Last time: Phase Transformations: Overview
Metastability, instability, and mechanisms
First-order and second-order transitions
Free energy functions; conserved and nonconserved variables
Today: Spinodal decomposition--I.
Background
Spinodal decomposition--I.
Diffusion within the spinodal (see notes from Lecture 27)
Free energy of an inhomogeneous system
Cahn and Hilliard 1958 laid important groundwork for the theory of spinodal
decomposition by working out a theory for the "gradient-energy" contribution
to the free energy of a solution that has composition gradients. The
usual fre energy vs. composition curves that we draw plot the free
energy per unit volume 
for a homogeneous solution in
which the composition is uniform. When there are compositional
inhomogeneities, there is an extra contribution to the total free
energy that is proportional to the square of the gradient of
composition.
This is readily extended to develop an expression for the free energy of equilibrium, diffuse interfaces between coexisiting phases.
An improved diffusion equation
The diffusion potential for classical diffusion (in which the
gradient-energy contribution to 
can be ignored) is modified by
gradient-energy effects. This introduces additional terms into the
flux and continuity equations for diffusion (Fick's first and second
laws). It's important to note that the modified diffusion equation is
an improvement over Fick's second law because it includes additional
physicsal phenomena. The modified equation differs from Fick's second
law only when the spatial scale in which the composition variations
take place is very small--in most materials this involves
compositional modulations on the order of 1-50 nm. For larger length
scales, the traditional Fick's second law is generally adequate.