Diffusion coefficients Lennard-Jones potential

Xenon has been considered as the diffusing species in simulations of microporous frameworks other than faujasite (10-12, 21). Pickett et al. (10) considered the silicalite framework, the all-silica polymorph of ZSM-5. Once again, the framework was assumed to be rigid and a 6-12 Lennard-Jones potential was used to describe the interactions between Xe and zeolite oxygen atoms and interactions between Xe atoms. The potential parameters were slightly different from those used by Yashonath for migration of Xe in NaY zeolite (13). In total, 32 Xe atoms were distributed randomly over 8 unit cells of silicalite at the beginning of the simulations and calculations were made for a run time of 300 ps at temperatures from 77 to 450 K. At 298 K, the diffusion coefficient was calculated to be 1.86 X 10 9 m2/s. This... 

Eor more careful (and more tedious) work, the model of Hirschfelder-Bird-Spotz [4] is recommended. It involves the Lennard-Jones potential between diffusing species, and that potential can be corrected slightly when both species are polar, using the method of Brokaw [5]. For engineering work. Equation (8.2) is normally adequate. According to the kinetic theory, the gas diffusion coefficient is independent of concentration. 

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 ... 

A model of immiscible Lennard-Jones atomic solvents has been used to study the adsorption of a diatomic solute [71]. Subsequently, studies of solute transfer have been performed for atoms interacting through Lennard-Jones potentials [69] and an ion crossing an interface between a polar and a nonpolar liquid [72]. In both cases the potential of mean force experienced by the solute was computed the results of the simulation were compared with the result from the transition state theory (TST) in the first case, and with the result from a diffusion equation in the second case. The latter comparison has led to the conclusion that the rate calculated from the molecular dynamics trajectories agreed with the rate calculated using the diffusion equation, provided the mean-force potential and the diffusion coefficient were obtained from the microscopic model. 

Li WZ, Chen C and Yang J. (2008) Molecular dynamics simulation of self-diffusion coefficient and its relation with temperature using simple Lennard-Jones potential. Heat Transfer - Asian Research, 37, pp. 86-93. 

The molecular dynamics results found from parameter set 1 are almost the same as those found for the Williams potential using his set VII, and reported elsewhere (10). The results for the rescaled Lennard-Jones potential used here are substantially better, as seen from Table II. We plan to study further refinements of this potential model, including the addition of an octupole moment, and we are also in the process of investigating other properties, including the orientational correlation functions, self-diffusion coefficients, and time correlation functions. 

The Lennard-Jones (6-12) potential has served very well as an inter-molecular potential and has been widely used for statistical mechanics and kinetic-theory calculations. It suffers, however, from having only two adjustable constants, and there is no reason why it should not gradually be replaced by more flexible and more realistic functions. Recently a number of applications have been made of the Buckingham (6-exp) potential [Eq. (82)], which has three adjustable parameters. For this potential the first approximation to the coefficient of diffusion is written by Mason (M3) in the form... 

The original Gay-Beme potential forms a nematic phase and a Smectic B phase, which is more solid like than liquid like. Ellipsoidal bodies do usually not form smectic A phases because they can easily diffuse from one layer to another layer. However, if one increases the side by side attraction it becomes possible to form smectic A phases [6]. When one calculates transport coefficients very long simulation runs are required. Therefore one sometimes re-places the Lennard-Jones core by a purely repulsive 1/r core in order to decrease the range of the potential. Thereby one decreases the number of interactions, so that the simulations become faster. The Gay-Beme potential can be generalised to biaxial bodies by forming a string of oblate ellipsoids the axes of which are parallel to each other and perpendicular to the line joining their centres of mass [35]. One can also introduce an ellipsoidal core where the three axis are different [38]. 

After the pioneering quantum mechanical work not much new ground was broken until computers and software had matured enough to try fresh attacks. In the meantime the study of intermolecular forces was mainly pursued by thermodynamicists who fitted model potentials, often of the Lennard-Jones form [10] 4e[(cr/R) — (cr// ) ], to quantities like second virial coefficients, viscosity and diffusion coefficients, etc. Much of this work is described in the authoritative monograph of Hirschfelder et al. [11] who, incidentally, also gave a good account of the relationship of Drude s classical work to that of London. 

Schoen, M. and C. Hoheisel. 1984. The mutual diffusion coefficient-Dij in binary-liquid model mixtures—Molecular-dynamics calculations based on Lennard-Jones (12-6) potentials. 1. The method of determination. Molecular Physics. 52, 33. 

In MD modeling, the molecular adsorption concept is used to interpret the Pt-C interactions during the fabrication processes. The Pt complexes are mostly attached to the hydrophilic sites on the carbon particles, viz. carbonyl or hydroxyl groups (Hao et ah, 2003). The adsorption is based on both the physical and chemical adsorptions. Many efforts have been done on the MD simulations of Pt nano-particles adsorbed on carbon with or without ionomers (Balbuena et ah, 2005 Chen and Chan, 2005 Huang and Balbuena, 2002 Lamas and Balbuena, 2003 2006). The Pt-Pt interactions are modeled with the many-body Sutton-Chen (SC) potential (Rafii-Tabar et al, 2006), whereas a Lennard-Jones (LJ) potential is used to describe the Pt-C interactions. The SC potential for Pt-Pt and Pt-C interactions provides a reasonable description of the properties for small Pt clusters. The diffusion of platinum nano-particles on graphite has also been investigated, with diffusion coefficients in the order of 10 cm s (Morrow and Striolo, 2007). 

Little is known concerning three body contributions to dynamic fluid properties. Fisher and Watts have calculated the self diffusion coefficient using the BFW pair potential without including the Axilrod-Teller interaction and obtained results similar to those for a Lennard-Jones fluid at the same densities (23). Schommers two dimensional Lennard-Jones plus Axllrod Teller simulations show significant three body effects on the velocity autocorrelation function however, the two dimensional self diffusion coefficient is little affected (16). Schommers is careful to point out, however, that results found in two-dimensional fluids do not necessarily extrapolate to three dimensions. 

Two parallel simulations for 108 particles interacting with Lennard-Jones plus Axllrod-Teller potentials have been performed. The first calculation utilized the CMD method in which the forces were explicitly evaluated at each time step. In the second run the two body forces were determined in the standard way and the LMTS method described above was applied to the three body forces. Both runs were started from the same initial particle positions and velocities and both were continued for 1650 time steps. A comparison of the properties obtained from the two calculations is given in Table 1. In addition to the properties listed in Table I, radial distribution functions, velocity, speed, and force autocorrelation functions, and atomic mean squared displacements (from which diffusion coefficients were obtained) were calculated. For all of these properties, the LMTS values were within 0.1% of the values obtained by the CMD method. Figure 4 shows the per cent deviation in the instantaneous total energy of the two calculations. 

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