# Ternary Eutectic System

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Mixtures of three distinct materials often melt and solidify in a ternary eutectic relationship. A good and very important geological example is the system quartz-anorthite-K-feldspar. If you understand binary eutectics well, ternary eutectics are very straightforward. Since ternary systems are plotted on triangle diagrams, a little review of triangle diagrams is in order first.

## Triangle Diagrams

 In a triangle plot, each vertex represents 100% of a component and the opposite edge represents zero. Point X is 28% of the way from the bottom to the top of the diagram so it has 28% A. It's 11% of the way from edge A-B to C, so it has 11% C. Percentages are measured perpendicular to the zero edge. A horizontal line through X represents compositions with 28%A and varying amounts of B and C. A line representing a constant ratio of two components starts at the third vertex. The red line represents a 4/1 ratio of C to B. All possible mixtures of compositions P and Q lie on the line between them. R is 3/4 of the way toward P, so it represents 75% P and 25% Q.

## Ternary Eutectics

 A eutectic diagram involving three components actually represents four dimensions: the three components plus temperature. Consider the three binary systems A-B, B-C, and C-A. We can arrange the three diagrams in three dimensions as shown here. As long as we only consider two components, we can work with the binary diagrams as usual. If a system just consists of two components, we can ignore the triangle diagram and just consider the system as a binary eutectic. So in three dimensions, a ternary eutectic is like three binary eutectics enclosing a triangular prism. The space inside the triangular prism represents systems with all three components present. There will be a field where A crystallizes first, a field where B crystallizes first, and one where C crystallizes first. In reality, there is a liquidus surface covering the triangle and as the system cools, it will slide down the surface. This diagram shows the relationship between temperature in the binary systems and temperature on the triangle diagram. Very often, temperature contours are omitted. This is not a problem if the system is simple, but if there are maxima or minima on the liquidus surface, the diagram becomes completely useless without contours.The "valleys" between fields are called cotectics.

## What Might We Expect?

 Here's our initial situation. A, B and C are eutectic compounds. We have a melt whose composition is given by the hollow square: richest in B, then in C, and poorest in A. Since this is a simple eutectic system, there's no mixing of the solid components. So we expect that once we cool to a certain point, one of the three components will begin to form. In this case it's B, no surprise considering our initial composition is richest in B. The composition of the melt begins to migrate away from the initial composition (hollow square). Since we're only forming B at this point, the solid composition stays at B. The relative amounts of melt and solid are shown by the relative position of the melt, solid, and overall system. At this point the system is already a little more than half solid At some point, another compound will begin to form. Given that the melt is next richest in C, we'd expect C to form. When the melt is near one vertex, we can more or less predict the sequence of events, but in the middle of the diagram, we can't be sure without actual data. Since both B and C are forming, the melt composition has to move away from both B and C, that is, in the general direction of A. Also, since we are now forming C as well as B, the solid composition shifts toward C. There is, as yet, no A, so the solid remains on B-C. At some point, A will begin to form as well. The melt remains at that composition while A, B and C crystallize. The solid composition moves toward A until it reaches the initial system composition. At that point all the melt has crystallized. The actual locations of the field boundaries depend on experimental data. They are not necessarily in the middle of the diagram.

## Evolution of Ternary Eutectics

In the diagrams below, the overall system composition is a hollow white square and  the melt and solid compositions are shown in red and blue, respectively. The history of the melt composition is shown in magenta and the solid in green. Blue is used to show geometric relationships between the solid, system, and melt compositions.

 The first thing that will happen is that one component will begin to crystallize. In this case it is B. As we remove B from the melt, the melt composition will migrate straight away from the B corner as shown. Eventually the melt will hit a field boundary and then a second component will begin to form. In this case it is C. The composition of the melt will migrate away from both B and C in the general direction of A. The path of the melt is shown in magenta. Since C is now forming along with B, the solid composition migrates in the direction of C.  The melt, system, and solid compositions always lie on a straight line as shown. Finally the melt reaches the ternary eutectic and all three components begin to form. The eutectic is usually a temperature minimum so the melt does not move once it reaches the eutectic. Since all three components are now forming, the solid composition moves into the interior of the triangle. The melt, system, and solid compositions always lie on a straight line as shown. Since the melt stays at the eutectic, the solid migrates toward the system composition as shown in green. Once it reaches the system composition, the entire system is solidified. Given the rules above, we can divide the triangle into six fields with crystallization orders as shown. Note that any order is possible.

One important ternary system is quartz-K feldspar-plagioclase. According to Bowen's reaction series, we expect to find plagioclase forming first in igneous rocks, then potassium feldspar, and last of all quartz. Since a large fraction of igneous rocks lie in the plagioclase rich corner of the diagram, we find that order in many cases and it's a useful rule of thumb. However, in silica rich magmas, quartz can crystallize first, and the three minerals can form in any order depending on the composition of the magma.

Created November 22, 1999, Last Update 14 December 2009

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