This appendix focuses on some recent theoretical and computational developments that are changing the very nature of materials science and engineering. Two complementary forces are driving these changes. First, there is the unprecedented speed, capacity, and accessibility of computers. Problems in mathematics, data analysis, and communication that seemed untouchable just a few years ago now can be solved quickly and reliably. Second, there is the growing complexity of materials research. The latter change has occurred in large part because we now have instruments with which to make highly detailed and quantitative measurements and we have the computational ability to deal with the resulting wealth of data. Complementing these technology push factors is the pulling force of the technological demand for increasingly complex materials.

Underlying all of these developments are advances in our theoretical understanding of the properties of materials and in our mathematical ability to devise accurate numerical simulations. In short, materials research is evolving into a truly quantitative science.

Analysis and modeling in materials research traditionally has been divided into roughly three different areas of activity—areas that can be characterized by the length scales at which the properties of materials are being considered. The most fundamental models, those used primarily by condensed-matter physicists and quantum chemists, deal with microscopic length scales, where the atomic structure of materials plays an explicit role. At a more phenomenological level, much of the most sophisticated analysis is carried out at

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APPENDIX E
Analysis and Modeling
OVERVIEW
This appendix focuses on some recent theoretical and computational developments that are changing the very nature of materials science and engineering. Two complementary forces are driving these changes. First, there is the unprecedented speed, capacity, and accessibility of computers. Problems in mathematics, data analysis, and communication that seemed untouchable just a few years ago now can be solved quickly and reliably. Second, there is the growing complexity of materials research. The latter change has occurred in large part because we now have instruments with which to make highly detailed and quantitative measurements and we have the computational ability to deal with the resulting wealth of data. Complementing these technology push factors is the pulling force of the technological demand for increasingly complex materials.
Underlying all of these developments are advances in our theoretical understanding of the properties of materials and in our mathematical ability to devise accurate numerical simulations. In short, materials research is evolving into a truly quantitative science.
Analysis and modeling in materials research traditionally has been divided into roughly three different areas of activity—areas that can be characterized by the length scales at which the properties of materials are being considered. The most fundamental models, those used primarily by condensed-matter physicists and quantum chemists, deal with microscopic length scales, where the atomic structure of materials plays an explicit role. At a more phenomenological level, much of the most sophisticated analysis is carried out at

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intermediate length scales, where continuum models are appropriate. Examples of topics in the latter category include fracture mechanics and microstructural pattern formation in alloys. Finally, there is work at macroscopic length scales, in which the bulk properties of materials are used as inputs to models of manufacturing processes and performance. Historically, research in each of these three areas has been carried out by separate communities of scientists—applied mathematicians, physicists, chemists, metallurgists, ceramists, mechanical engineers, manufacturing engineers, and so on. One of the committee’s principal theses is that the distinction between these areas of research is properly being blurred by modern developments.
The principal recommendations of the committee are the following:
Analysis, modeling, and computation should play a significant role in both the educational and the research components of academic programs in materials science and engineering. Renewed attention should be paid to mathematical analysis (as distinct from—and in addition to—computer programming) in educational curricula.
New support is needed both to make high-level computational facilities available for materials research and to develop validated data bases, algorithms, and numerical simulations.
Special attention should be paid to the need for accurate models of nonequilibrium phenomena, particularly processes relevant to manufacturability and performance of materials. Work toward the development of integrated approaches, combining science-based simulations with optimization of features regarding quality and cost, should be strongly encouraged.
PROPERTIES OF MATERIALS AT MICROSCOPIC LENGTH SCALES
Particularly interesting developments of the past few years are apparently feasible schemes for carrying out “first-principles” computations of complex atomic arrangements in materials starting with nothing more than the identities of the atoms and the rules of quantum mechanics. To put these developments in perspective, it will be best to mention some more conventional—and still very productive—approaches to atomistic modeling of materials before turning to this remarkably ambitious new point of view.
Statistical Mechanics
One conventional picture of how huge numbers of atoms collectively determine the bulk properties of materials is the classical statistical mechanics of Boltzmann, Gibbs, Einstein, and others, dating back to the turn of the century. Quantum theory plays only a subsidiary role in this picture. In

