Materials science is broadly concerned with the nature, properties, and use of materials. One materials science area of great significance is solid-state physics, which focuses on the physical properties of solid materials. It addresses such concerns as, for example, the properties that result from the distribution of electrons in metals, semiconductors, and insulators; oscillations in crystals; energy bounds; magnetic phenomena; dielectrics; ferromagnetics; and dislocations. Another materials science area of great importance is that of polymers, for which an additional concern about liquid flow arises. Whatever the context, be it solid, liquid, or some transitionary setting, materials science seeks an understanding of a material's macromolecular structure and properties by drawing on knowledge of its atomic and molecular constituents.

Until recently, the term ''materials science'' was used primarily to denote empirical study, fundamental research, synthesis, and production in metallurgy and ceramics. Today, the term is used in a broader context that involves interdisciplinary interactions among scientists, engineers, mathematicians, physicists, chemists, and biologists who are concerned with one or more of the four basic elements of modern materials science: properties, structure and composition, synthesis and processing, and performance (National Research Council, 1989). Traditional boundaries between disciplines sometimes needlessly constrain the development of unorthodox ideas and new theories. The National Science Foundation recognized this in establishing a number of interdisciplinary science and technology centers, many of which include several institutions and often several disciplines (National Science Foundation, 1993, 1992). In particular, materials science is today a vast and growing body of knowledge that is based on the physical sciences, engineering, and mathematics but is not obliged to conform to their limits. Interdisciplinary studies encompass all classes of materials, including biomaterials and biomolecular materials, ceramics, composites, electronic materials, magnetic materials, metals, optical and photonic materials, polymers, and superconducting materials (Federal Coordinating Council for Science, Engineering and Technology, 1992). Physics, chemistry, mechanics, and other traditional disciplines are now viewed as an arsenal of complementary scientific approaches serving the common goal of increased knowledge about and understanding of all aspects of materials, from discovery and synthesis to products and uses.

The mathematical sciences, for comparison, originally were developed as an integral part of the physical sciences. Often the same individuals contributed to both areas of research. With the exponential growth of all these areas has come increased specialization such that, in particular, the mathematical sciences and materials sciences are separate disciplines with too little contact between them.

The basic objectives of materials science are the synthesis and manufacture of new materials, the modification of materials, and the understanding and prediction of materials properties and their evolution over time. In each of these areas, the disciplines mentioned above play a role. The mathematical sciences, as a common language for the quantitative description of processes and phenomena, have their own unique role. They provide a

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Mathematical Research in Materials Science: Opportunities and Perspectives
1
SUMMARY AND OVERVIEW
Materials science is broadly concerned with the nature, properties, and use of materials. One materials science area of great significance is solid-state physics, which focuses on the physical properties of solid materials. It addresses such concerns as, for example, the properties that result from the distribution of electrons in metals, semiconductors, and insulators; oscillations in crystals; energy bounds; magnetic phenomena; dielectrics; ferromagnetics; and dislocations. Another materials science area of great importance is that of polymers, for which an additional concern about liquid flow arises. Whatever the context, be it solid, liquid, or some transitionary setting, materials science seeks an understanding of a material's macromolecular structure and properties by drawing on knowledge of its atomic and molecular constituents.
Until recently, the term ''materials science'' was used primarily to denote empirical study, fundamental research, synthesis, and production in metallurgy and ceramics. Today, the term is used in a broader context that involves interdisciplinary interactions among scientists, engineers, mathematicians, physicists, chemists, and biologists who are concerned with one or more of the four basic elements of modern materials science: properties, structure and composition, synthesis and processing, and performance (National Research Council, 1989). Traditional boundaries between disciplines sometimes needlessly constrain the development of unorthodox ideas and new theories. The National Science Foundation recognized this in establishing a number of interdisciplinary science and technology centers, many of which include several institutions and often several disciplines (National Science Foundation, 1993, 1992). In particular, materials science is today a vast and growing body of knowledge that is based on the physical sciences, engineering, and mathematics but is not obliged to conform to their limits. Interdisciplinary studies encompass all classes of materials, including biomaterials and biomolecular materials, ceramics, composites, electronic materials, magnetic materials, metals, optical and photonic materials, polymers, and superconducting materials (Federal Coordinating Council for Science, Engineering and Technology, 1992). Physics, chemistry, mechanics, and other traditional disciplines are now viewed as an arsenal of complementary scientific approaches serving the common goal of increased knowledge about and understanding of all aspects of materials, from discovery and synthesis to products and uses.
The mathematical sciences, for comparison, originally were developed as an integral part of the physical sciences. Often the same individuals contributed to both areas of research. With the exponential growth of all these areas has come increased specialization such that, in particular, the mathematical sciences and materials sciences are separate disciplines with too little contact between them.
The basic objectives of materials science are the synthesis and manufacture of new materials, the modification of materials, and the understanding and prediction of materials properties and their evolution over time. In each of these areas, the disciplines mentioned above play a role. The mathematical sciences, as a common language for the quantitative description of processes and phenomena, have their own unique role. They provide a

