 is limited to interaction potentials that decay more rapidly than r. Thus it can be applied to the Lennard-Jones potential but not to a longer range interaction such as dipole-dipole interaction. Where does this limitation come from, and what does it imply for interfacial tensions of various liquids ... 

One fascinating feature of the physical chemistry of surfaces is the direct influence of intermolecular forces on interfacial phenomena. The calculation of surface tension in section III-2B, for example, is based on the Lennard-Jones potential function illustrated in Fig. III-6. The wide use of this model potential is based in physical analysis of intermolecular forces that we summarize in this chapter. In this chapter, we briefly discuss the fundamental electromagnetic forces. The electrostatic forces between charged species are covered in Chapter V. 

We have two interaction potential energies between uncharged molecules that vary with distance to the minus sixth power as found in the Lennard-Jones potential. Thus far, none of these interactions accounts for the general attraction between atoms and molecules that are neither charged nor possess a dipole moment. After all, CO and Nj are similarly sized, and have roughly comparable heats of vaporization and hence molecular attraction, although only the former has a dipole moment. 

Molecular dynamics calculations have been made on atomic crystals using a Lennard-Jones potential. These have to be done near the melting point in order for the iterations not to be too lengthy and have yielded density functioi). as one passes through the solid-vapor interface (see Ref. 45). The calculations showed considerable mobility in the surface region, amounting to the presence of a... 

Molecular dynamics and density functional theory studies (see Section IX-2) of the Lennard-Jones 6-12 system determine the interfacial tension for the solid-liquid and solid-vapor interfaces [47-49]. The dimensionless interfacial tension ya /kT, where a is the Lennard-Jones molecular size, increases from about 0.83 for the solid-liquid interface to 2.38 for the solid-vapor at the triple point [49], reflecting the large energy associated with a solid-vapor interface. 

Refinements were made by Lennard-Jones, Taylor, and Dent [51-53], including an allowance for surface distortion. Their value of for (100) planes of sodium chloride at 0 K was 77 ergs/cm. Subsequently, Shuttleworth obtained a value of 155 ergs/cm [43]. 

The behavior of insoluble monolayers at the hydrocarbon-water interface has been studied to some extent. In general, a values for straight-chain acids and alcohols are greater at a given film pressure than if spread at the water-air interface. This is perhaps to be expected since the nonpolar phase should tend to reduce the cohesion between the hydrocarbon tails. See Ref. 91 for early reviews. Takenaka [92] has reported polarized resonance Raman spectra for an azo dye monolayer at the CCl4-water interface some conclusions as to orientation were possible. A mean-held theory based on Lennard-Jones potentials has been used to model an amphiphile at an oil-water interface one conclusion was that the depth of the interfacial region can be relatively large [93]. 

Table A2.3.2 Halide-water, alkali metal cation-water and water-water potential parameters (SPC/E model). In the SPC/E model for water, the charges on H are at 1.000 A from the Lennard-Jones centre at O. The negative charge is at the O site and the HOH angle is 109.47°.

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

Figure A3.1.1. Typical pair potentials. Illustrated here are the Lennard-Jones potential, and the Weeks-Chandler- Anderson potential, which gives the same repulsive force as the Leimard-Jones potential.

Larson R S and Lightfoot E J 1988 Thermally activated escape from a Lennard-Jones potential well Physica A 149 296-312... 

Straub J E, Borkovec M and Berne B J 1988 Molecular dynamics study of an isomerizing diatomic Lennard-Jones fluid J. Chem. Phys. 89 4833... 

Lennard-Jones J E and Devonshire A F 1936 Diffraction and seiective adsorption of atoms at crystai surfaces Nature 137 1069... 

Lennard-Jones J E 1932 Prooesses of adsorption and diffusion on soiid surfaoes Trans. Faraday Soc. 28 333... 

Figure B3.3.4. Lennard-Jones pair potential showing the and r eontributions.

There are various, essentially equivalent, versions of the Verlet algoritlnn, including the origmal method employed by Verlet [13, 44] in his investigations of die properties of the Lennard-Jones fluid, and a leapfrog fonn [45]. Here we concentrate on the velocity Verlet algoritlnn [46], which may be written... 

Verlet L 1967 Computer experiments on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules Phys. Rev. f59 98-103... 

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... 

McDonald I R and Singer K 1967 Calculation of thermodynamic properties of liquid argon from Lennard-Jones parameters by a Monte Carlo method Discuss. Faraday Soc. 43 40-9... 

Lotfi A, Vrabeo J and Fisoher J 1992 Vapour liquid equilibria of the Lennard-Jones fluid from the NpT plus test partiole method Mol. Phys. 76 1319-33... 

Agrawal R and Kofke D A 1995 Thermodynamio and struotural properties of model systems at solid-fluid ooexistenoe. II. Melting and sublimation of the Lennard-Jones system Mol. Phys. 85 43-59... 

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