Lennard-Jones potential

E(LJ) = 4 epsilon [(sigma/r)¹² - (sigma/r)⁶]

E(LJ) = (a/r)¹² - (b/r)⁶

• The zero of the Lennard-Jones potential is obtained at infinite separation of the two particles.
• The depth of the weakly attractive well (energy smaller than kT under normal conditions) is denoted by epsilon, which is sometimes cited with the units of temperature, epsilon / k.
• The attractive term b is proportional to the product of the polarizabilty of the pair of atoms. The attraction arises from induced dipole-induced dipole interactions, produced by motions of the electrons in the atom so that there is an instantaneous difference in the centers of the positive and negative charges. The inverse sixth power dependence of the attractive term arises because the attraction is produced by the interaction of these transitory dipoles.
  • Hydrogen has the smallest epsilon: epsilon/k = 8.6 K
  • Epsilon tends to increase as one moves down a column of the periodic table, due to the increase in the polarizability of the outer electrons as the atom becomes larger: epsilon/k = 52.8, 173.5, and 257.2 K for F, Cl, and Br, respectively.
• A steeply repulsive wall is encountered when r < sigma. Sigma is the separation at which the Lennard-Jones potential changes sign.
  • Hydrogen has the smallest sigma: sigma = 0.281 nm
  • Sigma tends to increase as one moves down a column of the periodic table: sigma = 0.283, 0.335, and 0.354 nm for F, Cl, and Br, respectively.
  • Sigma tends to decrease as one moves across a row of the periodic table, because the outer electrons (in the same shell) are drawn toward the nucleus by its increasing positive charge: sigma = 0.331, 0.295, and 0.283 nm for N, O, and F, respectively.
  • Since sigma is small (roughly 0.3 nm for atoms commonly found in polymers), the attraction is of short range.
• The inverse 12th power dependence of the repulsive term is empirical. Alternative descriptions of the repulsive part exist in other potential energy functions. They may use a somewhat different exponent in the repulsive part (such as 9 instead of 12), or a different mathematical form, based on an exponential function. There is no strong theoretical reason to prefer any one of these forms. All of them produce a strongly rising potential as r falls below sigma.

Simpler Approximations

Two simpler, discontinuous potential energy functions capture some of the features seen in the continuous Lennard-Jones potential.

• The hard sphere potential represents only the repulsive part of the Lennard-Jones potential, and uses an "infinite-or-none" representation of the repulsion. It contains one parameter, sigma.

  E^HS = infinity (r < sigma)

  E^HS = 0 (r greater than, or equal to, sigma)

  This potential can represent excluded volume, but it cannot account for cohesion, because it contains no intermolecular attraction.

• The square well potential combines a weak attraction at small separation with the hard sphere potential. It can produce a cohesive system. There are three parameters.

  E^HS = infinity (r < sigma₁)

  E^HS = - epsilon (r between sigma₁ and sigma₂)

  E^HS = 0 (r greater than, or equal to, sigma₂)

Lorentz/Berthelot Mixing Rules

For the interaction of heteroatomic pairs, the effective values of sigma and epsilon are calculated from those for the homoatomic pairs using the Lorentz-Berthelot mixing rules: arithmetic mean for sigma, geometric mean for epsilon.

sigma_AB = (1/2)(sigma_AA + sigma_BB)

epsilon_AB = (epsilon_AAepsilon_BB)^1/2

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