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The Schreinemakers Method B. Mishra

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PART – 3

THE SCHREINEMAKERS METHOD: A GEOMETRIC APPROACH FOR CONSTRUCTING PHASE EQUILIBRIA

Derivation of the Gibb’s phase rule

Important feature of the Gibb’s phase rule is to remember that the phase rule is simply an

accounting of the number of equations and unknowns and identities of the unknown are

totally lost in the process of such accounting. Here, we first try to identify the unknowns

(variables) and the equations, necessary for deriving the phase rule.

Unknowns

If we represent the compositions of all the phases in the system in theirs of mole fractions of

their components (x1, x2, ……. etc.), then there will a mole fraction term for each

component in every phase, so that there will be ‘cp’ compositional variable.

In addition, considering P and T as the other two variables, the total number of unknowns =

cp + 2.

Equations

We can write equations relating compositional variables in each phase as follows

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=∑=

C

iiX (1)

For ‘p’ no. of phases, there will be ‘p’ equations of the above type. Furthermore, since

transfer of chemical components takes place in the direction of decreasing chemical

potential, at equilibrium, chemical potentials of every component is the same in every phase

in which it appears.

Thus, pii 1............... μμμ βα === (2)


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There can be (p–1) number of equations (of eqn. 2) for ‘p’ no. of phases, that is (p–1) equal

signs and considering ‘c’ no. of component, the no. of equations of the above kind will be

c(p–1)

∴Total no. of equations = p + c×(p–1) (3)

The number of degrees of freedom (f) or the variance of the system

f = no. of unknowns – no. of equations

= cp + 2 – [p + c×(p–1)]

= cp + 2 – p – cp + c

= c – p + 2 (4)

For a P-T diagram (see Fig. 1), the phase rule states that

• Invariant points (f = 0) occur at fixed P and T

• Univariant (f = 1) reaction lines occur over a range of P and T, i.e., one is free to

change either P or T, but once we do that, the value of the other is fixed, and we

remain on the line.

• Divariant (f = 2) fields between reactions have two degrees of freedom, i.e., both P and

T can be varied within limits.

Special case: If one (or more) of the reactions degenerate, the reaction will include

fewer than n + 1 phases and there will be fewer than (i) n + 2 reaction curves, and (ii)

than n + 2 divariant fields. Degenerate reactions are those that can be described with

Corollaries to the phase rule, for a n component

system (c = n)

If f = 0 ⇒ no. of phases = n +2

If f = 1 ⇒ no. of phases = n +1 ⇐ n + 2

univariant reactions.

If f = 2 ⇒ no. of phases = n ⇐ n + 2 divariant

reactions.

Fig. 1 P-T diagram showing asimple one-component system –the Al2SiO5 triple point.


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fewer components than the overall system. When they are involved at any invariant

point – it may appear that there are too few reaction curves because two or more curves

may (a) be collinear on the opposite side of the invariant point and thus appear to be

same curve, or (b) may be superimposed on top of each other, so a single curve may

represent reactions with two or more phases absent.

Degenerate reactions may or may not pass through an invariant point. As because

these reactions actually depict two or more phase-absent curves, they can be either (i)

stable-on-stable (two reactions are superposed on top of each other and terminate at the

invariant point, with two phases absent levels on the end of the curve) or, (ii) stable on

metastable (each reaction is superposed on the metastable reaction of the other) so the

appearance is that the reaction curve passes directly thus the invariant point, but are

designated with different absent phases at the end. (H2O)/(Prl) - stable on metastable

(see Fig. 2).

Multi-component system:

In multi-component systems when two univariant reactions in a given system intersect

they create an invariant point - provided the total number of phases does not exceed c +

2. Again, if two reactions belong to different systems, they may cross without creating

an invariant point.

Fig. 2 An example of degenerate reaction (H2O)/(Prl). Note that the absent phases (marked within parentheses) are different on the opposite sides of the invariant point.


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An example of two different chemical systems is shown in Fig. 3a, where the reactions

are

System CaO-Al2O3-SiO2: Ca3Al2Si3O12 + SiO2 = CaAl2Si2O8 + 2CaSiO3 (5)

System CaO-MgO-SiO2: 2CaSiO3 + MgSiO3 = Ca2MgSi2O7 + SiO2 (6)

Fig. 3b shows an example of pseudo-invariant point in the CaO-Al2O3-SiO2 system where

six (6) phases (An, Wo, Gr, Qtz, Gh, Ky) coexist, as against five (5) phases, necessary, for

an invariant point, in a 3-component system, as expected from the phase rule. The gehlenite

(Gh)-forming reaction in Fig. 3b is

2CaSiO3 + Al2SiO5 = Ca2Al2SiO7 + 2SiO2 (7) Wo Ky Gh

3-Component System (MgO-Al2O3-SiO2): There are n + 2 (= 5) univariant reactions and

they intersect at the invariant point, having 5 phases (En, Co, Fo, Crd, Spl). The reactions

are given below and their dispositions are shown in Figure 4.

a b

Fig. 3 Examples of pseudo-invariant points: different chemical systems (a) and less number of phases (b).

