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A statistical approach to the study of phase equilibria in multicomponent systems
A S T A T I S T I C A L A P P R O A C H TO T H E S T U D Y OF P H A S E E Q U I L I B R I A IN M U L T I C O M P O N E N T SYSTEMS SURENDRA KUMAR SAXENA
Saxena, S. K. 1969: A statistical approach to the study of phase equilibria in multicomponent systems. /a't&n 3, 25--36. Eigenvalues and eigenvectors obtained from correlation coefficient matrix of a multicomponent multiphme system are used for the diagrammatic representation of the system. On such representations, it is possible to plot certain relative positions of all ~ minerals as well as of the components. Tie lines joining points representing a pair of coexisting minerals in such figures are sign:f~Lnt in the same way as they are in the concentration diagrams (Gibbs triangle for instance) but with the added advantage that they now represent the influence of all the components in the system. Diagrammatic representation of phase equilibria in charnockites from Varberg, Madras, Uusimas and in certain w . e t a m o r p ~ iron formations from Quebec is made by using this sufistical approach.
S. K. Saxena, Planetology Branch, NASA, Goddard Space Fh'ght Center, Greenbelt, Marylmut 20771, USA
Introduction To represent a multicomponent chemical system on paper, we have to study the interrelationships of the coexisting phases forming as a result of interactions of the various substances. If we use the concentrations of the several components as variables, we require space of several dimensions. Various methods in use which represent systems with two, three, four, or by using certain simplifications, several components (see Palatnik & Landau 19Or), are of limited value, particularly in representing rock systems. Therefore a different approach to this problem is necessary. Thermodynamic considerations indicate that in a multiphase system at equilibrium, all the components are distributed systematically in the coexisting phases. This distribution is a function of P, T and chemical pot.~nials of the components. As the latter are significant only in a relative sense, it should be possible in principle to use in the study of phase equilibria, certain statistical parameters, such as the partial correlatior', coefficients of the components distributed among the various phases of the system. Fog this purpose the method developed here uses the algebra: of eigenvalues and eigenvectors which was found ~ be useful in mineral, chemical problems (Saxena 1969 a, b). The advantage of using the sta~.*.~cal parameters for representing a chemical system over using original variables (concentrations) is that the N dimensional concentration space is no longer easen'.ial.
26
$URENDRAKUMAR 8AXENA
Table 1. Concentration matrix of a sample of Varberg charnockite
Q Or Pl Opx Cpx Horn Bi Gar Mt llm
SiOs
AlsO.
TiOs
'FeO'
MnO
MgO
CaO
NasO
KsO
100.00 63.01 63.29 46.50 48.50 38.40 33.50 36.30 0.48 0.06
0.00 19.73 22.70 1.00 1.90 10.70 14.40 19.00 0.02 0.00
0.00 0.00 0.00 0.07 0.14 1.84 3.86 0.04 3.00 51.80
O.00 0.62 0.17 40.00 20.20 25.40 26.50 30.70 95.87 46.00
0.00 0.00 0.00 0.77 0.34 0.19 0.06 1.56 0.00 1.90
0.00 0.19 0.00 7.30 6.10 4.50 5.60 1.40 0.40 0.03
0.00 0.26 3.00 0.90 19.60 10.70 0.00 7.10 0.23 0.00
0.00 1.51 9.95 0.08 0.60 1.50 0.08 0.00 0.00 0.00
0.00 14.53 0.20 0.00 0.00 1.50 8.60 0.00 0.00 0.00
Chemical analysis for Or, Mt and llm taken from Howie (1955, S. No. 2270). 'FeO' is total Fe calculated as FeO.
Abbreviations and symbols Opx - orthopyroxene, Cpx - clinopyroxene, Bi - biotite, Gru - grtinerite, Horn - hornblende, Gar - garnet, Q - quartz, Or - orthoclase, Cal - calcite, PI - plagioclase, Mt - magnetite, Ilm - ilmenite, Act - actinolite, Hm hematite,/~l -chemical potential of a component i, P - pressure, T - temperature.
Concentration matrix The matrix formed by the oxide weight percent of coexisting minerals in a rock sample is called a concentratiion matrix. The concentration (oxide weight percent in the present work)of i th component in the j th phase is denoted by Xl,l. The composition of a j th mineral is expressed by X~j+ X2j+ X.~j+ . . . + X n j = tO0 where n is tile number of components !n the system. The concentration matrix is set up in the following way; in the rows of the matrix the concentrations of different compot~ents are placed in the same phase and in the columns the concentrations of the same component in different phases. Thus the concentration matrix in the present work is taken as X1t A,, •
*
X,r
X21 X22 •
•
.... .... •
,
Xnl Xn2 •
•
....
.
.
Xnr
where r is the number of phases.
Statistics a n d t h e r m o d y n a m i c c o n s i d e r a t i o n s The quantity Xq is a function of P, T and the chemical composition of the system. The concentration matrices in two or more samples are generally
STATISTICAL APPROACH, PHASE EQUILIBRIA
27
Table 2. Principal component analyah uaing correlations. A sample of Varberg charnoddte. Variances for principal components (Eigenvalues) Variables
First
Second
3.198
1.796
0.443 0.379 -0.385 -0.442 -0.374 -0.032 0.035 0.321 0.26)
-0.109 0.275 0.369 -0.038 0.170 -0.599 -0.531 0.230 0.224
Third
Fourth
Fifth
1.242
0.978
0.920
Sixth to Ninth not listed
Prindpal Components (Eigenvectora)
SiOI
AlsOs TiO~ 'FeO' MnO MgO CaO Na,O K~O
0.080 --0.245 0.107 0.452 0.484 0.192 -0.244 -0.232 0,357 0.428 --0.139 0.350 0.415 0.304 0.473 -0.162 --0.587 0.468
0.543 -0.368 0.252 -0.497 0.250 -0.052 -0.120 -0.420 0.033
Transformed variables Quartz Orthoclase Plagioclase Opx Cpx Hornblende Biotite Garnet Magnetite llmenite
44.36 -10.97 39.43 1.84 39.99 0.05 2.80 -11.03 13.80 -19.39 10.17 -9.22 9.16 -0.69 9,32 --4.24 --43.38 -2,95 -41.0~ 17,68
8.039 --0.698 13.320 --6,25 6.87 2.07 -7,29 0,77 -22,~0 -1,04
Fourta to ninth Pot computed
different from each other. Therefore we may perform certain statistical computations on the two or more matrices to obtain certain characteristic values for the rock samples. The eigenvalues and eigenvec.'.ors would be such characteristic values. The writer uses a computer program written by Prof. R. R-.yment. From the concentration matrix, we obtain in turn the covariance nmtrix and the correlation matrix. The latter i~ a matrix of the correlation coefficients which has all its diagonal elements a,3 unity. From such a correlation matrix, the eigenvalues and eigenvectors a.'e computed. The algebra of eigenvalues and eigenvectors may be followed ill Hadley (1961). A brief introduction was also given by Saxena (1969b). As an example let us considel a charnockite sample from Varberg, Sweden, which contains the minerals Q, Or, PI, Opx, Cpx, Horn, Bi, Oar, Mt and Ilm. Nine of these are of variable composition. The concentration matrix for this sample is presented in Table 1, and the data obtained by the computations in Table 2. The first and second eigenvalues represent the maximum percent of total variation in the sample. The corresponding eigenvectors in columns 1 and 2 are used to compute the first and second transformed variables.
