Pair Potentials in Atomistic Computer Simulations
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found in calculations using empirical schemes is best guaranteed if they can be related to fitted material properties and are not sensitively dependent on the details of the fittings and functional forms employed.6 In this article I first describe some general aspects of the pair-potential description of atomic interactions, in particular stressing the important difference between potentials that fully determine the total energy of a system and those that only describe changes associated with structural variations. I then proceed to briefly review pair potentials employed in atomistic studies of defects in metallic materials and finally assess the usefulness of pair potentials in present developments. General Features of Pair Potentials There are two types of pair potentials that differ by their contribution to the total energy. Potentials belonging to the first category are those that are assumed to determine fully the total energy of the system studied. Such potentials have usually been constructed in analogy w i t h b i n d i n g c u r v e s of diatomic molecules and adequately describe, for example, the energetics of systems in which van der Waals-type interactions dominate. The pair potentials belonging to the second category describe the energy changes associated with the variation of atomic configurations at constant average density of the material but do not determine the total energy.7 Such potentials are suitable for metallic systems and were derived from first principles for spvalent metals in the framework of weak pseudopotentials. 8 ' 9 However, many short-range empirical potentials of this type have been constructed for both pure metals and alloys.10"12
The total energy of a system composed of N particles is in both cases (1)
where ik is the pair potential, rik is the separation between atoms / and k, and 17 = 0 for the potentials of the first category, while U is the cohesive contribu- j tion to the energy for the potentials of the ] second category and depends in this ! case on the average volume per atom ft or the average density of the material. In the latter case U provides the major contribution to the cohesion. Furthermore, O,V( may also be a function of ft.7~9 The following requirements, which in empirical schemes are usually the object of fitting, must be satisfied: (1) In equilibrium, when the force acting on any particle is zero, the structure must also be stable with respect to the application of a small, homogeneous strain tensor eap. When expanding the energy into a Taylor series with respect to eu/8, the linear term represents the tensor of internal stresses dik(r,k)
dU (2)
where r\ is the a component of the vector r,* (a = 1,2,3). The hydrostatic component of this tensor represents the pressure, and it is the only nonzero component in the case of cubic symmetry. However, in equilibrium all the stress components must vanish, and therefore, crap = 0. The quadratic term in the Taylor series determines the tensor of elastic moduli Cafiys. Employing the condition of zero stresses, C
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