Atomistic simulations are playing an increasingly prominent role in materials science. From relatively conventional studies of point and planar defects to large-scale simulations of fracture and machining, atomistic simulations offer a microscopic view of the physics that cannot be obtained from experiment. Predictions resulting from this atomic-level understanding are proving increasingly accurate and useful. Consequently, the field of atomistic simulation is gaining ground as an indispensable partner in materials research, a trend that can only continue. Each year, computers gain roughly a factor of two in speed. With the same effort one can then simulate a system with twice as many atoms or integrate a molecular-dynamics trajectory for twice as long. Perhaps even more important, however, are the theoretical advances occurring in the description of the atomic interactions, the so-called “interatomic potential” function.

The interatomic potential underpins any atomistic simulation. The accuracy of the potential dictates the quality of the simulation results, and its functional complexity determines the amount of computer time required. Recent developments that fit more physics into a compact potential form are increasing the accuracy available per simulation dollar.

This issue of MRS Bulletin offers an introductory survey of interatomic potentials in use today, as well as the types of problems to which they can be applied. This is by no means a comprehensive review. It would be impractical here to attempt to present all the potentials that have been developed in recent years. Rather, this collection of articles focuses on a few important forms of potential spanning the major classes of materials bonding: covalent, metallic, and ionic.

Computer modeling of crystal defects ranges at present from generic empirical investigations to first-principle quantum-mechanical calculations (see, for example, References 1–4). Descriptions of atomic interactions in terms of pair potentials dominated such studies until the early 1980s, and many fundamental features of lattice defects and interfaces were revealed in these calculations. The generic results of these studies withstood the test of time, and calculations employing more sophisticated schemes usually confirmed their validity. An early example goes back to the late 1950s when Vineyard and co-workers pioneered the very first computer simulations in their studies of radiation damage. Empirical pair potentials were used in these investigations in which many fundamental, generic aspects of the effect of irradiation of crystalline materials by energetic particles were discovered.

Such simple treatments of atomic interactions may appear totally inadequate from the point of view of pure physics. However, it must be recognized that the purpose of the majority of atomistic studies of lattice defects has been to elucidate atomic structure and atomic-level properties in materials with given: (a) crystal structure, (b) elastic properties and possibly phonon spectra, (c) values of certain material parameters such as vacancy formation energies and stacking fault energies, and (d) in alloys, alloying and ordering energies, and possibly antiphase boundary energies. This is in contrast with ab initio studies, the objective of which is to determine all these properties from first principles. These goals of atomistic studies are, of course, the same for all semi-empirical approaches discussed in this collection of articles. In general, the validity of the structural features of lattice defects 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 deails of the fittings and functional forms employed.

This article discusses simple but realistic models of the bonding in metallic systems. As is discussed in the introduction, such methods are valuable both to study complex systems that are intractable with more rigorous methods and to study generic properties that do not depend on the fine details of the energetics. In addition, simple models are useful for gaining a physical understanding of a system.

As discussed in the article by Vitek, a great deal of progress in the understanding of defect structures has been gained by the use of constant-volume pair potentials, that is, pair-potential models that include a volume-dependent term. In this picture, the energy is assumed to have two parts, a large density-dependent but structure-independent part and a structure-dependent part that is represented by pair interactions. This view follows from a physical picture where the metal is viewed as a uniform electron gas and the interactions between the atoms are obtained by performing perturbation theory on this reference system. This approach has a serious limitation, though, in that it is restricted to situations where the system is essentially uniform such as when the bulk or defects do not introduce significant changes in the local density. This is true for two reasons. First, there is not a clear prescription for how the structure-independent part of the energy should be treated in an inhomogeneous region. This is a serious problem since a large part of the binding energy is included in this term. Second, the pair-potential term in this picture depends on the overall density. In the vicinity of a inhomogeneity such as a surface, there is no prescription for how the pair-potential term should vary.

In a classification of solids according to their bonding character (into metals, ceramics and glasses, polymers, and semiconductors), the ceramic class includes an enormous range of industrially important materials. From the archetypal ionic solids through oxides to silicates, and to covalently bonded materials such as SiC, they exhibit a rich variety of structures and properties. They occur as structural materials, either on their own or as composites such as SiC/Al2O3. They are important functional materials, such as fast-ion conductors as electrolytes in fuel cells (for example ZrO2/Y2O3 for hydrogen combustion) or batteries (β-alumina in the sodium-sulfur battery), ferroelectric materials such as BaTiO3 and piezoelectrics such as PZT—a solid solution of PbTiO3 and PbZrO3. The high-temperature superconductors (for example, YBa2Cu3O7) are ceramics above the superconducting transition temperature. The products of corrosion and oxidation are ionic materials, and the properties of oxide coatings are vital to the survival of high-temperature alloys in gas turbines or fuel-element claddings in nuclear reactors.