28
SURENDRA KUMAR SAXENA
*/,0.0-
TiO 2 o
o AIzO3 1,20
oNa20
D
+20.0
Itmen,te 0
~.nO 0
Orthociase
~0 o u
o Biatite
Magnetite o''FeO"
°° "1 ¢ la ga io s e .
Garnet o
Opx
tn
oHornblende
Quartz o S; 0 2
O
o
Cpx
-20.0
-60.0
O
CaO
MgO O
"40.0
I
i
-20 0
0
w
*20.0
l
*40.0
First v a r i a b l e
Fig. 1. Relative positions of various phases and components in It sample of Varberg charnockite. The first an(~ second variables are calculated by using the eigenvectorl in the first and second column of Table 2 respectively. The concentrution matrix of the sample on which the matrix ,}f correlation coefficients and the computation of the eigenvalues and eigenvectora are based is presented in Table 1.
First variable: 0.443 Xtt+0.379 Xat--0.385X~t--....
+ 0.260 X,,t
(1)
Second variable: --0.109 Xtt+0.275 X2 +0.369 X3t--
. . . . . . +0.224. Xgt
(2)
Fig. 1 is a plot of these two transformed variables against each other. We obtain on such a plot the position of different minerals. Further by putting 100 percent of each oxide in (1) and (2) we can also obtain the coordinates for different oxides. There is no a priori thermodynamic argument for the use of eigenvalues and eigenvectors in the representations of a multicomponent system in Fig. 1. A difference of P, T and/or chemical composition results in different concentration matrices, different correlation matrices and, finally, different numerical values of the transformed variables. Therefore the position of
STATISTICAL
Table
ReL
3. M i n e r a l
No.
assemblage
APPROACH,
in ten samples of Quebec
PHASE
I
~
~
A
iron formation
1
2
3
4
5
6
7
8
9
188
192
195
199
205
206
207
209
215
Q
x
•
x
x
x
x
•
x
x
x
Opx
x
x
-
-
x
-
x
-
-
-
Specimen
No.
10 217A
Cpx
x
x
x
x
x
x
•
•
x
•
Gru
x
•
•
-
•
•
•
-
-
-
Act
-
-
-
•
-
-
-
•
•
•
Calc
•
•
•
•
•
x
•
x
•
•
Mt
•
x
•
•
•
x
•
•
x
x
Hm
-
-
-
x
-
-
-
x
x
x
x means
'present'
- means
'not noted'.
the points representing oxides and minerals in plots such as Fig.l is a function of the chemistry and the intensive variables of the system. T h e N dimensional tie lines A straight line connecting points for two coexisting minerals in a concentration triangle (Gibbs triangle for example) or in any other concentration diagram may be referred to as a tie line. On the basis of thermodylJamic arguments, the following is contended (see Greenwood et al. 1964, Greenwood 1967). In a simple ternary phase diagram for a system with only two coexisting minerals, one could conclude that when the tie lines do not cross and shift laterally the composition of the minerals in the rocks is at least partly a comequence of the differing bulk chemistry. When the tie lines cross, it is due to crystallization under different physical conditions. This may or may not be accompar,ied by a change in bulk chemintry. Similar arguments are valid for the eigenvector diagrams. We may expect the following. Tie lines joix,ing points for pairs of mineral phases in equilibrium in a sample may be nearly parallel or slightly inclined to similar tie lines from other samples. In such a case the shift in the tie lines should be due to a change in the composition of the rock system under consideration. If two tie lines actually intersect, the composition of the minerals in the two samples is a consequence of the change in the inunsive variables, namely, P, T and chemical potential of any mobile component. There may or may not be a change in the bulk composition. If two lines do not actually intersect but are of very different slopes, the composition of the coexisting phases is a result of both variation in bulk compoeition a~ well as in the intensive variables. Phase equilibria in a m e t a m o r p h o s e d iron formation from Quebec Butler Jr. (1969) presents mineral-chemical data in the system SiOz--FeO-MgO--CaO--H20--CO2--O2. Ten samples (Table 3) are used here to
30
flURENDRAKUMAR flAXI~A
40
Fig. 2. Tie line. for the pairs of amphibole (open circle.) and cfinopyroxene (crone.) in Quebec iron formation. Chemical data from Butler Jr. (1969).
t.
g
20
lO
_2ol
• 2'0
0
First
variable
illustrate the present method of representation. Samples 1, 2, 5, 6 and 7 contain the assemblage Q--Opx--CpxmGru--CalcmMt and 4, 8, 9 and 10, QmCpx--ActmCalcnMt--Hm. From Butler's discassion, it is understood that these assemblages formed at similar P and T but under a variable lzFt2o/~tco2. The tie lines for coexisting Gru-Cpx or Act-Cpx pairs are shown in ,-rig. 2. Act-Cpx tie lines for samples 4, 8, 9 and 10 shift !aterally and the slope changes gradual),y. From phase rule considerations (as presented by Butler Jr.), the only possible reason for such a change in the slope could be the change in the intensive variables tLrtzo • t~coz. The tie lines for Gru-Cpx pairs are nearly parallel for samples 1, 2 and 7. For sample 3, it is not significantly different. The tie lines for samples 5, 6 and 7 have different slopes. The three samples are collected from within a distance of l I0 feet. It is probable that tznao/t~co2 varied within the area. According to Butler Jr. the compositional changes between No. 205 (sa:nple 5) and No. 207 (sample 7) (0.62-0.70 --Fe~'°ox~jindicate gradients of both tzHzO and ~co2 over a distance of 110 feet. Phase equilibria in charnockitic rocks Mineral-chemical data on basic rocks from Uusimaa, Finland, were presented by Snxena (1969c). Mineral assemblages in five of these samples are listed in Table 4. Or, Mt and Itm were not analysed and plagioclase was amtlysed partially. There would not be any significant difference if we substitute theoretical (formula) values for Or, Mt and Ilm in the concen-
STATLSTXCAL.,,UPVI~C)A~ V~SE ~ Table
4. Mineral ~semb~
in U u s ~
~
31
rocks
S.N.
