Equations Lennard-Jones potential

For most gas species the adsorbate-adsorbate interaction is given by the Lennard-Jones potential equation... 

FIGURE 3.9 Comparison of the actual interatomic potential for two iodine atoms (solid line) with the approximate Lennard-Jones potential, Equation 3.32. The Lennard-Jones potential... 

Fig 1114. Comparison of scaled and unsealed Lennard-Jones potentials (Equation (11 31)) for the case where a particle disappears at A = 0 As A decreases the curves get progressively closer to the x axis... 

SOLUTION We can sdve this problem by substituting the Lennard-Jones potential (Equation 4.10) into the molecular formulation for the second virial coefficient. Equation (4.29), as fiJlows ... 

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. 

An approximate value for dc in the equation for tire Lennard-Jones potential, quoted above, may be obtained from the van der Waals constant, b, since... 

In these equations, a and e are parameters in the Lennard-Jones potential function for interactions between unlike molecules, the customary mixing rules were used ... 

When we consider a van der Waals system, we can start with the pair interaction as shown in Figure 2.2. The equation giving the pair potential is the 6-12 or Lennard-Jones-Devonshire equation ... 

Person 1 Use Eq. (1.16) with the values of m and n of the Lennard-Jones potential to solve for the constant a in terms of b and the equilibrium bond distance, ro. Now perform the determination of F ax as given by Eq. (1.15) substitute this value of a back into Eq. (1.12), differentiate it twice with respect to r (remember that ro is a constant), and set this equal to zero (determine, then maximize the force function). Solve this equation for r in terms of tq. The other constant should drop out. 

Equation (1.12) represents the well-known Lennard-Jones potential. 

An efficient method of solving the Percus-Yevick and related equations is described. The method is applied to a Lennard-Jones fluid, and the solutions obtained are discussed. It is shown that the Percus-Yevick equation predicts a phase change with critical density close to 0.27 and with a critical temperature which is dependent upon the range at which the Lennard-Jones potential is truncated. At the phase change the compressibility becomes infinite although the virial equation of state (foes not show this behavior. Outside the critical region the PY equation is at least two-valued for all densities in the range (0, 0.6). 

The equation will be solved under the assumption that the particles in the fluid are interacting through a truncated Lennard-Jones 12-6 potential. The Lennard-Jones potential has been used by several workers, including Khan and Broyles,5 Throop and Bearman,6 and Levesque.7 These workers studied the PY equation above the reduced critical temperature T c in considerable detail. Their results in that region provided a good check on the accuracy of the method described here. 

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). 

There have been several studies of the iodine-atom recombination reaction which have used numerical techniques, normally based on the Langevin equation. Bunker and Jacobson [534] made a Monte Carlo trajectory study to two iodine atoms in a cubical box of dimension 1.6 nm containing 26 carbon tetrachloride molecules (approximated as spheres). The iodine atom and carbon tetrachloride molecules interact with a Lennard—Jones potential and the iodine atoms can recombine on a Morse potential energy surface. The trajectives were followed for several picoseconds. When the atoms were formed about 0.5—0.7 nm apart initially, they took only a few picoseconds to migrate together and react. They noted that the motion of both iodine atoms never had time to develop a characteristic diffusive form before reaction occurred. The dominance of the cage effect over such short times was considerable. 

The Lennard-Jones potential (so-called 6-12 equation) commonly holds for nonpolar molecules having no permanent dipole moment such as helium, argon, and methane [39-41]. Nevertheless, this potential can be expected to give an accurate description of long-range forces only for sufficiently long distance between two bodies [27,42]. 

Similar to Muller et al. [103], several authors performed self-consistent numerical computations of the non-linear equations based on the Lennard-Jones potential instead of dealing with analytical approaches. Greenwood [108, 109] and Feng [112] calculated the dependence of the pull-off force on the transition parameter A. For fixed y and R, softer spheres require less force to separate, though they deform more significantly. At the extreme limits of small and large values of A, however, the pull-off force becomes virtually independent of A. On a local scale, this is consistent with the prediction of the JKR and the DMT model that the pull-off force is independent of elastic moduli of spheres [112]. 

In both equations, d is the separation between the atoms. The Lennard-Jones potential is simpler and computationally less demanding and is therefore favored for models of macromolecules such as proteins and DNA. The Buckingham function more closely resembles the energy relationship and is preferred when higher accuracy is required. The latter function is available in MOMEC and we will concentrate on this. 

Wood, W. W and F. R. Parker Monte Carlo Equation of State of Molecules Interacting with the Lennard-Jones Potential. I. A Supercritical Isotherm at about Twice the Critical Temperature. J. Chem. Phys. 27, 720 (1957). 

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