Review of Phase Diagrams: Binary Systems

(Note: Much of this material is reproduced with permission from Russ McDuff's Oceanography 540 course)

Introduction: The Binary Eutectic

To introduce phase diagrams, we will consider the characteristics of a series of phase diagrams of increasing complexity, beginning with a binary mixture of diopside and anorthite.

Figure 5-1

Some important elements of this phase diagram include its:

• Solidus: boundary below which no liquid phase exists.
• Liquidus: boundary above which there are no solid phases.
• Two solid+liquid fields between the solidus and the liquidus in which one of the two solids plus a liquid is present.
• Eutectic point: a point at which both of the solids and a liquid (three phases) coexist
• general shape: The freezing point of each end-member is depressed by a foreign substance

As an example consider the path of melting, starting from low temperature, of material of composition An₆₀. As heat is added the two solid phases are present until a temperature of 1274°C is reached. At this point, melting produces a liquid of composition An₄₂. The system remains at this temperature until all the diopside present in the original rock melts. Why? Because the eutectic is the only point at which three phases (two solids plus the liquid) can coexist. The mass balance for melting at the eutectic point requires that the composition of the residual solid be given by:

Eq 5-1:

Eq 5-2:

In these and the following equations f_l is the fraction of liquid and is the composition with the subscripts E, s and l denoting the eutectic liquid, the solid and the liquid, respectively. Melting will continue until the composition of the residual solid reaches the value 1, i.e., the diopside component has melted completely and the residual solid is pure anorthite. At this point the system is no longer restricted to the eutectic point and the heat applied to system will again increase temperature. For this example, the critical fraction of liquid is 0.69.

As temperature rises again, once this critical fraction is reached, the liquid composition follows the liquidus until the liquid is of starting composition An₆₀. The position of the liquidus indicates the composition of liquid at a particular temperature. Thus when one of the two solids is present in equilibrium with a liquid, mass conservation requires that for an arbitrary composition :

Eq 5-3:

Eq 5-4:

In geometric terms:

Figure 5-2

and so this mass balance is the basis of the lever rule, i.e.

Eq 5-5:

We call the situation just described closed system, equilibrium melting. What if melt were continuously separated from residual solids instead? We call this open system or fractional melting. Until the critical fraction of liquid is reached, the system behaves in exactly the same way as before--the liquid composition is determined by the eutectic. However once diopside is gone what remains is the single phase anorthite which melts at 1553°C. Obviously the energetics of melting are quite different.

Binary reactions

A reaction occurs in a system such as Figure X, where phase AB is at a more A-rich composition than the reaction that produces it. At compositions in between, the first phase formed on crystallization will be A, but this early formed phase will react at the peritectic point (which is an invariant) until all A is exhausted and only B and AB remain. Below, the natural system Fo-Qz is given as an example of a binary reaction system.

Figure X: Idealized binary reaction system.

The system Fo-Qz, with enstatite as an intermediate phase.

Example 1: Melting of a Fo-En rock (Harzburgite). The first liquid generated is at the eutectic composition at the right. The melting reaction is thus En -> Liq + Fo. In the equilibrium case, the liquid remains to react with the forsterite and enstatite. The higher the temperature, the more the liquid composition wanders upward along the liquidus. When the reaction point is reached, any unreacted enstatite is converted to forsterite + liquid and the melting proceeds as if it were a simple binary eutectic.

In the fractional case, the melt composition and the temperature remain fixed by the eutectic until all en is dissolved. Then the remaining forsterite stays melt-free until its melting point is reached -- the instantaneous liquid composiiton makes a "liquid hop" over the reaction point.

Binary Solid Solution

A number of solids exhibit solid solution, for example plagioclase, so that there is compositional variation in a single phase. The phase diagram for plagioclase is:

Figure 5-3

The solidus is no longer a horizontal line--the solid composition at equilibrium depends on the bulk composition of the system composition and the temperature. For a given composition, one finds the solid and liquid proportions by applying a lever ruler:

Figure 5-4

As an example consider the cooling of material of composition An₆₀. The first crystals form at 1450°C, of composition ~An₈₂. With continued cooling the liquid composition follows the liquidus and the solid composition follows the solidus until the original liquid is fully crystallized at a temperature of about 1333°C. The last liquid composition is about An₂₀.

It would be unusual at best for a natural solid to follow this closed system, equilibrium behavior. The solid initially formed will either be segregated by mechanical settling or protected from further reaction with the liquid by crystals formed later on the surface (a phenomenon called zoning). We call this path the path of fractional crystallization. In the following figure focus on the curves labeled TSC. These represent the integrated total solid composition obtained as a function of temperature when the solid produced at any instance is in equilibrium with the liquid composition at that temperature--more calcic solids produced early in the crystallization process do not back react with the liquid. The endpoints for the TSC are known, the initial value determined by the gap between the solidus and the liquidus and the final state by the bulk composition of the original liquid. The key differences from equilibrium crystallization are that the last liquid is much more sodium rich and liquid remains in a system to a much lower temperature.

Figure 5-5

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