The mathematics of microstructure and the design of new materials.

The ‘‘pathological’’ energy function E(u) 5 u2 for u Þ 0, E(0) 5 1, has no minimizer. As u decreases to 0, the energy also decreases, but there is no way to achieve the value 0. Although examples like this might seem to be unimaginably far from scientific thought, they are at the heart of a new approach (1) to understand the complex microstructure and macroscopic response of materials that undergo phase transformations. The free energy of such materials typically has no minimizer, and the observed microstructures (complex, fine-scale patterns of domains of different atomic lattice structure as shown below in a micrograph of CuAlNi by C. Chu and R.D.J.; Fig. 1) have their origin in the material’s ultimately futile attempt to find the minimum energy state (2). The lack of a ground state prohibits prediction of the macroscopic response from microscopic data via the standard procedure: determine the free energy, find the minimizing state, and evaluate its macroscopic properties. Emerging mathematical methods, linked to profound work in the 1940s by L. C. Young and recently surveyed in (3), nevertheless deliver well defined macroscopic quantities, obtained via averaging over all lowenergy states. One area where predictions obtained in this new way have played a role is the recent synthetization of a new magnetostrictive material (4, 5) whose magnetostrictive strain is 50 times larger than that of giant magnetostrictive materials (formerly those with the largest strain). Energy Functions and Energy Wells. The materials on which the new coarse-graining methods have been brought to bear are alloys exhibiting a martensitic transformation. In this transformation, below a critical temperature, the unit cell of the crystal undergoes a bifurcation into different, lower symmetry unit cells. (Not just thousands of alloys but also ceramics and proteins undergo this transformation.) The relevant microscopic parameters (transition temperature, symmetry changes, and lattice parameters) can be regarded as known from atomic measurements and can be subsumed into a cell energy function F(F, u) depending on temperature and on the 3 3 3 matrix F 5 (e1, e2, e3) of lattice vectors of the cell. A macroscopic sample can be described by a continuous vector field y(x) indicating the position of the lattice site formerly at x (martensitic transformations are coherent, i.e., the atomic bonds stay intact). The complicated cooperative effects between the cells, which result in structures like the one shown above, can be explained via minimization of the total free energy (1)