Spinodal Decomposition

For those cases in which the molar free energy of mixing has regions of negative curvature (as in the example of the regular solution model), the mechanism by which the system decomposes into its equilibrium phases is different than the mechanism when the curvature is positive. This distinction between mechanisms when the curvature is negative (called the spinodal decomposition mechanism) when the curvature is positive (called the nucleation and growth mechanism) is important for kinetics. It will be useful to discuss the spinodal mechanism in the context of free energy curves.

Consider a part of the free energy curve where the curvature is negative:

Figure: $\frac{\partial^2 \ensuremath{\overline{G^{mix}}}}{\partial X^2_B} < 0$

Suppose that a material is manufactured with a composition $\ensuremath{{X}_\circ}$ that is a function of some spatial coordinate $z$:

Figure 32-6: Composition as a function of a spatial coordinate with a small fluctuation about a fixed value of composition.

Suppose that a very small fluctuation occurs and consider what happens to the free energy for the small fluctuation:

Figure 32-7: The Gibbs free energy construction for a small decomposition fluctuation.

Apparently, the free-energy charge is negative for an arbitrarily small fluctuation in composition such that one part of the system gets more concentrated at the expense of another. The system is inherently unstable and and phase separation will proceed as illustrated:

Figure 32-8: Composition profiles drawn at different times during decomposition.

This process is called spinodal decomposition and it occurs spontaneously when

$$\input{equations/24-7A}$$ (32-17)

Consider the part of the curve where the curvature is positive but inside the miscibility gap (miscibility gap is another way of saying the two-phase region):

Figure: The molar free energy change for the case $(\partial^2 \ensuremath{\overline{G^{mix}}})/(\partial X_B^2) > 0$.

Apparently, the free energy increases. Therefore, the system is ``stable'' with respect to small fluctuations. In other words, it is metastable with respect to infinitesimal composition fluctuations.

Such a system is clearly unstable to the separation into the limiting compositions given by the common tangent construction. How does the system phase separate?

Figure 32-10: Illustrating that a large composition difference is required to nucleate a the stable phases.

Apparently an average composition within the two phase region, but outside of the spinodal curves requires large composition fluctuations to decrease the energy. Therefore, the system phase separates as illustrated in the following cartoon:

Figure 32-11: Illustration of the nucleation of an unstable phase. For nucleation--by contrast to phase separation by spinodal decomposition--the new phase must initiate with a composition that is not near that of the parent phase. Nucleation is a phase transition that is large in degree (composition change) but small in extent (size); whereas spinodal decomposition is small in degree but large in extent.

A process requiring a large composition fluctuation is called ``nucleation.'' After the nucleus forms, the new phase grows. Together, the process is called nucleation and growth.

This kinetic information can be graphically codified into the phase diagram:

Figure 32-12: A phase diagram with a spinodal miscibility gap.