Mathematics and Materials

Materials science is not one of the popularly celebrated sciences — in bookstores, there is hardly anything on materials among the evolution, cosmology, and AI books — but ever since our ancestors baked bricks we have relied upon our mastery of the stuff that stuff is made of. And mathematics has been a part of that ever since the Egyptians discovered that if they built their pyramids too steeply they would start to fall down. Now that we are beginning to look at and manipulate materials at the nanoscale, we rely on increasingly sophisticated mathematics to understand what we are looking at and manipulating.

This anthology consists of a selection of snapshots of contemporary materials research, primarily from the point of view of statistical mechanics. In the Introduction, the editors contrast this approach with that of approaching materials science via continuum mechanics, and they write that the articles stress “the interplay between geometry and statistical mechanics” (although the geometry here is largely occasional). The seven presentations, divided into 29 bite-sized lectures, are write-ups from the Park City summer school in July, 2014.

In statistical mechanics, there is a phase space containing all relevant configurations the system could be in, which we could parametrize by the amount of energy in the system (different configurations have different energies). One fundamental consideration is this: given a particular energy, what proportion of the phase space consists of configurations of that energy? From such considerations, we obtain a theory relating temperature, pressure, entropy, and other properties of material systems, and we can generate predictions of how various systems will behave under certain conditions. The original approach was to treat the particles as point particles, but if we treat them as hard spheres, as rods, as ellipsoids, etc., we impose restrictions on the configurations that can appear in the phase space, and obtain more refined (and hopefully more accurate) predictions of the material’s behavior.

Many people, seeing a book on materials, might expect a book on solid materials. Three of the articles seem ultimately to be about solids, although even these solids may not be all that solid. Many (not necessarily mathematical) materials science books start with crystals, and one of the articles addresses a related subject: packing. Henry Cohn addresses the problem of how many hard balls one can squeeze into some space. If the space is the \(n\)-sphere of radius \(1\) and the balls are \((n-1)\)-dimensional and of radius \(r\), one has a coding problem that can be addressed using spherical harmonics; if the space is \(n\)-dimensional Euclidean space, then the question is the maximal density of any packing of \(n\)-dimensional balls, a problem that can also be addressed using harmonics. In both cases, it is striking how different the problem is in different dimensions — and how wide open the open problems are.

The other two articles on (relatively) solid materials are quite different. Roman Kotecký considers deformations of surfaces, which he represents using the “masses with springs” picture, which allows him to use an Ising model, obtaining a mathematical description of a phase transition in free energy. Peter Bella, Arianna Giunti, and Felix Otto, on the other hand, consider materials composed of heterogeneous components, and propose a computational method for such materials in which the various components are somewhat evenly mixed: set the problem up as if it was for a homogeneous solid, and then add a “corrector” term to correct for the error.

Three of the articles concern movable particles, perhaps suspended in a medium. For example, DNA tiles are engineered pieces of DNA which one places in solution in the hopes that they will assemble into a particular target object, such as a robot or gizmo able to carry out (medical) tasks. A liquid crystal, on the other hand, consists of non-spherical (often rod-shaped) molecules able to shift or rotate individually so that it has several possible “phases,” which we could oversimplify as follows. Starting at the solid bottom, it could be in a crystalline phase, in which all molecules are regularly placed and oriented as in a crystal, then a smectic phase, in which molecules are somewhat aligned and somewhat in layers roughly perpendicular to their alignment, then a nemetic phase in which the layering disappears but the (rough) common alignment remains, and finally the somewhat liquid isotropic phase, in which molecules are randomly ordered. By manipulating temperature and other parameters, one can induce a liquid crystal to go from one phase to another, encountering a phase transition en route, and this makes them a useful example for studying phase transitions.

Alpha Lee and Daan Frankel present an overview of phase transitions as “entropy-driven” events. This entails an extended and (compared to the other articles) relatively accessible account of statistical mechanics. While the article starts with hard balls and ends with powders, most of it is concerned with phase transitions in liquid crystals. P. Palffy-Muhoray, M. Pevnyi, E. G. Virga, and X. Zhang focus largely on liquid crystals, and describe how the geometry of the molecules result in attractive and repulsive interactions, all of which results in a complex phase diagram. Moving into particles in a medium, Michael Brenner considers a colloid or solution of particles (such as DNA tiles) and explores the problem of yield: given a solution of building blocks that are supposed to assemble into a target structure and not something else, how much of the building blocks wind up in target structures?

Finally, since this book is essentially an anthology of applications of statistical mechanics, it starts with an introductory article on statistical mechanics by Veit Elser. A 40-page article to bring novices up to speed on a variety of frontier applications of statistical mechanics is a bit of a challenge, so a number of articles start with their own introduction. (Unfortunately, the articles are not particularly integrated with Elser’s introduction.) Lee and Frankel’s may be the most accessible of the lot, and in fact, the novice may prefer to read their introductory “lectures” before reading Elser’s introductory article. Still, the authors tend to assume considerable familiarity with statistical mechanics — often very advanced topics in the subject — and a novice may be advised to review a standard introduction before attempting this book. (It may be useful to have Wikipedia handy while reading some of these articles.)

This book is a collection of snapshots of active work in materials science from the point of view of statistical mechanics, often with geometric considerations. Some articles may be useful for graduate students interested in finding out what’s out there; some may be useful for researchers interested in familiarizing themselves with this part of the frontier. But it is not a textbook and not a handbook; it is more of an advertisement for an active and interesting enterprise.

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Greg McColm is an associate professor of mathematics & statistics at the University of South Florida–Tampa. He worked in mathematical logic but is now active in mathematical crystallography.