The mathematics of microstructure and the design of new materials

Abstract

The "pathological" energy function E(u) = u^2 for u ≠ 0, E(0) = 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 low-energy 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).

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