Fig. 4 An invariant point in the 3-component system MgO-Al2O3-SiO2, with five univariant reactions.


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(En) Mg2Al4Si5O18 + 8MgAl2O4 = 5Mg2SiO4 + 10Al2O3 (8) Crd Spl Fo Co

(Co) Mg2Al4Si5O18 + 5Mg2SiO4 = 10MgSiO3 + 2MgAl2O4 (9) Crd Fo En Spl

(Fo) Mg2Al4Si5O18 + 3MgAl2O4 = 5MgSiO3 + 5Al2O3 (10) Crd Spl En Co

(Crd) MgSiO3 + MgAl2O4 = Mg2SiO4 + Al2O3 (11) En Spl Fo Co

(Spl) Mg2Al4Si5O18 + 3Mg2SiO4 = 8MgSiO3 + 2Al2O3 (12) Crd Fo En Co

4 -Component System (CaO-Al2O3-SiO2-H2O): There are six (n + 2) univariant reactions.

Each reaction involve 5 (n + 1) phases, excepting two (Mrg = Dsp + Zo + Ky and Ky + H2O

= Prl+ Dsp) that contain 4 phases. There are 6 (n + 2) phases at the invariant point and 4 (=

n) phases are stable in the divariant fields. There are two degenerate reactions, each

containing 4 phases. These are the (i) near-vertical (Mrg)/(Zo) reaction, and (ii) near-

horizontal (H2O)/(Prl) reaction. Details of these two reactions are given below and their

dispositions are shown in Figure 5.

Fig. 5 An invariant point in the 4-component system CaO-Al2O3-SiO2-H2O, containing six phases including pyrophyllite (Prl), margarite (Mrg), kyanite (Ky), diaspore (Dsp), zoisite (Zo) and H2O. There are twodegenerate reactions, (H2O)/(Prl) and (Zo/Mrg).


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(Mrg)/(Zo) 4Al2SiO5 + 4H2O = Al2Si4O10(OH)2 + 6AlO(OH) (13) Ky Prl Dsp

(H2O)/(Prl) 2CaAl4Si2O10(OH)2 = 3AlO(OH) + Ca2Al3Si3O12(OH) + Al2SiO5 (14) Mrg Dsp Zo Ky

The other two reactions are:

(Ky) 8CaAl4Si2O10(OH)2 + 4H2O = Al2Si4O10(OH)2 + 18AlO(OH) + 4Ca2Al3Si3O12(OH)

Mrg Prl Dsp Zo (15)

and

(Dsp) 4CaAl4Si2O10(OH)2 + Al2Si4O10(OH)2 + 6Al2Si2O5 + 4H2O (16)

Mrg Prl Ky

5-Component System (CaO-MgO-SiO2-H2O-CO2): This is a different example, compared

to the previous two, since it is an isobaric (dP = 0) T-XCO2 diagram, for which the phase rule

becomes f = c – p + 1. However, H2O and CO2 are together counted as one fluid phase.

Hence, it actually becomes a 4-component system and a normal univariant reaction involves

5 (n + 1 = 4 + 1) phases. For example, the (Di) reaction involves four minerals and a fluid.

Invariant points involve 6 phases, and 4 phases are stable in each divariant field. The

(Tr)(Cc) reaction is degenerate. The reactions are shown in Fig. 6 and are listed bellow.

Fig. 6 T-XCO2 diagram in the CaO-MgO-SiO2-(H2O-CO2) system, showing an invariant point and one degenerate reaction, i.e., (Tr)(Cc).


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(Qtz) Ca2Mg5Si8O22(OH)2 + 3CaCO3 = CaMg(CO3)2 + 4CaMgSi2O6 + H2O + CO2 (17) Tr Cc Dol Di

(Dol) Ca2Mg5Si8O22(OH)2 + 2SiO2 + 3CaCO3 = 5 CaMgSi2O6 + H2O + 3CO2 (18) Tr Cc Di

(Di) 5CaMg(CO3)2 + 8SiO2 + H2O = Ca2Mg5Si8O22(OH)2 + 3CaCO3 + 7CO2 (19) Dol Tr Cc

(Tr)(Cc) CaMg(CO3)2 + 2SiO2 = CaMgSi2O6 + 2CO2 (20)

Dol Di

Compatibility Diagrams: e.g. system Al2O3-SiO2-H2O

The triangular compatibility diagrams contain reaction lines dividing them into further

triangular fields, each containing stable mineral assemblages. Since change in the mineral

assemblage takes when a reaction is crossed, the tie lines change as well. Crossing tie line

Fig.7 Invariant point in the Al2O3-SiO2-H2O system involving pyrophyllite, diaspore, kaolinite (Ka), quartz and H2O. Triangular compatibility diagrams show changes in mineral assemblages from one fieldto the next.

can be either because (i) a tie line flips

(e.g. Ka + Qtz = Prl + H2O) or (ii) of a

terminal reaction, i.e., a reaction that

has a single phase on one side (e.g. Prl

= Dsp + Qtz). Flipping-tie line

reactions results in one line on the

compatibility diagrams, disappearing

and being replaced by a different tie

line e.g., Ka + Qtz line disappearing

and being replaced by the Prl + H2O

line.

Al2Si2O5(OH)4 + 2SiO2= Ka Al2Si4O10(OH)2 + 2H2O (21) Prl

Crossing a terminal reaction results in a

phase disappearing completely and

involves several tie line disappearing.


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The following points have geometric rationale for depicting phase relations.

1. If a reaction is missing a particular phase, that phase is present on the opposite side of all

other reactions. For example, since the (Dsp) reaction is at the bottom of the diagram (Fig.

7) and Dsp is stable on the top side of all other reactions.

2. The metastable extension of any reaction (the dashed part) occurs in the divariant field

bounded by the univariant curves that have the absent phases facing each other. For

Fig. 8 The (Qz), (Dsp) and (Prl) 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 (Qtz) reaction: it cannot be stable up and to the left of the invariant point because the reaction involves pyrophyllite, andpyrophyllite is not stable to the left of the (Ka)(H2O) reaction. Note that the degenerate (Ka)(H2O) reaction passes through the invariant point. Degenerate reaction often, but not always, are stable on both sides of an invariant point.


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example, the metastable extension of the (Qtz) reaction extends into the divariant field

bounded by Dsp + Qtz + H2O and Dsp + Qtz (see Fig. 8).

4. The Morey-Schreinemakers (180°) Rule states that no sector (the wedge between two

reaction curves) around the invariant point, in any divariant field, can go more than 180°. In

other words, a divariant assemblage always occurs in a sector that makes an angle ≤ 180°

about the invariant point. For example, Ka + Qtz = 120o and Prl enjoys the full 180o

stability, i.e., three right lower sectors (see Fig. 8)

5. As we move around the invariant point, 2-phase assemblage breaks down before either

the phase (in the assemblage) breaks down itself. Similarly, a 3-phase assemblage breaks

down, before any 2-phase assemblage (containing two of the original three phases). In the

above example (Fig. 8), the (Qtz) reaction limits the assemblage Prl + Dsp + H2O. Moving

clockwise, the (Dsp) reaction limits Prl + H2O. Then the (Ka)(H2O) reaction limits Prl. Here,

we must have a word of caution. It does NOT mean that the sequence 3-phase, 2-phase, 1-

phase is followed for all assemblages. It just states that if 3-, 2-, and 1-phase reactions (that

involve the same phases) are present, then they always follow in order. For complex

systems, other reactions may interfere, but the order must be followed.

Practical Steps for Creating Schreinemakers 'Bundles' Let us recall the corollaries of the phase rule. For an n-component system, (i) if there are n +

2 phases, an invariant point is generated, and (ii) there are n + 2 univariant curves that

radiate from this point, unless one or more of the reactions degenerate. The first task is to

make a list of all possible reactions. The best way to do this is to think about the phases

NOT involved (absent phases), systematically make a list of reactions and label them by the

absent phases, as shown below.

