around a dislocation. In 1972, Cahn worked with David W. Hoffman to formulate vector-based thermodynamics to describe the thermodynamics of interfaces, a formulation which is necessary to account for anisotropic materials. This is also known as the capillary vector formulation of interface energies. The mathematics of this treatment involves the concept of norms, although Cahn and Hoffman were unaware of it at the time. In 1975, Cahn worked with his graduate student Sam Allen on phase transitions in iron alloys, including order-disorder transitions. This work led to the Allen–Cahn equation. From 1984, he was an affiliate professor at the University of Washington. Work The spinodal In 1957, Cahn worked with John E. Hilliard to develop the Cahn–Hilliard equation which describes the thermodynamic forces driving phase separation in many systems, and developed the joint theory of spinodal decomposition. Solidification In the theory of crystal growth, Cahn concluded that the distinguishing feature is the ability of the surface to reach an equilibrium state in the presence of a thermodynamic driving force (typically in the form of the degree of undercooling). He also concluded that for every surface or interface in a crystalline medium, there exists a critical driving force, which, if exceeded, will enable the surface or interface to advance normal to itself, and, if not exceeded, will require the lateral growth mechanism. Thus, for sufficiently large driving forces, the interface can move uniformly without the benefit of either a heterogeneous nucleation or screw dislocation mechanism. What constitutes a sufficiently large driving force depends upon the diffuseness of the interface, so that for extremely diffuse interfaces, this critical driving force will be so small that any measurable driving force will exceed it. Alternatively, for sharp interfaces, the critical driving force will be very large, and most growth will occur by the lateral step mechanism. Droplets and surfaces In 1977, Cahn published a simple mathematical treatment of the thermodynamics of wetting: the interaction between a liquid in contact with a solid surface. This paper laid out a simple formulation for describing the wetting transition—the point at which a liquid changes from forming a droplet on a surface to spreading out evenly as a liquid film over the surface. This theory had wide-ranging implications for many materials processing techniques. Quasicrystals In 1982, Dan Shechtman observed a new crystalline structure with puzzling features. Cahn contributed to the theory of how such a structure could be