Question

At which composition the critical wave length for spinodal decomposition is the smallest? Use regular solution model. How the answer depends on temperature?

Answer 1

Answer:

Spinodal decomposition is an example of a homogenous isostructural transformation that is diffusion driven and occurs for unstable states. The global composition during the transformation is constant, but increasing composition fluctuations lead to formation of domains separated by diffuse interfaces. As shown, any wavelengths larger than the critical wavelength (indicated by an arrow in the figure) that grows in amplitude will decrease the free energy. No nucleus is needed, as shown by the dotted states at zero amplitude​​​​​​​

The initial stage of spinodal decomposition is:

δfM=12fM(x+δx)+12fM(x−δx)−fM(x),

where f^M are the free energies of mixing of the homogeneous solution, and x + δx and x − δx are the compositions of the first and second formed domains. The free energy of mixing of the homogeneous solution is written as:

fM(x)=αA−Bx(1−x)+RT[xlnx+(1−x)ln(1−x)],

The variation of the free energy of mixing obtained by using the Taylor's series expansion is:

δfM=12d2fM(x)dx2(δx)2

The regular solution losses stability if the condition:

δfM=0

is fulfilled, since, if δf^M < 0, then the negligibly small phase separation perturbation decreases the free energy. Accordingly, the boundary of the spinodal decomposition range of a regular solution is obtained from the equation:

d2fMdx2=−2αA−B+RTx(1−x)=0...

The critical wavelength does not only vary with composition, but also with temperature. The critical wavelength decreases with temperature, resulting in a maximum driving force for decomposition at the bottom of the miscibility gap, for the composition at the peak of the free energy.