Construction of PressureTemperature Diagrams for 

Multicomponent Systems After 

the Method of SchreinemakersA Geometric Approach 

GEOLOGICAL SURVEY BULLETIN 1225 

Construction of PressureTemperature Diagrams for 

M ulticomponent Systems After 

the Method of SchreinemakersA Geometric Approach 

ByE-AN ZEN 

GEOLOGICAL SURVEY BULLETIN 1225 

A discussion of the topological relations 

between the compositions of phases and 

the p-T representation of the invariant, 

univariant, and divariant equilibria 

in which these phases participate 

Corrected and 

reprinted 1984 

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1966 
DEPARTMENT OF THE INTERIOR 

WILLIAM P. CLARK, Secretary 

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director 

First printing 1966 

Second printing 1976 

Third printing 1984 

For sale by the Distribution Branch, U.S. Geological Survey, 

604 South Pickett Street, Alexandria, VA 22304 
CONTENTS 

Page 

Abstract-------------------------------------------------~-------- 1 

Introduction______________________________________________________ 2 

Acknowledgments_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 4 

General d~cus~on_________________________________________________ 4 

~omenclature_________________________________________________ 4 

The combinatorial formula______________________________________ 5 

Intersection of free-energy surfaces______________________________ 7 

The fundamental axiom________________________________________ 9 

The Morey-Schreinemakers rule___ , _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 10 

Arrangement of univariant curves_______________________________ 12 

Systematic discussion _______________ • _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15 

One-component system_________________________________________ 15 

Two-component system__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 18 

Three-component system_______________________________________ 21 

Systems of more than three components__________________________ 27 

The pencil theorem_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 27 

Degenerate systems____________________________________________ 30 

One-component system _______ ._____________________________ 30 

Two-component system_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 30 

Three-component system___________________________________ 33 

Application of Schreinemakers' method_______________________________ 41 

Examples as applied to mineral ,-. ""lflmblages_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 41 

Extension of Ostwald's rule to binary systems ________ N _ _ _ _ _ _ _ _ _ _ _ _ 52 

References________________________________________________________ 55 

ILLUSTRATIONS 

Page 

FIGURE 1.. Coefficients of binomial expansion________________________ 6 

2. The meeting of two Gibbs free-energy surfaces, G= G (p, T) _ 8 

3. Graphical representation of the Morey-Schreinemakers rule__ 11 

4. Two enantiomorphic forms of succession of univariant lines__ 14 

5. The geometric relations of univariant lines in a one-component 

system______________________________________________ 15 

6. Three isobaric G-T sections (lines) of the G-p-T surfaces of 

three polymorphs_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1() 

7. The p-T diagram for ice, water, and steam________________ 17 

8. The designation of the four phases in a binary system_______ 18 

9. The relation of the univariant lines about the quadruple point 

in a binary system____________________________________ 19 

lll 
IV CONTENTS 

Page 

FIGURE 10. Derivation of the geometric relations of a binary system from 

that of a unary system________________________________ 20 

11. The p-T diagram in a ternary system where the five phases 

form a pentagon-------------------------------------- 21 

12. The p-T diagram in a ternary system where four of the five 

phases form a quadrilateraL___________________________ 22 

13. The p-T diagram in a ternary system where three of the five 

phases form a triangle_________________________________ 23 

14. Derivation of the three types of p-T diagrams for ternary 

systems from the configuration for a binary system_ _ _ _ _ _ _ 26 

15. Demonstration of the pencil theorem______________________ 29 

16. Four possible sets of degenerate relations in a binary, fourphase system_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 3'1 

17. The p-T diagrams corresponding to the degenerate relations 

of figure 16_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 32 

18. Four possible sets of degenerate relations among the compositions of five phases in a ternary system, effected by the colinearity of three phases, and their corresponding p-T 

diagram types________________________________________ 34 

19. Three possible sets of compositionally degenerate relations in 

ternary, five-phase systems effected by a single compositional coincidence (dimorphism) of two phases and their cor-· responding p-T diagram types_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 36 

20. Three possible sets of compositionally degenerate relations in 

ternary, five-phase systems effected by two sets of three 

colinear phases and their corresponding p-T diagram types__ 38 

21. Three possible sets of compositionally degenerate relations in 

ternary, five-phase systems effected by the colinearity of 

three phases and a compositional coincidence of two phases, 

and their corresponding p-T diagram types___ _ _ _ _ _ _ _ _ _ _ _ 39 

22. The single possible set of compositionally degenerate relations 

in ternary, five-phase systems effected by one compositional 

coincidence of three phases, and its corresponding p-T diagram type __________________________ .. ________________ 39 

23. The single possible set of compositionally degenerate relations 

in ternary, five-phase systems effected by two compositional coincidences of two phases each, and its corresponding 

p-T diagram type__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 40 

24. The single possible set of compositionally degenerate relations 

in ternary, five-phase systems effected by the colinearity of 

four phases, and its corresponding p-T diagram type______ 40 

25. The ternary system Al20 3-SiOrH20, with the .five phases 

water, andalusite, quartz, kaolinite, and pyrophyllite_____ _ 41 

26. The ternary system Mg0-Al20rSi02, with the five phases 

enstatite, forsterite, spinel, kyanite, and sapphirine_ _ _ _ _ _ _ 44 

27. The ternary system Mg0-Al20 3-Si02, with the five phases 

clinoenstatite, forsterite, cordierite, sillimanite, and spineL_ 45 

28. The ternary system Al20a-SiOrH20, with the five phases 

kaolinite, pyrophyllite, andalusite, kyanite, and quartz____ 46 

29. The ternary system Ca0-Mg0-Si02, with the five phases 

merwinite, diopside, monticellite, wollastonite, and A.kerrnanite______________________________________________ 47 
CONTENTS 

FIGURE 30. The ternary system Al,Oa-Si02-H20, with the five phases 

corundum, diaspore, gibbsite, water vapor, and pyrophylv 

Page 

lite_________________________________________________ 49 

31. The ternary system CaO-Ti02-Si02, with the five phases 

wollastonite, quartz, sphene, rutile, and anatase__________ 50 

32. The quaternary system Ca0-Mg0-Si02-C02, with the six 

phases calcite, dolomite, diopside, enstatite, quartz, and 

vapor_______________________________________________ 51 

33. Schematic representation of Gibbs free-energy relations of the 

phases a.-ound an invariant point in a binary four-phase 

systenl______________________________________________ 54 

TABLES 

Page 

TABLE 1. Possible pencjls for n $7 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 30 

2. Entropy and volume data at 25°C and 1 bar pressure for se- lected minerals_________________________________________ 43 

CONSTRUCTION OF PRESSURE-TEMPERATURE 

DIAGRAMS FOR MULTICOMPONENT SYSTEMS 

AFTER THE METHOD OF SCHREINEMAKERSA GEOMETRIC APPROACH 

ABSTRACT 

For an n-component system at chemical equilibrium, an assemblage of 1!-+ 2 

phases is thermodynamically invariant, an assemblage of n+ 1 phases is univariant, and an assemblage of n phases is divariant. On a pressure-temperature 

diagram, an invariant point could occur only at some specific temperature and 

pressure; from this point there radiate n+2 univariant curves, and from each 

curve there emanate n+ 1 divariant regions. How these univariant curves and 

divariant regions are related to one another and how these relations reflect the 

relative compositions of the n+2 phases are the subjects of this paper; the 

treatment is based largely on the method discovered by F. A. H. Schreinemakers. 

Because each univariant reaction involves n+ 1 phases, one of the n+ 2 phases 

that occur at the invariant point is missing along this reaction line. The univariant line, therefore, can be uniquely identified by the missing phase; 

by convention this missing phase label is given in parentheses. This paper 

demonstrates that for any given univariant reaction, there exists a corresponding 

set of relations among the univariant curves. Thus, if in a ternary system the 

heterogeneous reaction 

A+ B = C + D (phase E missing) 

is univariant, then on the p-T diagram the univariant curves bearing the labels 

(A) and (B) must occur on one side of the curve bearing the label (E), whereas 

the curves bearing the labels (C) and (D) must occur on the other side of curve 

(E). Because each univariant curve contains n+ 1 phases, any two univariant 

curves contain all the n+2 phases, barring compositional degeneracy. Knowledge of two nondegenerate univariant reactions in the system thus suffices to 

determine all the other univariant reactions and therefore the succession of 

univariant curves around the invariant point. When the succession of univariant curves is known, it is a simple matter to determine next the relative 

locations of the divariant assemblages on the p- T (pressure-temperature) 

diagram. 

The Schreinemakers method can be applied to systems containing any number 

of components. For both unary and binary systems, there is but one corresponding topological arrangement of the univariant curves; for ternary systems, 

there are three possible topological arrangements. The number of possible 

arrangements increases rapidly for systems beyond the ternary; however, all 

the possibilities can be readily enumerated by means of the "pencil theorem" 

of Schreinemakers, which is discussed in detail. 

1 
2 p--T DIAGRAMS. MULTICOM.PONENT SYSTEMS 

An n-component system is degenerate if one or more of the associated univariant curves can be described by fewer than n components. Chemographically, this corresponds to the coincidence of phases (n=2,3,4, ... ), colinearity 

of pha~es (n=3,4,5, ... ), coplanarity of phases (n=4,5,6, ... ) and so on. 

The phases participating in degenerate univariant reactions are the singular 

phases; those which do not participate are the indifferent phases. In a degenerate system, two or more univariant lines assume the same value of slope at 

the invariant point. If the indifferent phases are chemographically on opposite 

sides of the singular phases, the degenerate curves coincide stable to stable (so 

that fewer than n+ 2 univariant lines obtain); if t}le indifferent phases are on the 

same side of the singular phases, the degenerate curves coincide stable 

to metastable. All the possible cases of degeneracy for binary and ternary 

systems are described in this paper. 

The method of Schreinemakcrs is useful in analyzing petrologic problems. 

Thus, for a given mineralogical system, if only some of the n+2 univariant 

curves about an invariant point have been determined experimentally or by 

calculation from available thermodynamic data, the application of the method 

enables one to compute approximately the remaining curves. Because the 

chemographic relations of the phases determine the sequence of univariant 

curves about the invariant point, the method is useful in evaluating the consistency of existing thermochemical data on minerals of known compositions. Even 

when no experimental or thermochemical data are available, knowledge of the 

chemical compositions of the phases alone often allows one to calculate and 

predict the relative p-T dispositions of mineral assemblages in specified chemical 

and mineralogical systems; such knowledge is frequentlylhelpful in the study of 

mineral paragenesis. Examples of the application of the Schreinemakers method 

to mineralogical systems are given. 

INTRODUCTION 

The methods of representation of heterogeneous phase equilibria 

on p-T (pressure-temperature) diagrams have been developed along 

two complementary lines. One of these is mathematical representation, perhaps most notably known through the work of Morey and 

Williamson (1918) and Morey (1936). The second main line of 

analysis is geometric, the study being based on graphical analysis. 

One of the cla:ssics in this latter method is a·series of 29 articles by 

SchreinemaJters (1915-25), remarkable for its originality. This 

graphical approach to phase diagrams has been adopted by most 

modern texts on the phase rule, for instance those by Ricci (1951) 

and Zernike (1955); it is especially familiar to petrologists through 

such works as those of Niggli (1930, 1954) and Korzhinskiy (1959). 

No satisfactory summary of the method of Schreinemakers· is 

readily available in the literature; the summary by Niggli (1954) is 

so succinct that many logical steps are omitted or obscured. The 

present paper is intended to' fill this gap and is addressed primarily to 

students. The approach is largely that of Schreinemakers and is 

axiomatic. Emphasis, however, is placed on the direct relation 

between p-T diagrams and the Gibbs free-energy surfaces of phases 
INTRODUcriON 3 

that participate in the equilibria. A mathematical background will 

be helpful but hardly necessary; the degeneracy of phase equilibria, 

fop example, can be represented with equal validity either by the 

linear dependence of algebraic equations or by the linear, planar, or 

hyperplanar coincidence of phase compositions in diagrams. 

The present study considers only phases of fixed compositions; this 

procedure is justified because, even for phases of variable compositions, these compositions are constant at an invariant point. As 

conditions depart from those of the invariant point, of course, effects 

due to compositional variability may become manifest; however, 

these effects do not concern us here. Some of these effects have been 

studied by Schreinemakers (1915-25). 

A knowledge of the theory of graphical representation of invariant, 

univariant, and divariant equilibria in n-component, n+2 phase 

systems is useful in many mineralo~cal and petrological proplems; a 

few illustrations must suffice here. In recent years many univariant 

equilibrium curves ·on mineralogical systems have been determined 

experimentally. Seldom, however, have all the n+2 possible univariant curves about a given invariant point been determined, so it 

may be necessary to calculate relative locations of the remaining 

curves from those available. The Schreinemakers method is eminently suited for this purpose. 

Even if no experimental data exist for a mineralogical system, data 

on the entropies and volumes of the phases allow one to calculate the 

slopes of the univariant curves by means of the Clapeyron equation. 

Information on these slopes, in turn, permits one to calculate the 

relative dispositions of the curves about the invariant point. However, the available thermodynamic d.ata are not always mutually 

consistent, although these inconsistencies may be difficult to discover, 

inasmuch as they involve small numerical differences. 'Because the 

sequence of univariant curves must obey the restrictions imposed by 

the relative compositions of the phases, failure of the computed curves 

to obey these restrictions immediately reveals that the thermodynamic 

data are mutually inconsistent. 

Many mineralogical systems have not been studied experimentally, 

and the thermodynamic data on them are absent or incomplete. 

Knowledge of the relative dispositions of the univariant curves, however, allows one to gain insight into the possible conditions of formation of the corresponding mineral assemblages. This situation is 

particularly true for reactions that involve devolatilization, because 

such reactions are in general favored by rising temperanure. Similarly, because density data are available for most minerals, good estimates of the probable locations of mineral assemblages relative to the 

pressure axis often can be made with the aid of the density data and 

218-934 0-66-2 
4 p-T DMGRAMS, MULTIOOl\fPONENT SYSTEMS 

of a knowl~dge of the sequence of succession of the univariant curves. 