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Mathematical Research in Materials Science: Opportunities and Perspectives
unifying force, reveal underlying structure, offer an avenue for knowledge transfer among disciplines, and serve as the vehicle for computational modeling of the processes and phenomena. Both the mathematical and materials sciences have much to gain from each other. Modern mathematical methods can aid in solving significant problems in materials science, while problems in materials science can suggest fruitful areas for mathematical research. In the larger perspective, however (National Research Council, 1989), it is clear that the scientific vigor, technological strength, and economic health of the nation all argue in favor of universities, government, industry, and professional societies stimulating and facilitating new collaborations between mathematical scientists and materials scientists.
The committee's task was to prepare a broad survey that (1) identifies and describes areas where the mathematical sciences have significantly aided materials research, (2) identifies areas of mathematical research in which increased progress would accelerate materials research, (3) identifies obstacles, if any, to increased collaborative research, and (4) makes recommendations for facilitating this type of cross-disciplinary work, including how to attract students and young researchers to this area. Chapters 2 through 8 of this report address items (1) and (2). This chapter and Chapter 9 address item (3), and Chapter 9 addresses item (4). This report is written for both mathematical and materials science researchers with an interest in advancing research at this interface, for federal and state agency representatives interested in encouraging such collaborations, and for any persons wanting information on how such cross-disciplinary, collaborative efforts can be successfully accomplished.
Concerning obstacles, it should first be noted that despite existing impediments to interdisciplinary work in these areas, there are nonetheless many successful interactions and collaborations between materials scientists and mathematical scientists (some are referred to in the technical Chapters 2 through 8 that follow). However, those impediments do need to be recognized and addressed.
One obstacle to increased collaborative research between the mathematical sciences and materials science concerns the differences in education of researchers in the two disciplines. The education of materials scientists exposes them to varying amounts of mathematics, but it mainly involves classical mathematics and therefore little knowledge of modern mathematics, especially tools that might be beneficial for exploring problems in materials science. In their education, mathematical scientists rarely take physical science courses beyond the elementary undergraduate classical courses, and therefore they generally have little feeling for current research areas and the applications of their expertise to materials sciences. Another impediment is a large jargon barrier that exists between the two disciplines. Similar jargon barriers exist even among different subdisciplines of materials science (and subdisciplines of the mathematical sciences); continual efforts are needed to eliminate jargon as a barrier to interdisciplinary research. Some negative attitudes constitute another barrier and are perhaps best summarized by the comments of, on the one hand, the great mathematician G. H. Hardy, who expressed his pleasure that none of his work had practical applications and, on the other hand, the materials scientist who cautions his students that "too much rigor leads to rigor mortis." University departmental structures that often discourage mathematical research by materials scientists and materials-oriented research by mathematical scientists present another obstacle, as this (and other) cross-

