# Phase separation

Entry by Emily Redston, AP 225, Fall 2011

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

$G_{mix}= RT (x _{A}\ln x _{A}+x _{B}\ln x _{B})+\epsilon x _{A} x _{B}$

$\epsilon = zN[V_{AB} - {1 \over 2}(V_{BB}+V_{BB})]$

$V$ 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. Figure 1 shows a plot of $G_{mix} \over RT$ versus composition for various values of $\epsilon \over RT$. For $\epsilon \over RT$ values < 2, we can see that the curve only has a single minimum. This means that, for any concentration, the system is at a lower free energy when fully mixed than it would be if it separated into two different compositions. Thus full mixing will occur and we will only have a single phase. However, for $\epsilon \over RT$ values > 2, something quite different occurs.

he curve has a single minimum. Consider graph (a). This energy surface shows that for any concentration, \varphi _{0}, the system is at lower free energy fully mixed than separated into two different concentrations.

Figure 1 Free energy versus composition for various values of $\epsilon \over RT$ (made by Emily Redston)

Phase separation occurs when

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

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).

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

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). This will occur at $x_A = 0.5$.

$T_C = {\epsilon \over 2R}$

Figure 2 Temperature versus composition phase diagram showing phase separation (made by Emily Redston)

From here, we can go on to consider the spinodal lines.