Spinodal Decomposition - Dynamics in K-space

Dynamics in K-space

In the spinodal region of the phase diagram, the free-energy can be lowered by allowing the components to separate, thus increasing the relative concentration of a component material in a particular region of the material. The concentration will continue to increase until the material reaches the stable part of the phase diagram. Very large regions of material will change their concentration slowly due to the amount of material which must be moved. Very small regions will shrink away due to the energy cost in maintaining an interface between two dissimilar component materials.

To initiate a homogeneous quench a control parameter, such as temperature, is abruptly and globally changed. For a binary mixture of -type and -type materials, the Landau free-energy

is a good approximation of the free-energy near the critical point and is often used to study homogeneous quenches. The mixture concentration is the density difference of the mixture components, the control parameters which determine the stability of the mixture are and, and the interfacial energy cost is determined by .

Diffusive motion often dominates at the length-scale of spinodal decomposition. The equation of motion for a diffusive system is

where is the diffusive mobility, is some random noise such that, and the chemical potential is derived from the Landau free-energy:

We see that if, small fluctuations around have a negative effective diffusive mobility and will grow rather than shrink. To understand the growth dynamics, we disregard the fluctuating currents due to, linearize the equation of motion around and perform a Fourier transform into -space. This leads to

which has an exponential growth solution:

Since the growth rate is exponential, the fastest growing angular wavenumber

will quickly dominate the morphology. We now see that spinodal decomposition results in domains of the characteristic length scale called the spinodal length:

The growth rate of the fastest growing angular wave number is

where is known as the spinodal time.

The spinodal length and spinodal time can be used to nondimensionalize the equation of motion, resulting in universal scaling for spinodal decomposition.