# 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 shows a plot of $G_{mix}$/$RT$ versus composition for various values of $\epsilon$/$RT$.

Figure 1 Free energy versus composition for various values of $\epsilon$/$RT$ (made by Emily Redston)
Figure 2 Schematic free energy curves (Jones, Fig. 3.3)

For $\epsilon$/$RT$ values < 2, we can see that the curve only has a single minimum. Let's look at the schematic of this in Figure 2(a). This shows that homogeneous mixture of the components is the stable equilibrium state. We can immediately see this by comparing the free energy of the homogenous solution of composition $\phi_0$ with that of a heterogeneous mixture of two phases of arbitrary compositions $\phi_1$ and $\phi_2$. The free energy of the heterogeneous mixture is the sum of the free energies of the volume fractions of the constituent phases, corresponding to the point $F_{sep}$. It is clear that the free energy of the mixture $F_{sep}$ is higher that that of the solution $F_0$. Thus the homogeneous phase is thermodynamically stable; for any composition, the system is at a lower free energy when fully mixed than it would be if it separated into two different compositions. However, for $\epsilon$/$RT$ values ≥ 2, something quite different occurs because the curve has two minima. Now we consider the schematic diagram in Figure 2(b). The solution can lower its energy from $F_0$ to $F_{sep}$ by separating into two phases $\phi_1$/$\phi_2$. The state of lowest energy $F_{sep}$ corresponds to the common tangent to the curve at the points $\phi_1$/$\phi_2$. Thus any composition in this region will spontaneously separate into two phases --- this is known as phase separation!

We can define this more explicitly for our regular solution model by saying that phase separation occurs when

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

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 3).

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

We can also define a critical point in the 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 3 Temperature versus composition phase diagram showing phase separation (made by Emily Redston)

We can find the proportions of both phases in our 2-phase region using the "lever rule". From here, we can also go on to consider spinodal lines.