Well depth

The feature that distinguishes intemrolecular interaction potentials from intramolecular ones is their relative strengtii. Most typical single bonds have a dissociation energy in the 150-500 kJ mol range but the strengdi of the interactions between small molecules, as characterized by the well depth, is in the 1-25 kJ mor range. 

Despite the complexity of these expressions, it is possible to hrvert transport coefficients to obtain infomiation about the mtemiolecular potential by an iterative procedure [111] that converges rapidly, provided that the initial guess for V(r) has the right well depth. 

The venerable bireciprocal potential consists of a repulsive tenn A/t and an attractive temi -B/r with n > m. This potential fiinction was introduced by Mie [118] but is usually named after Leimard-Jones who used it extensively. Almost invariably, / = 6 is chosen so that the attractive tenn represents the leading dispersion tenn. Many different choices of n have been used, but the most connnon is n = 12 because of its computational convenience. The Leimard-Jones (12,6) potential can be written in tenns of the well depth (s) and either the minimum position or the zero potential location (a) as... 

The fomier is iisefiil but the latter tends to overestimate the well depth. A hamionic mean rule... 

A more natural way to account for the anisotropy is to treat tire parairreters in an interatomic potential, such as equation (A 1.5.64). as fiurctioirs of the relative orientation of the interacting molecules. Comer [131] was perhaps the first to use such an approach. Pack [132] pointed out that Legendre expansions of the well depth e and equilibrium location of the interaction potential converge more rapidly tirair Legendre expansions of the potential itself... 

The relative separation is scaled by a, the distance at which the Leimard-Jones first passes tlirough zero. The energy is scaled by the well depth, s. 

Approximating the real potential by a square well and infinitely hard repulsive wall, as shown in figure A3.9.2 we obtain the hard cube model. For a well depth of W, conservation of energy and momentum lead [H, 12] to the very usefiil Baule fomuila for the translational energy loss, 5 , to the substrate... 

The Lennard-Jones 12-6 potential contains just two adjustable parameters the collision diameter a (the separation for which the energy is zero) and the well depth s. These parameters are graphically illustrated in Figure 4.34. The Lennard-Jones equation may also be expressed in terms of the separation at which the energy passes through a minimum, (also written f ). At this separation, the first derivative of the energy with respect to the internuclear distance is zero (i.e. dvjdr = 0), from which it can easily be shown that v = 2 / cr. We can thus also write the Lennard-Jones 12-6 potential function as follows ... 

Hential is an exponential-6 potential with just two parameters the mirrimum energy 5 and the well depth e [Hill 1948] ... 

This is similar in spirit to the arithmetic-mean rule but with each individual r,) being weighted according to the square of its value. The well depth in this function starts with a formula proposed by Slater and Kirkwood for the Cg coefficient of the dispersion series expansion ... 

From this the well depths s are then obtained as follows ... 

Here, k is a factor which converts to units (kcal/mol in this case where the distances are in A and the polarisabilities in A ). G, and Gj are constants chosen to reproduce the well depths for like-with-like interactions. The atomic polarisability values are obtained from an examination of appropriate molecular experimental data (such as measurements of molar refractivity). 

The attraction for two neutral atoms separated by more than four Angstroms is approximately zero. The depth of the potential wells is minimal. For the AMBER force field, hydrogen bonds have well depths of about 0.5 kcal/mol the magnitude of individual van der Waals well depths is usually less. 

Figure 2.10. Part of the better description of the Morse and Exp.-6 potentials may be due to the fact that they have three parameters, while the Lennard-Jones potential only employs two. Since the equilibrium distance and the well depth fix two constants, there is no additional flexibility in the Lennard-Jones function to fit the form of the repulsive interaction.

Class G Intended for use from surface to 8,000 ft (2,440 m) depth as manufactured, or can be used with accelerators and retarders to cover a wide range of well depths and temperatures. No additions other than calcium sulfate or water, or both, shall be interground or blended with the clinker during... 

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