1
2
3
4
5
(~
X
--
•
--
•
•
PI
x
x
x
•
Opx
•
•
•
•
•
Cpx Horn
• •
• =r
• •
• •
• -
Bi
•
x
•
•
•
Mt
x
•
•
•
•
Ilm
-
•
•
•
Or
.
x means 'present', - m e ~ =
• .
.
.
•
'not observed'. Samples I to 5 are the same as samples 1, 3, 7, 4 a n d
6 respectively in Table 1 (Saxena 1969c).
tration matrix. Plagioclase analyses except for Na, K and Ca could be completed likewise. Fig. 3 shows the plot of the first transformed variables against the second for the five samples. Except for sample 5, no tie lines for the same mineral pairs are significantly different in slope from each other. The tie lines for sample 5 have different slopes and they cross tie lines for other samples. Opx-Horn and Opx-Bi tie lines for samples 2 and 4 cross each other. The crossing of the tie lines in Fig. 3, particularly for those of sample 5, should be attributed to the variation of the intensive parameters. P and T could not have varied appreciably. There might have been sharp gradients in FH2O and FH2 (or Fo2) within ~;mall distances (Saxena 1969c). Sample 5 5
&
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,
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:'
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,, ,
m
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/,
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,.
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~ •
3x X
•
o e
0
&
t
y
•
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•
o
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•
"~.'
m
~_...--.---"-~=
.
o.15 o
03
-20 .,,
.
15 First
;
20
i
25
variable
F i g . 3. T i e lines for the pairs o f f e r r o m a g n e s i a n m i n e r a l s in charnockites f r o m U u s i m a a T r i a n g l e - - biotite, cross - - h o r n b l e n d e , o p e n circle - - o r t h o p y r o x e n e , cross inside circleclinopyroxene. C h e m i c a l data f r o m S a x e n a (1Q69c).
32
SUSL~DRAKUMAR8AXENA
Table 5. Mineral assemblage and chemical composition of four charnockite samples from Madras. Data from Howie (1955) 1 (4642A)
2 (2270)
3 (2941)
4 (3709)
48.50 14.88 1.18 2.38
45.01 8.82 1.32 4.36 12.38 19.90 0.22 7.28 0.17 0.02 0.02 0.21
37.8 9.3
SiO2 AlsO, TiO, Fe2Os
51.55 13.86 1.12 1.51
FeO
11.63
MgO MnO CaO Na~O KaO H~O ÷ HgO -
5.80 0.23 10.28 3.08 0.54 0.25 O. 12
54.00 15.52 1.61 3.97 5.95 4.37 0.15 7.11 4.36 1.53 0.37 0.20
45.0 0.9
6.1 3.8 52.9 1.4
11.48
8.66 0.23 9.44 2.01 0.34 0.38 O. 15
Modes Q Or PI Horn Bi Opx
Cpx Mt lira Spinel
-
7.9
-
.8.7 -
25.2 24.8 3.7
5.5 13.7 4.6
30.0 19.4 3.2
56.4 25.4 5.1
-
-
-
4.2
contains Bi and Or. Or is missing from the rest of the four samples. The following reactions are possible: KAISi3Os + 3 FeSiO3 + H20 ~ KF%A1Si3010(OH)2 + 3 SiO2 . . . . . (a) KAISi3Os-F Fe304 + H2 ~ KFe3AISi3OI0(OH)z . . . . . . . . . . . . . . . . . . . . (b) At a certain equilibrium value of ~H20 or ~-H2and of other intensive variables, we may have orthod, ase and biotite coexisting as in (a) or as in (b). If ~H20 changes, orthoclase raay be consumed in the reaction (a). It would require a ditferent value of btI~,o (or tZH2) to get the assemblage which includes biotite and the other minerals. To illustrate further the present method of representation and interpretation, tie lines for :four charnockite samples from Madras (Table 5) are shown in Fig. 4. Opx-Cpx tie lines either cross each other or are inclined sigaificantly toward each other. It is inferred that some of the intensive variables were different during the crystallization of these assemblages. B,:sides, the total composition in sample 4 should also be very different from that of the other samples. Opx-Cpx tie lines for samples 2 and 3 cross. In this example it is noted that they are different in bulk chemistry also.
33
STATISTICAL A P P R O A L ~ , PHASE I ~ R J I L I B R I A
These samples are from widely separated localities, and it is possible that P and T might have varied. If Opx and Cpx are dose to the ideal solution for Fe and Mg end-members, the distribution coefficient KD is useful in ,~timating P and T. Kretz (1963) found that KD for these four samples are rather similar (0.53 to 0.56). If P and T were similar, the disposition of the tie lines in Fig. 4 must be due to variations in the chemical pontentials of certain mobile components. Mineral-chemical data on Varberg charnockites, presented by Saxena 1968, are reconsidered here. The data for coexisting Or, Mt, Ilm, and partially for PI, are assumed to be as in the standard chemi~l formulae. Fig. 5 shows the plot of the first variables agains*~ the second for five saml~Yes which contain Q~ Or~PI--Opx--Cpx~Horn--Bi--Mt--Ilm. Three of the samples 1, 2 and 5 also contain garnet in addition. Opx---Cpx tie lines for samples 1, 3 and 5 do not cross. Opx--Cpx tie line for sample I has a somewhat different slope from that of samples 3 and 4. This is traced to the low SiO2 (46 percent, probably wrong analysis) in Opx. The tie line fi)r sample 2 makes a very different angle with the rest of the group. If we consider the tie lines for all pairs of ferromagnesian minerals, it should be noted that the tie lines for samples 2 and 5 (in particular Op~mHorn, Opx--Bi and Opx-Cpx) are different in slope from the rest. The composition of the minerals in
-5"
4 q*
, @
I
/
f
-10"
2
1
a/
o ,1o
*
/
.m L
,
/
>
c uo o u')
3
A
///
/
~-15
A
7
A
2
!
