# Difference between revisions of "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 Free energy versus composition for various values of $\epsilon$ (made by Emily Redston)

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).

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

Figure 1 (from Haasen)