The estimate of the p-T relations of the three polymorphs of aluminum silicate, andalusite, kyanite, and sillimanite, based on incomplete thermodynamic data (Miyashiro, 1949), exemplifies such an 

application and an early triumph of the use of theoretical analysis of 

plausible p-T diagrams to geologically important systems. 

Eight examples of the application of the Schreinemakers method to 

the construction of p-T diagrams for systems of geologic interest are 

given on page 41. These examples are designed to illustrate the 

construction of geometrically distinct types of diagrams in ternary and 

quaternary systems, for which the selected phases have known 

compositions as well as partial or complete thermodynamic dat~. 

The extension of the so-called "Ostwald's step rule" to binary systems is discussed on page 52. • :rhe extension is a direct outcome of 

the concept of the relative heights of Gibbs free-energy surfaces in 

G-p-Tspace, and predicts the ·possible sequences of binary, two-phase 

equilibrium assemblages for given values of p, T, and the bulk composition of the system. 

ACKNOWLEDGMENTS 

This paper was critically reviewed by E. H. Roseboom and Priestley 

Toulmin III, U.S. Geological 'Survey; for their many penetrating 

comments and constructive suggestions I am deeply grateful. The 

idea of applying the Ostwald rule to two-component systems resulted 

from a discussion wi_th Toulmin; for this stimulation I am particularly 

indebted to him. The manuscript, during various stages of preparation, was also read by other colleagues in the U.S. Geological 

Survey: Fred Barker, R. 0. Fournier, C. C. Silber, and R. I. Tilling. 

To these colleagues, I extend my thanks. 

GENERAL DISCUSSION 

NOMENCLATURE 

In this study, the term "component" will be used to mean independent chemical component, that is, one that is distinct and cannot 

be formed by combinations of other components in the set. The 

component, thus defined, need not be physically real and may even 

be a negative quantity. For instance, the components C03 and -CO 

could be used to describe all compositions in the binary system C-0, 

because 

C=-i(COa+3(-CO)) 

O=itCOa+ (-CO)) 

In practice, we are mainly interested in the number of components 
GENERAL DlSCUSSION 5 

needed to describe a given system and only incidentally in their 

precise identity, because components have physical meaning only in 

reference to phases, which are part of the physical world. Special 

significance can be attached to components which coincide with 

compositions of real phases; such components have been called 

"composants" by Schreinemakers (1923, 24, p. 719) 1 and "inactive 

components" by Thompson (1959, p. 451), who discussed the subject 

in careful detail. · 

The letter n, without subscript, will be used to designate the 

number of components in a system. The labels n11 n2, ... , will be 

used to indicate individual components. All symbols for components 

are in italics except for the chemical formulas which are in roman type. 

The term "phase" refers to a physical entity. The symbol rp will 

be used to denote the number of phases in a system; the symb6Is 

1, 2, 3, ... , will denote specific phases in the system. All phase 

sym bois are in roman type. 

The symbol w will be used to denote the variance or number of 

thermodynamic degrees of freedom as given by the Gibbs phase rule, 

w=n+2-rp 

A system is invariant, univariant, divariant, and sodorth, according 

to the situations where w=O, 1, 2, and so forth. 

Schreinemakers (1915, 1, p. 118) first introduced the useful notation 

whereby a univariant or divariant -equilibrium is designated by the 

phase or phases that do not participate in the reaction; the absent 

phases are given in parentheses. The number of phase labels appearing in given parentheses immediately indicates the variance of the 

equilibrium: thus (3, 5) is a divariant equilibrium which does not 

include the phases 3 and 5. 

THE COMBINATORIAL FORMULA 

The phase rule is a formula that relates the number of phas,es, the 

number of components, and the variance; only indirectly does it 

identify these tuantities with entities of the physical world. Thus 

it is a formula for reckoning, equally applicable to stable, metastable, 

or unstable assemblages of real or imaginary phases. The reckoning 

process is facilitated by the use of the combinatorial formula, which 

gives the number of ways, C, of combining k objects, taken m at a 

time, provided that the m chosen objects and the (k-m) residual 

objects are not distinguished within each subgroup. The formula 

states that 

C(k, m)=k!fm!(k-m)! 

k(k-1)(k-2) ... 2X 1 

m(m-1)(m-2) ... 2X 1X (k-m)(k-m-1) ... 2X 1 

1 Reference to Schreinemakers (1915-25) will be given hereafter by the serial number or the article and 

the page number of the journal. 
6 p-T DLAQ:RAMS, MULTIOOMPONENT SY-STEMS 

In terms of the phase-rule notation, the oombinatorial formula 

may be written as 

C(n+2, cp) = (n+2) !jcp!(n+2- cp)! 

=(n+2) !fcp!w! 

Figure 1 shows the number of C(k, m)=C(n+2, cp) for integer values 

of k between 1 and 10, although only those rows with k~3(n~l) 

have physical meaning. The diagonal rows group systems with the 

same variance. 

k=n+2 2 3 4 5 6 7 8 9 10 

2 

3 

4 

5 

6 

7 

8 

0 

FIGURE I.-Coefficients of binomial expansion, 

k! (n+2)! 

1 1 , where w=n+2-tp. m!(k-m)! tp!W. 
GE!NERAL DISCUSSION 7 

From the· combinatorial formula, it is easy to compute the number 

of possible assemblages of any given variance (w=O, 1, 2, ... ) for 

a system of any given number of components. Thus, for example, in 

an equilibrium ternary five-phase system there is (5, 0) = 1 invariant 

point, 0(5, 1)=5 distinct univariant lines, 0(5, 2)= 10 distinct divariant 

assemblages. 

INTBBSBCTION OF FBBB-BNBB.GY SUBI'ACBS 

The change of the molar Gibbs free energy, G, of any phase of fixed 

composition, whether stable or metastable, can be written as 

so that 

iG=-SdT+ Vdp 

(

()G) - - =-S<O ()T , 

where V and S are the molar volume and entropy of the phase. 

Moreover, because the inequalities express intrinsic properties of 

phases, they are true also for any assemblage of phases in arbitrary 

but positive proportions. Inasmuch as V and S are always finite and 

positive quantities, and for any given phase assume unique values at 

any specified T and p, in a plot expressing Gas functions of T and p, 

for any specified assemblage of phases, jj is a single-valued continuous 

surface having positive and finite slopes in G-p sections and negative 

and finite slopes in G-T sections. These slopes are never parallel to 

the. T, p, or G axes, though sharp changes in the slopes (fig. 2B), 

corresponding to the occurrence of higher order phase transitions, are 

not precluded. 

The phase assemblages of particular interest to us are those which 

constitute divariant equilibrium assemblages. Consider two divariant 

assemblages that are derived from the same invariant assemblage in a 

system of n components, and that share n-1 phases in common. We 

further assume that the quantities of the two assemblages are such 

as to contain the same· number of moles of the components. Inasmuch 

as all the phases are fixed in composition, each assemblage may De 

described by a single surface in G-p-T space; these two surfaces will 

meet along a line, the univariant line of reaction. The orthographic 

projection of this line onto the p-T plane is the familiar univariant 

curve of p-T diagrams; because the G-surfaces are never parallel to 

the G-axis, the procedure of projection never reduces the line to a point. 
8 p-T DLAORAMS, MULTIOOMPONENT SYSTEMS 

G G 

FIGURE 2.-The meeting of two Gibbs free-energy surfaces, G=G(p, T). (A) 

By interpenetration: On one side of the line of intersection, one of the surfaces 

is higher than the other; on the other side of the line, it is lower. The two surfaces are differently ruled. For clarity, both surfaces are shown as planar. (B) 

By common tangency. The same surface is higher on both sides of the line of 

intersection. One surface is dotted and the other is ruled. 

~fost probably, two G-surfaces meeting in space will interpenetrate 

(fig. 2A). This, however, is not the only possibility. Under special 

circumstances, the two surfaces could meet along a common tangent 

(fig. 2B). Such a meeting of two surfaces is most likely when one 

of the surfaces, in the neighborhood of the meeting, happens to be 

undergoing strong changes in curvature consequent upon, for example, 

a high-order phase transformation with resultant discontinuities in 

the isothermal incompressibility and in the isobaric.;heat capacity. 

Univariant lines that arise from the tangential meeting of two surfaces, one of which is strongly curved, are instances of univariant 

lines which may be isolated in space and not related to any invariant 

point.2 

Because G-surfaces for divariant assemblages are never parallel 

to the coordinate axes, when two G-surfaces meet in space one surface can always be described as higher than the other, that is, at the 

same p and T, the Gibbs free-energy content of one divariant assemblage is greater, and this assemblage is therefore less stable than the 

other. A univariant curve that results from the common tangency 

2 Point-eon tact of two surfaces may be considered as a limiting case of the intersection of two G-surraces 

where the line of intersection forms a closed loop. In this general case, one divariant assemblage is completely surrounded by another, and the univariant line is not associated with any invariant point. It 

is clear that such a relation between the two G-surfaces could reduce either directly into tangency at a point 

or indirectly through tangency along a line. 
GENERAL DISCUSSION· 9 

of two G-surfaces has the same surface occupying thelowerposition 

on both sides of the curve, so the divariant assemblage represented 

by the surface is relatively more stable on both sides on the univariant 

curve. If the curve is isolated in space· and is not associated with 

some invariant point, then one divariant assemblage completely 

surrounds the univariant curve. 

In contrast, when two divariant G-surfaces intersect by penetration, then one surface will be lower on one side of the univariant line 

of intersection, and vice versa. The univariant line, in this instance, 

separates distinct divariant assemblages, each of which occupies a 

part of the p-_T plane. The univariant line continues until it is 

terminated at an invariant point. 

Meeting of divariant G-surfaces by tangency is very unlikely. 

The fact that such an occurrence is probably associated with higher 

order phase transformations makes it progressively more improbable 

as the phase assemblage becomes more complex, for the effect of 

anomalies in the higher order derivatives of the free energy of a 

single phase becomes less pronounced as more and more free-energy 

functions are summed together. Meeting of G-surfaces by common 

tangency is not known empirically. For these reasons, we will 

hereafter disregard this remote possibility, and consider only the 

situation of intersection of G-surfaces. 

THE FUNDAMENTAL AXIOM 

A G-surface that represents a given divariant-phase assemblage 

cannot possess singularities because all the constituent G-functions 

of individual phases are continuous functions of T and p. The 

univariant curve that results from the intersection of two G-surfaces, 

therefore, must also be smooth and continuous. ·'This fact enables 

us to speak unambiguously about "one side" and "the other side" 

of a univariant curve at each and every point along the curve. 

We then arrive at the following obvious, but extremely powerful, 

conclusion (first pointed out in a slightly different form by Schreinemakers, 1915, 1, p. 119): When two divariant assemblages, each 

of n phases, meet along a univariant curve of n+ 1 phases, then on 

one side of the univariant curve the divariant assemblage I is relatively less metastable than assemblage II, whereas on the other side 

of the curve assemblage II is relatively less metastable than assemblage I. 

Much of the following discussion depends on the verity of this 

statement. The statement is true so long as only univariant curves 

resulting from intersection of G-surfaces are considered. We will 

refer to the statement as the "fundamental axiom." 
10 p-T Dl!AGRAMS, MULTIOOMPONENT SYSTEMS 

THB KOBBY-SCHB.BINBKAJCBB.S BULB 

Consider an n-component system which at equilibrium consists of 

n+2 phases, which we designate as 1, 2, 3, .. . , n+1, n+2. These 

phases can coexist only at an invariant point. From this invariant 

point, n+2 univariant lines emanate, bearing the labels (1), (2), (3), 

.. . , (n+1), (n+2); each of these lines represents equilibrium among 

n+ 1 phases and the absent phase is identified by the phase-label in 

parentheses. _Now consider the equilibrium divariant assemblage 

(n+1, n+2) 3 which consists of phases 1, 2, 3, .. . , n; phases n+l 

and n+2 are missing. If to this assemblage the phase n+2 is added, 

we recover the univariant assemblage (n + 1) of n+ 1 phases. By the 

fundamental axiom, the stable divariant assemblage (n+1, n+2) can 

extend only to this univariant curve; beyond this curve some other 

divariant assemblage, for example (n, n+2), will be relatively more 

stable. Thus the stable part of assemblage (n+1, n+2) is bound by 

the univariant curve (n+1); likewise, it is bound by the curve (n+2). 

The two curves that bound the divariant assemblage bear phase labels 

that are precisely those that identify the divariant assemblage itself. 4 

The two curves (n+l) and (n+2) divide the p-T plane into two 

sectors. Because in general these two curves do not possess the same 

slope value at the invariant point, one sector will subtend a~ angle 

greater than 180°, whereas the other sector will subtend the explementary angle less than 180° (fig. 3). In which sector does the 

assemblage (n+l, n+2) occur? The answer follows directly from 

the fundamental axiom. 

Because in general the two curves have distinct slopes at the invariant point, the metastable extension of, let us say, (n + 1) beyond 

the invariant point will be within the larger sector defined by the two 

curves (fig. 3). Suppose this is the sector in which the assemblage 

(n+1, n+2) occurs stably throughout. By the fundamental axiom, 

the curve (n+l), whether stable or metastable, necessarily separates 

that part of the sector in which (n+1, n+2) is stable from the part 

in which it is metastable. Thus in the region denoted by 8 in figure 3, 

3 We will assume that the equilibrium relations discussed above pertain to stable relations; however, 

the arguments are the same for metastable equilibrium relations. What is significant is the relative stabili· 

ties of two competing divariant assemblages at given values of p and T; the absolute stabilities of the phase 

assemblages are less significant properftes in the present context and commonly cannot be defined by the 

Schreinemakers method. 

c Because the two unfvarfant reaction equations that bound a given dfvarfant assemblage are different, 

they involve different amounts of the phases of this divariant assemblage. Consequently, the Gibbs 

free-energy surface of this assemblage at the two boundaries cannot be directly compared. This apparent 

inconsistency can be resolved by the following consideration: As long as all the phases of the given divariant 

as..cemblage are present at either boundary ·curve regardless of their proportions, these phases will always 

tend to react to form the alternative assemblage that is more stable beyond the boundary curve, with a net 

decrease in the free-energy content of the total system. Therefore, regardlesll of the proportions of the phases 

present, the Gibbs free-energy surface of the given divariant assemblage is always higher across the boundary 

curves relative to another divariant assemblage of phases having the same bulk composition. 
GENERAL DISCUSSION 

(n+ 1, n+2) 

Stable 

(n+ 1, n+2) 

Metastable 

11 

FIGURE 3.-The Morey-Schreinemakers rule. The assemblage (n+l, n+2) 

must be stable in the smaller sector, for otherwise in the sector marked by the 

angle 8 this assemblage must be simultaneously stable and metastable. 

the assemblage (n+l, n+2) must be both stable and metastable, and 

we arrive at a contradiction. 

We therefore conclude: a divariant assemblage (n + 1, n + 2) always 

occurs in a sector which makes an angle about the invariant point no 

greater than 180°. The limiting value of 180° may be assumed when 

there is compositional coincidence or compositional degeneracy (see 

p. 30); it is clearly still compatible with the fundatnental axiom. 

This statement is the Morey-Schreinemakers rule, deduced geometrically by Schreinemakers (1915, 1, p. 121) and analytically by 

Morey and Williamson (1918, p. 66). It is the cornerstone of the 

method of geometric construction of p-T diagrams. 

A direct consequence of the Morey-Schreinemakers rule is the 

following: The p-T region in the neighborhood of each invariant 

point is divided by univariant curves into n+2 sectors (n~l), each 

of which is ~ 180° in angular extent and is occupied by one or more 

divariant assemblages, one of which is unique for the sector. 

The above statement is true for the stable univariant and divariant 

assemblages as well as for their metastable reflections about the 

invariant point. 

Except for one-component systems, more than two divariant 

equilibria can be generated by a given univariant equilibrium. Therefore, in general at least on one side of the univariant curve there 

218-934 0-66--3 
12 JrT DM:GRAMS, MULTIOOMPONENT SYSTEMS 

could simultaneously exist two or more distinct divariant assemblages 

under identical values of p and T; which assemblages in fact prevail 

in a particular sector depends on the bulk composition of the system. 

The different divariant assemblages on the same side of the univariant 

curve will, of course, extend to different univa.riant curves. 

The following statement (Schreinemakers, 1915, 1, p. 121) governs 

the mutual relations of these divariant assemblages that overlap in 

the p-T field: Each divariant assemblage that extends across the 

univariant line (j) (stable or metastable) contains the phase j. 

The truth of this statement is easy to see. Consider, for example, 

the divariant assemblage (j, j + 1). This assemblage extends to, but 

does not cross, the stable part of the line (j). Because the assemblage (j, j + 1) occupies a sector in the p-T field that is ~ 180° in 

angle, the assemblage can at most extend to,. but not beyond, the 

metastable part of (j). Because the assemblage (j, j + 1) is arbitrary, 

any assemblage that does not contain phase j cannot extend over (j); 

only those assemblages that do contain j can do so. 

We shall refer to this statement as the "overlap rule." Notice, 

however, that the converse of the statement is incorrect; not all 

assemblages that contain j extend over (j). 

The overlap rule will be useful in confirming the consistency of 

arrangements of assemblages in the construction of phase diagrams. 

There are, however, other potent chemographic (Niggli, 1954, p. 380) 

canons which can be used for this purpose. 

ARRANGEMENT OF UNIVARIANT CURVES 

One elegant achievement of the Schreinemakers method is that 

correct arrangements of univariant curves on a p-T diagram can be 

constructed with minimum information on the chemography of the 

phases. The method may be explained best by an example. 

Consider a univariant reaction involving n+ 1 phases; we label 

this curve and the univariant assemblage (n+2). Suppose the 

reaction is 

1 + 2 + ... + k = k+1 + k+2 + ... + n + n+1 

where the stoichiometric coefficients of the phases, though not specified, are all positive. The divariant assemblages associated with the 

reaction (n+l) bear the labels (1, n+l), (2, n+l), (3, n+l), ... 

(n, n+l) and (n+2, n+l); each of these assemblages can be obtained 

by depleting the appropriate phase as a result of the univariant 

reaction. As a specific example, consider the divariant assemblage 

(n+1, n+2). This assemblage is bounded by the stable parts of the 

curves (n+l) and (n+2), and the stable part of the assemblage 

occupies the angular sector that is ~ 180°. Therefore, if we start 

out with the stable part of (n+2), and traverse in such a direction 
GENERAL DISCUSSION 13 

(let us suppose this direction to be clockwise) as to cross the stable 

part of the assemblage (n-t-1, n!f-2), we will meet the stable part of 

(n+l) before we meet the metastable extension of (n-t-2) at 180°. 

We say, then, that in relation to curve (n-t-2), curve (n+l) lies in the 

clockwise direction. Conversely, in relation to curve (n-t-1), curve 

(n +2) is in the counterclockwise direction. 

Divariant assemblages that include all the phases 1, 2, ... , 

k-1, k must occur stably only on one and the same side of the curve 

(n +2) because of the fundamental axiom. Therefore, it follows that 

the divariant assemblages (k-t-1, n-t-2), (k-t-2, n-t-2), ... , (n, n-t-2), 

(n-t-1, n-t-2) must all occur on the same side of the curve (n-t-2), 

which we have stipulated to be in the clockwise direction in relatiop 

to (n-t-2). By t~e same token, all the assemblages (1, n-t-2), (2, n-t-2), 

(3, n -t-2), ... , (k-1, n +2) and (k, n +2) must lie in the counterclockwise direction from (n-t-2). The curve (n-t-2) divides the p-T 

plane into two half planes, in each of which a distinct group of divariant 

assemblages is located. The only overlap is the divariant field into 

which the curve (n +2) extends metastably. 

Therefore, the curve (n +2) divides the univariant curves into two 

groups. One group, (1), (2), (3), ... , (k-1), (k), lies on the 

counterclockwise side of the curve (n-t-2), whereas the other group, 

(k-t-1), (k-t-2), (k-t-3), ... , (n), (n+l),lieson the clockwise side of 

(n-t-2). Schematically, we can represent the relations in the form 

(1)(2)(3) ... (k-l)(k)j(n+2)j(k+l)(k-t-2)(k-!-3) ... (n)(n+l) 

where the univariant curve enclosed between the vertical bars is the 

reference curve about which the grouping is effected. We will refer 

to representations such as this, relating the geometry of univariant 

curves, as the "univariant scheme." 

Comparison of the univariant scheme with the original univariant 

reaction equation shows a remarkable similarity. Thus 

1+2+3+ .. . +k-l+k (n+2) 

k-t-1 + k-t-2 + k+3 + . . + n + n+l 

and 

(1)(2)(3) ... (k)j(n+2)j(k+1)(k+2)(k+3) . . (n+l) 

The parallelism of course is no accident; it results from the fact that 

both expressions depict the same physical process. The reaction 

equation portrays the relations in terms of chemography; the univariant scheme portrays them in terms of the corresponding pressure 

and temperature coordinates. The formal one-to-one correspondence 

of the two representations is generally true. Any symmetry relation 

that exists among the compositions of the phases is reflected in the 

symmetry of arrangement of the univariant curves about the invariant 

point. As soon as a reaction is written, with- positive values for the 
14 p-T DlA'GRAMS, MULTIOOMPONENT SYSTEMS 

stoichiometric coefficients (that is, no phase is placed on the "wrong" 

side of the equation), the grouping of the corresponding univariant 

curves is determined. Clearly, if enough reactions can be specified 

to this extent, the relative arrangements of all the univariant curves 

about the invariant point can be determined. 

The present method, however, can carry the solution no farther. 

Three other categories of information about the univariant curves and 

their generative invariant points, important in the physical interpretation of phase diagran1s, cannot be derived by the method; these are: 

the location of the invariant point in the p-T diagram, the numerical 

values of the slopes of the univariant curves in the neighborhood of 

the invariant point, and the choice between a given sequence of the 

univariant curves (with specified sense of progression) and its mirror 

image ("enantiomorphic form") (see fig. 4; also Niggli, 1954, p. 408). 

The determination of these properties in general requires subsidiary 

data, either through determination of the thermodynamic properties 

of the phases, or through direct experimental determination of univariant equilibrium curves.5 Only for activity-activity diagrams can 

the choice of the enantiomorphic form and the numerical values of the 

slopes be detern1ined directly through the values of the stoichiometric 

coefficients of the univariant reaction equations (Korzhinskiy, 1959, 

p. 96; Zen, 1961a). 

(1) (1) 

(5) (5) 

(3) (3) 

FIGURE 4.-Two enantiomorphic forms of succession of univariant lines consistent with a given univariant scheme. The correct choice from the two 

forms for a given p-T diagram cannot be determined by the chemographic 

relations alone. 

3 Actually, only n+I slope values at the invariant point are independent; the (n+2)th slope is determined 

by the other n+I values because, for a given bulk composition, the net change in .!1Sand .11 Vare both zero 

for the system after it undergoes a complete p-T circuit around the invariant point. 
p-T DMGRAM!S, MULTICOMPONENT SYSTEMS 

SYSTEMATIC DISCUSSION 

ONE-COMPONENT SYSTEM 

15 

An invariant assemblage for a one-component system co~sists of 

three phases; the iriv.ariant point is surrounded by three univariant 

curves of two ph~es each. These curves separate three divariant 

fields of one phase each. The invariant point is a triple point. 

The three reaction curves and their corresponding uni variant 

schemes are: 

and 

1=2 

0) I (3) I (2) 

1=3 

(1) 1(2) 1(3) 

2=3 

(2}1(1) j(3) 

The only arrangements of the curves consistent with the requirements 

of the univariant schemes are given by the two possibilities of figure 5. 

The result is geometrically reasonable: because there are only three 

curves while two are needed to define a sector, no more than one 

metastable extension can be found within a sector subtended by 

~ 180°, defined by two stable curves. Therefore, no fewer than one 

metastable extension must exist within this sector. The stablemetastable succession around the in~ariant point results. 

(1) (2) (3) 

TEMPERATURE 

A 

TEMPERATURE 

8 

FIGURE 5.-The general geometric relations of the three univariant lines 

about the triple point in a one-component system. Two enantiomorphic 

forms arc shown; both are consistent with the relations depicted in 

figure 6. 
16 p-T DMO:RAMS, MULTIOOMPON·ENT SYSTEMS 

Figure 5 may also be instructively derived by considering the 

Gibbs free-energy surfaces for the three phases (see Darken and 

Gurry, 1953, p. 311). At the pressure of the invariant point, Pa, the 

G-T section of the surfaces for the three phases will be as given in 

figure 6A. (The lines for phases 1, 2, and~ may be thought of as for 

steam, water, and ice, respectively, in the system H20; the slopes 

are all negative and the steepest curve represents the phase of greatest 

molal entropy content.) At slightly higher (or lower) pressures the 

three curves cease to nieet at a single point; the relations will be as 

given in figure 6B. Correspondingly, at a lower (or higher) pressure 

than Pa the relations will be as given in figure 60. 

At a given pressure, the sequence of stable phase-assemblages 

with increasing temperature is traced by the lowest set of free .. energy 

curves in the system. Thus for figure 6B, the sequence with rising 

temperature is 3 and 1; 2 is metastable. At the pressure pb, the only 

stable univariant line (appearing as a piercing point in the figure) 

that is intersected with rising temperature is (2); the metastable 

~nivariant lines intersected are (3) at a temperature below the stable 

intersection with (2), and (1) at a temperature above the stable 

interSection with (2). Correspondingly, at Pc the stable sequence of 

phases is 3-2-1; the intersections with stable univariant lines are 

(1) and (3), in that order with rising temperature. The intersection 

with (2) is metastable and at a temperature between the two stable 

intersections. 

Figure 6A shows that at the pressure of the invariant point, p~, 2 is 

metastable throughout except at the point itself. This fact is expressed in figure 5; clearly two and only two divariant fields could 

straddle the isobaric line Pain the neighborhood of the invariant point. 

A B 

TEMPERATURE ~ TEMPERATURE~ TEMPERATURE---+ 

FIGURE 6.-Three isobaric G-T sections (lines) of the G-p-T surfaces of three 

polymorphs. A, At p=p., the pressure of the triple point, the three lines 

intersect at a single point. B, At p=pb, slightly higher (or lower) than p •. 

C, At p=pc, slightly lower (or higher) than p.. In each diagram, the sequence 

of stable univariant and divariant assemblages with rising temperature is 

given by the lowest set of free-energy lines. Note labels refer to phases, not phases 

absent. 
SYSTEMATIC DISCUSSION 17 

Moreover, in both figures 5A and 5B, 1 is the hi~h-temperature phase, 

3 the low-temperature phase, and 2 the intermediate phase, being 

more stable than 3 but less stable than 1 at higher temperatures. 

This relation reflects the assumption that the entropies of the phases 

are such that 81>82>83• 

However, to choose between the two mirror images of figure 5, we 

need information on the molal volumes of the phases: geometric 

considerations alone do not suffice. Let us suppose that V1 > V3> V2, 

so that at a given temperature rising pressure favors 2 relative to both 

1 and 3, and 3 relative to 1. These conditions cannot be fulfilled by 

figure 5A, but are compatible with figure 5B. The diagram thus 

might look like figure 7. The numerical values of the slopes can be 

determined by the Clapeyron equation, 

dP/dT= !lSI~ V; 

(1) 

2 

(Water) 

t 3 

(2) 

TEMPERATURE -

FIGURE 7.-The p-T diagram for the phases ice, water, and 

steam in the unary system H20; the diagram is consistent 

with figure 58. 
18 p-T Dr.AGRAMB, MULTIOOMPONENT SYSTEMS 

however, to locate the curves and the triple point, additional thermodynamic data (for example, enthalpies of the :phases) are needed. 

TWO-COKPONBNT SYSTBK 

Chemographically, all phases in a binary system can be represented 

along a straight line. The invariant point is a quadruple point, 

involving four phases. Four univariant curves emanate from the 

invariant point, and, according to the combinatorial formula, there 

are 4!/2!2!=6 distinct divariant assemblages consisting of two phases 

each. 

We consider here only the situation where the compositions of all 

four phases are distinct and fixed; the situation where two or more 

phases have the same composition is treated in the section on degenerate systems. The chemography of the phases, therefore, may be 

represented by figure 8. Phases 1 and 4 are related by the symmetry 

of their chemographic locations, as are phases 2 and 3. 

The four univariant reactions and their corresponding univariant 

schemes are: 

(1): 2+4=3 

(2)(4) 1(1) 1(3) 

(2): 1+4=3 

(1) (4) 1(2) 1(3) 

(3): 1+4=2 

(1) (4) 1(3) 1(2) 

(4): 1+3=2 

(1) (3) 1(4) 1(2), 

With (1) as the reference line, (2) and (4) are on one side and (3) is 

on the other. However, the diagram is not yet determined because 

the mutual relation of (2) and (4) and their relation to the metastable 

part of (3) are not known. The third univariant scheme, however, 

shows that, relative to (3), (1) and (4) are on the same side whereas 

(2) is on the opposite side; (4) must therefore lie between (1) and (2), 

2 3 4 

n~.----~------------~------~----~-----n2 I I I 

FIGURE 8.-The designation of the four phases in a binary 

system. 
SYSTEMATIC DISCUSSION 19 

and the metastable extension of (3) must lie between the stable parts 

of (2) and (4). These considerations uniquely fix the diagram, as in 

figure 9; it is readily established that the arrangement ' agrees with the 

second and fourth univariant schemes. 

The four sectors of the p-T surface are occupied by the assemblages 

1 +4, 1 +3, 2+3, and 2+4. These four assemblages completely~ 

define the possible relations, for the two remaining divariant assemblages, 1 +2 and 3 +4, are thereby implied. 

The assemblage 1 +2 lies between (3) and (4) and straddles (1); 

the assemblage 3+4 lies between (1) and (2) and straddles (4). 

Similarly, the assemblage 1 +4 straddles the metastable extensions 

of (1) and ( 4), the assemblage 2 +4 straddles the metastable extension of (2), and so forth, so that the overlap rule is obeyed. Notice, 

however, that the converse of the overlap rule is emphatically invalid: 

for example the assemblage 2+3 does not straddle the curves (2) or 

(3), either stably or metastably. 

The chemographic symmetry between 1 and 4, and also between 2 

and 3, are reflected in the mirror symmetry of the disposition of the 

univariant curves. 

Note that in the sector (2, 3) (assemblage 1+4), there exist two 

metastable extensions; this sector is flanked by two sectors each with 

(1) 

(3) 

(4) 

(2) 

FIGURE 9.-The unique geometric relation of the four univariant lines 

about a given invariant (quadruple) point in a binary system. 

218-934 0-66-4 
20 p-T DI•AGRAMS, MULTIOOMPON·ENT SYSTEMS 

one metastable extension, and finally the sector (1, 4) contains no 

metastable extension. This arrangement is unique in a binary system. 

As the phase reactions are traced around the invariant point, only 

one event occurs each time a univarht:nt curve is crossed; for example, 

3 appears between 1 and 4 across (2). This is as it should be, for 

each event represents a distinct reaction; simultaneous occurrence of 

two or more events requires a most unlikely coincidence of two or 

more independent slopes of univariant equilibria. 

On a purely geometric basis, the sequence of stable and metastable 

curves around an invariant point in a binary system can be derived 

from the corresponding diagram for the unary system. In a unary 

system, there is a st.able curve between two adjacent metastable 

curves, and vice versa. Let these be labeled aa', bb', and cc' (fig. 10). 

When the invariant point generates four univariant curves, as in a 

binary system, one more curve, dd', must be added. This added 

curve inevitably divides one of the three stable sectors into two parts, 

one of which can contain no metastable extension (sector cod in fig. 10), 

a 

b 

FIGURE 10.-Derivation of the geometric relations of a binary system from 

that of a unary system. If aa', bb', and cc' arc the three univariant lines of 

a unary system, the addition of a fourth line, dd', required by a binary system, necessarily results in one sector having no metastable extension and 

its opposite sector having two metastable extensions. Each of the two 

flanking sectors has one metastable extension. 
SYSTEMATIC DISCUSSION 21 

and the other of which must contain one metastable extension (sector 

doa)... The metastable part of the added curve, od', extends into one 

of the adjacent stable sectors which initially contains one metastable 

extension. By the lVIorey-Schreinemakers rule, both sectors bod and aod 

are ~ 180°, therefore the sector that now contains two metastable 

extensions cannot be adjacent to the sector that now contains none. 

THREE•COMPONENT SYSTEM 

A system of three components can be depicted chemographically by 

a plane triangle. The invariant point in a three-component system 

consists of five phases; it is a quintuple point. From the invariant 

point, five univariant lines of four phases each emanate; from these 

curves, 10 divariant regions of three phases each develop. 

Barring accidents of colinearity in compositions, which will be 

discussed in the section on degenerate systems, the compositions of 

the five phases that occur at the quintuple point are related to one 

another in one of three ways: (a) the five phase compositions can be used 

to define a pentagon without reentrant angles (fig. 11A); (b) four of 

the five phase compositions can be used to ~fine a quadrilateral 

without reentrant angles, whereas the fifth phase composition falls 

within the limits of the quadrilateral (fig. 12A); and (c) three of the 

five phase compositions can be used to define a triangle, whereas the 

remaining two phase compositions fall within the limits of the triangle 

(fig. 13A). These three different chemographic relations will be separately analyzed. 

(1) 

(4) 

n2'------------~n3 

A 8 

FIGURE 11.-The p-T diagram in a ternary system where the five phases form a 

convex pentagon. A, The chemographic relations; B, the corresponding p-T 

diagram. Notice that the univariant curves follow in numerical sequence 

when the phases are labeled in diagonal sequence. 
22 p-T DI!A:GRAMS, MULTICOMPONENT SYSTEMS 

3~1 (S) 

5 2 

"2 "3 (4) 

A 8 

FIGURE 12.-The p-T diagram in a ternary system where four of the five 

phases form a quadrilateral and the fifth phase is an interior point. See 

also caption for figure 11. 

Case (a) Five phase compositions define a pentagon.-For reasons 

shown below, the phases are labeled in the diagonal <;>rder shown in 

figure 11A. The five uriivariant reactions and their univariant 

schemes are: 

(1): 2+3=4+5 

(2){3)1(1)1(4)(5) 

(2): 1+5=3+4 

(3): 

(1)(5)1(2)1(3)(4) 

1+2=4+5 

(1)(2)1(3)1(4)(5) 

(4): 1+5=2+3 

(1)(5)1(4)1(2)(3) 

(5): 1+2=3+4 

(1)(2)1(5)1(3)(4) 

Relative to curve (1), (2) and (3) lie on one side and (4) and (5) on 

the opposite side. Relative to (2), (1) and (5) must be"'on one side 

and (3) and (4) on the opposite side. The sequence of curves therefore 

must be (1), (2), (3), (4), and (5) around the invariant point. Because the chemographic relations of the five phases are entirely 

symmetrical, so that upon cyclic rotation of the phase indices the 

mutual relations of the phases are topologically indistinguishable 

from the original, the five univariant lines also must be symmetrically 

disposed with respect to one another, and there must be one metastable 

extension between each pair of adjacent stable curves. The diagram 

must be as shown in figure llB. The univariant curves succeed one 

another in numerical order when the phases are labeled diagonally 

in the chemographic representation. 
SYSTEMATIC DISCUSSION 23 

(2) 

(5) 

8 

FIGURE 13.-The p-T diagram in a ternary system where three of the five phases 

form a triangle and the remaining two points fall inside the triangle. See also 

caption for figure 11. 

The chemography in the five sectors of the p-T diagram can be 

readily ascertained. For instance, between (1) and (2) lies the assemblage (1,2) =3+4+5, which defines the chemography of the sector. 

Each such divariant assemblage is indicated by a triangle of threephase coexistence. Clearly, a triangle indicating an assemblage 

such as (1,2) gives rise to two other triangles which complete the 

five-phase chemography. In the example used, these are (1, 3)=2+4 

+5 and (2, 5) = 1 +3+4. These three divariant assemblages could 

exist under the same set of physical conditions for different bulk 

compositions. The chemography of the five sectors is indicated in 

figure llB; notice that in crossing each univariant curve only one 

pair of tie lines is switched. To switch mote than one pair of tie 

lines under a given set of p and T values requires coincidence, provided 

only at the invariant point; extension of such a coincidence over a 

continuous set of p-T conditions, as along a univariant curve, is 

extremely improbable. 

Although in figure llB each sector has three possible divariant 

assemblages, depending on the bulk composition, the total number of 

divariant assemblages is only 10 because each of the 5 nondefining 

assemblages (all of which have two "outside" edges of the pentagonal 

figure) occurs in two sectors. Thus we have (1, 3) =2+4+5 between 

(1) and (3) and straddling (2). The overlap rule is obeyed. 

The succession of univariant reaction curves in figure liB can be 

deduced also by symmetry considerations. As has been discussed 

(p. 22), the five phases of figure llA are symmetrically disposed to 

one another, so between each adjacent pair of stable univariant curves 
24 }.}-T DI•AGRAMS, MULTICOMPON'ENT SYSTEMS 

there must be one metastable extension, and vice versa. Next, we 

note that each defining divariant assemblage must possess two inside 

diagonals as its boundary, for only such triangles unequivocally 

delimit the entire five-phase region. There are just five such triangles, 

one for each sector. One of these defining triangles, for example, is 

3+4+5 = (1, 2); for this assemblage the stable parts of (1) and (2) 

must be adjacent univariant curves. Similarly, (2) and (3) must be 

adjacent. By symmetry, (1) and (5) must be adjacent on the side 

away from (2), and (4) must be next to (5) on the side away from (1). 

Case (b) Four phase compositions define a quadrilateral and the 

fijth composition point is inside the quadrilateral.-The phases can be 

labeled in the following manner: 

Construct the two diagonals in the quadrilateral. The fifth 

phase, barring degeneracy, will fall within one of the four sectors. 

The resulting figure can be made into the pentagonal figure of figure 

11A if the "inside" phase is "pulled" outside through the adjacent 

leg of the quadrilateral, that is, without crossing more than one line. 

Next, designate the five phases in the same way as before, using the 

diagonal rule; then restore the "inside" phase to the original position 

within the quadrilateral. We can then complete all the diagonals 

for the five-sided figure; the result is shown in figure 12A. 

The five univariant reactions and the univariant schemes are: 

(1): 2+3=4+5 

(2) (3) 1(1) 1(4) (5) 

(2): '1+5=3+4 

(1) (5) 1(2) 1(3) (4) 

(3): 1+2+5=4 

(1)(2)(5)1(3)1(4) 

(4): 1+5=2+3 

(1){5)1(4)1(2) (3) 

(5): 1+2+3=4 

(1) (2) (3) I (5) I c 4) 

The curves (1), (2), and (4) are formally identical with the corresponding curves for the example in case (a). The schemes (1), (3), 

and (5), on the other hand, are formally identical with case (c) in 

the next section. Neither of these triplets of lines can be used to 

define the p-T diagram. However, if the curves (2), (3), and (4) are 

selected, for example, the unique result of figure 12B is obtained, in 

which the progression of the labels for the univariant curves is again 

sequential. 

We observe again that each univariant curve brings about one 

shift of tie lines, or a single disappearance of a phase that results 

from the intrinsic instability of this phase relative to a divariant 

assemblage of the same bulk composition. Where 4 appears or 
.SY1STEMATIC DISCUSSION 25 

disappears, thus, it does so in or into a field of three phases so that 

the divariancy of the assemblage is not violated: 4 does not disappear 

across (1) or (2). 

There are again 10 divariant assemblages altogether, but the 

distribution of the 5 nondefining assemblages is different from that 

of case (a). Three of the sectors have four assemblages each, and 

two sectors have only two assemblages each. This distribution 

reflects the fact that 4 is intrinsically not stable in these two latter 

sectors which share the boundary (4). The situation is different from 

case (a), in which no phase is intrinsically unstable. We notice that 

the overlap rule and also the Morey-Schreinemakers rule are observed. 

The arrangement of the stable and metastable segments of the 

univariant curves reflects the bilateral symmetry of the chemography: 

2 and 5 have their symmetry counterparts in 1 and 3, and 4 occupies 

a special position. 

Case (c) Three phase compositions define a triangle, and the two 

remaining composition points fall inside the triangle.-The relation 

corresponds to that of figure 13A. In order to number the phases 

according to the diagonal rule, all the possible connecting lines are 

drawn among the phases, and, as in case (b), each inside phase is 

mentally transferred to the outside in such a way as to cross only 

one line. 

The univariant reaction curves and their univariant schemes are: 

(1): 2+3=4+5 

(2) (3) I< 1) I (4) (5) (2): 1+4+5=3 

(1)(4)(5)1(2)1(3) (3): 1+2+5=4 

(1)(2)(5)1(3)1(4) (4): 1+2+5=3 

(1)(2)(5)1(4)1(3) (5): 1+2+3=4 

(1)(2)(3)1(5)1(4) 

Because (1), (3), and (5) are formally identical with their counterparts 

in case (b), this triplet of univarian t schemes cannot be used to define 

the succession of curves. We can use any other combination, however, as for example (2) and (5), and the Morey-Schreinemakers rule. 

The result is shown in figure 13B. 

The succession of univariant curves follows in sequential order 

of their labels. Each sector contains one defining divariant assemblage as well as zero, two, or fo11r additional divariant assemblages 

which lap over one or more univariant curves, so that, in all, 10 

distinct divariant assemblages exist. The nondefining assemblages 

obey the overlap rule as well as the Morey-Schreinema!ters rule. 
26 p-T DM.GRAMJS, MULTICO:MPONENT SYSTEMS 

Chemographical1y, phase 1 occupies a special symmetry position, 

where 3 and 4 as well as 2 and 5 are symmetrically disposed towards 

each other. These facts are fully expressed by the symmetry relations 

of the univariant curves in figure 13B, as well as by the symmetry of 

the chemography of the divariant assemblages in the sectors. The 

sectors having the fewest phases have the greatest number of metastable extensions, and vice versa. 

The three types of succession of univariant curves in a ternary 

system can be appreciated also from a purely geometric point of view. 

We begin with the unique arrangement for a binary system (fig. 9). 

To change to a ternary system, one univariant curve must be added 

to the invariant point. As figure 14 shows, there are only three 

ways to do this, which lead to the distinctodiagrams of cases (a), (b), 

and (c) above. Each case can be derived by adding a stable curve to 

the correspondingly labeled sector; for example, if a stable curve 

were added to the sector labeled (a), case (a) results. Thus the three 

types of ternary p-T relations, which were derived from the chemographic relations and the Morey-Schreinemakers rule, are also direct 

consequences of geometry. 

FIGURE 14.-Derivation of the three types of p-T diagrams for ternary systems 

from the unique configuration for a binary system. Addition of a fifth curve 

in the sectors labeled a, b, or c results in figures 11, 12, or 13, respectively, by 

geometric considerations alone. 
·SYSTEMATIC DISCUSSI'ON 27 

SYSTEMS OF MORE THAN THREE COMPONENTS 

Graphic portrayal of systems which have more than three components is difficult and will not be attempted. If the compositions 

of the phases are kno\vn, however, the appropriate balanced univariant reaction equations can be written, and these equations define 

the correct sequence of univariant curves about the invariant point. 

Two independent and nondegenerate univariant equations ultimately 

suffice, because all the other equations can be derived from linear 

combinations of these two equations. 

The balancing of univariant reaction equations of a system of n 

components can be done by solving a set of simultaneous linear 

algebraic equations that relate n + 1 unknowns with n equations. 

When n is a large number (in pra,ctice when n ~4), the use of determinants may be helpful. A simple illustration of the use of determinants is given on page 46; .additional examples may be found 

in Korzhinskiy (1959, p. 103). 

The pencil theorem, discussed below, predicts the total possible 

number and geometric arrangements of distinct univariant curves 

about an invariant point for any given number of n, so that for systems 

of more than three components, the construction of p-T diagrams 

can be readily accomplished. 

THE PENCIL THEOREM 

The p-Tplane in the neighborhood of an invariant point is separated 

by adjacent pairs of stable univariant curves into a number of sectors. 

Within each sector, there may be zero, one, two, or more metastable 

extensions of univariant lines. The topological arrangement of the 

stable and metastable curves about an invariant point, thus, can be 

uniquely specified by giving the number of metastable extensions 

within each sector and by listing the sectors sequentially around the 

invariant point. For example, in a one-component system the succession of lines can be given as "1,1,1," which indicates that there is 

just one metastable extension in each sector and that there are three 

sectors altogether. Likewise, for a binary system the statement is 

"1,2,1,0," which indicates that there are four sectors, that one sector 

contains two metastable extensions, that this sector is flanked by two 

sectors each containing one metastable extension, and that the fourth 

and last sector contains no metastable extension. 

In the three-component system, the three distinct types of relations 

may be similarly summarized. Thus case (a) corresponds to the 

statement "1,1,1,1,1"; case (b) corresponds to the statement 

"1,0,2,2,0"; and case (c) corresponds to the statement "1,3,1,0,0." 

The nota.tion, however, can be further simplified by means of the 

concept of a "pencil." Let us define a "pencil" of stable univariant 
28 p-T DI·AGRA.MlS, MULTICOMPONENT SYSTEMS 

curves in the following manner (the following statement will remain 

true if the terms "stable" and "metastable" are interchanged): A 

pencil of stable univariant curves about an invariant point consists 

of all those curves that are not separated from one another by 

metastable extensions of univariant curves.6 

Thus a pencil is limited on both sides by metastable curves. A 

pencil may be designated by the number of stable univariant curves 

it contains; the sequence of pencils around an invariant point is 

preserved by the sequential listing of these numbers. Such a. sequence of numbers, enclosed in brackets, completely specifies the 

topology of univariant lines about an invariant point. 

Thus, the pencils about a unary invariant point may be designated 

[1,1,1]. The pencils about a binary invariant point are [2,1,1] because the sector previously designated as "0" does not constitute a 

separate pencil. A ternary invariant point may be in [1,1,1,1,1], 

[1,2,2], or l1,1,3] arrangement; the sum of the numbers in all the 

pencils is n+2. 

We can now give the remarkable pencil theorem of Scbreinemakers 

(1915, 5, p. 1027): The number of pencils about a given invariant 

point is always odd. The proof of the theorem is simple. In figure 

15 let each solid line, P1, P2, ••• , Pk, represent schematically a pencil 

of unspecified number of stable univariant lines. Similarly each 

dashed line represents the pencil of metastable extensions. There 

are k pencils in a:U. Let the metastable part of P1 fall between the 

stable parts of P r+t and P r+2 (by definition, there must be one and 

only one metastable pencil between any two adjacent stable pencils). 

In traversing from tbe stable part of P1 to its metastable part clockwise around the invariant point, r stable pencils will be crossed. As 

between each adjacent pair of stable pencils there is a metastable 

one, the same number, r, of metastable pencils are crossed. However, each metastable pencil in the clockwi-se traverse implies a stable 

pencil in the counterclockwise traverse, so that the total number of 

stable pencils, including P1 itself, is k= (2r+ 1) where r is an integer 

51 (by the Morey-Schreinemakers rule, the minimum number of 

pencils is 3). 

With the aid of the pencil theorem, all the possible geometric 

arrangements of univariant curves about an invariant point in any 

polycomponent system can be listed; each arrangement corresponds 

to a unique set of chem~graphic relations. These are given in table 

1 for n to 7. In a complex system, the sequence of numbers is im-

• This definition is Schreinemakers' (1915, 5, p. 1026). Schreinemakers called these pencils "bundles." 

Korzhinskiy, however, used the term "Schreinemakers' bundles" (1959, p. 96) to designate the totality of 

univariant curves about a given invariant point. To avoid possible confusion, therefore, the term "pencil," 

introduced by Palatnik and Landau (1964, p. 12), is used for tM "bundle" in the original sense of 

Schceinemakers. 
SY'STEMATIC DISCUSSION 29 

Figure 15.-Demonstration of the pencil theorem. Each stable line labeled P,, 

P2, ... , represents a pencil of stable univariant lines which are not separated 

from one another by metastable extensions of other univariant lines. The 

total number of pencils is always odd. 

portant; thus the sequence of pencils [1,2,1,2,1] in a quinary system 

is different from the sequence l1,2,2,1,1], but the latter is identical 

with [1,1,1,2,2] or [2,1,1,1,2]. 

By analogy with the Morey-Schreinema.kers rule, we can state: the 

angle subtended by a. pencil about the invariant point is no greater 

than 180°. This rule is useful in the construction of p-T grids that 

consist of many invariant points in the same chemical system (Zen, 

1966). 

Knowledge of the pencil theorem clearly aids in the construction 

of invariant points and associated univariant curves in systems of 

many components. 
30 p-T DFAGRAMS, MULTICOMPONENT SYSTEMS 

TABLE !.-Possible pe·acils for n -~7 

Number of Number of Topological arrangement of the pencils co.~ponents, n pencils 

1_-- __ . ____ -------

2_- -------------- 3 _______________ _ 

4_--- ------------

5_- --------------

6_- --------------

7----------------

3 [1, 1, 1] 

3 [1,1,2] 

5 [1, 1, 1, 1, 1] 

3 [1, 2, 2], [1, 1, 3) 

5 [1, 1, 1, 1, 2] 

3 [1, 1, 4], [1, 2, 3], [2, 2, 2] 

7 [1, 1,1, 1,1, 1, 1] 

5 [1, 1, 1,1, 3], [1, 1,1, 2, 2], [1, 1, 2, 1, 2] 

3 [1, 1, 5], [1, 2, 4], [1, 3, 31, [2, 2, 3] 

7 [1, 1, 1, 1, 1, 1, 2] 

5 [1, 1, 1, 1, 4], [1, 1, 1, 2, 3], [1, 1, 2,1, 3], [1, 1, 2, 2, 2], [1, 2, 1, 2, 2] 

3 [1, 1, 6], [1, 2, 5], [1, 3, 4], [2, 2, 4], [2, 3, 3] 

9 [1, 1, 1, 1, 1, 1, 1, 1, 1) 

7 [1,1, 1, 1, 1, 1, 3], [1, 1,1, 1,1, 2, 2], [1, 1, 1, 1, 2, 1, 2], [1, 1, 1, 2, 1, 1, 2) 

5 [ 1,1 ,1.1.5]. [ 1.1,1,2,4]. [ 1.1,2.1,4]. [ 1.1.1.3,3 J. [ 1.1,3,1,3·]. 

3 [ 1.1.2.2.3]. [ 1,1,2,;{,2]. [ 1.2.1,2.:-q' [ 1.2.2,1,:-q. [] .2,2.2.2] 

[ 1.1.1 J, [ 1,2,61. [ 1.3,51. [ 1.4.41. [ 2,2,51, [2.3,41, L3.a.aJ 

DEGENEBATE SYSTEMS 

A system is said to be degenerate if either of the following coincidences exists among the phases participating at a given invariant 

point: 

(a) Two or more phases have the same composition, that is, are 

polymorphs of one another. 

(b) In a n-component system, three or more phases are compositionally colinear (for n=3, 4, 5, ... ) ; four or more phases are compositionally coplanar (for n=4, 5, 6, ... ), and so forth. 

In both instances, certain assemblages of phases, because of compositional coincidences, participate in univanant equilibria which can 

be described by fewer than n components. 

ONE-COMPONENT SYSTEM 

All ppases in an one-component system are degenerate in a trivial 

sense. 

TWO-COMPONENT SYSTEM 

Four chemographically distinct types of degeneracy are possible 

in a binary system as shown in figure 16, where a heavy dot is used to 

denote degenerate compositions. Clearly, compositional colinearity 

can play no part in binary degeneracy. 

Those phases which are related to one another by fewer than n 

components (and therefore fewer than n+ 1 phases in the corresponding univariant equilibria) are called the "singular phases;" the remaining phases are called the "indifferent phases" (Schreinema.kers, 

1916, 10, p. 515). These compositional coincidences lead to special-
1,2,3 

• 

1,2 

• 

3 

I 

A 

c 

SY'STEMATIC DISCUSSI'ON 

4 

I 

4 

I 

1,2 

• B 

2,3 

• D • 

3.4 

• 

4 

I 

FIGURE 16.-Four possible sets of degenerate relations in a 

binary, four-phase system. A heavy dot indicates that 

the composition point is shared by two or more phases. 

31 

ized arrangements of the univariant curves around the invariant 

point (see Niggli, 1930) as is discussed below. 

Two kinds of indifferent phases can be distinguished: (1) those 

which are absolutely indifferent and participate in none of the univariant equilibria of the system and (2) those which are relatively 

indifferent and participate in some of the univariant equilibria (Sobel'-

man, 1964, p. 545). Relative indifference leads to the coincidence of 

two or more univariant curves about the invariant point. Absolute 

indifference leads t~ the coincidence of the curves bearing the labels 

of these phases with all other univariant curves in the system. 

Figure 16A depicts a situation of unary invariant and univariant 

equilibria involving phases 1, 2, and 3; 4 cannot participate in any of 

the reactions. The p-T diagram therefore is figure 17 A; 4 may be 

present throughout the region. 4 is an absolutely indifferent phase; 

1, 2, and 3 are the singular phases. 

In figure 16B, reactions between 1 and 2 and also between 3 and 4 

describe unary univariant equilibria; there can be no participation of 

1 or 2 in reactions involving 3 and 4, and vice versa. The p-T 

diagram must correspond to figure 17 B. 

In contrast, figures 160 and D lead to univariant equilibria which 

involve both the singular and the relatively indifferent phases. In 

figure 160, the four univariant equilibria and their univariant schemes 

are: 

{1): 2+4=3 

{2){4) 1(1) 1(3) 

{2): 1+4=3 

(1){4) 1{2) 1(3) 

(3): 1=2 

(4): 

(1) 1(3) 1(2) 

1=2 

(1) 1(4) 1(2) 
32 p-T DI•AGRAMS, MULTICOMPONENT SYSTEMS 

2 2 4 '--------' '--------' 

3 

L..----...J 

1 

L..----...J 

A 8 

1 2 L..----...J ~ 

c D 

FIGURE 17.-The p-T diagrams corresponding to the four sets of degenerate 

relations of figure 16; A refers to figure 16A and so forth. Lines with 

multilabels are degenerate, that is, each of them represents the coincidence of 

two univariant lines. The coincidence may be stable to stable (A and D) or 

stable to metastable (B and C), depending on whether the indifferent phases are 

on opposite sides or the same side, respectively, of the singular phases. In all 

cases, however, three distinct slope values obtain at the invariant point. 

Curves (3) and (4) refer to the same reaction, so the two curves must 

coincide; curves (1) and (2) lie on opposite sides of this combined 

curve. As (2) and (4) lie on the same side of (1), the disposition of 

curves must conform to figure 17 0. Note that although the combined curve (3)(4) courses through the quadruple point continuously, 

the curve does change its label; the metastable part of one curve 

coincides with the stable part of the other. 

The above example illustrates a general rule, the "coincidence rule," 

which states that when the two indifferent phases chemogra.phica.lly 
SY'STEMATIC DISCUSSION 33 

lie on the same side (about a point in a binary system, about a line 

in a ternary system, about a plane in a quaternary system, and so 

forth) of the singular phases, then the univariant curves bearing the 

labels of the indifferent phases coincide stable to metastable; when 

the two indifferent phases lie on opposite sides of the singular phases, 

then the univariant curves bearing the labels of the indifferent phases 

coincide stable to st;able (Schreinemakers, 1916, 10, p. 522; Morey 

and Williamson, 1918, p. 69). 

In figure 16D the univariant equilibria are: 

(1): 

(2): 

(3): 

(4): 

2=3 

(2) I 0) I (3) 

1+4=3 

0) (4) I (2) I (3) 

1+4=2 

(1) (4) I (3) I (2) 

2=3 

(2) J<4) 1<3) 

Curves ( 1) and ( 4) refer to the same reaction and they must coincide; 

curves (2) and (3) lie on opposite sides of th~ combined curve. (1) 

and (4), however, must be on the same side of (2) or (3), so that the 

curves (1) and (4) must coincide stable to stable and metastable to 

metastable, the stable part terminating at the invariant point. The 

coincident curve possesses the double label (1)(4). According to (2), 

the metastable part of (2) must lie between the stable parts of (3) 

and (1)(4); the disposition of the univariant curves and the divariant 

chemographic relations therefore must b~ as given in figure 17D. 

The relations obey the second part of the coincidence rule. 

In figure 17 0 four curves radiate from the invariant point, in figure 

17 D three ..curves do so; however, only th:w;ee distinct slope values are 

associat.ed with the curves at each invariant point. In a nondegenerate system, as we have seen, there are n+2 curves having n+2 

distinct slope values at each invariant point. Figures 17 0 and D are 

examples of singly degenerate systems, each showing n+2-1=n+1 

distinct slope values. 

THREE-COMPONENT SYSTEM 

Different sets of degenerate relations are possible in a ternary system 

because degeneracy could result both from polymorphism and from 

compositional colinearity, and because for five phases as many as two 

independent sets of degenerate relations could occur simultaneously. 

In all, 16 possibilities exist, as given in figures 18-24. 
n3 

A B 

n3 

c 

n3 

D 

§ 

E F G H 

FIGURE 18.-The four possible sets of degenerate relations among the compositions of five phases in a ternary system (A-D) 

effected by the colinearity of three phases and t:tJ.eir corresponding p-T diagram types (E-H). Note that five stable univariant 

lines obtain when the indifferent phases lie on the same side of the singular phases; four stable univariant lines obtain when 

the indifferent phases lie on opposite sides of the singular phases. In all events, four distinct slope values obtain at the invariant 

point. 

~ 

~ 

l 

t::l 

~ 

_I 

I( 

~ 

§ 

~ ~ ~ 

~ 

~ 

~ t.ZJ 

~ 
SYSTEMATIC DISCUSSI'ON 35 

One compositional colinearity of three phases (jig. 18).-We give 

illustrations below. The reactions corresponding to figure 18A are: 

(1): 2+3=4+5 

(2): 1+5=3+4 

(3): 1+2=4 

(4): 1 +5=2+3 

(5): 1+2=4 

These relations show that curves (3) and (5) are identical but for the 

label; 3 and 5 are the indifferent .phases for the singular reactions 

depicted by these two curves. Inasmuch as (3) and (5) occur on 

opposite sides of (1), (2), and (4), the univaria.nt curves must be 

disposed as shown in figure 18E, which verifies the coincidence rule: 

the two indifferent phases occur on the same side of the singular 

phases, and the curves (3) and (5) coincide stable to metastable. 

Five curves radiate from the invariant point, but they have only four 

distinct slope values. The symmetry of the chemographic relations 

is carried over, as it should be, into the p-T·diagra.m. 

The five univariant reactions corresponding to figure 18D are: 

(1): 2+3+4=5 

(2): 1+5=4 

(3): 1+5=4 

(4): 1 +2+3=5 

(5): 1+2+3=4 

The indifferent phases are 2 and 3, and curves (2) and (3) describe 

the same reaction and are coincident. Because (2) and (3) occur on 

the same side of the other three reactions, the combined curve terminates at the invariant point. The p-T diagram must be as given in 

figure 18H, having only four distinct curves and fow slope values. 

The p-T diagrams corresponding to figures 18B and 0 are figures 

18F and G, the derivation of which is straightforward. 

The coincidence rule can be given a simple physical explanation. 

If the two indifferent phases chemographically lie on the same side 

of the singular phases (for example, fig. 18A), any univariant reaction 

that involves these two phases must have them on opposite sides of 

the reaction as a direct consequence of the compositional degeneracy 

of the singular phases, together with the fact that the bulk composition 

must be the same for the two sides of the univariant equation. In the 

set of univariant schemes, the two corresponding curves must lie on 

opposite sides of the three curves whose labels are those of the singular phases. These indifferent curves, therefore, must occur on 

both sides of the invariant point, and the coincidence is stable to 

metastable. 
36 p-T DMGRAMS, MULTICOMPONE:;NT SYSTEMS 

Conversely, if the two mdifferent phases chemographically lie on 

opposite sides of the three singular phases (for example, fig. 180), they 

must be on the same side of the reaction equation because of the 

compositional degeneracy of the singular phases, combined with the 

fact that the bulk compositions for the two .sides of the reaction must 

be equal. Hence, the curves that bear the labels of these indifferent 

phases lie on the same side of the three curves bearing the labels of the 

singular phases, and the two indifferent curves must terminate at the 

invariant point. The coincidence must be stable to stable. 

One compositional coincidence of two phases (jig. 19).-Figure 19A 

will illustrate the relations. The five univa.riant reactions are: 

(1): 4+5=2+3 

(2): 1=4 

(3): 1=4 

(4): 1 +5=2+3 

(5): 1=4 

Here we have three indifferent and two singular phases. Accordingly, 

curves (2), (3), and (5) coincide. Because (2) and (3) lie on the 

D E F 

FIGURE 19.-The three possible sets of compositionally degenerate relations 

in ternary, five-pha.Se systems effected by a single compositional coincidence 

(dimorphism) of two phases (A-C) and their corresponding p-T diagram 

types (D-F). Note that four stable lines obtain when the indifferent 

phases are on the same side of the singular phases; three stable lines obtain 

when the indifferent phases are on opposite sides of the singular phases. 

In all cases, three distinct slopes values obtain at the invariant point. On 

the p-T diagrams, only those phases are labeled whose identities, because 

of compositional coincidences, would otherwise be ambiguous. Heavy 

dots indicate phases with coincident compositions. 
SYSTEMATIC DISCUSSOON 37 

same side of (1) or (4) and (5) on the other side, (2) and (3) coincide 

stable to stable but (5) coincides with (2) and (3) stable to xp.etastable. The disposition of the curves and their met astable extensions 

must be as shown in figure 18D; four curves radiate from the invariant 

point, but these have only three distinct slopes. 

The p-T diagrams corresponding to figures 19B and 0 are, respectively, figures 19E and F. In both instances, there exist three distinct 

slope values, and one of the curves bears a triple label. For figure 19F, 

the three indifferent curves all coincide stable to stable, reflecting the 

chemographic symmetry of phases 1, 2, and 3 in their mutual relations 

and in their relations to 4 and 5. In figure 19E, (2) and (3) coincide 

stable to stable, but (5) coincides stable to metastable, again reflecting 

the chemographic symmetry. 

Figures 19D and E are similar. In figure 19D, the smaller angle 

between (1) and (4) is on the side of (5), but the converse is true in 

figure 19E. This difference reflects the fact that in figure 19A, 5 is 

outside the compositional triangle of phases 1-4-2-3, but it is inside 

the same triangle in figure 19B. 

Two compositional colinearities of three phases each (fig. 20).-

Corresponding to figure 20A, we have 

(1): 2+3=4+5 

(2): 1+5=3 

(3): 1+2=4 

(4): 1+5=3 

(5): 1+2=4 

Curves (2) and (4) are coincident; according to (1) the coincidence is 

stable to metastable. Similarly for curves (3) and (5). Moreover, 

relative to (1), the pair of curves (4) and (5) is on one side but the 

pair of curves (2) and (3) is on the other side; relative to (3), (1) and 

(2) are on the same side but (4) is on the opposite side. The p-T 

diagram must be as given in figure 20D. The five curves emanating 

from the invariant point possess three distinct slope values, agreeing 

with the existence of two independent single degeneracies, Figure 20D 

may be regarded as a special case of figure 11 where 3 becomes colinear 

with 1 and 5, and 4 with 1 and 2, so that curves (3) and (5) coincide 

and curves (2) and ( 4) coincide. It is clear that the relations can also 

be derived from figure 13 ·by analogous reasoning. The coincidences 

are all stable to metastable because the indifferent phases, in both 

instances, are on the same side of the respective singular phases. 

The p-T diagrams corresponding to figures 20B and 0 are derived 

likewise; they are given, respectiv~ly, in figures 20E and F. In each 

case there are three slope values at the invariant point. In figure 20E 

one coincidence is stable to stable and the other is stable to meta-
38 p-T DIIAGRAMS, MULTICOMPONEN'l' SYSTEMS 

D F 

FIGURE 20.-The three possible sets of compositionally degenerate relations in 

ternary, five-phase systems each effected by two independent sets of three 

colinear phases (A-C), and their corresponding p-T diagram types (D-F}. 

Depending on the chemographic relations of the indifferent and singular phases, 

five, four, or three stable univariant lines may obtain, but in all cases only 

three distinct slope values obtain at the invariant point. 

stable, so that four curves exist. In figure 20E, both coincidences are 

stable to stable, so that only three curves obtain. These relations 

fully reflect the chemographic arrangements. 

One compositional coincidence of two phases (dimorphism) and one 

colinearity of three phases (jig. 21).-Two situations may be distinguished: either these two degeneracies are coupled, that is, share one 

phase in common (fig. 21A and B), or not coupled (fig. 210). 

The chemographic relations to figure 21A or B are directly related 

to binary degeneracies depicted in figures 160 and D, w1th a fifth 

phase, 5, now added, which, however, is absolutely indifferent to all 

the binary univariant equilibria. The p-T relations are therefore 

given in figures 21D and E, strictly similar to figures 170 and D. 

Both figures display three distinct slope values. 

For the chemographic relation of figure 210, two entirely independent univariant equilibria, one unary and one binary, are obtained; 

the p-T diagram is given by figure 21F. 

011e compositional coincidence of three phases.-The chemographic 

relations, figure 22A, are analogous to the binary degeneracy of figure 

16A, with a fifth phase added which is absolutely indifferent to the 

equilibria in the system. The p-T diagram is given in figure 22B, 

displaying three distinct slope values. 
SYSTEMATIC DISCUSSroN 39 

FIGURE 21.-The three possible sets of compositionally degenerate relations in 

ternary, five-phase systems effected by the colinearity of three phases and a 

compositional coincidence of two phases (A -C), and their corresponding 

p-T diagram types (D-F). If the two degeneracies are coupled (A, B), 

three distinct slope values exist at the invariant point; if the two degeneracies 

are not coupled (C), only two distinct slope values exist at the invariant 

point. 20A and B correspond to the binary degeneracies of figures 17C 

and D, with an absolutely indifferent fifth phase added. On the p-T diagrams, only those phases are labeled whose identities, because of compositional coincidences, would otherwise be equivocal. Heavy dots 

indicate phases having coincident compositions. 

3 

D 

1 

D n2.~------------------------~n3 

A 8 

FIGURE 22.-The single possible set of compositionally degenerate relations 

in ternary, five-phase systems effected by one compositional coincidence of 

three phases, and its corresponding p-T diagram type. Phases 4 and 5 

are absolutely indifferent phases. On the p-T diagrams, only those phases 

are labeled whose identities, because of compositional coincidences, would 

otherwise be ambiguous. Heavy dots indicate phases with coincident compositions. 
40 p-T D!oAGRAMIS, MULTICOMPONENT SYSTEMS 

Two compositional coincidences of two phases each.-The chemographic relations (fig. 23A) are analogous to binary degeneracies 

depicted in figure 16B, with a fifth phase added which is absolutely 

indifferent to the equilibria in the system. The p-T diagrams therefore must correspond to figure 23B, displaying two distinct slope values. 

One compositional colinearity of four phases (f'tfl. 24). -The relations correspond to a binary four-phase system with an absolutely indifferent fifth phase 

added; the p-T diagram is given by figure 24 B. 

A 8 

FIGURE 23.-The single possible set of compositionally degenerate relations in 

ternary, five-phase systems effected by two compositional coincidences of two 

phases each, and its corresponding p-T diagram type. Phase 5 is an absolutely 

indifferent phase. On the p-T diagram only those phases are labeled whose 

identities, because of compositional coincidences, would otherwise be equivocal. 

Heavy dots indicate phases with coincident compositions. 

"~-~------------------------------------~n3 

A 8 

FIGURE 24.-The single possible set of compositionally degenerate relations in 

ternary, five-phase systems effected by the colinearity of four phases, and its 

corresponding p-T diagram type. The diagram is that of a binary fo~r-phase 

system, with an absolutely indifferent fifth phase added. 
p-T DLAGRAMlS, MULTIOOMPONENT SYSTEMS 41 

APPLICATION OF SCHREINEMAKERS' METHOD 

BXAMPLBS AS APPLIED TO :MINBBAL ASSBMBLAGBS 

The following examples of the actual application of the results of 

the preceding discussions to rock-forming minerals illustrate the 

different chemographic relations and show how even an incomplete 

set of thermodynamic data on the phases can be useful in delimiting 

the arrangement of the p-T diagrams. Although the mineral assemblages may not all be physically realistic, Schreinemakers' method is 

not limited in its use to systems in stable equilibrium. The phases are 

presumed to have fixed compositions specified in each example. 

Example 1.-The five-phase assemblage andalusite (A; Al2Si05), 

pyrophyllite (P; Al2Si.010(0Hh), kaolinite (K; Al~i20s(OH).), quartz 

(Q; Si02), and water (W; assumed to be pure H20) belongs to a ternary 

system whose components may be chosen as Al20 3, Si02, H20. The 

chemographic relations are given in figure 25A; they correspond to 

the triangular configuration with two interior points given by figure 

13. The five univariant lines that emanate from the quintuple point 

therefore must have the relative dispositions shown in figure 25B. 

The determination of the numerical values for the slopes of the 

univariant lines, by means of the Clapeyron equation, requires both 

entropy and volume data for all the phases; unfortunately the molar 

AI203'-------A-----~ 

A 

t 

LIJ 

a:: 

:::1 

(/) 

(/) 

LIJ 

a:: 

Cl. 

TEMPERATURE 

B 

FIGURE 25.-The ternary system Al20 3-Si02-H20, with the five phases water 

(W; H20), andalusite (A; Al2Si05), quartz (Q; Si02), kaolinite (K; 

Al2Si20s(OH).), and pyrophyllite (P; Al2Si40 10 (0Hh). A, The chemographic 

relations in gram-formula proportions of the oxide components; B, the 

p-T diagram. The diagram is properly oriented with respect to the pressure 

axis, but not to the temperature axis. In this and the succeeding figures, 

an arrow following the label "temperature" or "pressure" shows that the 

diagram is properly oriented with respect to that coordinate axis. 
42 p-T Dl!A:GRAMS, MU'LTIOOMPONENT SYSTEMS 

entropies for many minerals are not known. However, molar volume 

data are commonly available for minerals and can be used to orient 

the univariant curves properly with respect to the pressure axis. 

To orient the univariant curves with respect to the pressure axis, 

the balanced univariant reaction equations must first be derived. 

Figure 25A shows that both pyrophyllite and kaolinite break down 

into water, andalusite, and quartz; the properly balanced equations 

are: 

(K): A+3Q+W=P 

(P): A+Q+2W=K 

By eliminating in turn A, Q, and W from the two equations, we 

obtain: 

(A): 2Q+K=W+P 

(Q): 3K=P+5W+2A 

(W): A+5Q+K=2P 

The molar volumes are given in table 2; AV for the reactions (right 

minus left) are, respectively, -11.1cc (cubic centimeters), -11.1cc, 

-0.2cc, +22.1cc, and -ll.lcc.7 The assemblage on the right-hand 

side of each reaction is favored by high pressure if AVis negative, 

and vice versa. Moreover, the reaction that produces liquid water 

probably leads to an entropy increase, and therefore is favored by 

rising temperature. Figure 25B is constructed with these restrictions 

in mind; with the exception of (W), the slopes of the unh~ariant curves 

probably have correct signs; reaction (W) involves only solids, and 

its entropy change, hence the sign of the slope, cann<)t be predicted. 

Example 2.-The five-phase assemblage enstatite (E; MgSi03), 

forsterite (F; Mg2Si04), kyanite (K; Al2Si05), spinel (Sp; MgAl204), 

and sapphirine (8; Mg~10Si 23 ), having the specified compositions, 

belongs to the ternary system Mg0-Al20 3-Si02• The chemography 

of the phases is given in figure 26A. This is an example of a quadrilateral configuration with one interior point; the p-T diagram therefore must correspond to figure 12 and is given by figure 26A. The 

diagram is not oriented with respect to the temperature or pressure 

axis. 

The compositions of both enstatite and sapphirine are variable 

even within this ternary system. Enstatite can dissolve significant 

amounts of Al20 3 (Boyd and England, 1964, p. 157) under high 

' Although these values refer to 26° C and 1 bar pressure, for reactions involving only condensed phases 

the error caused by this approxbnation is likely negligible. 
APPLICATION OF SCHRE'INEMAKERS' METHOD 43 

TABLE 2.-Entropy and volume data at S5°C and 1 bar pre88Ure for 8elected mineral8 

(Data are from Robie (1962) except as indicated] 

Mineral Composition Entropy 

(decijoule 

per deg 

per mole) 

Akermanite. _ ---------------------- CazMgShOr. _ --------------------- 12,903 

Anatase. _____________ --------------- Ti02. ----------------------------- 499 

Andalusite ____ --------------------- AhSiOa. ___ ----------------------- 932 

Calcite ____________ ------------------ CaCOa ___ ------------ ___ ---------- 928 

Clinoenstatlte. ___ ------------------ MgSiOa. __ • ----------------------- 679 

Cordierite ••• ----------------------- Mg,AltSiaOu. --------------------- 14,072 

Corundum.------------------------- AhOa. __ • ------------------------- 510 

Diaspore_.------------------------- HAl02. --------------------------- 352 

Diopside .• ------------------------- CaMgShOe------------------------ 1, 430 

Dolomite __ --------- ____ ------------ CaMg(COa)J. --------------------- 1, 551 

Forsterite ••. _ -- __ ------- _ •• ____ •. _ •• Mg2Si0t .• _____ • ____ . __ --- ____ . _ _ __ 952 

Gibbsite ••. ------------------------- Al(OH):JL ------------------------- 700 

Kaolinite •• ------------------------- Al,SizOa(Olt}t .• ------------------- 2, 031 

Kyanite. --------------------------- AhSi05 •• _. ----------"--~--------- 838 

Merwinite. ___ ---------- ___ --------- CaaMgSiaOs- _ --------------------- 12,532 

Monticellite ••. __ --- __ •.• __ • ___ ___ ___ CaMgSiOt----------. __ . ______ --- __ • -------••• ___ --. _ 

Pyrophyllite •• ---.----------- _ -- __ __ AltSitO!o(OH) 2-.--- _ -----------. _. ------- __ . __ ...• __ Quartz ••• --.--. __ ---.---. __ .------__ Si02-----. ________ --- ____ ___ ___ ___ _ 414 

Rutile •• -------_----- ___ ----_. _____ • Ti02 •• ____ .• ______ •• __ •••••• ___ --- 503 

Sillimanite •• ----------------------- AhSiOa •••• ----------------------- 960 

Sphene.---------------------------- CaTiSiOa-------------------------- 1, 291 

Spinel,----------------------------- MgAltOt. ------------------------- 806 

Water----------- ________ ._. _______ •• HsO ••••.•• ----- _________ --- ___ ---- 700 

Wollastonite.----------------------- CaSiOa .• ------------------------- 819 

1 Weller and Kelley (1963). 

z Robie and Bethke (1963). 

a Volume data for dickite. 

• American Society for Testf.I::g Materials (1963). 

Volume (cc 

per mole) 

92.82 

20.49 

51.54 

36.94 

31.47 

2 233.50 

25.57 

17.76 

66.10 

64.35 

43.67 

31.96 

3 99.31 

44.11 

'98,22 

51.37 

2126.6 

22.69 

18.80 

49.91 

55.70 

39.72 

18.07 

39.94 

pressure, and the composition of sapphirine ma.y depart from the 

oxide ratios of 4:5:2 (Keith and Schairer, 1952), apparently reaching 

2:2:1( Deer and others, 1962, p. 176-177). These variations must 

be considered when one analyses the p-Tbehavior of univariant curves 

pertaining to actual mineral assemblages. 

Example 3.--...The five phases clinoenstatite (E), forsterite (F), 

sillimanite (8), cordierite (0), and spinel (Sp), having the specified 

compositions of table 2, belong to the ternary system Mg0-Al20 3-

Si02. Figure 27 A shows the chemographic relations; the configuration is pentagonal, and thus the five univariant curves are disposed 

as given in figure 27 B corresponding to the type of figure 11. 

For the five phases given, entropy and volume data are available 

(table 2) so that the actual slopes of the univariant lines can be 

calculated. Two of the five reactions are: 

(E): 5F+108=30+4Sp 

(F): 5E+5S=2C+Sp 
44 ~T DIAGRAM'S, MULTIOOMPONENT SYlSTEMS 

MgO ~------a.----~AI 3 L....-.-----------

Sp TEMPERATURE 

A B 

FIGURE 26.-The ternary system Mg0-Ah03-Si02, with the five phases 

enstatite (E; MgSi03), forsterite (F; Mg2SiO.), spinel (Sp; MgA120 4), 

kyanite (K; Al2Si05), and sapphirine, S, whose composition is taken to be 

Mg4Al10Si20 2a. A, The chemographic relations in gram formula proportions 

of the oxides; B, the p-T diagram. The p-T diagram is not properly. oriented 

with respect to either the temperature or the pressure axis. 

These equations suffice to determine all the others: 

(S) (=(E) -2(F)): 10E+2Sp=5F+C 

(C)(= (E) -3(F)): 3E+Sp=S+2F 

(Sp)(=(E)-4(F)): 2S+4E=F+C 

The univariant slopes are given by the Clapeyron equation, 

dP/dT=!lS/ilV 

and have the values, for (E), (F), (S), (C), and (Sp), of +7.6, +7.5, 

+7.5, + 6.7, and +7.5, respectively, in bars per degree. As the 

reactions are written above, the assemblages on the right-hand sides 

are relatively more stable at higher temperatures and lower pressures. 

The slope values are so similar that the differences are well within 

the uncertainties of the data. Thermodynamic data, in this instance, 

cannot be used to verify the qualitative results of the Schreinemakers 

method, although they are not contradictory. Figure 27 B therefore 

was constructed mainly on the basis of the results of the qualitative 

analysis, where the signs of the slopes were taken from the numerical 

data. 

In general, disagreements between the qualitative and quantitative 

considerations can stem from one of three sources: (1) erroneous 

thermodynamic values are used; (2) the approx.imation introduced 
APPLICATION OF SCHRE'INEMAKERS' METHOD 45 

t 

L1J 

0::: 

::::> 

en 

en 

L1J 

0::: 

a.. 

MgO '-------='-----~AI

3 ...._ __________ _ 

TEMPERATURE~ 

A B 

FIGURE 27.-The ternary system MgO-Al20rSi02, with the five phases 

clinoenstatite (E; MgSi03), forstei'ite (F; Mg2SiO.), cordierite (C; Mg2Al4-

Si5018), sillimanite (S; Al2Si05), and spinel (Sp; MgA120 4). A, The chemographic relp.tion in gram formula proportions of the oxides; B, the p-T 

diagram. The univariant curves are properly oriented with respect to 

both the temperature and the pressure axis. 

by using the entropy and volume data for 25°0 is invalid, or (3) the 

phases for which the values were measured do not have the compositions stipulated by the chemographic diagrams. 

Example 4..-The five phases andalusite (A), kyanite (Ky), 

kaolinite (K), pyrophyllite (P), and quartz (Q), constitute assemblages 

found in some North Carolina pyrophyllite deposits (Zen, 196lb). 

The chemography is given in figure 28A. A single degeneracy occurs 

between kyanite and andalusite; the other phases are relatively indifferent. The p-T relations are shown in figure 28B, corresponding 

to the type of figure 19B. The entropy of pyrophyllite is unknown; 

however, the diagram is correctly oriented with respect to the pressure 

axis, and the volume data are given in table 2. The balanced reactions are: 

(A): Ky+5Q+K=2P 

(Ky): A+5Q+K=2P 

(Q)=(K)=(P): A=Ky 

Example 5.-The five phases merwinite (Me), monticellite (Mo), 

wollastonite (W), diopside (Dp), and A.kermanite (A) belong to the 

ternary system Ca0-Mg0-Si02• The chemography is given in figure 

29A; there are two separate sets of compositional colinearities. The 

p-T diagram must have the relative dispositions of figure 29B, cor-
46 p-T DIAGRAMS, MULTIOOMPONENT. SYSTEMS 

A 

TEMPERATURE 

8 

FIGURE 28.-The ternary system Al20 3-SiOrH20, with the five phases 

kaolinite (Ky: Al2Si20 5(0H) 4), pyrophyllite (P; Al2Si40 10(0Hh), andalusite 

(A; ~1 Si0s), kyanite(KyrAhSi05), apd quartz (Q; Si02). A, The chemographic 

relations in gram formula proportions of the oxides; B, the p-T diagram which 

is properly oriented with respect to the pressure axis. An instance of compositional degeneracy is introduced by the polymorphism of andalusite and 

kyanite. The indifferent phases lie on the same side of these two singular 

phases; as a result the curve bearing the labels .of the indifferent phases courses 

through in invariant point, and there are only three distinct slope values at the 

invariant point. 

responding to figure 200. The molar volume data for the phases are 

given in table 2; the diagram has been properly oriented with respect 

to the pressure axis. 

Although the univariant reaction equations can be easily balanced 

by inspection, we will use this system to illustrate the use of determinants. Korzhinskiy (1959) discussed many aspects 6f the procedure 

as ap:plied to univariant reactions; a particularly lucid presentation 

of the mathematical theory has been given by Aitken (1951). 

A univariant reaction involves n+ 1 phases in an n-component 

system. Therefore, the determinant can be formulated in the 

following way: Each phase is assigned to one of the n+ 1 rows of the 

determinant, and each component is assigned to one of the n+ 1 

columns. The number of gram-formulas of component i, as it appears 

in the formula of phase j, is entered in position ij of the determinant. 

However, there is one more row than column, so that in the position 1j 

we write the label of phase j. The (n+ 1) by (n+ 1) determinant can 
APPLICATION OF SCHREmEMAKERS' METHOD 47 

CaO !fl 

l 

LIJ 

0:: 

:> 

fJ (/) 

(/) 

LIJ 

0:: @ a.. 

MgO Si02 

TEMPERATURE 

A 8 

FIGURE 29.-The ternary system Ca0-Mg0-Si02, with the five phases merwinite 

(Me; CaaMgSi20 8), diopside (Dp; CaMgSh06), monticellite (Mo; CaMgSi04), 

wollastonite (W; CaSiOa), ~nd Akermanite (A; Ca2MgSi207). A, The chemographic relations in gram formula proportions of the oxides; B, the p-T 

diagram which is properly oriented with respect to the pressure axis. Two 

distinct cases of compositional colinearity exist; they are reflected.in thefact 

that only three distinct univariant curves emanate fi·om the invariant point. 

The univariant curves all terminate at the invariant point, because for both 

cases of degeneracy the indifferent phases lie on opposite sides of the singular 

phases. 

now be expanded m terms of the minors and the stoichiometric coefficients of the phases in the univariant reaction calculated directly. 

