Teaching Phase Equilibria
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Method of Schreinemakers – A Geometric Approach to Constructing Phase Diagrams

Dexter Perkins, University of North Dakota and Dave Mogk, Montana State University

If you would like an 13-page long summary of the information presented on this page, perhaps to use as a class handout, click link text (Acrobat (PDF) 421kB Aug11 07) to obtain a pdf version that contains most of the same information. Please note that, unless otherwise noted, all images on this page are courtesy of Dave Mogk and Dexter Perkins.
Schreinemakers' method is a geometric approach used to determine the relationships of reaction curves that intersect at an invariant point in multicomponent systems. This method produces topologically correct bundles or sequences of reactions around an invariant point, and can be applied to a wide variety of phase diagrams such as P-T, T-X, activity-activity, etc. The method is fully described in Zen (1966, Construction of Pressure-Temperature Diagrams for Multicomponent Systems after the Method of Schreinemakers – A Geometric Approach. U.S. Geol. Surv. Bull. 1225, 56p.), and the basics are presented below.

Examples of Invariant Points and Bundles of Reactions


This example of an invariant point in PT space includes reactions involving enstatite, corundum, cordierite, forsterite, and spinel. In this example, all reactions have two phases on each side, but that is not always the case. As is traditionally done, the reactions have been labeled by putting the "missing" phase(s) in parentheses at the end of the reaction curve.
This example of an invariant point in PT space includes reactions involving pyrophyllite, diaspore, zoisite, margarite, kyanite and H2O. As is traditionally done, the reactions have been labeled by putting the "missing" phase(s) in parentheses at the end of the reaction curve. Note that there are two degenerate reactions at this invariant point (reactions involving only 4 phases). They are the nearly vertical (Mg)/(Zo) reaction, and the nearly horizontal (H2O)/(Py) reaction.
This example of an invariant point in TX space includes reactions involving tremolite, calcite, dolomite, diopside, quartz, CO2 and H2O. As is traditionally done, the reactions have been labeled by putting the "missing" phase(s) in parentheses at the end of the reaction curve. Analyzing TX diagrams using the Schreinemakers approach is a bit different than analyzing a PT diagram because TX diagrams imply the presence of H2O and CO2. Note that the (Tr)(Cc) reaction is degenerate; it involves fewer phases than the others.


Start with the phase rule (P + F = C + 2). P, F, and C refer to the number of phases present, the degrees of freedom, and the number of system (chemical) components. An invariant point (where reactions intersect) has 0 degrees of freedom (F=0), a reaction line has 1 degree of freedom (F=1), and a divariant field between reactions has 2 degrees of freedom (F=2).

For a PT diagram, what the phase rule tells us is that:

Corrolaries to the phase rule are that, for an "n" component system (C = n):

Other Observations:

Click here to learn more about identifying possible reactions...


One Component Systems


Most of the bulleted relationships (above) can be readily seen in the simple one component system (C = n = 1) Al2SiO5.

The aluminosilicate phase diagram: one component (Al2SiO5), three univariant reactions involving three phases (andalusite, kyanite, sillimanite)

The phases andalusite, sillimanite and kyanite may coexist (n+2 phases) at an invariant point. The phases are related by 3 (n+2) univariant reaction curves, each involving 2 (n+1) phases, and these curves limit 3 (n+1) divariant fields, each of which contains either andalusite, kyanite or sillimanite (n phases).

All three univariant reactions stop at the invariant point because they become metastable beyond the point. Their extensions beyond the invariant point could have been plotted as dashed lines but it serves no useful purpose in this case.


Multicomponent Systems


These two examples show reactions crossing, but they do not make an invariant point. In the first example (left), the two reactions take place in different chemical systems. One includes Mg, the other does not. One includes Al, the other does not. The second example involves two reactions in the CaO-Al2O3-SiO2 system, but there are 6 phases between the two. The phase rule says that an invariant point for a 3 component system may only involve 5 phases. These are both examples of what are called "indifferent crossings."

In multicomponent systems, when two univariant reactions in a given system cross, they create an invariant point provided the total number of phases involved does not exceed C+2 (the number of system components + 2). If, however, the two reactions belong to different systems, or if there are too many phases between the two, the reactions may cross without creating an invariant point (see Figure, at left).

When two univariant reactions intersect at an invariant point, additional reactions also must pass through that point. Some may be stable on both sides of the point, some may be metastable on one side only, and some may be entirely metastable. The sequence of reactions around the invariant point, and their stability or metastability, depend on the compositional relationships of the phases involved.