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Mathematical Research in Materials Science: Opportunities and Perspectives
disciplinary research is generally not rewarded in considerations of tenure, promotion, and salary. A minimal level of cross-discipline education couples with the large jargon barrier, some negative attitudes, and a lack of motivating rewards to inhibit cross-fertilization between the mathematical and materials sciences. Chapter 9 presents the committee's conclusions regarding the main obstacles to increased collaboration between the materials science and mathematical sciences communities. It then addresses item (4) with the committee's recommendations to universities, government, industry, and professional societies on how to enhance and further increase collaborative efforts between the two communities.
In making these recommendations, the committee was aware of serious difficulties that exist to developing cross-disciplinary work, difficulties that are not unique to materials science. Progress in cross-disciplinary research, as well as in research within a discipline, can come either incrementally or following breakthroughs. Breakthroughs are often the result of progress on prototype problems, examples being the Ising model of a ferromagnet (the inspiration for the renormalization group work cited in each of Chapters 3 through 6 and 8) or Edward Lorenz's chaos sequence (xn+1 = axn − xn2) that Mitchell Feigenbaum studied for his breakthrough on chaos (Lorenz, 1963, 1979, 1984; Feigenbaum, 1978, 1981). These prototype problems provide an excellent vehicle for cross-disciplinary communication and by their elegance and ability to challenge can attract excellent researchers. Areas where such breakthroughs could take place might include, for example, the long time scale behavior of protein folding or the transitions from microscale to mesoscale and from mesoscale to macroscale. Specific candidates for such problems probably already exist, but this committee is not aware of prototypes that have been truly simplified to an irreducible essence.
Therefore, the committee's recommendations focus on enabling incremental advances in cross-disciplinary research that involves mathematical sciences applied to materials science. Some of the main obstacles to cross-disciplinary efforts, such as the reward system, have been and are being addressed in a number of other reports (for example, Joint Policy Board for Mathematics, 1994; National Research Council, 1993, 1991c, 1990, 1989; Boyer, 1990; Sigma Xi, 1988; and Institute for Mathematical Statistics, 1988). The committee attempted to go beyond those general difficulties and offer specific suggestions in its recommendations.
The mathematical challenges in materials science vary with length scale, time scale, and temperature regime (Pantelides, 1992; Baskes et al., 1992). For example, quantum mechanics is the discipline that governs processes and phenomena at the most microscopic level where electronic effects are important. Continuum mechanics and thermodynamics govern macroscopic deformations. Statistical mechanics is one area that connects the microscopic with the macroscopic regime. The basic laws of these disciplines have very different mathematical structures and pose distinct challenges. In each case, the classical mathematical areas such as geometry, differential and integral calculus, and statistics play distinct and mutually reinforcing roles. These different ways of viewing materials science, as well as the breadth and the scope of the modern field, can lead to different ways of breaking up the subject into categories so as to address items (1) and (2). For example, although there are not, say, ten "outstanding problems" in materials science for mathematical scientists, one can identify ten areas with great promise for mathematical applications to

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Mathematical Research in Materials Science: Opportunities and Perspectives
materials science: block copolymers, dynamic fracture, effective moduli of composites, grain boundaries, grain growth, martensite and shape-memory materials, mushy regions, processing of semiconductor chips, stokesian dynamics for complex fluids, and superconductivity. As a basic framework for description, the committee's choice of general categories of focus or themes became the titles of this report's Chapters 2 through 8. This organization of subjects by themes is in contrast to the committee's phase-one survey (National Research Council, 1991a), which was organized around various classes or applications of materials (such as ceramics, electronic and semiconductor materials, polymers, and so on). Extensive cross-referencing has been provided between subjects or chapters when the same or a related topic is discussed elsewhere in this report. Subjects within chapters were chosen for illustration; the lists of subjects are not meant to be comprehensive. Further, the descriptions of those subjects and associated mathematical research opportunities present only a part of the much more wide-ranging totality (for example, cf. Langer, 1992; National Research Council, 1991c, 1989; Psaras and Langford, 1987). In the same way, the references given are intended to help the reader search further into the literature, with no attempt made to be complete. What appears in this report reflects the committee members' expertise and knowledge, that of the cross-section of individuals (see appendix) who were kind enough to provide information to the committee, and the project limits on time and funding.
The committee hopes that this report will help encourage research in the mathematical sciences that complements vital research in materials science, will generally raise awareness of the value of quantitative methods in materials science, and will spur researchers to explore the interface between the mathematical sciences and materials science. The committee also wishes to repeat the perspective expressed in its previous short report on the subject (National Research Council, 1991a): cross-disciplinary collaborations require long-term commitments.