While much of the work on structural materials has concentrated on metals, there are important issues involving nonmetallics for coatings, corrosion, and fuel. In this section, we review other types ofpotentials.

1.10.10.1 Covalent Potentials

Empirical potentials for covalent materials have been much less successful than for metals. As with the NFE pair potentials, the bulk of the energy is contained in the covalent bond, and potentials which well describe distortions from fourfold coordination tend to fail when applied to other bond situations, such as sur­faces, high pressure phases, or liquids.

A commonly used example, the Stillinger-Weber potential,29 is written as

Uj = e V (rj )+EI3

j j k V3 /

where V(rj) and F(rj) are short-ranged pair poten­tials. The form of the three-body term, with its mini­mum at 109°, ensures the stability of a tetrahedrally bonded network. The stacking fault energy (equiva­lently, the difference between cubic and hexagonal diamond) is zero, so the ground state crystal structure is not unique; however, this is not far from correct, and hexagonal diamond can be found in carbon and silicon. Moreover, it has little effect in many simulations since the large kinetic barrier against cubic-hexagonal phase transitions prevents them occurring in simulations. Solidification and recovery from cascade damage are counterexamples.

Stillinger-Weber works well for fourfold — coordinated amorphous networks, and vibrational properties of the diamond structures. It gives too low

density (and coordination) for the liquid and high — pressure phases, because it fails to reproduce the rebon­ding of atoms at the surface to remove dangling bonds.

An alternate approach to stabilizing diamond via the 109° angle is to do so through its tetravalent nature. The simplest type is the restricted bond pair potential30:

и = X A(r/) — X B(r/)

j j=M

where the attractive part of the potential is summed over at most four neighbors (one per electron). This formalism describes well the collapse of the network under pressure or melting, but lacks shear rigidity (only the repulsion of second neighbors provides shear rigidity). There is also some ambiguity over which four neighbors should be chosen, which makes implementation difficult.

An embellishment on this is the bond-charge model, in which the electrons in the bonds repel one another. This adds a three-center term of the form

X C(J)

i

where 1 and k are bonded neighbors ofi. This approach avoids explicitly introducing the tetrahedral angle into the potential. Note that although this term is asso­ciated with atom i (and is often interpreted as a bond­bending term at i), in the simplest form forces derived from this term are independent of i.

The problem of defining ‘bonded neighbors’ can be circumvented, in the spirit of the embedded atom method, by having an embedding function that effec­tively cuts off after the bonding reaches four neigh­bors worth31 as in the Tersoff approach:

U = X A(rJ )-X B(r/)

j j

where

B(riJ ) = f (ri/) X G(ri’b J; rij)

The bond ij is weakened by the presence of other bonds ik and jk involving atoms i and j. The pro­tetrahedral angular dependence is still necessary to stabilize the structure, and further embellishment by Brenner32 corrects for overbinding of radicals.

These potentials give a good description of the liquid and amorphous state, and have become widely used in many applications, in addition to elements such as Si and C, as well as covalently bonded compounds such as silicon carbide33 and tungsten carbide.34