# Spinodal

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Entry by Emily Redston, AP 225, Fall 2011

Please see phase separation for an introduction to the regular solution model phase diagram we will be focusing on.

Figure 1 Schematic free energy curve (Jones, Fig. 3.4)
Figure 2 Schematic diagram of stability (http://en.wikipedia.org/wiki/Metastability)

Figure 1 shows the free energy of mixing as a function of composition for one portion of the two-phase region. Let's look at composition $\phi_a$ with free energy $F_a$. Small fluctuation in composition create a higher energy state ${F_{a}}^'$, and this compostion is therefore considered a metastable state. On the other hand, composition $\phi_b$ is unstable since small fluctuations in composition create a lower energy state. A quick way to visualize stability can be seen in Figure 2. Point (1) is metastable because small fluctuations lead to higher energy states, but if it gets enough energy to make it over the barrier, it can move to a lower energy state. Point (2) is unstable because any fluctuations will cause it to move to a lower energy state. Point (3) is stable.

For every free energy curve, we can find two compositions in the 2-phase region corresponding to the transition from the metastable compositions to unstable compositions -- that is to say, the inflection points on the free energy versus composition curves. Let's consider what the spinodal looks like for the regular solution model. We can find the spinodal line by looking at

${{\partial^2 G_{mix}} \over \partial {x_A}^2} = 0$

By taking this derivative and solving for $T_{s}$ (below which the homogeneous mixture passes from a unstable to metastable state), we will define a phase boundary on our temperature versus composition phase diagram (Figure 3).

$T_{s} = {{2 \epsilon x_A} \over R} (1-x_A)$

Figure 3 is the phase diagram of a two-component liquid system whose free energy of mixing is described by the regular solution model. The outer blue/purple curve is the phase boundary; within this curve, phase separation occurs. The inner pink curve is the spinodal, and it is where the homogeneous mixture passes from a metastable to unstable state. In summary: (1) outside both curves, the homoegenous mixture is stable, (2) the homogenous mixture is metable once we are in-between our two derived curves, and (3) the homoegeneous mixture is unstable once we are within the spinodal line and phase separation will always occur. As a brief aside, the name spinodal, which means "thorn" in Latin, refers to their shape in one-component systems [1].

Figure 3 Temperature versus composition phase diagram showing phase separation (made by Emily Redston)

See also:

## References

[1] Spaepen, Frans. Applied Physics 282: Solids: Structure and Defects. Harvard University

[2] Haasen, Peter. Physical Metallurgy. Cambridge: Cambridge UP, 1996.