f
// i
///
//o
°
,/
,
First
variable
"/
/.'~3
//
-20 l
-25
5
' 10
' 15
' 25
3'o
Fig. 4. T i e lines for the pairs of ferromagnesian minerals in charnockites from Madras. Symbols as in Fig. 3. Chemical data from Howie (1955). 3 -
Lithm 3:1
34
8URENDRA KUMAR SAXENA
these two samples is probably due to change in some intensive variable. Saxena (1968) found that the distribution of Ca between coexisting garnet and plagioclase in samples 2 and 5 was different from that in sample 1. One possible reaction representing equilibrium among the phases was written as 2 Ca(Mg, Fe)Si206 + CaAl2Si2Os ~ Ca3A[2Si3012 -~- 2 (Mg, Fe)SiO3 q- SiO2
!
O.
A
I a
o
I
•
ii o
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o • •
•
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s
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*
.
e* /
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,'
-
. . . . .
.
.
.
.
.
.
.
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[
i
......x
.-- "" ~"
\ - ~ _L\ t ~ \\\
L
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2
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ot •
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,
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s
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.
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.
C
B -T5
\-~...
';
~
-20
o
g
io
;s First
10
2'5
variable
Fig. 5. Tie lines for the pairs of ferromagnesian minerals in charnockites from Varberg. S:'mbols as in Fig. 3. Chemical data from Saxena (1968). ~,Pl/yOar
c,/-, c, was suggested to change with change in P. Such a relationship in the composition of coexisting plagioclase and garnet would be according to experimental observations (Cohen et al. 1967). Fig 6 shows tie lines for pairs of PI--Gar, Pl--Horn and PI--Cpx. The tie lines for samples 2 and 5 are crossing over all similar tie lines for other samples. It is probable that rocks represented by samples 2 and 5 crystallized at different P (tectonic overpressure ?). Samples 1, 2 and 5 contain 10 phases (excluding water and certain accessories such as zircon and apatite) and 11 components (including H20 and excluding Zr and P). If water is not regarded both as a component and a phase, the system in these rocks is trivariant. Therefore besides P and T, there may be one more independently variable [z. From the study of phase
API~OACa, PHA~.~mLmmA
~rA~nCAL
35
equilibria in Quebec iron formations and in Uusimaa, it would seem ~a', FH2O may show sharp variations within small distances. It is probable that FH2O was not uniform in the Varberg rocks as suggested by the writer before (8axena 1968). This aspect will be discussed in a subsequent work. 2
+10" B*°7 sLew*
.'"
®
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..
1
O"
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x
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......
P l - G.ir
1'0
-
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2() ............... variable
30'..................
40i'
Fig. 6. Tie lines for the pain of minerals with high calcium in the same samples as represented in Fig. 5. A C K N O W L E D G E M E N T S . - The writer thanks Prof. R. A. Reyment for permitting the use of a computer programme. Thanks are also due to Prof. H. Ramberg for providing various facilities and to the Swedish Board for Technical Development for financial support.
Institute of Mineralogy and Geology, Box 555, Uppsala, Sweden
;~EFERENCES i]utler, P. Jr. 1969: Mineral compositions and equilibria in the metamorphmed iron formation of the Gagnon Region, Quebec, Canada..y. Petrol. 10, 56-101. Cohen, L. H., Ito, K. & Kennedy, G. C. 1967: Melting and phase relatioim in an anhydrous basalt to 40 kilobars. Am..7. So/. 26E, 5~9-38. Greenwood, H. J., Doe, B. R. & Phinney, W. C. 1964: A disctt, ion. Phase equilibria in the metamorphic rocks of St. Paul Island and Cape North, Nova Scotia..7. Petrr' 5, 189-94.
36
8URENDIL~ KUMAR $AXKNA
G:eenwood, H. J. 1967: The N-dimensional tie-line problem. Geodn'm. et Cosmocln'm. Acta 31, 465-90. Hadley, G. 1961: Linear Algebra. Addison-Wesley. Howie, R. A. 1955: The geochemistry of the charnockite ser~es of Madras, India. Trans. Royal Soc. Edinburgh 62, 725--68. Kretz, R. 1963: Distribution of magnesium and iron betwec 1 orthopyroxene and clinopyroxene in natural mineral assemblages, aT. Geol. 71, 773°-85. Palatnik, L. S. & Landau, A. I. 1964: Phase Equilibria in Msdtieomponent Systems. Holt, Rinehart & Winston. Saxena, S. K. 1968: Chemical study of phase equilibria in cha~'nockites, Varberg, Sweden. Am. Mineral. 53, 1674-95. Saxena, S. K. 1969a: Silicate solid solutions and geothermometry. 3. Distribution of Fe and Mg in coexisting garnet and biotite. Contr. Mineral. and Petrol. 22, 259-67. Saxena, S. K. 1969b: Silicate solid solutions and geothermometry. 4. Statistical study of chemical data on garnets and clinopyroxene. Contr. Mineral. and Petrol. 23, 140-56. Saxena, S. K. 1969c: Distribution of elements in coexisting minerals and the problem of chemical disequilibrium in metamorphosed basic rocks. Con;r. Mineral. and Petrol. 20, 177-97. Revised manuscript accepted June 1969
Printed January 1970
Saxena, S. K. 1969: A statistical approach to the study of phase equilibria in multicomponent systems. /a't&n 3, 25--36. Eigenvalues and eigenvectors obtained from correlation coefficient matrix of a multicomponent multiphme system are used for the diagrammatic representation of the system. On such representations, it is possible to plot certain relative positions of all ~ minerals as well as of the components. Tie lines joining points representing a pair of coexisting minerals in such figures are sign:f~Lnt in the same way as they are in the concentration diagrams (Gibbs triangle for instance) but with the added advantage that they now represent the influence of all the components in the system. Diagrammatic representation of phase equilibria in charnockites from Varberg, Madras, Uusimas and in certain w . e t a m o r p ~ iron formations from Quebec is made by using this sufistical approach.