Applying this procedure and listing the components in the order 

CaO-MgO-Si02 from the left-hand column to the right-hand column, 

we have 

(A): Me 3 1 2 

Mo 1 1 1 

w 1 0 1 

Dp 1 1 2 

=0 

Expanding by minors, and recalling that each odd permutation of 

the row (or column) introduces a minus sign in front of the determinant, we have 

Me I ~ ~ ~ 1-Mo I t ~ ~ I + W I ! ~ ~ 1-Dp I t l11 =O 

2Mo+2W=Dp+Me 
48 ~T DI~G'RAMS, MULTIOOMPONENT SYSTEMS 

Similarly, we can calculate 

(Me): Mo 

w 

Dp 

A 

1 1 1 

1 0 1 =0 

1 1 2 

2 1 2 

Mo I ~ ~ ~ 1-WI ~ i ~ I +Dp I ~ 6 ~ 1-A I ~ 6 ~ I =0 212 212 212 112 

A=Mo+W 

In this example, the coefficient of Dp is zero, and diopside does not 

appear in the reaction equation. This fact immediately signifies that 

(Me)= (Dp) and that the system is degenerate. 

Finally, we have 

(W): !Me 3 1 2 

Mo I 1 1 1 

Dp 1 1 2 I =0; 

A 2 1 2 

Me 11111 1 1 2 - Mo 13121 1 1 2 + Dp 13 1 121 1 1 -A 13121 1 1 1 =0 

212 212 2 1 2 1 1 2 

Me+Dp=2A 

This result shows that (W) and (Mo) must refer to the same reaction 

and that there is a second degeneracy involving the colinearity of 

three phases. Because (Me) and (Dp) appear on the same side of 

the univariant scheme (Mo) = (W), the curves (Me) and (Dp) coincide 

stable to stable; similar consideration indicates that (W) and (Mo) 

also coincide stable to stable. 

Example 6. -The five phases diaspore (D; HA102), gibbsite (G; 

Al(OH)a), corundum (C; Al20 3), pyrophyllite (P; Al2S4010 (0Hh), 

and vapor (V; H20) belong to the ternary system Al20 3-Si02-H20. 

fhe chemography is given by figure 30A. The four phases, corundumdiaspore-gibbsite-vapor are colinear in composition, and pyrophyllite 

is, therefore, an absolutely indifferent phase. The p-T diagram is 

that for a binary four-phase system, with a fifth phase present throughout; the situation corresponds to figure 24. Because, however, a 

vapor high in entropy and low in density is present along three of 

the univariant equilibria, the divariant assemblage that includes the 

vapor must be on the high-temperature and low-pressure side of each 

associated univariant curve; such curves also have positive slopes. 

The fourth reaction, C+G=3D, involves only solids, and the 

p-T slope can be calculated from thermodynamic data (table 2); the 

value is +36 .bars per degree, and diaspore is the low-temperature 
APPLICATION OF SCHREII.NEMAKERS' METHOD 49 

A 

TEMPERATURE'~ 

8 

FIGURE 30.-The ternary system Ah03-Si02-H20, with the five phases corundum 

(C; .t\.1203), diaspore (D; HA102), gibbsite (G; Al(OH)a), vapor (V; H20), and 

pyrophyllite (P; AhSi40 10 (0Hh). ·A, The chemographic relations in gram 

formula proportions of the oxides; B, the p-T diagram which is properly 

oriented with respect to both the temperature and the pressure axes. Because 

pyrophyllite is an absolutely indifferent phase,' the diagram corresponds to 

that of a binary four-phase system with a fifth phase present throughout. 

and high-pressure assemblage. The p-T diagram must look something like figure 30B. 

Example 7.-The five-phase assemblage wollastonite (W), sphene 

(S), quartz (Q), rutile (R), and anatase (A) belong to the ternary 

system CaO-Ti02-8i02. The chemography is given in figure 31A. 

Quartz is an absolutely indifferent phase~ whereas sphene and wollastonite are relatively indifferent with respect to the transition between 

rutile and anatase. The situation thus corresponds to figure 21A, and 

only three distinct univariant lines exist. The univariant reactions 

are: t 

(A)=(Q): R+W=S 

(R)=(Q): W+A=S 

(W) = (S) = (Q): R=A 

The slopes of the univariant curves are calculated from the data of 

table 2; these slope values are, respectively, + 10.5, +5.8, and -2.7, 

in bars per degree; the assemblage on the right-hand side of each 

reaction is the low entropy and therefore low-temperature assemblage. 

The quantitatively correct p-T diagram is given in ~aure 31B. 
50 p--T DlAGR.A.MB, MULTIOOMPONENT S'Y'STEMS 

1° c 

TEMPERATURE ~ 

A B 

FIGURE 31.-The ternary'system CaO-TiOrSi02, with the five phases wollastonite {W; Ce.Si03), quartz (Q; Si02), sphene (S; CaTiSiOs), rutile (R; Ti02), 

and anatase (A; Ti02). A, The chemographic relations· in gram formula 

proportions of the oxides; B, the p-T diagram which is properly oriented with 

respect to both the temperature and the pressure axes and for which the numerical values of the univariant slopes, calculated from thermodynamic data, are 

indicated by the scale. Two separate cases of compositional degeneracy exist: 

the polymorphism of rutile and anatase, and the colinearity of either of these 

phases with wollastonite and sphene. Quartz is an absolutely indifferent phase 

and sphene and wollastonite are relatively indifferent phases. Only three 

distinct slope values exist at the invariant point, while four univariant curves 

emanate therefrom. The curve marked (Q) (8) (W) courses through the 

invariant point because the three phases bearing these labels lie on the same 

side of the singular phases A and R. 

Example 8.-The six phases clinoenstatite (E), diopside (Dp), 

dolomite (D), calcite (C), quartz (Q), and vapor (V), all presumed to 

be stoichiometric and to have the compositions given in table 2, belong 

to a quaternary system whose components may be taken to be CaO, 

MgO, Si02, and C02. The chemographic relations are given in perspective in figure 32A. From the phase compositions two univariant 

reactions are immediately obvious: 

(Q)=(V): 2C+2E=D+Dp 

(C)= (E): D+2Q=Dp+2V 

By linear combinations of these we can eliminate Dp and D in turn to 

get, respectively, 

and 

(Dp): D+Q=C+E+ V 

(D): C+Q+E=Dp+ V 
APPLICATION OF SCHREIINEMAKERS' METHOD 51 

A 

VEDDp 

CDVDp 

VQEDp 

CDpQV 

t 

TEMPERATURE_. 

8 

FIGURE 32.-The quaternary system Ca0-Mg0-Si02-

C02, with the six phases calcite (C; CaCOa), dolomite 

(D; CaMg(COa)2), diopside (Dp; CaMgSi20e), enstatite (E; MgSi03), quartz (Q_; SiD,), and vapor 

(V, C02). A, The chemographic relations drawn in 

perspective, in gram formula proportions of the 

oxides; B, the p-T diagram, which is properly oriented 

with respect to both the temperature and the pressure 

axes. Two separate cases of compositional degeneracy (coplanarity) exist: In one, Q and V lie on the 

same side of the singular phases, and the curve (Q) (V) 

courses through the invariant point; in the other, E 

and C lie on opposite sides of the singular phases, and 

the curve (E)(C) ends at the invariant point. As a 

result, five distinct curves emanate from the invariant 

point having but four distinct slope values. Within 

each sector of the p-T diagram, the compositional 

wedge defined by the six phases are divided into three 

adjoining tetrahedra, one of which characterizes and 

defines the phase assemblages of the sector. This 

defining assemblage is shaded and is given at the tOp 

of the list of assemblages for that sector. 

Because there are two sets of degeneracies in the system, only four 

distinct univariant schemes exist. We note that (0) and (E) always 

occur on the same side of the other four schemes, but (Q) and (V) 

always occur on opposite sides; these relations reflect the facts that 

C and E occur on opposite sides, but Q and Von the same side, of the 

respective singular phases. The curve (C)= (E) therefore must 

tenninate at the univariant point (a sextuple point), but the curve 

(Q)=(V) must course through the point. 

On the basis of the thermodynamic data of table 2, the p-T slope 

of the vapor-free reaction (Q) = (V) can be co;mputed to be 36 bars per 

degree; diopside and dolomite are favored by low temperature and 
52 tr-T DI!AGRAM'S, MULTIOOMPON·ENT SYSTEMS 

high pressure. The numerical values of the slopes for the other univariant equilibria cannot be determined because in these equations 

the entropy and volume values of C02 gas dominate, and these values 

depend sensitively on the location of the curves on the p-T diagram. 

However, the reactions leading to the evolution of gas are favored by 

high temperatures and low pressures, and so tend to have positive 

slopes on the p-T plot, thus the curves must be qualitatively 

describable by figure 32B. 

EXTENSION OF OSTWALD'S RULE TO BINARY SYSTEMS 

There are many different versions of the so-called "Ostwald's 

step rule" in the geochemical literature. One original statement by 

Ostwald (1902, p. 447) is as follows: 

In allgemeiner Weise findet sich der Satz, class beim Verlassen eines unbestandigen Zustandes ein gegebenes chemisches Gebilde nicht den bestandigsten Zustand 

aufsucht, sondern den nachstliegenden, d.h. den (Vortibergehend oder dauernd) 

bestandigen, welcher von dem augenblicklichen aus mit dem geringsten Verlust 

von freier Energie erreicht werden kann. [Freely translated: In general, one 

finds the rule that, by the breakdown of an unstable state [=phase], a given 

chemical composition will form not the most stable state, but rather the adjacent, 

that is, a transitory state, which can be reached directly from the existing one with 

the least loss of free energy.] 

Ostwald's rule was originally proposed for the polymorphic transition 

in one-component systems. The stepwise progression from phase 

assemblages of relatively higher free energy to those of lower free 

energy, however, can be applied equally well to systems of more than 

one component. The application to binary systems is immediate, 

and the results allow the prediction of possible relative sequences 

of formation of metastable assemblages. 

As explained in an early section of this paper, the invariant point, 

univariant curves, and divariant fields of ap-T diagram are the projections of points, curves, and areas of Gibbs free-energy surfaces of 

phase assemblages from the G-p-T space onto the p-T plane. It is 

therefore possible to construct a qualitatively correct U-X (X= 

composition) diagram for each divariant field, univariant curve, and 

invariant point, depic.ting the free-energy relations among the phases, 

directly from the p-T diagrams. For divariant fields, no more than 

two phases (represented as points in the G-X plot) are colinear; for 

univariant curves, the three phases in equilibrium are colinear; for 

invariant points, all four points are colinear. These colinearity 

relations are valid regardless of whether an assemblage is stable or 

metastable; the stable set of assemblages is given by the geometrically 

lowest set of straight lines connecting phase points in the G-X plot. 
APPLICATION OF SCHREINEMAKERS' METHOD 53 

On the G-X plot, therefore, the three colinear points occur on the 

lowest set of straight lines along the stable part of a univariant curve, 

but they lie above the lowest set of lines along the metastable part of 

the same univariant curve. The lowest set of straight lines remains 

the same between any two adjacent stable univariant curves; the 

phase points, however, move relatively up and down at different 

p-T values between these univariant curves in such a way that the 

higher sets of straight lines, which apply to the metastable assemblages, 

change their relations as successive metastable extensions of univariant 

curves are crossed. 

By using these criteria, the straight lines connecting the free-energy 

values of phases are constructed for every part of the p-T diagram 

around an invariant point; these are shown in figure 33. This diagram immediately allows one to predict the possible sequence of metastable assemblages according to the step rule. The sequence of 

assemblages depends not only on the location of a given p-T point 

relative to the univariant curves, but also on the precise bulk composition of the system. Thus, for instance, for some specific bulk composition lying between phases 2 and 3 and in the p-T sector between the 

stable parts of (1) and (4), one possible set of successively more stable 

divariant assemblages is: 1 +4, 2 +4, 1 +3, ·and finally 2 +3, which is 

the stable assemblage. In this example, it is interesting that phase 2, 

which is stable, may initially form in equilibrium with the metastable 

phase 4, but may then disappear as 2 +4 is replaced by 1 +3; this 

latter assemblage may in turn be replaced by the stable assemblage, 

in which phase 2 reappears. The mere appearance of a stable phase 

in an equilibrium assemblage, thus, does not guarantee that the 

assemblage is stable. Conversely, the disappearance of a phase is no 

assurance that the phase is not intrinsically stable under the prevailing 

p-T-X conditions. 

At particular values of temperature and pressure, certain bulk 

compositions, X', could lead to two alternative metastable divariant 

assemblages that have the same total Gibbs free energy and therefore 

the same degree of metastability; for example, between lines (1) and 

( 4) in figure 33 where the composition X' is indicated by the crossing 

of the two lines that connect the phases of the divariant assemblages. 

At the bulk composition X', the chemical potential of each component 

is the same in both phases of a given assemblage, however, it is not 

equal in the two assemblages. The two alternative assemblages, 

therefore, are not in true chemical equilibrium, nevertheless they do 

not tend to interreact because aG of reaction is zero. As the p-T 

values sweep the sector, the bulk composition X' also shifts, spanning 

the entire range of compositions between phases 2 and 3. 
54 trT DLA:GRAM'S, MULTIOOMPONENT SY:STEMS 

b:J ~/ 

I 

2 3 4 

FIGURE 33.-Schematic representation of Gibbs free-energy relations of the 

phases around an invariant point in a binary four-phase system. The metastable univariant lines are extended in full, but are shown as dashed lines. 

Each G-X plot is topologically correct for the entire sector as shown; the Gibbs 

free-energy values increase upward so .that the straight lines connecting the 

lowest set of points depict the stable univariant or divariant assemblages. In 

each G-X plot, the successively higher lines connect univariant or divariant 

assemblages which are successively more metastable. Where two such lines 

cross, for instance between univariant curves (1) and (4), a metastable fourphase assemblage could form fortuitously; such an assemblage, however, is 

not invariant. 
p-T DI!AORAMIS, MULTICOMPONENT SYSTEMS 55 

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