# Interdisciplinary Applied Mathematics

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(16.4)

-ea/re—z(r/a—r),    r > a,

where e is an attractive term, a is the hard-sphere diameter, and z is an adjustable parameter. The inverse power dependence of this potential means that it can be applied to ionic systems (Rowlinson, 1989). It has been shown that when z = 1.8, the potential behaves very similarly to the Lennard-Jones (12-6) potential (Duh and Mier-Y-Teran, 1997).

3. Lennard-Jones Potential. Lennard-Jones potential is one of the most widely used potentials for nonpolar molecules. The generalized form of a Lennard-Jones potential is

V (r)

e

m

—n

n

X

n — m

n — m

x

—m

(16.5)

where n and m are constants, x = r/rm, and rm is the separation corresponding to minimum potential energy. The “hard-sphere” diameter is related to the energy-minimum separation rm by

a

rm

1

TTt n—m

n

(16.6)

The most common form of the Lennard-Jones potential is obtained when n =12 and m = 6, i.e.,

V (r) = 4e

<7 12 r

6

(16.7)

The first term in equation (16.7) represents a short-range repulsive force, which prevents overlap of the atoms, while the second term represents an attractive interaction. Figure 16.2 shows the variation of the Lennard-Jones potential and the corresponding force (F—VrV(r)). The advantage of the Lennard-Jones potential is that it combines a realistic description of the intermolecular interaction with computational simplicity.

4. WCA Potential. The WCA (Weeks-Chandler-Andersen) potential is a modification of the Lennard-Jones potential, where the atoms

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