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principle, it is needed to tell us how the atoms interact with one another, but, in practice, these interactions often are replaced by phenomenologically determined pairwise forces between the atoms. Perhaps the most visible success of this classical approach has been the understanding of phase transitions, for example, the crystalline ordering of solids, the compositional ordering of alloys, or the magnetization of ferromagnets. For the most part, however, these are only “in principle” successes, because the details for most cases have yet to be worked out. A promising start has been made in the computation of phase diagrams for multicomponent metallic alloys. (Here the quantum theory of metallic binding does turn out to be of practical importance, but the basic statistical methods remain relevant.)
Research in the area of phase transitions was placed on a much firmer footing in the mid-1970s, when the classic problem of critical phenomena was solved by means of what has come to be known as the “renormalization group” method. With the new understanding that has emerged from this theoretical development, we are now able to classify a wide variety of different kinds of phase transitions and to determine what analytic or numerical approaches might be appropriate for carrying out accurate calculations in various cases of practical interest.
Systems Far from Equilibrium
The remarks in the preceding paragraphs pertain only to what are known as “equilibrium states of matter,” that is, to states of matter that have been allowed to relax all the way to thermal and mechanical equilibrium with their surroundings. Many of the most fundamentally challenging and technologically important problems, however, have to do with systems that are far from equilibrium.
For example, the distribution of the chemical constituents in metallic alloys or multicomponent ceramic or polymeric materials is usually very far from what it would be if those materials were allowed to relax by annealing or indefinitely long aging. Thus the problems of predicting relaxation rates and the structural changes that materials undergo on their way toward equilibrium are of great practical importance if one is interested in either the manufacturability or the performance of such materials. Another example occurs during the processing of materials. In processing, substances are almost always driven away from their states of equilibrium. In casting, welding, extrusion, melt spinning, and so on, complex patterns are being made to emerge from relatively simple structures, which means that the natural trend toward equilibrium is actually being reversed. Clearly, the standard techniques of equilibrium statistical mechanics are not adequate in such cases.
Much progress has been made during the past decade or so in the theory of nonequilibrium processes and in techniques for modeling such processes

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numerically. But many deep problems remain unsolved. A prime illustration of the unsettled state of this field is that we do not yet have a satisfactory way of characterizing the intrinsically nonequilibrium, amorphous—i.e., glassy—states of matter.
Quantum Mechanical Calculations
The modern quantum theory of the structure of materials has its origins in the calculation of the cohesive energy of metals by E.Wigner and F.Seitz in the 1930s. With the advent of large computers during the past two decades, such calculations have achieved quantitative predictive capabilities when applied to regular (or very nearly regular) crystalline solids. Recent developments open the possibility of similar accuracy in describing irregular configurations such as crystalline deformations near defects, surfaces, or grain boundaries. It is even possible that the new methods will allow studies of metastable or strongly disordered states of matter.
In order to predict the structure of a solid, in principle, it is first necessary to calculate the total energy of the underlying many-body system of interacting electrons and nuclei for an arbitrary configuration of these constituents, and then to find the specific configuration that minimizes this energy. A typical computation of the kind that has been tested carefully during the past decade might proceed by, first, fixing the positions of the ion cores, then using what are known as self-consistent “density-functional” and “pseudopotential” methods to find the electronic ground-state energy in this configuration, including ion-ion interactions to compute the total energy, and finally comparing this energy with that of other configurations in order to determine the equilibrium state of the system as a whole. In recent applications of this technique, total energy differences between alternative crystalline structures have been obtained accurately to within a few tenths of an electron volt per atom, structural parameters to within tenths of angstroms, and bulk moduli and phonon frequencies to within a few percent. Note, however, that the method described above pertains only to zero-temperature ground states of regular crystalline arrays of atoms and not to irregular configurations or to alternative phases that might occur at higher temperatures.
Very recently, new methods for performing ab initio total energy calculations have been suggested that provide a novel way of carrying out the above procedure and that also can deal with arbitrary configurations of fairly large arrays of atoms—50 to 100 atoms in a supercell geometry using currently available computers. The basic idea is to minimize the total energy of the system by allowing both the electronic and the ionic degrees of freedom to relax toward equilibrium simultaneously. The useful computational scheme is known as “simulated annealing,” a recent development in mathematical optimization theory that has been borrowed

