Crystal Shapes and Phase Equilibria:
A Common Mathematical Basis
J. W. Cahn, and W. C. Carter
MSEL, NIST, Gaithersburg, MD 20899
Submitted to Met. Trans. A, March 1995
for the special issue honoring Hub Aaronson
"Atomistic Mechanisms of Nucleation and Growth in Solids"
Geometrical constructions, such as the
tangent construction on the molar free energy for determining
whether a particular composition of a solution, is stable, are related to
tangent constructions on the orientation-dependent interfacial energy for
determining stable interface
orientations and on the orientation dependence of the crystal growth rate
tests whether a particular orientation appears
on a growing crystal.
Subtle differences in the geometric constructions for the three fields
arise from the choice of a metric (unit of measure). Using results from
studies of extensive
and convex functions we demonstrate that there is a common mathematical
structure for these three disparate topics, and use this to find new uses for
well-known graphical methods for all three topics.
Thus the use of chemical potentials for solution thermodynamics is very
known vector formulations for surface thermodynamics, and the method of
characteristics which tracks the interfaces
of growing crystals; the Gibbs-Duhem equation
is analogous to the Cahn-Hoffman equation.
The Wulff construction for equilibrium crystal shapes can be
modified to construct a ``phase shape'' from solution free energies that is a
potentially useful method of numerical calculations of phase diagrams from