Spinodal mechanism

In phase separating systems, at low temperatures, the Gibbs free energy consists of regions of concave curvature (as shown) and in such regions the binary alloy separated into a mechanical mixture of A and B-rich regions instead of remaining a solid solution. This is because such a phase separation into mechanical mixture reduces the free energy of the system.

Figure 2: The free energy of a system with concave curvature. In the concave curvature region, the system becomes a mechanical mixture of A rich and B rich $\beta$ phases with the given compositions marked in this figure.

[scale=0.6,angle=0]Figures/freeenergy.pdf

Further, in such systems with concavity of free energy, for certain compositions, the mechanism of phase separation changes from the classical nucleation and growth to spinodal decomposition. The change over in the mechanism is related to the curvature of the free energy curve as shown in Fig. 3. In Fig. 4 we show the phase separation region along with the points at every temperature at which the curvature of the free energy versus composition plot changes its sign; the locus of these points is as shown and is known as chemical spinodal.

Figure 3: Positive curvature (nucleation) and negative curvature (spinodal) regions of the free energy versus composition diagram; phase separation mechanism changes from nucleation to spinodal at the point of zero curvature.

[scale=0.6,angle=0]Figures/spinodal.pdf

Figure 4: Phase diagram and chemical spinodal using free energy versus composition diagrams

[scale=0.6,angle=0]Figures/PhaseDiaChemSpinodal.pdf

In the case of spinodal phase separation, any small composition fluctuation grows leading to A-rich regions becoming richer in A and B-rich regions becoming richer in B. This is because such a process leads to a decrease in free energy as shown in Fig. 5.

Figure 5: In regions of negative curvature, A-rich (and B-rich) regions spontaneously become richer in A (and B), because, the free energy decreases in such a process.

[scale=0.6,angle=0]Figures/spinodal2.pdf

Thus, the process of spinodal decomposition is in contrast to the classical diffusion equation scenario which predicts that regions with positive curvature for composition profile grow in time while those with negative curvature decay leading to homogenisation. This difference in shown in Fig. 6.

Figure 6: Evolution of composition during homogenisation and phase separation. While homogenisation obviously follows from the classical diffusion equation (Fick's second law), phase separation implies D $<$ 0.

[scale=0.3,angle=0]Figures/classicaldiffusion.pdf
[scale=0.3,angle=0]Figures/nonclassicaldiffusion.pdf

The reason why classical diffusion equation fails to describe the spinodal decomposition process can also be understood by looking at the free energy reduction. The driving force for any process is the free energy reduction; in the case of mass flow, this driving force manifests itself as equalising chemical potential. As energy flows till temperature becomes equal, and volumes change till pressures become equal, the mass flow takes place till chemical potentials become equatl. Even though in the classical cases such free energy reduction is also accompanied by decay of compositional heterogeneities, in the case of spinodal it is not so. Hence, the Fick's first law should be given in terms of chemical potentials and not composition. The chemical potential driving force is shown pictorially in Fig. 7

Figure 7: Flux is a compositionally heterogeneous system is always driven to even out chemical potential differences. In some cases this also results in the evening out of compositions. But in the case of spinodal shown here, the homogenisation of chemical potential leads to heterogenities in composition.

[scale=0.6,angle=0]Figures/chemicalpotential.pdf

If we modify the Fick's first law as

$$J = -N_V M \nabla \mu$$ (3)

where $N_V$ is the Avogadro number, M is the mobility and $\mu$ is the chemical potential difference given by $\mu = \frac{\partial G}{\partial N_B}$ where $N_B$ is the number of B atoms and G is the free energy, then, one can show that the diffuson equation becomes

$$\frac{\partial x_B}{\partial t} = \left(\frac{M}{N_V} \right) \left[\frac{\partial^2 G}{\partial x_B^2} \right] \nabla^2 x_B$$ (4)

In other words, in addition to the curvature of the composition profile one should also consider the curvature of the free energy with respect to composition. Once both the curvatures are considered, the spinodal behaviour is easy to understand. This is shown schematically in Fig. 8.

Figure 8: Evolution of composition during homogenisation and phase separation. By combining the signs of the curvature of the composition profile and free energy versus composition curve, we can explain both.

[scale=0.3,angle=0]Figures/classicaldiffusion2.pdf
[scale=0.3,angle=0]Figures/nonclassicaldiffusion2.pdf