S. K. Saxena, Planetology Branch, NASA, Goddard Space Fh'ght Center, Greenbelt, Marylmut 20771, USA
Introduction To represent a multicomponent chemical system on paper, we have to study the interrelationships of the coexisting phases forming as a result of interactions of the various substances. If we use the concentrations of the several components as variables, we require space of several dimensions. Various methods in use which represent systems with two, three, four, or by using certain simplifications, several components (see Palatnik & Landau 19Or), are of limited value, particularly in representing rock systems. Therefore a different approach to this problem is necessary. Thermodynamic considerations indicate that in a multiphase system at equilibrium, all the components are distributed systematically in the coexisting phases. This distribution is a function of P, T and chemical pot.~nials of the components. As the latter are significant only in a relative sense, it should be possible in principle to use in the study of phase equilibria, certain statistical parameters, such as the partial correlatior', coefficients of the components distributed among the various phases of the system. Fog this purpose the method developed here uses the algebra: of eigenvalues and eigenvectors which was found ~ be useful in mineral, chemical problems (Saxena 1969 a, b). The advantage of using the sta~.*.~cal parameters for representing a chemical system over using original variables (concentrations) is that the N dimensional concentration space is no longer easen'.ial.
26
$URENDRAKUMAR 8AXENA
Table 1. Concentration matrix of a sample of Varberg charnockite
Q Or Pl Opx Cpx Horn Bi Gar Mt llm
SiOs
AlsO.
TiOs
'FeO'
MnO
MgO
CaO
NasO
KsO
100.00 63.01 63.29 46.50 48.50 38.40 33.50 36.30 0.48 0.06
0.00 19.73 22.70 1.00 1.90 10.70 14.40 19.00 0.02 0.00
0.00 0.00 0.00 0.07 0.14 1.84 3.86 0.04 3.00 51.80
O.00 0.62 0.17 40.00 20.20 25.40 26.50 30.70 95.87 46.00
0.00 0.00 0.00 0.77 0.34 0.19 0.06 1.56 0.00 1.90
0.00 0.19 0.00 7.30 6.10 4.50 5.60 1.40 0.40 0.03
0.00 0.26 3.00 0.90 19.60 10.70 0.00 7.10 0.23 0.00
0.00 1.51 9.95 0.08 0.60 1.50 0.08 0.00 0.00 0.00
0.00 14.53 0.20 0.00 0.00 1.50 8.60 0.00 0.00 0.00
Chemical analysis for Or, Mt and llm taken from Howie (1955, S. No. 2270). 'FeO' is total Fe calculated as FeO.
Abbreviations and symbols Opx - orthopyroxene, Cpx - clinopyroxene, Bi - biotite, Gru - grtinerite, Horn - hornblende, Gar - garnet, Q - quartz, Or - orthoclase, Cal - calcite, PI - plagioclase, Mt - magnetite, Ilm - ilmenite, Act - actinolite, Hm hematite,/~l -chemical potential of a component i, P - pressure, T - temperature.
Concentration matrix The matrix formed by the oxide weight percent of coexisting minerals in a rock sample is called a concentratiion matrix. The concentration (oxide weight percent in the present work)of i th component in the j th phase is denoted by Xl,l. The composition of a j th mineral is expressed by X~j+ X2j+ X.~j+ . . . + X n j = tO0 where n is tile number of components !n the system. The concentration matrix is set up in the following way; in the rows of the matrix the concentrations of different compot~ents are placed in the same phase and in the columns the concentrations of the same component in different phases. Thus the concentration matrix in the present work is taken as X1t A,, •
*
X,r
X21 X22 •
•
.... .... •
,
Xnl Xn2 •
•
....
.
.
Xnr
where r is the number of phases.
Statistics a n d t h e r m o d y n a m i c c o n s i d e r a t i o n s The quantity Xq is a function of P, T and the chemical composition of the system. The concentration matrices in two or more samples are generally
STATISTICAL APPROACH, PHASE EQUILIBRIA
27
Table 2. Principal component analyah uaing correlations. A sample of Varberg charnoddte. Variances for principal components (Eigenvalues) Variables
First
Second
3.198
1.796
0.443 0.379 -0.385 -0.442 -0.374 -0.032 0.035 0.321 0.26)
-0.109 0.275 0.369 -0.038 0.170 -0.599 -0.531 0.230 0.224
Third
Fourth
Fifth
1.242
0.978
0.920
Sixth to Ninth not listed
Prindpal Components (Eigenvectora)
SiOI
AlsOs TiO~ 'FeO' MnO MgO CaO Na,O K~O
0.080 --0.245 0.107 0.452 0.484 0.192 -0.244 -0.232 0,357 0.428 --0.139 0.350 0.415 0.304 0.473 -0.162 --0.587 0.468
0.543 -0.368 0.252 -0.497 0.250 -0.052 -0.120 -0.420 0.033
Transformed variables Quartz Orthoclase Plagioclase Opx Cpx Hornblende Biotite Garnet Magnetite llmenite
44.36 -10.97 39.43 1.84 39.99 0.05 2.80 -11.03 13.80 -19.39 10.17 -9.22 9.16 -0.69 9,32 --4.24 --43.38 -2,95 -41.0~ 17,68
8.039 --0.698 13.320 --6,25 6.87 2.07 -7,29 0,77 -22,~0 -1,04
Fourta to ninth Pot computed
different from each other. Therefore we may perform certain statistical computations on the two or more matrices to obtain certain characteristic values for the rock samples. The eigenvalues and eigenvec.'.ors would be such characteristic values. The writer uses a computer program written by Prof. R. R-.yment. From the concentration matrix, we obtain in turn the covariance nmtrix and the correlation matrix. The latter i~ a matrix of the correlation coefficients which has all its diagonal elements a,3 unity. From such a correlation matrix, the eigenvalues and eigenvectors a.'e computed. The algebra of eigenvalues and eigenvectors may be followed ill Hadley (1961). A brief introduction was also given by Saxena (1969b). As an example let us considel a charnockite sample from Varberg, Sweden, which contains the minerals Q, Or, PI, Opx, Cpx, Horn, Bi, Oar, Mt and Ilm. Nine of these are of variable composition. The concentration matrix for this sample is presented in Table 1, and the data obtained by the computations in Table 2. The first and second eigenvalues represent the maximum percent of total variation in the sample. The corresponding eigenvectors in columns 1 and 2 are used to compute the first and second transformed variables.
