Lennard-Jones potential coefficient

Values of the Lennard-Jones potential well depth F (J/mol) and characteristic radius R (nm) for calculating the Lennard-Jones potential coefficients Cl i = 2FR and B = FR for the interaction of alkane atomic species with various surface atomic species... 

For many practically important interaction functions, the Fourier coefficients in Eq. (D.9) have finite analytic forms, for example, the Lennard-Jones potential, the Yukawa potential, the Morse potential, and functions that can be derived from those functions. For a power-law interaction... 

Equations (1) and (4) or other variations of the 12-6 power law are often called the Lennard-Jones potential. The numerical values of the constants in the Lennard-Jones potential may be obtained from studies of the compressibility of condensed phases, the virial coefficients of gases, and by other methods. A summary of these methods and other expressions for the molecular interaction energy can be found in the book by Moelwyn-Hughes (1964). 

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... 

Fig. 54. The friction coefficient or memory function of a spherical particle of mass fifty times that of the solvent particles. The particles interact by means of a Lennard-Jones potential. After Brey and Ordonez [529 J.

All of the transport properties from the Chapman-Enskog theory depend on 2 collision integrals that describe the interactions between molecules. The values of the collision integrals themselves, discussed next, vary depending on the specified intermolecular potential (e.g., a hard-sphere potential or Lennard-Jones potential). However, the forms of the transport coefficients written in terms of the collision integrals, as in Eqs. 12.87 and 12.89, do not depend on the particular interaction potential function. 

The temperature-dependent second and third virial coefficient describe the increasing two- and three-particle collisions between the gas molecules and their accompanying increase in gas density. The virial coefficients are calculated using a suitable intermolecular por-tential model (usually a 12-6 Lennard-Jones Potential) from rudimentary statistical thermodynamics. 

Monte Carlo simulation techniques are used for calculating the distribution coefficients of benzene between supercritical C02 and slitpores at infinite dilution. The Lennard-Jones potential model is used for representing the pair interactions between C02, benzene, and graphite carbon. The effects of temperature, slitwidth, and benzene-surface interaction potential on the distribution coefficients are explored at constant density and constant pressure. 

FIGURE 9.1 Dependence of solubility coefficient of various hydrocarbons in natural rubber on hydrocarbon boiling temperature, 7], (a) and on diameter of their molecules calculated from Lennard-Jones potential, ctu (b). (From Semenova, S.I., Membranes (in Russian),... 

Here the superscripts (BC) and (LJ) refer to the Buckingham-Corner and Lennard-Jones potentials, respectively is the distance between the m-th atom of the molecule and the s-th atom of the surface. The coefficients C, D, B, Q depend on microscopic characteristics, such as static polarizability, ionization potential, etc. For a detailed discussion of atom-atom interaction potentials, the reader is referred to [12]. The subscripts M and S denote atomic species of the adsorbed molecule and the adsorbent surface note that in summations like (11), it is implied that for any value of m (or s) there is the definite M (or S) value which corresponds to a particular atomic species. Therefore, the internal summation in equation (11) can be performed to give the sum of atomic contributions ... 

By use of these formulas accurate calculations of the transport coefficients can thus be performed provided that the Lennard-Jones potential parameters like i and ci are known. Extensive lists of these parameters are given for many substances by Hirschfelder et al [39] and Bird et al [5], among others. 

Figure 10.2 Normalized gas-solid virial coefficients for a Lennard-Jones potential, as a function of the reduced temperature T/T, for different values of the correlation length Tq and for a given value of the standard deviation of the adsorptive potential kg T,. Adapted from Ref. 25.

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. 

Fig. I. The differential cross section of neon. The squares represent experimental data and the error bars, average deviations. The solid line represents the values calculated from the scattering potential of Fig. 3. The dashed curve was calculated using the Lennard-Jones potential obtained from virial coefficient data.

Fig. 7. The second virial coefficient of argon. The results calculated from the scattering potential of Fig. 4 are indicated by the solid line and those of the Lennard-Jones potential, by the dashed curve. The experimental data are represented by squares, open circles, triangles, and solid circles. ... 

In the case of pure liquids numerical computations for the transport coefficients in argon, krypton, and xenon have been carried out by Palyvos et al. using a modified Lennard-Jones potential and the radial distribution function of Kirkwood, Lewinson, and Alder. The results, for instance for argon, represent percentages betw een 60 and 90% of the experimental values in a wide range of temperatures and densities. Besides, they agree with experiment better than the results derived from the Kirkwood of Rice-AIInatt types of theories. 

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 ... 

Figure A2.1.7. The second virial coefficient 5 as a function of temperature T/T-g. (Calculated for a gas satisfying the Lennard-Jones potential [8].)...

This derivation, going back to van der Waals [1], has a disturbing flaw. If the expansion is continued to the next non-vanishing order (fourth), the expression for the respective coefficient, computed analogous to Eq. (21), diverges when the common Lennard-Jones potential is used. 

A force sensor in an AFM can only work if the probe interacts with the force field associated with a surface. The interaction force between the probe and the surface in ambient air is illustrated in Figure 9.2. The total intermolecular pair potential is obtained by assuming one attractive (-Ci/z ) and another repulsive potential (C2/z )-Superimposing the two gives an expression for the well-known Lennard-Jones potential, where Ci and C2 are the corresponding coefficients for the attractive and repulsive interactions, respectively, and z is the distance between the sample surface and rest position of the cantilever. 

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