Maps of Phase Equilibrium - Binary Eutectic Phase Diagrams

In the previous section we introduced the simplest of binary phase diagrams: the isomorphous binary phase diagram. Now, what if we have a more complicated phase diagram in which the two chemical species don't nicely mix? As we'll see, these can get complex!

The good news is that we can use what we've learned so far with regards to navigating and interpreting phase diagrams almost precisely as we've applied it to a binary isomorphous phase diagram, with a few twists. So, as we proceed here, remember your foundations: at any coordinate on a $T$-$X$ 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 skteched microstructure at Point B in the two-phase region of our binary isomorphous phase diagram (@ref()). At the point remember we can define 1. the phases present ($L$ and $\alpha$), their compositions (about 32 wt% Ni and about 43 wt% Ni, respectively), and their weight fractions ($W_{L} = 0.73$ and $W_{\alpha} = 0.27$). So, we can sketch the microstructure, more or less, as shown on the right-hand side of @ref(). We have two phases there - which would be identifiable if we looked at the material with a microscope. Since they are distinct, 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 phases can also be a microconstituent.

The 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 10.7.2. 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 component $X$ on the $x$-axis in wt% and temperature on the $y$-axis. We phase fields separated by phase boundaries, including liquidus lines. We have regions of two-phase equilibrium (i.e., a phase field with water ice and saltwater solution) and single-phase regions (saltwater).

However, there's some new features. Instead of having completely miscible chemical components at low temperature like we did with Ni and Cu, we have we have phase separation in which NaCl crystals and ice water crystals do not dissolve into each other in solution. We also have 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. 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 $-21.1 ^{\circ}\text{C}$. This is useful - solid water ice is slippery, but liquid saltwater is not (as much) and can flow. This is "freezing point depression" as chemists call it. It's also refered 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 and a eutectic temperature.

Let's look at another binary eutectic phase diagram (Figure 10.7.3). 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 10.7.3 is very similar to that in Figure 10.7.2 in that we're mixing two chemical components (Ag and Cu), and yielding a behavior in which we extend the liquid part of the phase diagram to lower temperatures compared to either of the pure Ag or Cu constituents. This is the so-called eutectic invariant point ($T = 779 ^{\circ}\text{C}$ and $X = 71.9 \text{wt% Ag}$). 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.

We now see two new single-phase fields on the composition extrema of the phase diagram: the $\alpha$ phase, or 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 this temperature we have so-called solvus lines which define maximum 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 10.7.4, an important solder) and then practice ourselves.

Let's look at the temperature $T = 150^{\circ}\text{C}$ and composition $X = 40 \text{wt% Sn}$:

1. What phases are present at this conditions?

Here, we just navigate to the point on the phase diagram (a black dot in Figure 10.7.4 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 $C_{\alpha} \approx 11 \text{wt% Sn}$ 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 $S$ and $R$ on the tie line above. We can constructor our lever rule equations:

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

$$W_{\alpha} = \frac{99-40}{99-11} = \frac{59}{88} = 0.67$$

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

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

$$W_{\beta} = 0.33$$

Not bad! Now try yourself in exercise @ref().