Spinodal Decomposition - Diffusion Equation

Diffusion Equation

The mathematical theory of spinodal decomposition is based largely on the development of a generalized diffusion equation. A diffusion equation relates a spontaneous flux of material to a gradient in composition. Fundamental thermodynamic principles dictate that in order for the flux to be spontaneous, it must be associated with a net decrease in the free energy of the system. Consider the following diffusion equation relating the flux of two species ( J_A and J_B ) to the gradient of the chemical potential difference:

As pointed out by Cahn, this equation can be considered as a phenomenological definition of the mobility M, which must by definition be positive. It consists of the ratio of the flux to the local gradient in chemical potential.

The quantity ( μ_A - μ_B ) is the change in free energy when we reversibly add a unit amount of A atoms ( ΔF = + μ_A ) and simultaneously remove an equal number of B atoms ( ΔF = - μ_B ). This term may include factors such as composition, compositional gradients, stresses, and magnetic fields. For a homogeneous system:

The quantity f is the free energy of that number of lattice points in the crystal which initially occupied a unit volume. Substituting,

and defining the interdiffusion coefficient D by:

We can then define the interdiffusion coefficient D as follows:

Note that since M must always be positive, D takes its sign from the sign of f", which is negative within the spinodal. This has often been referred to as "uphill diffusion".

The above derivation of the diffusion coefficient is valid for concentration gradients that are so small that, for all practical purposes, each atom finds itself in surroundings which are similar to that which it would have in a homogeneous material of identical composition. If, however, concentration gradients are so large that within the range of interaction of an atom the average concentration has changed appreciably, then the atom will be aware of its inhomogeneous environment. This leads to a change in its chemical potential, and for fluids:

Substitution yields:

By taking the divergence, we obtain the new diffusion equation:

Alternatively, since:

the flux equation can be written as:

For a system in equilibrium, the chemical potentials, and hence their difference, are constant throughout the system. Thus this equation for the flux satisfies the physical requirement that the net flux should go to zero as equilibrium is approached. For the time dependence of the composition we obtain on differentiation:

Comparing this equation with the usual statement of Fick's second law

it is seen that the mobility is related to the interdiffusion coefficient by the following:

It then follows from the solution to be described next that a particular solution to this new diffusion equation is given by:

in which c_o is the average composition and A(β,t) is the amplitude of the Fourier component of wavenumber β at time t. In terms of the initial amplitude at time zero:

where R(β) is an amplification factor given by: