When a number of metamorphic phase reactions meet at a point, Schreinemaker analysis enables us to say something about the order of the curves and even something about their angles.
The first step is to list all the reactions whose curves meet at a point and enumerate all the phases in all the reactions. If P is the number of phases in all the reactions then:
For example, consider these reactions:
Is the water liquid or vapor? Both, and neither. Since we have silicate melt, we are way above the boiling point of water, but under pressure, boiling point increases. If we follow the boiling curve of water, the liquid gets less dense and the vapor gets more dense until they become indistinguishable. This point, called the critical point, happens at 374 C and 220 bars for water, which are geologically pretty moderate conditions. So in almost all metamorphic and igneous settings, water is supercritical and can best be considered a very dense vapor.
Altogether, six phases are involved. From here on, phases will be abbreviated as shown.
The fundamental principle is shown below:
We label the reactions with the phase or phases that do not participate in the reaction:
We'll begin with one curve, say:
Muscovite + Quartz == Orthoclase + Sillimanite + Water (L)
Label one side of the line with the reactants and the other with the products. For now, we can pick the sides arbitrarily.
Here's the logic of labeling the curves as we did above. Consider the curve labeled M, for example. Along the M reaction curve, muscovite is not involved in any chemical reactions. Obviously, then, this curve cannot be on the side of the (L) curve where muscovite is involved in chemical reactions. It can't be on the reactant side of the line (M+Q); it has to be on the product side (O+S+W). In general, labeled curves have to go on the side of the line where the labeled phase is not involved. If we label the curve with reactants on one side and products on the other, then the labels for the remaining curves go opposite the respective phases.
In this example, Muscovite and Quartz are involved as reactants above the line, so the reaction curves where neither phase is involved cannot be on that side of the line. The M and Q curves have to be on the opposite side of the line.
Note that we have one reaction:
Orthoclase + Quartz + Water === Silicate Melt (M, S)
that only involves only four phases.
The fact that one curve has two labels is very significant. We can see that the two labels lie on opposite sides of the (L) line. Thus, the (M,S) line simply crosses the (L) line - it does not end at the intersection. Let's begin with this line. We can put the labels on the proper sides of the (L) line as shown, but what about the reactants? The liquid phase must occur on the opposite side of the (M,S) line from the (L) line. Thus we have O+Q+W on the left and L on the right as shown below. For compactness, we'll omit the plus signs: MQ means M + Q, and so on.
|Note that we have no information about the angles between the
lines. Also we could just as easily have put the reactants on opposite sides of the (L)
line and gotten a mirror image diagram. What this method gives us is the topology
of the phase diagram - what lines intersect and in what relative order. Actual angles and
left- or right-handedness have to be derived from actual temperature and pressure data.
The dashed line labeled L' is called the metastable extension of line (L)
The metastable extension means this. The vertical line is the boundary where the reaction O+Q+W = L occurs. Now suppose we don't have any excess silica. Then the reaction can't go. That means we could still go to the right of the vertical curve and have the reaction on the horizontal line occur. If we approach the metastable extension from below, we would have O+S+W reacting to produce M+Q. But now we have quartz present, so the reaction O+Q+W = L can go. There could also be kinetic reasons why the reaction M+Q=O+S+W might go instead of some more thermodynamically favorable reaction. The metastable extension will turn out to be important, as we shall see below.
Let's consider another line, like (O). It has to occur above (L) and to the right of (M,S) to be consistent with their labeling (at left, below). The reactants have to be placed so that S and L are on the opposite side of (O) from (S) and (L), that is, with M + Q + L on the left of (O) and S + L on the right (below, right). The metastable extension of (O) is indicated by O'.
The real power of this method now becomes apparent.
The final remaining curves are (W): Muscovite + Quartz === Orthoclase + Sillimanite + Silicate Melt and Q: Muscovite + Silicate Melt === Orthoclase + Sillimanite + Water. Q' has to extend into a sector where Q is a reactant, somewhere between (L) and (O).
Created December 1, 1997, Last Update January 27, 2012
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