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from statistical mechanics. (Note the profitable two-way interaction between materials science and other areas of research.) In effect, the system is made to find its minimum energy accurately and efficiently by cooling from a state of high temperature.
If the new ab initio methods can, in fact, be made to work with the anticipated precision, a wide range of materials problems will become open for quantitative investigation. The following examples should provide some sense of the scope of these opportunities.
In the area of surface and interface science, there is interest in predicting the electronic and geometric structure of clean surfaces, grain boundaries, and adsorbed and chemisorbed surfaces. Questions include: How and why does a surface or grain boundary reconstruct? What is the nature of the atomic relaxations? Where might chemisorbed atoms and molecules be attached on a reconstructed surface? What are their binding energies? What are the effects of steps, defects, and impurities? Quantitative answers to such questions now appear to be obtainable, and some relevant investigations have been started. For example, it recently has turned out to be possible to perform an ab initio calculation of the atomic positions at a twist grain boundary in germanium. In comparison to what would have happened if the atoms had been kept frozen in their bulk crystalline positions, the relaxed configuration determined computationally exhibits substantial distortions and the formation of many new covalent bonds.
A closely related area involves interfaces between chemically different substances and the manufacture of artificially structured materials. Here the interest lies in constructing heterojunctions and superlattices of semiconductors and metals in order to obtain special electronic properties. Overlayers and material sandwiches can lead to new structural phases, new electronic states, and new magnetic states. Quantitative predictions concerning interfacial bonding and electronic and magnetic structure of ideal interfaces would be extremely valuable. Present modeling limitations seem primarily to be associated with the long-range stresses and the misfit dislocations that are generated by mismatch of lattice constants across interfaces. Here one can see the need for combining atomistic calculations—the interface structure—with continuum theories—elasticity, plastic deformation, and so on—in a way that may well characterize much of the future work in this area.
Finally, the committee suggests that there is a major opportunity for the application of ab initio methods in the modeling of nonequilibrium processes. For example, in epitaxial growth using molecular beam or chemical vapor deposition techniques, it might be very useful to model the kinetics of atoms interacting with a substrate during growth. One could imagine that the new “annealing” algorithms would be suitable for modeling a wide variety of such growth processes at the surfaces of materials.

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CONTINUUM MODELS OF THE PROPERTIES OF MATERIALS
In this section attention is focused on problems in analysis and modeling in which the relevant length scales are on the order of microns or more, that is, much larger than the distances between neighboring atoms. When modeling the behavior of materials at such large length scales, one generally does not need to keep track of the positions of the individual atoms. Rather, it suffices to deal with local average properties—e.g., density, temperature, strain, and magnetization—and to describe the behavior of these quantities by continuum equations in which it is assumed that all variations are extremely slow when viewed on atomic length scales. Thus one uses diffusion equations to describe the transport of heat, composition, or chemical reagents; hydrodynamic equations to describe the motion of fluids; and elasticity theory to relate strains in solids to applied stresses. Of course, many of the ingredients of such models—the transport coefficients, for example—ultimately are determined by fundamental, atomistic principles. But the basic point of view is macroscopic in the sense that it pertains to length scales that are much larger than atomic, and classical in the sense that it makes no explicit use of quantum mechanics.
The advent of the computer has brought about an important change in the perspective from which scientists view continuum analysis. Because continuum models in principle derive from atomistic theories, they often have been viewed as less fundamental, less of a venture into uncharted territory, less apt to produce surprises. It now seems that just the opposite may be true, at least as regards many of the questions that are most relevant to materials research. Now that we actually can explore the consequences of the continuum models—whose ingredients have been known and trusted for decades or longer—we are discovering a wealth of unexpected phenomena and challenging mathematical problems.
In order to describe the implications of some of these recent developments, two broad classes of continuum problems that are part of the traditional core of materials research—microstructural solidification patterns in alloys and fracture mechanics—are discussed below. These are by no means the only areas of materials research where analysis and modeling at continuum length scales are appropriate. Note, for example, the wide variety of materials processing problems in which hydrodynamics is important, or the yet more complicated problems in which fluid motion is coupled to diffusion and chemical reactions. Some of these more complex modeling problems will be referred to toward the end of this section in the discussion of integrated approaches to materials technology.
Microstructures in Alloys
When a molten alloy is solidified by quenching, its chemical constituents tend to segregate. This happens even in situations where equilibrium ther-

