Pair potential Lennard-Jones

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

Rare-gas clusters can be produced easily using supersonic expansion. They are attractive to study theoretically because the interaction potentials are relatively simple and dominated by the van der Waals interactions. The Lennard-Jones pair potential describes the stmctures of the rare-gas clusters well and predicts magic clusters with icosahedral stmctures [139, 140]. The first five icosahedral clusters occur at 13, 55, 147, 309 and 561 atoms and are observed in experiments of Ar, Kr and Xe clusters [1411. Small helium clusters are difficult to produce because of the extremely weak interactions between helium atoms. Due to the large zero-point energy, bulk helium is a quantum fluid and does not solidify under standard pressure. Large helium clusters, which are liquid-like, have been produced and studied by Toennies and coworkers [142]. Recent experiments have provided evidence of... 

The simplest choice for the interaction potentials between the particles (at distance r) is the assumption of Lennard-Jones pair potentials,... 

Earlier work on systems such as Ni-Al-Cr reported in Sanchez et al. (1984b) used FP methods to obtain information on phases for which there was no experimental information. In the case of Ni-base alloys, the results correctly reproduced the main qualitative features of the 7 — 7 equilibrium but cannot be considered accurate enough to be used for quantitative alloy development. A closely related example is the work of (Enomoto and Harada 1991) who made CVM predictions for order/disorder (7 — 7 ) transformation in Ni-based superalloys utilising Lennard-Jones pair potentials. 

In the case of a square lattice and for systems with the lateral interaction represented by the truncated Lennard-Jones pair potential with = 2.5cr, the dependence of Us m upon a is represented by the following equation... 

Fig. 4.3. Lennard-Jones pair potential plotted in dimensionless units.

Figure 7.3 A typical Lennard-Jones pair potential (a), is split into two parts a soft part (b) and a hard part (c).

Figure 5. Weeks-Chandler-Andersen pair potential (solid line). The Lennard-Jones pair potential is shown for comparison (dashed line).

The solubility parameter approach is a thermodynamically consistent theory and it has some links with other theories such as the van der Waals internal pressure concept, the Lennard-Jones pair potentials between molecules, and entropy of mixing concepts of the lattice theories. The solubility parameter concept has found wide use in industry for nonpolar solvents (i.e. solvent selection for polymer solutions and extraction processes) as well as in academic endeavor (thermodynamics of solutions), but it is unsuccessful for solutions where polar and especially hydrogen-bonding interactions are operating. 

Figure A2.2.5 shows a sketch off(r) for Lennard-Jones pair potential. Now if AA is the excess Helmholtz free energy relative to its ideal gas value, then (-PAT) = log(2/2ideal AU/N= [5(PAzl/A)/(5p)]. Then, integrating with respect to P, one obtains...

Figure A2.2.5. Sketch off(r) for the Lennard-Jones pair potential u r) = 4s[(a/r) - (cj/r) ] full curve -Ps =...

Figure 3.3 The Lennard-Jones pair potential is a good approximation for the potential energy between two approaching particles...

During the simulation, the force on each molecule is calculated as the sum of the forces of interaction with all of the surrounding molecules. These are the dispersion forces, also referred to as the London and van der Waals forces, which depend on the separation distance, r, between two molecules, as shown in Figure 2.1. These are represented by the dimensirm-less form of the commonly used Lennard-Jones pair potential, U r]/e ... 

Figure 2.1 Lennard-Jones pair potential and force.

The diameters of solvents play a role in theoretical considerations, such as the application of the scaled particle theory. For gaseous solvent molecules, the collision diameter, o, is related to the Lennard-Jones pair potential energy,... 

Equations (9.2) represent a set of second-order differential equations coupled in time, which can be solved numerically on a large computer for N 1000 and a certain small time period, if the pair potential is not too complicated. Here, we restrict ourselves to the application of the Lennard-Jones pair potential of the form... 

As an example of the practical utility of MEMO simulations of the thermal conductivity, a summary of a calculation by Ravi et al. (1992) is given. The object was to explore the contribution of internal degrees of freedom to the conductivity of a polyatomic molecule. This contribution has not been quantitatively demonstrated until very recently (Murad et al. 1991). In fact, it is usually assumed that this contribution is independent of number density and is given by the dilute-gas value at the corresponding temperature. In the paper of Ravi et al. (1992), heat flow is discussed for a model benzene-like liquid with a six-centered Lennard-Jones pair potential. 

Luckhurst and Simmonds [68] have used Lennard-Jones pair potentials to characterise the molecular interaction potential f/av(H], 2 for two liquid crystal molecules with molecular long axes in the direc-... 

In one series of calculations the anharmonic Lennard-Jones pair potential was replaced by a truncated harmonic pair potential of the form... 

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