Maps of Phase Equilibrium - Binary Eutectic Phase Diagrams

While the previous section may have seemed complex, the isomorphous phase diagram is in fact one of the simplest of binary phase diagrams. Now, what if we have a more complicated phase diagram in which the two chemical species (e.g like Ni and Cu in the previous section) don't nicely mix? As we'll see, these can get even more complex!

The good news is that we can, with a twist or two, use what we've learned to interpret more complex phase diagram! So, as we proceed here, remember your foundations: at any coordinate on a temperature-composition ($T$-$C$) phase diagram you can find the 1. The phases present in the mixture, 2. the composition of those phases, and 3. the phase fraction of each of those phases.

As we proceed to these new phase diagrams we'll also find that we can make predictions about microstructure (the arrangement of phases and microconstituents in a mixture) - as well as the phase fraction of those microconstituents.

Let's briefly define a microconstituent before we continue. Let's look at the sketched microstructure at Point B in the two-phase region of our binary isomorphous phase diagram (Figure 10.7.1). Our analysis from the previous section yielded:

1. The phases present ($\ell_{\mathrm{s}}$ and $\alpha$)
2. The compositions of each phase: $C_{\ell_{\mathrm{s}}} = 61 \text{ wt% Ni}$ and $C_{\alpha} = 71.5 \text{ wt% Ni}$
3. Their weight fractions ($W_{\ell_{\mathrm{s}}} = 0.62$ and $W_{\alpha} = 0.38$).

So, we can sketch the microstructure, more or less, as shown on the right-hand side of Figure 10.7.1. We have two phases there. These would be identifiable if we looked at the material with a microscope. Since they are distinct and identifiable microstructures we call these phases microconstituents. As well see coming up, a phase can be a microconstituent (as we show here) but a specific structural arrangement of multiple phases can also be a microconstituent.

Let's briefly return to our food analogy. In the traditional chocolate chip cookie, we're calling the chocolate one phase and the cookie another phase. These are analogous to the microconstituents we see in real materials at the microstructure.

Let's return to our cake analogy. Let's define the white cake and chocolate chocolate cake as different "phases". However, we can process white and chocolate cake batter to produce a new, marbled structure (Figure 10.7.2) which is distinct from a cake that is simply one chunk of white cake and one chunk of chocolate cake. Marbled cake is distinct and identifiable in this way. (And, better, in my opinion!)

When we're referring to a microconstituent in MSE, this is what we're talking about. a structure comprised of one or multiple phases that is distinct and identifiable, like marbled cake is. We'll see soon that with a little thermodynamics we can control the presence of these structural microconstituents, which can, of course, influence properties

A Familiar(?) Binary Eutectic Phase Diagram

When we have two chemical components that do not mix well, sometimes we get a new phase diagram with new features called a binary eutectic phase diagram. You should be very familiar with one of these binary eutectics if you've experienced an Evanston winter! The classic example is the $\ce{NaCl}-\ce{H2O}$ phase diagram, shown in Figure 10.7.3. This is an important phase diagram for de-icing of roads.

This phase diagram has a few familiar features and a few new features. We are plotting a chemical composition $C$ on the $x$-axis in wt% of NaCl and temperature on the $y$-axis. The phase fields separated by phase boundaries, including liquidus lines. We have regions of two-phase equilibrium (i.e., a phase field with crystalline ice denoted by as $I_{\mathrm{h}}$ and saltwater solution denoted as $\ell_{\mathrm{s}}$) and single-phase regions (saltwater).

However, there are some new features we didn't see in isomorphous phase diagrams. Instead of having completely miscible chemical components at low temperature like we did with Ni and Cu, we have phase separation in which NaCl crystals and ice water crystals do not dissolve into each other in solution. We also observe a horizontal line below which we only have solid phases, but - unlike the solidus boundary seen in the previous section - this line only exists at a single temperature $T_{\mathrm{E}}$.

We also have a really interesting feature - the liquid solution region for saltwater extends much below the melting temperature for pure water ice! This implies that if we mix together $\ce{NaCl}$ and $\ce{H2O}$ with precisely 23.3 wt% $\ce{NaCl}$, we can sustain a pure liquid phase all the way down to $T_{\mathrm{E}} - 21.1 ^{\circ}\text{C}$. This is "freezing point depression" as chemists call it. It's also referred to as the "eutectic point". The word "eutectic" is derived from the Greek - "eu" means "good" or "well", and "tectic" means "melting". The "good" melting temperature. We therefore call this a "eutectic" point, and it exists at a eutectic composition $C_{\mathrm{E}}$ and a eutectic temperature $T_{\mathrm{E}}$.

These eutectic points exist in many chemical mixtures and are very useful in engineering. Consider Exercise 10.7.1 with regards to the utility of a eutectic point for an engineer - maybe focusing on your specific field of study.

Another Binary Eutectic Phase Diagram

Let's look at another binary eutectic phase diagram (Figure 10.7.4). This is the Ag-Cu phase diagram, which is the basis for Sterling silver, an ancient alloy system used to improve the strength and durability of silver while maintaining its appearance and luster.

The phase diagram in Figure 10.7.4 is very similar to that in Figure 10.7.3 in that we're mixing two chemical components (Ag and Cu) that are largely immiscible in the solid solution. Again, we see that this mixture has a region in which the liquid solution is in equilibrium (i.e., stable) to a temperature that is lower than either constituent's melting temperature. The point of lowest melting temperature is that special eutectic point, existing at a specific melting temperature and composition ($T_{\mathrm{E}} = 779\,^{\circ}\text{C}$ and $C_{\mathrm{E}} = 71.9 \text{ wt% Ag}$). This point is labeled E in Figure 10.7.4.