To understand the behavior of ceramic materials, and to optimize their production, processing, and application, it is often necessary to model their behavior at an atomic level. In some cases this is obvious. Ionic diffusion in a solid electrolyte is a self-evidently atomic process. In other cases the need for atomistic simulation is less clear. Oxidation, for example, is a subtle blend of atomic diffusion (often along grain boundaries), metal-ceramic bonding, stress relief, and grain growth. The course of oxidation can be spectacularly affected by impurities and alloying, and this can only be understood by considering the atomicscale processes involved.

Tremendous advances in computing speed, increasingly clever algorithms, and more accurate interatomic-force models have made atomistic simulation a powerful tool in many areas of materials science and engineering. Indeed, from traditional areas such as the study of grain boundaries, fracture, and point defects to somewhat less traditional areas such as nanometer-scale engineering and device fabrication, atomistic simulations are providing exciting new data and insights that cannot be obtained in any other way. Central to the success of an atomistic simulation is the use of an appropriate force model. For simulations requiring large systems and/or long times, the computational efficiency offered by classical potential-energy functions is necessary. However, obtaining quantitative results requires a model that can both accurately describe an appropriate database of physical properties and that is transferable to structures and dynamics beyond those to which it is fit. The latter property is especially critical if an atomistic simulation is to have useful predictive capabilities. While an extensive and well-chosen database from which parameters are determined is important, transferability ultimately depends on the chosen functional form. The definitive mathematical form, however, has yet to be developed.

This article covers my attempts to develop classical potential-energy functions based on an empirical bond-order formalism that put qualitative and semiquantitative features of chemistry into large-scale condensed-phase simulations. To help explain this work, the next section gives a brief tutorial of the “chemistry” of few-body potentialenergy surfaces. Although familiar to chemists, the concepts discussed in this section may not be as familiar to mate rials scientists and engineers. A discussion of the bond-order formalism and how it can be related to both solid-state structure and chemical dynamics follows. The article ends with a discussion of two applications of this formalism, modeling chemical dynamics near a shock front and the reactive chemistry of diamond deposition.

At the heart of any atomistic simulation is a description of the atomic interactions. A whole hierarchy of models of atomic interactions has been developed over the last twenty years or so, ranging from ab initio density-functional techniques, to simple empirical potentials such as the embedded-atom method and Finnis-Sinclair potentials in metals, valence force fields in covalently bonded materials, and the somewhat older shell model in ionic systems. Between the ab initio formulations and empirical potentials lies the tight-binding approximation: It involves the solution of equations that take into account the electronic structure of the system, but at a small fraction of the cost of an ab initio simulation, because those equations contain simplifying approximations and parameters that are usually fitted empirically.

Tight binding may be characterized as the simplest formulation of atomic interactions that incorporates the quantum-mechanical nature of bonding. The particular features that it captures are as follows: (1) the strength of a bond being dependent not only on the interatomic separation but also on the angles it forms with respect to other bonds, which arises fundamentally from the spatially directed characters of p and d atomic orbitals, (2) the filling of bonding (and possibly antibonding) states with electrons, which controls the bond strengths, and (3) changes in the energy distribution of bonding and antibonding states as a result of atomic displacements. These features enable one to obtain considerable improvements in accuracy compared to the simple “glue models” of bonding since use is made of the physics and chemistry of bonding.

Several 40-year anniversaries of the modern area of the synthesis of diamond at high pressures and temperatures (HPHT) have come and gone. If you are a proponent of the Norton Company, this first synthesis would be somewhere between 1948 and 1950. ASEA's Feb. 15, 1953, accomplishment was noted at an international meeting in 1993 on high pressure (AIRAPT) with a small session that included papers by ASEA and General Electric (GE) representatives. The GE success in the Hall experiment of Dec. 16, 1954 (announced in February 1955), was marked only in the personal memories of the remaining members of that diamond team. DeBeers recognizes the ASEA date, but if it celebrates the 40th anniversary of its own in-house accomplishment, it will do so late in 1998; the Russians will do so on or about the year 2000. In any case, the basic, muchcopied GE HPHT process is repeated thousands of times each day at sites all over the world (Figure 1) to make a product that has replaced 90% of natural diamond as an abrasive and has given to technology new superhard products in forms unavailable from the earth (Figure 2). All manufacturers are in fierce competition for a total business that probably is in the range of $1 billion per year, and engineers spend their time primarily in finer tuning for faster growth, greater yields, lower costs, and higher quality crystals. The HPHT process seems to have survived the threat of the low-pressure chemical vapor deposition (CVD) process with respect to abrasive grain and gem-quality stones. However, the low-pressure process has some unique capabilities that obviously cannot be achieved by HPHT, and the product is finding niches in specialized applications (Figures 3 and 4).