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modynamics predicts stable, homogeneous solid solutions. Segregation occurs because, during the solidification process, the liquid and solid phases fall out of equilibrium with each other, and the chemical constituents are rearranged by being driven across the moving solidification front. Thus the last bits of liquid to solidify may be compositionally very different from those that solidified first. The result is often an intricate pattern of cellular or dendritic (treelike) structures, on the scale of tens or hundreds of microns, that are easily observed through an optical microscope.
Control of these microstructural patterns has long been understood to be essential in materials technology. The processes by which the microstructures form are important in determining the grain structure of the solidified material. The microstructures themselves, within each grain, affect the mechanical properties of the material; for example, they may pin or impede the motion of dislocations. They also determine the way the material will behave under further processing such as heating, deformation, aging, or exposure to corrosive environments. The strength of a weld depends on the microstructure in the resolidified material and on microstructural changes in the heat-affected zone. The suitability of a semiconductor crystal for use in electronic devices depends on careful suppression of microstructural solute segregation. Other examples of the technological importance of microstructural properties appear frequently, either explicitly or implicitly, throughout this report.
To the aspiring modeler, the basic ingredients of the microstructural problem may seem pedestrian. Generally, one is being asked simply to solve well-understood diffusion equations subject to apparently simple boundary conditions. The trouble is that the boundaries are moving; in fact, the essence of the problem is to compute their motion. To make matters yet more interesting, microstructural patterns are caused by morphological instabilities of these boundaries. Initially smooth shapes naturally develop grooves and fingers, the fingers split or develop side branches, the side branches split or develop tertiary side branches, and so on. What has been discovered only very recently is that the patterns generated by this process are controlled by an extremely delicate interplay between a basic diffusional instability and a number of ostensibly much weaker effects, most notably surface tension, but also crystalline anisotropy, interfacial attachment kinetics, and even ambient noise. In more technical terms, the instabilities produce intrinsically “nonlinear” behavior, and the weak, controlling effects are “singular perturbations.”
The recent developments in this area have been brought about by an interactive combination of mathematical analysis, numerical computation, and careful experimentation. At the time that this report is being written, there is a sense among workers in the field that they may be narrowing in on a new understanding of fundamental principles—for example, that it may finally be possible to compute the growth rates and geometries of dendritic

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or cellular solidification patterns. But the situation is not yet clear; in fact, the recent history of the field makes it seem likely that further surprises are in store.
Let us assume, however, that the fundamental principles of solidification theory are about to come under control. What happens next? Historically, it has not been easy for advanced concepts in solidification theory to have much impact on applied materials technology, presumably because neither the principles nor the computers have been powerful enough for such work to have quantitative predictive capability. The phenomenologist has managed generally to stay a step or two ahead of the theorist in this field. If this situation is indeed changing, the next step will be to make the new methods of analysis and modeling accessible to process engineers, welding specialists, and the like. Note how difficult this step is going to be. Our aspiring modeler may start out as a computational physicist or a theoretical metallurgist, but he or she will have to master some very subtle mathematical concepts in order to write sensible computer codes, and then will have to learn the language of technology in order to translate numerical results into useful rules of procedure for the processing of materials.
Fracture Mechanics
Research in fracture mechanics is described in Appendix C. The reader should refer to that discussion for a broad view of the importance of this field and a summary of outstanding problems. Fracture mechanics is revisited briefly here in order to focus on certain aspects that pertain specifically to analysis and modeling.
In comparison with the solidification problems discussed above, analysis and modeling in fracture mechanics appear to be considerably more complicated in their physical ingredients and in the variety of their applications. Even the linear theory of elasticity for crystalline solids is intrinsically more complicated for computational purposes than theories of heat transport or solute diffusion. To describe real cracks in real solids, one needs a theory of nonlinear elasticity supplemented by models of plastic deformation and viscoelastic dissipative mechanisms. One also needs to take into account defects, inclusions, grain boundaries, and the overall shape of the object in which the crack is located. It even seems likely that the solution of important problems in fracture mechanics, such as the dynamics of decohesion at a crack tip, will require a quantum theoretic description of atomic bonding of the kind mentioned earlier in this appendix.
Despite its relative complexity, computational modeling in fracture mechanics is currently much more advanced in its impact on materials technology than the modeling of microstructural solidification patterns. This has happened because, as it turns out, much progress can be made in