There's something special about this eutectic point. At this point we cannot change the temperature of the mixture or the composition of the phases and maintain three-phase equilibrium. Try it by navigating the phase diagram. If I increase the temperature, we move into the $\ell_{\mathrm{s}}$ phase field. the $\alpha$ and $\beta$ phases are not equilibrium phases in these conditions. If I lower the temperature $\ell_{\mathrm{s}}$ is no longer stable. The equilibrium is invariant with regards to temperature.

If I change the composition of the phases, I must also move away from that point. At the eutectic point I can use a tie line to find that the $\alpha$ phase is about 8 wt% Ag and the $\beta$ phase is about 91.2 wt% Ag. If I wanted to increase the Ag composition in the $\beta$ phase but still maintain three-phase equilibrium, I'd be out of luck. We'd have to do so by moving into the $\beta + \ell_{\mathrm{s}}$ phase field or the $\alpha + \beta$ phase field. This means that at three-phase equilibrium, the compositions of the phases is also invariant.

So, we will call the exact point at ($T = 779\,^{\circ}\text{C}$ and $C = 71.9 \text{wt% Ag}$) the eutectic invariant point. We call this point invariant because you cannot change the temperature of the mixture or composition of the phases and still maintain three-phase equilibrium.

(Now, there's a bit of more advanced thermodynamic business that happens right on the eutectic isotherm that stems from something called the Gibbs Phase Rule. As such, if we're considering just equlibrium (no kinetics) it's best to assume that you can simply never be exactly on the isotherm unless you are at the eutectic composition itself. So, we'll leave that for upper-division courses and just talk about equilibrium states either side of the isotherm in the scenarios we introduce here.)

Beyond this new invariant point, we also now see two new single-phase fields on the composition extrema of the phase diagram. One is the $\alpha$ phase, or a solid solution with Cu as the solvent and Ag as the solute. One can identify the maximum solubility of Ag in $\alpha$ from the diagram: about 8. wt% Ag at $779 ^{\circ}\text{C}$. Similarly, we can identify the maximum solubility of Cu in $\beta$, or the solid solution with Ag as the solvent and Cu as the solute. This is at about 91.2 wt% Ag and $779 ^{\circ}\text{C}$.

Below the isotherm temperature we have so-called solvus lines which define maximum solid solubility. For example, if I take a Cu-rich $\alpha$ crystal at $600 ^{\circ}\text{C}$ and start diffusing in Ag, I'll reach the solubility limit at about 5 wt% Ag. At < 5 wt% Ag, I'll have a single phase of $\alpha$, but >5 wt% Ag I'll now have two phases of different compositions: $\alpha$ (rich in Cu) and $\beta$ (rich in Ag). Understand, we can use tie lines and lever rules in the same way here and will do so below.

Eutectic Invariant Reactions

At the invariant point something interesting happens. You might notice that, as you cool the mixture through the eutectic invariant point ($T = 779 ^{\circ}\text{C}$ and $X = 71.9 \text{ wt% Ag}$) we go from the liquid phase field $L$ to the two-phase field below it $\alpha$ and $\beta$, we have a distinct transition. Above $T_{E}$, we only have liquid phase. Below $T_{E}$ we have two solid phases. So, we have the "reaction" upon cooling of

The liquid of composition $C_{E}$ converts to $\alpha$ phase of composition $C_{\alpha, E}$ and $\beta$-phase of composition $C_{\beta, E}$. For the Ag-Cu system, we can write this out explicitly:

This is a very important reaction in eutectic systems, and will allow us to predict microstructures accurately. As well see in future chapters, there are other important types of invariant reactions, all with their own Greek-derived names.

Deriving Information from a Binary Eutectic

We promised you that we can use the same methods that we used with isomorphous phase diagrams in binary eutectic systems to derive relevant information from a phase diagram. Let's do an example for the Sn-Pb binary eutectic (Figure 10.7.5, an important solder) and then practice ourselves.

Let's look at the temperature $T = 100^{\circ}\text{C}$ and composition $C = 80 \text{ wt% Sn}$:. The work for this example is shown in Figure 10.7.6.

1. What phases are present at this condition?

Here, we just navigate to the point on the phase diagram (a red dot in Figure 10.7.5 and find we're in the mixed-phase region of both $\alpha$ and $\beta$.

1. What are the compositions of the phases at this condition?

We use our tie line and look at the intersections with the phase boundaries. Here, we find that $C_{\alpha} \approx 5 \text{wt% Sn}$ and $C_{\beta} \approx 99 \text{wt% Sn}$.

1. What are the phase fractions of the phases present?

Here, we use the lever rule. I've drawn spans $\overline{C_{\alpha}C_0}$ with length $R = C_0 - C_{\alpha}$ and $\overline{C_0C_{\beta}}$ with length $S = C_{\beta}-C_0$ on the tie line above. We can construct our lever rule equations:

$$W_{\alpha} = \frac{S}{R+S} = \frac{C_\beta-C_0}{C_{\beta}-C_{\alpha}}$$

$$W_{\alpha} = \frac{99-80}{99-5} = \frac{19}{94} = 0.20$$

We only have two phases so we can find $W_{\beta}$:

$$W_{\alpha} + W_{\beta} = 1$$

$$W_{\beta} = 0.80$$

Does this look right? A great way to do a quick sanity check this is to look at the length of the relative spans. $R$ is about 80% of the length while $S$ is about 20% of the length... seems about right. Further, we're closer to the $\beta$ solvus line, meaning we will expect that we have much more $\beta$ than $\alpha$. This agrees with our calculations.

Not bad! Now try yourself in exercise Exercise 10.7.2.