Lennard-Jones radius

According to Lennard-Jones (6), the attractive energy varies as the inverse sixth power of the distance while the repulsive energy varies approximately as the inverse twelfth power. The radius of a cesium atom is about 2.7A., and it can sink about O.sA.. below the surface. Hence its nucleus will be 2.2A. above the surface. According to Figure 9, (Pa = 2.4 volts. This locates the minimum in the atom curve in Figure 11. 

The computer simulations employed the molecular dynamics technique, in which particles are moved deterministically by integrating their equations of motion. The system size was 864 Lennard-Jones atoms, of which one was the solute (see Table II for potential parameters). There were no solute-solute interactions. Periodic boundary conditions and the minimum image criterion were used (76). The cutoff radius for binary interactions was 3.5 G (see Table II). Potentials were truncated beyond the cutoff. 

The form of the potential for the system under study was discussed in many publications [28,202,207,208]. Effective pair potentials are widely used in theoretical estimates and numerical calculations. When a many-particle interatomic potential is taken into account, the quantitative description of experimental data improves. For example, the consideration of three-body interactions along with two-particle interactions made it possible to quantitatively describe the stratification curve for interstitial hydrogen in palladium [209]. Let us describe the pair interaction of all the components (hydrogen and metal atoms in the a. and (j phases) by the Lennard Jones potential cpy(ry) = 4 zi [(ff )12- / )6], where Sy and ai are the parameters of the corresponding potentials. All the distances ry, are considered within c.s. of radius r (1 < r < R), where R is the largest radius of the radii of interaction Ry between atoms / and /). 

In our model, a PFPE molecule is composed of a finite number of beads with different physical or chemical properties [Fig. 1.41]. For simplicity, we assume that all the beads, including the endbeads, have the same radius. Lennard-Jones... 

It has been traditional to define a van der Waals potential (which combines Coulomb s law and the Lennard-Jones 6-12 potential function) and thereby subsume electronic shell repulsion, London forces, and electrostatic interactions under the term van der Waals interaction. Unfortunately, the resulting expression is an oversimplified treatment of the electrostatic interactions, which are only calculated between close neighbors and are considered to be spatially isotropic. Both of these implicit assumptions are untrue and do not represent physically realistic approximations. We prefer to use the term van der Waals distance for the intemuclear separation at which the 6-12 potential function is a minimum (see Fig. 6), the van der Waals radius being one-half this value when the two interacting atoms are identical, and explicitly treat the Lennard-Jones and electrostatic terms separately. While the term van der Waals interaction may have some value as a shorthand in structure description, it should be avoided when energetics are treated quantitatively. 

Fig. 6. Schematic drawing of the shape of the Lennard-Jones 6-12 potential energy versus interatomic distance (r). The equilibrium separation distance occurs at the potential energy minimum and is defined to be twice the van der Waals radius if the two interacting atoms are identical.

For the interaction potential between hydrogen and carbon, we introduce a new procedure to derive the Lennard-Jones parameters from existing parameters that are appropriate for carbon atoms with sp2 and sp3 hybridizations. These parameters may come from existing force fields, and may have been obtained using either experimental or ab initio results. The L-J parameters a and s are made explicitly dependent on the radius of the nanotube, r, using the following equations ... 

The principal tools have been density functional theory and computer simulation, especially grand canonical Monte Carlo and molecular dynamics [17-19]. Typical phase diagrams for a simple Lennard-Jones fluid and for a binary mixture of Lennard-Jones fluids confined within cylindrical pores of various diameters are shown in Figs. 9 and 10, respectively. Also shown in Fig. 10 is the vapor-liquid phase diagram for the bulk fluid (i.e., a pore of infinite radius). In these examples, the walls are inert and exert only weak forces on the molecules, which themselves interact weakly. Nevertheless,... 

Figure 10. Vapor-liquid equilibria for an argon-krypton mixture (modeled as a Lennard-Jones mixture) for the bulk fluid (R = >) and for a cylindrical pore of radius R = / /Oaa = 2.5. The dotted and dashed lines are from a crude form of density functional theory (the local density approximation, LDA). The points and solid lines are molecular dynamics results for the pore. Reprinted with permission from W. L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc.

In the common case of the site-site interaction potential comprising the Lennard-Jones and Coulomb terms, it is reasonable to choose the weighting radius for site a as half its Lennard-Jones size, r = o /2. The convolution (32) is conveniently calculated in the reciprocal space as... 

As a modep) we assume an ideal infinite pore with radius a (fig. 1.32a). The liquid in the pore is not identical to that of the bulk because it is influenced by the interaction with the wall. The molar energy depends in some way on the distance to the wall, say = U r). e.g. according to a Lennard-Jones relationship. The pressure difference across the interface is given by the Laplace equation... 

The simulations conducted model the flow of Lennard-Jones (LJ) methane at ISO K and 170 K in a cylindrical silica pore of radius 1.919 nm, having infinitely thick pore walls comprising spherical LJ sites. For methane we use the established LJ parameter values... 

Arkhipov et al. have taken multiscale coarse graining to an extreme in their recent work, in which they use atomistic models to parameterize CG models of a complete virus capsids. Each CG particle in the simulation represented about 200 atoms. Each CG particle interacted via a Lennard-Jones potential, which was parameterized to match the size of the domain the CG particle represented, as calculated from the radius of gyration of that domain measured from an atomistic model. The resulting CG model was able to simulate complete virus capsids (of dimensions 10 nm to 100 nm) over timescales of 1 ps to 10 ps. 

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