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fracture mechanics without directly confronting mathematical problems as difficult as the free boundary problems encountered in solidification. One of the earliest applications of digital computers was the numerical calculation of elastic stresses in irregularly shaped objects subject to forces applied at their (fixed) boundaries. Such calculations, combined with estimates of elastic limits and yield stresses, continue to be immensely important in the design of structural components such as pressure vessels and airplane wings. The same techniques, brought down to micromechanical length scales, are used in fracture mechanics to compute the stress intensity at the tip of a crack and, from this, to determine whether the crack will grow. Such calculations and their extensions are now providing useful, quantitative predictions of the performance of materials. But much important work remains to be done.
A look at new opportunities and outstanding problems in fracture mechanics indicates that the next stage in the development of this field may be just as hard as—and perhaps very much harder than—the problems now (apparently) being solved in solidification. Consider a few examples of important unsolved problems: Microcracks form spontaneously in stressed or heterogeneous materials; ordinarily, it is these defects whose coalescence or growth leads to crack propagation and macroscopic failure. We do not yet have a quantitative understanding of the mechanisms by which microcracks form. Much less is known about the dynamics of crack propagation than the statics of crack initiation. We have no quantitative understanding of what controls the extent of plastic deformation near the tip of a moving crack, how fast the crack will move under given external stresses, or what governs the path along which the crack will move in a heterogeneous medium. Note that all of these problems, like solidification theory, inescapably involve nonequilibrium phenomena; irreversible, dissipative processes play essential roles. Moreover, these are all free boundary problems. Their solutions almost certainly will require detailed analyses of the actual shapes of voids, crack tips, zones of plastic deformation, and so on.
The conclusions to be drawn from these remarks about fracture mechanics are essentially the same as those of the preceding paragraphs regarding microstructural solidification patterns. These conclusions are equally relevant to a much broader class of opportunities for analysis and modeling in materials research, including work at both microscopic and macroscopic length scales. In short, there is a real chance that recent theoretical and computational developments will lead to a new level of predictive capability. However, achieving this level is not going to be easy. The mathematical and scientific problems are hard, and the problems of translating new quantitative capabilities into technologically useful information seem particularly challenging. The latter challenge is a starting point for the remarks that follow.

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INTEGRATED APPROACHES TO DESIGN AND MANUFACTURING
To complete this summary of opportunities for analysis and modeling in materials research, ways in which modern computational capabilities might have a direct impact on manufacturing technology are considered. The combination of science-based numerical simulations with new methods for storing, retrieving, and analyzing information ought to make it possible to optimize not just the properties of specific substances but entire processes for turning raw materials into useful objects.
Materials considerations are important throughout the life cycles of most products from design, through manufacturing, to support and maintenance, and finally to disposal or recycling. Significant improvements in quality, reliability, and economy might be realized at all stages of this cycle if quantitative models of processing and performance could be used at the beginning.
Consider, as a simplified example, the design of a turbine disk for an aircraft engine. Under ordinary conditions, the designer starts by attempting to achieve given performance specifications in a way that satisfies a few basic constraints, perhaps minimization of weight in the present circumstance. Fabrication engineers are then asked to see whether the part can be produced. If problems exist, or if the cost appears excessive, some design modifications may be required. Finally, maintenance and inspection specialists are consulted, but it is usually too late by this stage in the procedure for their needs to have a major impact on design. This serial approach emphasizes mechanical aspects of the design at the expense of production and maintenance considerations.
A far better approach is to incorporate all or most of these considerations into computer simulations carried out during the initial stages of the design process. To start, one might examine possible processing paths in order to optimize metallurgical microstructures for different properties in different regions of the component. In the example of the turbine disk, the microstructure could be optimized for creep strength at the rim and for low cycle fatigue and ultimate tensile strength in the bore. The same detailed simulations might also address issues of technical feasibility and economics. Models that relate microstructural properties to processing paths might also be used to examine manufacturing options and even to optimize for ease of maintenance.
It is quite likely that such an integrated approach to materials design eventually can lead to small but significant changes in technology that, in turn, will produce large improvements in performance and cost over the lifetimes of products. It is also possible, because complex problems in systems analysis are involved, that the results of these integrated simulations

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occasionally will turn out to be very different from what was expected. When that happens, the technological impact is apt to be very great indeed. The limiting factors in this program are the availability of analytic and numerical models and the availability of the specially trained scientists and engineers who are needed to develop these models and bring them to bear on technology.

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