Let us consider the CaO-MgO-SiO2 system as an example, and the phases are wollastonite

(Wo), quartz (Qtz), diopside (Di), enstatite (En), and akermanite (Ak). A 3-component

system means that there are four phases in each potential reaction and five possible reactions

in all. These are (Wo), (Qtz), (Di), (En), (Ak), with one that degenerates, leaving:


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A. (Wo) 2Di = Ak+Qtz+En

B. (Di) Ak+Qtz=2Wo+En

C. (En) Ak+Qtz=Di+Wo

D. (Qtz)(Ak) Di=Wo+En

In principle, while creating a Schreinemakers bundle, it does not matter which two

curves we start with, or which sides we label (as 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 we have some. We

can draw the two reactions intersecting, and label them with products/reactants, and missing

phases. If portions of the reaction curves are clearly metastable, those portions can be

represented as dashed lines. Once we "fix" the positions of the first two curves, all the other

curves will fall into the appropriate sequence. If we find difficulty, we can try starting with

two different curves.

Two Solutions

(1) The Schreinemakers method produces bundles of reactions that are topologically correct.

Depending on how we orient, i.e., label reactants and products in the first two reactions, we

can get two different solutions that are mirror images of each other. In other words, 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.

(2) There is no priori way to determine which of the two the correct solution is. Deciding the

correct one requires some geologic intuition and knowledge of the types of reactions in

orienting the Schreinemakers bundles on a phase diagram. For example, in a P-T diagram,

(i) high density phases tend to be present on the high-P side of a reaction, (ii)

devolatilization reactions tend to have a steep, positive slope, and the volatiles are liberated

on the high-T side of the diagram. Additional in-depth understanding can be gained from


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thermodynamic principles. The slopes of individual reactions may be calculated using the

Clausius-Clapeyron equation ⎟⎠⎞

⎜⎝⎛

ΔΔ

=VS

dTdP .

Step 1: In our example, reactions (A) and (D) are terminal reactions. They are plotted in

Figure 9a. Their orientation and labels have been placed arbitrarily. The metastable parts are

shown dashed.

Now we can consider an additional reaction. With reference to the first two curves, we can

determine the quadrant in which the new reaction may be stable, draw the reaction in that

quadrant and continue it through the invariant point and make the line dashed where

metastable. We must label each curve with the absent phase, and also label curves with

reactants and products on the correct side of each reaction, strictly in accordance with the

180°-rule, and other hints in the points # 1 through 5, mentioned above (especially #5).

There is only one correct way to place the products and reactants. If we get them reversed,

we are asking for confusion and trouble.

Fig. 9 (a) Step 1: arbitrarily start with two terminal reactions [(Wo) and (Qtz)(Ak)]. (b) Step 2: addition of the third reaction (Di), which must lie in the lower right quadrant as it involves Ak-Qtz-Wo-En, and those phases can only be stable together in that quadrant because of the first two reactions. "Wo+En" is placed on the right (high-T) side of the (Di) reaction becausethere is another reaction limiting Wo+En. The two must "face" each other, so that the anglebetween them cannot exceed 180°.

a: Step 1 b: Step 2


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Step 2: For our example, we will now plot reaction (b) (Fig. 9b). This reaction includes Ak,

Qtz, En, and Wo. Hence, if it is stable, it can only be stable in the bottom right quadrant. We

plot it there and extend it (metastably) through the invariant point. We then label the right

hand side "Wo+En" because the (Qtz)(Ak) reaction also limits Wo+En, and Wo+En- sector

can not be stable more than 180°. Note that, as expected, the metastable part of this reaction

(Di) lies between two curves that eliminate Di.

This way, we can continue to position all additional curves around the invariant point

using the conventions above, label each curve with the absent-phase, and also label curves

with reactants and products on the correct side of each reaction, with some guess. However,

in the extreme case, if we guess wrong, we will end up finding that most of the reactions

turn out to be metastable and we do not have a reasonable invariant point.

Step 3: We add the last reaction (c) in Fig.10. Here

we note that it contains Di, which means that it

must lie in the upper left quadrant, if it is stable

(since two other reactions limit Di to that

quadrant). We plot it there and extend it through

the invariant point. We label the left hand side

"Ak+Qtz" because the (Di) reaction also limits

Ak+Qtz, and again that assemblage cannot be

stable more than 180°. Finally, we complete the

diagram, by adding chemographic drawings

showing mineral assemblages in each of the

divariant fields.

Fig. 10 The invariant point with all four reactions plotted.

Step 3


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Step 4: We now appropriately eliminate the metastable reaction extensions (Fig. 11a).

Step 5: Finally, we add the triangular diagrams depicting stable mineral assemblages (Fig.

11b).

Fig. 11 Step 4: Eliminating the metastable extensions in Figure 10 (a) and Step 5: Construction of the Schreinemakers P-T bundles for the CaO-Mgo-SiO2 system with triangular diagrams showing stable assemblages in each field.

a b

P

T

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