28
SURENDRA KUMAR SAXENA
*/,0.0-
TiO 2 o
o AIzO3 1,20
oNa20
D
+20.0
Itmen,te 0
~.nO 0
Orthociase
~0 o u
o Biatite
Magnetite o''FeO"
°° "1 ¢ la ga io s e .
Garnet o
Opx
tn
oHornblende
Quartz o S; 0 2
O
o
Cpx
-20.0
-60.0
O
CaO
MgO O
"40.0
I
i
-20 0
0
w
*20.0
l
*40.0
First v a r i a b l e
Fig. 1. Relative positions of various phases and components in It sample of Varberg charnockite. The first an(~ second variables are calculated by using the eigenvectorl in the first and second column of Table 2 respectively. The concentrution matrix of the sample on which the matrix ,}f correlation coefficients and the computation of the eigenvalues and eigenvectora are based is presented in Table 1.
First variable: 0.443 Xtt+0.379 Xat--0.385X~t--....
+ 0.260 X,,t
(1)
Second variable: --0.109 Xtt+0.275 X2 +0.369 X3t--
. . . . . . +0.224. Xgt
(2)
Fig. 1 is a plot of these two transformed variables against each other. We obtain on such a plot the position of different minerals. Further by putting 100 percent of each oxide in (1) and (2) we can also obtain the coordinates for different oxides. There is no a priori thermodynamic argument for the use of eigenvalues and eigenvectors in the representations of a multicomponent system in Fig. 1. A difference of P, T and/or chemical composition results in different concentration matrices, different correlation matrices and, finally, different numerical values of the transformed variables. Therefore the position of
STATISTICAL
Table
ReL
3. M i n e r a l
No.
assemblage
APPROACH,
in ten samples of Quebec
PHASE
I
~
~
A
iron formation
1
2
3
4
5
6
7
8
9
188
192
195
199
205
206
207
209
215
Q
x
•
x
x
x
x
•
x
x
x
Opx
x
x
-
-
x
-
x
-
-
-
Specimen
No.
10 217A
Cpx
x
x
x
x
x
x
•
•
x
•
Gru
x
•
•
-
•
•
•
-
-
-
Act
-
-
-
•
-
-
-
•
•
•
Calc
•
•
•
•
•
x
•
x
•
•
Mt
•
x
•
•
•
x
•
•
x
x
Hm
-
-
-
x
-
-
-
x
x
x
x means
'present'
- means
'not noted'.
the points representing oxides and minerals in plots such as Fig.l is a function of the chemistry and the intensive variables of the system. T h e N dimensional tie lines A straight line connecting points for two coexisting minerals in a concentration triangle (Gibbs triangle for example) or in any other concentration diagram may be referred to as a tie line. On the basis of thermodylJamic arguments, the following is contended (see Greenwood et al. 1964, Greenwood 1967). In a simple ternary phase diagram for a system with only two coexisting minerals, one could conclude that when the tie lines do not cross and shift laterally the composition of the minerals in the rocks is at least partly a comequence of the differing bulk chemistry. When the tie lines cross, it is due to crystallization under different physical conditions. This may or may not be accompar,ied by a change in bulk chemintry. Similar arguments are valid for the eigenvector diagrams. We may expect the following. Tie lines joix,ing points for pairs of mineral phases in equilibrium in a sample may be nearly parallel or slightly inclined to similar tie lines from other samples. In such a case the shift in the tie lines should be due to a change in the composition of the rock system under consideration. If two tie lines actually intersect, the composition of the minerals in the two samples is a consequence of the change in the inunsive variables, namely, P, T and chemical potential of any mobile component. There may or may not be a change in the bulk composition. If two lines do not actually intersect but are of very different slopes, the composition of the coexisting phases is a result of both variation in bulk compoeition a~ well as in the intensive variables. Phase equilibria in a m e t a m o r p h o s e d iron formation from Quebec Butler Jr. (1969) presents mineral-chemical data in the system SiOz--FeO-MgO--CaO--H20--CO2--O2. Ten samples (Table 3) are used here to
30
flURENDRAKUMAR flAXI~A
40
Fig. 2. Tie line. for the pairs of amphibole (open circle.) and cfinopyroxene (crone.) in Quebec iron formation. Chemical data from Butler Jr. (1969).
t.
g
20
lO
_2ol
• 2'0
0
First
variable
illustrate the present method of representation. Samples 1, 2, 5, 6 and 7 contain the assemblage Q--Opx--CpxmGru--CalcmMt and 4, 8, 9 and 10, QmCpx--ActmCalcnMt--Hm. From Butler's discassion, it is understood that these assemblages formed at similar P and T but under a variable lzFt2o/~tco2. The tie lines for coexisting Gru-Cpx or Act-Cpx pairs are shown in ,-rig. 2. Act-Cpx tie lines for samples 4, 8, 9 and 10 shift !aterally and the slope changes gradual),y. From phase rule considerations (as presented by Butler Jr.), the only possible reason for such a change in the slope could be the change in the intensive variables tLrtzo • t~coz. The tie lines for Gru-Cpx pairs are nearly parallel for samples 1, 2 and 7. For sample 3, it is not significantly different. The tie lines for samples 5, 6 and 7 have different slopes. The three samples are collected from within a distance of l I0 feet. It is probable that tznao/t~co2 varied within the area. According to Butler Jr. the compositional changes between No. 205 (sa:nple 5) and No. 207 (sample 7) (0.62-0.70 --Fe~'°ox~jindicate gradients of both tzHzO and ~co2 over a distance of 110 feet. Phase equilibria in charnockitic rocks Mineral-chemical data on basic rocks from Uusimaa, Finland, were presented by Snxena (1969c). Mineral assemblages in five of these samples are listed in Table 4. Or, Mt and Itm were not analysed and plagioclase was amtlysed partially. There would not be any significant difference if we substitute theoretical (formula) values for Or, Mt and Ilm in the concen-
STATLSTXCAL.,,UPVI~C)A~ V~SE ~ Table
4. Mineral ~semb~
in U u s ~
~
31
rocks
S.N.
1
2
3
4
5
(~
X
--
•
--
•
•
PI
x
x
x
•
Opx
•
•
•
•
•
Cpx Horn
• •
• =r
• •
• •
• -
Bi
•
x
•
•
•
Mt
x
•
•
•
•
Ilm
-
•
•
•
Or
.
x means 'present', - m e ~ =
• .
.
.
•
'not observed'. Samples I to 5 are the same as samples 1, 3, 7, 4 a n d
6 respectively in Table 1 (Saxena 1969c).
tration matrix. Plagioclase analyses except for Na, K and Ca could be completed likewise. Fig. 3 shows the plot of the first transformed variables against the second for the five samples. Except for sample 5, no tie lines for the same mineral pairs are significantly different in slope from each other. The tie lines for sample 5 have different slopes and they cross tie lines for other samples. Opx-Horn and Opx-Bi tie lines for samples 2 and 4 cross each other. The crossing of the tie lines in Fig. 3, particularly for those of sample 5, should be attributed to the variation of the intensive parameters. P and T could not have varied appreciably. There might have been sharp gradients in FH2O and FH2 (or Fo2) within ~;mall distances (Saxena 1969c). Sample 5 5
&
e • 0
,
-5
A • 1
e
Aj o,
e
,
o
-10
•
, ,
•
/
../
•
',
.
',
/
,
/
/
/
/
~
/
o lw #
//
/~
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•
•
•
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~
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- .// =
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./. .
~ o /
~ ~ . ~
,/
.
~
•
•
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a
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/
on
•
. . •
•
• ..3 I>.
/."
,'. ~ , ~
/ /
•
/
0• 0
•
" . / ;, / • ,p
/
:'
/
o
/t.
'
e'
'
4/o
,, ,
m
t-
o
2
'
/,
•
,.
J
"o
,/
""
o
~ •
3x X
•
o e
0
&
t
y
•
• ._. •
•
o
" . "~ .~
~
•
"~.'
m
~_...--.---"-~=
.
o.15 o
03
-20 .,,
.
15 First
;
20
i
25
variable
F i g . 3. T i e lines for the pairs o f f e r r o m a g n e s i a n m i n e r a l s in charnockites f r o m U u s i m a a T r i a n g l e - - biotite, cross - - h o r n b l e n d e , o p e n circle - - o r t h o p y r o x e n e , cross inside circleclinopyroxene. C h e m i c a l data f r o m S a x e n a (1Q69c).
32
SUSL~DRAKUMAR8AXENA
Table 5. Mineral assemblage and chemical composition of four charnockite samples from Madras. Data from Howie (1955) 1 (4642A)
2 (2270)
3 (2941)
4 (3709)
48.50 14.88 1.18 2.38
45.01 8.82 1.32 4.36 12.38 19.90 0.22 7.28 0.17 0.02 0.02 0.21
37.8 9.3
SiO2 AlsO, TiO, Fe2Os
51.55 13.86 1.12 1.51
FeO
11.63
MgO MnO CaO Na~O KaO H~O ÷ HgO -
5.80 0.23 10.28 3.08 0.54 0.25 O. 12
54.00 15.52 1.61 3.97 5.95 4.37 0.15 7.11 4.36 1.53 0.37 0.20
45.0 0.9
6.1 3.8 52.9 1.4
11.48
8.66 0.23 9.44 2.01 0.34 0.38 O. 15
Modes Q Or PI Horn Bi Opx
Cpx Mt lira Spinel
-
7.9
-
.8.7 -
25.2 24.8 3.7
5.5 13.7 4.6
30.0 19.4 3.2
56.4 25.4 5.1
-
-
-
4.2
contains Bi and Or. Or is missing from the rest of the four samples. The following reactions are possible: KAISi3Os + 3 FeSiO3 + H20 ~ KF%A1Si3010(OH)2 + 3 SiO2 . . . . . (a) KAISi3Os-F Fe304 + H2 ~ KFe3AISi3OI0(OH)z . . . . . . . . . . . . . . . . . . . . (b) At a certain equilibrium value of ~H20 or ~-H2and of other intensive variables, we may have orthod, ase and biotite coexisting as in (a) or as in (b). If ~H20 changes, orthoclase raay be consumed in the reaction (a). It would require a ditferent value of btI~,o (or tZH2) to get the assemblage which includes biotite and the other minerals. To illustrate further the present method of representation and interpretation, tie lines for :four charnockite samples from Madras (Table 5) are shown in Fig. 4. Opx-Cpx tie lines either cross each other or are inclined sigaificantly toward each other. It is inferred that some of the intensive variables were different during the crystallization of these assemblages. B,:sides, the total composition in sample 4 should also be very different from that of the other samples. Opx-Cpx tie lines for samples 2 and 3 cross. In this example it is noted that they are different in bulk chemistry also.
33
STATISTICAL A P P R O A L ~ , PHASE I ~ R J I L I B R I A
These samples are from widely separated localities, and it is possible that P and T might have varied. If Opx and Cpx are dose to the ideal solution for Fe and Mg end-members, the distribution coefficient KD is useful in ,~timating P and T. Kretz (1963) found that KD for these four samples are rather similar (0.53 to 0.56). If P and T were similar, the disposition of the tie lines in Fig. 4 must be due to variations in the chemical pontentials of certain mobile components. Mineral-chemical data on Varberg charnockites, presented by Saxena 1968, are reconsidered here. The data for coexisting Or, Mt, Ilm, and partially for PI, are assumed to be as in the standard chemi~l formulae. Fig. 5 shows the plot of the first variables agains*~ the second for five saml~Yes which contain Q~ Or~PI--Opx--Cpx~Horn--Bi--Mt--Ilm. Three of the samples 1, 2 and 5 also contain garnet in addition. Opx---Cpx tie lines for samples 1, 3 and 5 do not cross. Opx--Cpx tie line for sample I has a somewhat different slope from that of samples 3 and 4. This is traced to the low SiO2 (46 percent, probably wrong analysis) in Opx. The tie line fi)r sample 2 makes a very different angle with the rest of the group. If we consider the tie lines for all pairs of ferromagnesian minerals, it should be noted that the tie lines for samples 2 and 5 (in particular Op~mHorn, Opx--Bi and Opx-Cpx) are different in slope from the rest. The composition of the minerals in
-5"
4 q*
, @
I
/
f
-10"
2
1
a/
o ,1o
*
/
.m L
,
/
>
c uo o u')
3
A
///
/
~-15
A
7
A
2
!
