Solutions to the tutorial

• Let us consider the coherency strain associated with composition fluctuations: the change in lattice paramter with composition is given by $\frac{1}{a} \frac{d a}{d x}$; thus, for a total composition change of $\Delta x$, molar volume $V_m$, Young's modulus $E$ and Poisson's ratio $\nu$, the coherency strain energy is given by

  
  

  $$\Delta G_{coh} = \eta^2 (\Delta x)^2 V_m E^{\prime}$$ (10)

  
  
  where
  

  $$\eta = \frac{1}{a} \frac{d a}{d x}$$ (11)

  
  
  and
  

  $$E^{\prime} = \frac{E}{1-\nu}$$ (12)

  
  

  Adding the above energy $\Delta G_{coh}$ to Eq. 7, one obtains the equation
  

  $$\Delta G = \Delta G_{chem} + \Delta G_{grad} + \Delta G_{coh}$$ (13)

  $$= \left(\frac{d^2 G}{d x_B^2} + \frac{2 \kappa}{\lambda^2} + 2 \eta^2 E^{\prime} V_m \right) \frac{(\Delta x)^2}{2}$$

  
  
  from which the condition for spinodal is derived as
  

  $$- \frac{d^2 G}{d x_B^2} > \frac{2 \kappa}{\lambda^2} + 2 \eta^2 E^{\prime} V_m$$ (14)