# Phase separation

Entry by Emily Redston, AP 225, Fall 2011

Figure 1 (from Haasen)
Figure 2 (from Haasen)

We typically talk about phase separation in terms of the regular solution model of liquids. The free energy of mixing can be written as,

$G_{mix}= RT (x _{A}\ln x _{A}+x _{B}\ln x _{B})+\epsilon x _{A} x _{B}$
where $\epsilon = zN[V_{AB} - {1 \over 2}(V_{BB}+V_{BB})]$

$V$ represents bond energies, $z$ is the number of neighbors, $N$ is the total number of atoms, and $x$ is the mole fraction. The sign of $\epsilon$ is essential in determining the solution's behavior as a function of temperature. If $\epsilon$ < 0 then the change in free energy upon mixing is always negative, so the atoms will always want to be fully mixed. For $\epsilon$ > 0, there is a temperature dependance, and either mixing or phase separation can occur. Phase separation occurs when

${G_{mix} \over x_B} = 0$

By taking this derivative and solving for $T_{P}$ (below which phase separation occurs), we will define a phase boundary on our temperature versus composition phase diagram (Figure 2).

$T_{P} = {-\epsilon (2 x_B -1) \over R[\ln (1-x_B) - \ln x_B]}$

We can also define a critical temperature $T_c$, which is the maximum temperature below which phase separation will occur (the maximum of the $T_P$ curve).

$T_C = \epsilon \over 2R$