Consider again the three examples above:

This example of an invariant point in PT space includes reactions involving enstatite, corundum, cordierite, forsterite, and spinel. In this example, all reactions have two phases on each side, but that is not always the case. As is traditionally done, the reactions have been labeled by putting the "missing" phase(s) in parentheses at the end of the reaction curve.
This example of an invariant point in PT space includes reactions involving pyrophyllite, diaspore, zoisite, margarite, kyanite and H2O. As is traditionally done, the reactions have been labeled by putting the "missing" phase(s) in parentheses at the end of the reaction curve. Note that there are two degenerate reactions at this invariant point (reactions involving only 4 phases). They are the nearly vertical (Mg)/(Zo) reaction, and the nearly horizontal (H2O)/(Py) reaction.
This example of an invariant point in TX space includes reactions involving tremolite, calcite, dolomite, diopside, quartz, CO2 and H2O. As is traditionally done, the reactions have been labeled by putting the "missing" phase(s) in parentheses at the end of the reaction curve. Analyzing TX diagrams using the Schreinemakers approach is a bit different than analyzing a PT diagram because TX diagrams imply the presence of H2O and CO2. Note that the (Tr)(Cc) reaction is degenerate; it involves fewer phases than the others.


Help From Compatibility Diagrams


One useful way to make sure that reactions are arranged appropriately is to use compatibility diagrams, as shown in the large figure below.

The triangular compatibility diagrams (C=3) contain lines dividing them into triangular fields. Each triangle represents a stable mineral assemblage. Because stable assemblages change when a reaction is crossed, the tie lines change as well.

This example shows an invariant point in the Al2O3-SiO2-H2O system involving pyrophyllite, diaspore, kaolinite, quartz and H2O. Triangular compatibility diagrams show changes in mineral assemblages from one field to the next.
The (Qz), (Dsp) and (Py) reactions are all stable on one side of the invariant point only. This diagram, however, shows metastable extensions of those reactions into divariant fields. Consider the (Qz) reaction: it cannot be stable up and to the left of the invariant point because the reaction involves pyrophyllite, and pyrophyllite is not stable to the left of the (Ka)(H20) reaction. Note that the degenerate (Ka)(H20) reaction passes through the invariant point. Degenerate reaction often, but not always, are stable on both sides of an invariant point.
Two examples demonstrating the Morey-Schreinemakers Rule. The assemblage kaolinite + quartz is limited to two sectors by the reactions (Py) and (Dsp). Pyrophyllite is stable over a full 180°, only being limited by the terminal reaction (Ka)(H2O).


Practical Steps for Creating Schreinemakers 'Bundles'


For an n-component system, a) if there are n + 2 phases an invariant point is generated, and b) there are n + 2 univariant curves that radiate from this point, unless one or more of the reactions is degenerate. The first step is to make a list of all possible reactions. The best way to do this is to think about the phases NOT involved, and systematically make a list of reactions and label them by the phases absent.

Click here to learn more about identifying possible reactions...


In principle, when creating a Schreinemakers bundle, it doesn't matter which two curves you start with, or which sides you label as the reactants and products of the reactions. However, some starting choices may make the analysis easier than others, so trial and error may be involved. In general, it is best to begin with terminal reactions if you have some. Draw the two reactions intersecting, and label them with products/reactants, and missing phases. If portions of the reaction curves are clearly metastable, make those portions dashed lines at this time.

Two enantiomorphic forms of the same invariant point discussed above. The correct choice cannot be determined without additional knowledge beyond what the the reactions are.


Note: Schreinemakers method will produce bundles of reactions that are topologically correct. Depending on how you orient (label reactants and products) the first two reactions, you can get two different solutions. They will be mirror images of each other. That is, if you go around the invariant point clockwise for one solution, you will hit reactions in the same order as going around the other solution counter-clockwise. The two possibilities are called enantiomorphic projections or enantiomorphic pairs, and one way to think of them is that there is a "right-handed" and a "left-handed" sequence of reactions.

There is no a priori way to determine which of the two is the correct solution. Deciding which is correct requires some geologic intuition and a knowledge of the types of reactions in orienting the Schreinemakers bundles on a phase diagram. For a P-T diagram, for example, high density phases tend to be present on the high pressure side of a reaction. Devolatilization reactions tend to have a steep, positive slope, and the volatiles are liberated on the high T side of the diagram. Additional insight can be gained from thermodynamic principles. The slopes of individual reactions may be calculated using the Clausius-Clapeyron equation.

Once you "fix" the positions of the first two curves, all the other curves will fall into the appropriate sequence. If you find difficulty, try starting with two different curves.

We start with two reactions. Their placement and orientation are arbitrary. We indicate the metastable parts of the curves with a dashed line.
Now we add another reaction. In this case, the (Di) reaction. We indicate the metastable part with a dashed line. The metastable part of this reaction must be in the upper left quadrant because that is where two "Di" reactions face each other. Anothe way to think of this is that the (Di) reaction, if stable, must take place in the lower right quadrant because it involves Ak-Qz-Wo-En, and those phases can only be stable together in that part of the diagram because of the first two reactions. "Wo En" must be on the right-top side of (Di) because there is another reaction limiting Wo+En. The two must "face" each other, and the angle between them cannot exceed 180°.
The invariant point with all four reactions plotted. The metastable extensions are useful because the help determine if we have violated some of the Schreinemakers "rules" listed above.