f
// i
///
//o
°
,/
,
First
variable
"/
/.'~3
//
-20 l
-25
5
' 10
' 15
' 25
3'o
Fig. 4. T i e lines for the pairs of ferromagnesian minerals in charnockites from Madras. Symbols as in Fig. 3. Chemical data from Howie (1955). 3 -
Lithm 3:1
34
8URENDRA KUMAR SAXENA
these two samples is probably due to change in some intensive variable. Saxena (1968) found that the distribution of Ca between coexisting garnet and plagioclase in samples 2 and 5 was different from that in sample 1. One possible reaction representing equilibrium among the phases was written as 2 Ca(Mg, Fe)Si206 + CaAl2Si2Os ~ Ca3A[2Si3012 -~- 2 (Mg, Fe)SiO3 q- SiO2
!
O.
A
I a
o
I
•
ii o
,
o
o • •
•
•
s
o
*
.
e* /
1
,'
-
. . . . .
.
.
.
.
.
.
.
~\\
"U
[
i
......x
.-- "" ~"
\ - ~ _L\ t ~ \\\
L
I
A,~,
,'
'
2
L
ot •
•
•
0
• e
o s
,
-10
i
s
s
-5"
I
•
o °"
,
5
'3
/'
. - - , - _I.* ~ .4 • ', . ,.,
/'. I
"~
2~~
'/ ,
/I ,
I
J,
"
"
.
,
.
C
B -T5
\-~...
';
~
-20
o
g
io
;s First
10
2'5
variable
Fig. 5. Tie lines for the pairs of ferromagnesian minerals in charnockites from Varberg. S:'mbols as in Fig. 3. Chemical data from Saxena (1968). ~,Pl/yOar
c,/-, c, was suggested to change with change in P. Such a relationship in the composition of coexisting plagioclase and garnet would be according to experimental observations (Cohen et al. 1967). Fig 6 shows tie lines for pairs of PI--Gar, Pl--Horn and PI--Cpx. The tie lines for samples 2 and 5 are crossing over all similar tie lines for other samples. It is probable that rocks represented by samples 2 and 5 crystallized at different P (tectonic overpressure ?). Samples 1, 2 and 5 contain 10 phases (excluding water and certain accessories such as zircon and apatite) and 11 components (including H20 and excluding Zr and P). If water is not regarded both as a component and a phase, the system in these rocks is trivariant. Therefore besides P and T, there may be one more independently variable [z. From the study of phase
API~OACa, PHA~.~mLmmA
~rA~nCAL
35
equilibria in Quebec iron formations and in Uusimaa, it would seem ~a', FH2O may show sharp variations within small distances. It is probable that FH2O was not uniform in the Varberg rocks as suggested by the writer before (8axena 1968). This aspect will be discussed in a subsequent work. 2
+10" B*°7 sLew*
.'"
®
.#*# f /
/
..
1
O"
A~
.I L.
2 "'" ..--'-:" /...-",,~..,,r.---~ 7" . . ~ /. - ~ , / / . . . - -x-....-'..-'... ..-..-...--f,,/
r0
"o 12
1
.,-":.::-'""'I__-/ /
..--'""
°o -lO-
/
.//i
o
/"
4x'"
/I i~/f /
I 1,r" I
I/I/
ii
// i/
/I
2~
Gar
• =it
i
• Cpx
-20-
x
Horn P I - Cpx PI- Horn
......
P l - G.ir
1'0
-
First
2() ............... variable
30'..................
40i'
Fig. 6. Tie lines for the pain of minerals with high calcium in the same samples as represented in Fig. 5. A C K N O W L E D G E M E N T S . - The writer thanks Prof. R. A. Reyment for permitting the use of a computer programme. Thanks are also due to Prof. H. Ramberg for providing various facilities and to the Swedish Board for Technical Development for financial support.
Institute of Mineralogy and Geology, Box 555, Uppsala, Sweden
;~EFERENCES i]utler, P. Jr. 1969: Mineral compositions and equilibria in the metamorphmed iron formation of the Gagnon Region, Quebec, Canada..y. Petrol. 10, 56-101. Cohen, L. H., Ito, K. & Kennedy, G. C. 1967: Melting and phase relatioim in an anhydrous basalt to 40 kilobars. Am..7. So/. 26E, 5~9-38. Greenwood, H. J., Doe, B. R. & Phinney, W. C. 1964: A disctt, ion. Phase equilibria in the metamorphic rocks of St. Paul Island and Cape North, Nova Scotia..7. Petrr' 5, 189-94.
36
8URENDIL~ KUMAR $AXKNA
G:eenwood, H. J. 1967: The N-dimensional tie-line problem. Geodn'm. et Cosmocln'm. Acta 31, 465-90. Hadley, G. 1961: Linear Algebra. Addison-Wesley. Howie, R. A. 1955: The geochemistry of the charnockite ser~es of Madras, India. Trans. Royal Soc. Edinburgh 62, 725--68. Kretz, R. 1963: Distribution of magnesium and iron betwec 1 orthopyroxene and clinopyroxene in natural mineral assemblages, aT. Geol. 71, 773°-85. Palatnik, L. S. & Landau, A. I. 1964: Phase Equilibria in Msdtieomponent Systems. Holt, Rinehart & Winston. Saxena, S. K. 1968: Chemical study of phase equilibria in cha~'nockites, Varberg, Sweden. Am. Mineral. 53, 1674-95. Saxena, S. K. 1969a: Silicate solid solutions and geothermometry. 3. Distribution of Fe and Mg in coexisting garnet and biotite. Contr. Mineral. and Petrol. 22, 259-67. Saxena, S. K. 1969b: Silicate solid solutions and geothermometry. 4. Statistical study of chemical data on garnets and clinopyroxene. Contr. Mineral. and Petrol. 23, 140-56. Saxena, S. K. 1969c: Distribution of elements in coexisting minerals and the problem of chemical disequilibrium in metamorphosed basic rocks. Con;r. Mineral. and Petrol. 20, 177-97. Revised manuscript accepted June 1969
Printed January 1970