periodic.16 For certain theoretical interaction models, the ground state has been shown to be quasi-periodic. In some of these models there are no equilibrium periodic states at any temperature. Many two-dimensional surface phase diagrams show widespread existence of quasi-periodic phases extending to low temperatures. There is no thermodynamic basis for requiring periodicity.

In 1977 an attempt to use Landau theory to predict crystallization of a liquid yielded the rather surprising conclusion that an icosahedral arrangement of density waves was a likely possibility, but since this was obviously inconsistent with translational periodicity, the result was an enigma.17

RAPID SOLIDIFICATION AND METASTABLE PHASES

Metastable phases abound in the natural world. Natural diamonds form as equilibrium crystals at high pressure, and they survive metastably at ambient pressure because the rate of conversion to the equilibrium state, graphite, is unmeasurably slow. More generally, metastable phases form and survive without ever being stable because the stabler phases form slowly. Materials scientists create nonequilibrium materials through processing cycles designed to capitalize on such kinetic differences, thereby vastly increasing the range of available materials. The heat-treatment cycles of steel discovered 3,000 years ago create carbide particles instead of the equilibrium graphite and thereby give steel its unusual properties. One of the more widely used processing techniques that create metastable materials is rapid solidification, developed by Pol Duwez in the late 1950s.18 Crystals vary by many orders of magnitude in their crystallization rates. Such diverse elements as nickel and phosphorus can crystallize from their melts at rates of 10 m/s. Quartz crystallization from its melt can be 10 orders of magnitude less. A rate-limiting factor in the crystallization of liquid mixtures is diffusion. If no crystal can form with the same composition as the melt, crystal growth rate is limited to about 10 cm/s. Rapid solidification thus favors phases with wide solid solution ranges over intermetallic compounds with narrow composition ranges. When during rapid solidification a liquid is cooled below its melting point, it becomes metastable and can crystallize to metastable solids that are more stable than the liquid. The degree of metastability is small, approximately 0.01 eV/atom, but there are an amazingly large number of different metallic phases with almost the same energy that become accessible. The simplest are the solid, supersaturated solutions, which play an important role in twentieth-century metallurgy.

Solid solubilities change with temperature, and the excess solutes can be made to precipitate from the solid state to give desirable precipitation-hardened alloys. Precipitation hardening, discovered this century,19,20 has been



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