Spinodal decomposition

Consider a system with an interdiffusion coefficient $D$ and undergoing spinodal decomposition. Within the spinodal region, the composition fluctuations grow as shown in Fig. 5; the fluctuations grow with a characteristic time constant $\tau = - \lambda^2 / 4 \pi^2 D$ where $\lambda$ is the wavelength of the composition modulation (assuming one-dimensional modulations). Thus, for smaller $\lambda$, the rate of transformation becomes high; however, there is a minimum $\lambda$ below which spinodal decomposition cannot occur; this is because, during spinodal decomposition, as A- and B-rich regions are forming, there are also interfaces between these regions where AB bonds are formed which are energetically costlier; these regions give rise to an increase in free energy; the `incipient' interfacial energy associated with the formation of these regions with large AB bonds are the ones which set the lower wavelength limit. The lower limit on the wavelength $\lambda$ can be obtained using the following argument.

Consider a homogeneous alloy of composition $x_B^0$ decomposing into two parts: one with composition $x_B^0 + \Delta x$ and another with composition $x_B^0 - \Delta x$. It can be shown that the total free energy change associated with this decomposition is

$$\Delta G_{chem} = \frac{1}{2} \frac{d^2 G}{d x_B^2} (\Delta x)^2$$ (5)

As noted earlier, the AB bonds in the incipient interface regions also contribute to the free energy; this free energy contribution, thus, is associated with the gradients in composition. Consider a sinusoidal composition modulation of wavelength $\lambda$ and amplitude $\Delta x$; the maximum composition gradient is thus $\Delta x / \lambda$ and the gradient energy contribution is

$$\Delta G_{grad} = \kappa \left( \frac{\Delta x}{\lambda} \right )^2$$ (6)

where $\kappa$ is a proportionality constant which is dependent on the difference between AB and AA and BB bond energies.

The total change in free energy associated with a composition fluctuation of wavelength $\lambda$ is thus given by the addition of the chemical and gradient terms (Eqns. 5 and 6):

$$\Delta G = \Delta G_{chem} + \Delta G_{grad} = \left(\frac{d... ...2} + \frac{2 \kappa}{\lambda^2} \right) \frac{(\Delta x)^2}{2}$$ (7)

From the above expression, it is clear that for spinodal decomposition

$$- \frac{d^2 G}{d x_B^2} > \frac{2 \kappa}{\lambda^2}$$ (8)

Or,

$$\lambda^2 > - \frac{2 \kappa}{\frac{d^2 G}{d x_B^2}}$$ (9)