Now consider an additional reaction. By considering the first two curves, you can determine in which quadrant this new reaction may be stable. Draw the reaction in that quadrant and continue it through the invariant point and out the other side; make the line dashed where metastable. Label each curve with the phase absent, and also label curves with reactants and products on the correct side of each reaction. To place the products and reactants on the correct side of the line, consider the 180° rule, and other hints in notes 1 through 5 above (especially note 5). There is only one correct way to place products and reactants. If you get them reversed, be prepared for confusion and bad juju.

Continue to position all additional curves around the invariant point using the conventions above. Label each curve with the phase absent, and also label curves with reactants and products on the correct side of each reaction. In rare instances, you will have to guess how to label the reactions. You continue to the end of the process and check for consistency with the notes above. Generally if you guess wrong, you will find that most of the reactions turn out to be metastable and you do not have a reasonable invariant point.

Continue adding reactions one-by-one until all are on the diagram. Check for consistency with all the rules listed above. Finally, complete the diagram, if you wish, by adding chemographic drawings showing mineral assemblages in each of the divariant fields.

This is the same as the previous figure, except that we have eliminated metastable reaction extensions.
Final diagram with triangular diagrams showing assemblages stable in each field.


Suppose We Have More Than C+2 Phases?


If we are considering a n-component system, and more than n+2 phases, they cannot all exist at a single invariant point. So, there must be multiple invariant points connected by a network of reactions.

This diagram show several possible invariant points in the SiO2 system and the reactions that emanate from each of them. All Schreinemakers rules are obeyed around each invariant point. The diagram is also consistent with the phase rule. This is a 1-component system. So, three phases coexist at each invariant point, two phases are related by each reaction, and 1 phase is stable in each divariant field.

In this and the following two diagrams, St=stishovite, Co=coesite, aQz=α-quartz, bQz=β-quartz, Trd=tridymite, Cr=cristobalite, and Liq = melt.
The previous diagram is unrealistic because some of the points at the bottom can only exist at negative pressure, and because melting occurs before some of the high-temperature points can be reached. This diagram shows the correct relationships, including the melt (Liq) field. In this diagram, the fields are labeled instead of the reactions. The reactions can be inferred. For example the line separating St and Co must be the St=Co reaction.
Just as some reaction curves are metastable, some invariant points can be metastable, too. This shows an enlargement of the bottom center part of the previous diagram (lines have been extended a bit into negative pressure to show the [Qz] invariant point). The [Cr] invariant point is metastable because it plots in the center of the Cr field. The three reactions that go to the [Cr] invariant point are also metastable.


The drawing above (left) shows phase relationships involving 6 different phases in the SiO2 system. Only 3 phases may coexist at an invariant point; there are 20 possible invariant points, each involving three of the 6 phases. Only 5 invariant points are shown. The others may exist out of view, or may not exist at all. For demonstration purposes, this diagram has been distorted and extended into negative pressure to include invariant points that do not exist in nature.

The middle diagram above is a "correct" phase diagram showing phase relationships in the SiO2 system. The invariant point at negative pressure is gone, and a high-temperature melt field is present (creating 3 new invariant points).

We often label reactions by putting the phase absent in normal parentheses. We sometimes label invariant point by putting the phase (or phases) absent in square brackets. The diagram above (right) shows portions of the previous diagram involving β-quartz, cristobalite, tridymite and liquid. Three stable invariant points are shown: [Liq], [β-Qz], and [Trd]. These are the points where NO reactions include liquid, β-quartz, or tridymite. The [Cr] invariant point, between the other 3, is metastable. It is the point where β-quartz, liquid and tridymite would coexist but cannot. They cannot coexist because cristobalite will form instead of any of the other 3.

Concluding Remarks


The beauty of Schreinemakers' Method is that it will always generate a topologically correct "bundle" of reactions for a given set of phases. If you know that two univariant reactions intersect to form an invariant point, then you automatically know the relative positions of all other stable (and metastable) reactions around that invariant point. The slopes of the reaction curves on a pressure-temperature diagram can be determined using the Clapeyron equation. And, because the thermodynamic properties of the phases involved are additive, you only need to obtain thermodynamic data for two reactions to generate the thermodynamic relations of the other reactions. This extends to experimental petrology–experiments need to be completed for only two reactions that intersect at an invariant point–a tremendous savings in time and energy! Application of the method of Schreinemakers is essential to the construction of phase diagrams and pseudosections using modern advanced modeling programs such as Thermocalc, PERPLEX, TWQ and Melts.

For Further Reading


Teaching Activities and Problem Sets



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