Thermodynamics

• Came from miscibility gap
• Gibbs free energy analysis

Spinodal decomposition: a) The formation of spinodal - Gibbs free energy $G$ vs composition $x_B$. b) solvus line ($\Delta G' = 0$, solid line) and chemical spinodal ($\Delta G'' = 0$, dashed line) vs $T$

Reverse diffusion

Cahn-Hillard equation

Interfacial gradient term (gradient energy)

Assume

$$ ⁍ $$

• $L = 0$ for crystal with symmetry
• Further assume $K = K_2 + dK_1/dc$

We have

$$ f_V = f_V^0 + K[c']^2 $$

We can further assume that the distribution $c(x) - c_0 = A \cos \beta x$ where $\beta = 2\pi/\lambda$.

Chemical gradient term

$f'' = \mathrm{d}f_V / \mathrm{d}c^2$

Analysis

$\beta$: modulation wavelength

• Analysis 1:

  $$ \frac{\Delta F}{V} = \frac{A^2}{4} \left[ f'' + 2K\beta^2\right] $$

  where $f'' = \partial^2 f / \partial c^2$

  $$ f'' < -2K\beta^2 $$

  and we have $\beta_\text{c}$, where $\Delta F / V = 0$

  $$ \beta_\text{c} = \sqrt{-\frac{f''}{2K}} $$