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Grain contacts in crystalline rocks
GRAIN DEREK
CONTACTS
IN
CRYSTALLINE
ROCKS
FLINN
FLINN, D. 1969: Grain contacts in crystalline rocks. Lithos 3, 361-370. The distributions of mineral grains in gneisses from Shetland and Siberia are studied. Statistical tests show that the grains, instead of being distributed at random, are specially arranged so that grains of the same phase tend not to occur in contact with each other. It is concluded that this arrangement arises from grain boundary migration leading to the insertion of grains of one phase between pairs of like grains of other phases. This process arises from the fact that the interfacial energies of contacts between like phases are greater than those between unlike phases.
Some years ago it was suggested that recrystallization of a rock might give rise to a distribution within the rock of the mineral grains of different type in a pattern minimizing the interfacial (grain boundary) energy, and further that to minimize the total interfacial energy of the rock would bring about not only an ideal arrangement of the mineral grains but also an ideal proportional relationship between the grains, which would result in rocks of certain compositions (De Vore 1956, p. 54; Laffitte 1957, p. 328). The second process does not necessarily follow from the first, and indeed seems unlikely from the field occurrence and appearance of rocks. However, the first process, the rearrangement of mineral grains into a pattern tending to minimize the interfacial energy, seems sufficiently possible to be worth seeking the results in natural rocks. The writers mentioned above did not test their hypotheses in this way but Rogers & Bogy (1958) did investigate grain contacts. In 1957-8 I had time and opportunity to investigate this problem, and working on granitic gneisses from Shetland, developed statistical tests of randomness of mineral grain distribution. In 1960 another opportunity arose and I spent much time testing a suite of granulites from Siberia, but I was unable to finish the project because of the multiplicity of tests that may be employed. However, in a recent issue of this journal there appeared a study of grain contacts in a metamorphic rock employing a test identical to one used by me (Kretz 1969, pp. 39-65). The very detailed paper by Kretz makes it possible for me to publish my results in a much briefer form than I could have otherwise employed. Furthermore, my results are made more interesting by the fact that although they are very similar to those of Kretz, they led me to diametrically opposite conclusions.
362
DEREKFLINN
The problem investigated Is the position of a mineral grain of a given type dependent to any extent on the positions of the same or other types of mineral grains ? If so, is this dependence due to interfacial energy effects ? Clearly, if a rock is metasomatised by the migration of material along grain boundaries a special arrangement of mineral grains could arise by chemical reaction. A special arrangement of mineral grains might also arise during penetrative deformation, due to mechanical differences in the different grains. An attempt was made to eliminate mineral grain patterns of the last tnvo types by special selection of the rocks studied. Selection of material Regional metamorphic rocks with the following characteristics were studied: uniform grain size, equidimensional grains, simple pattern of grain boundaries (triple point network), lack of signs of mechanical strain in grains, no obvious signs of preferred orientation, and no signs of reaction betnveen the grains. Such rocks are likely to have resulted from prolonged recrystallization under static conditions. If surface energy influences the relative positions of mineral grains of different type then that influence should be most apparent in such rocks, and the other processes mentioned above which tend to produce special arrangements of grains should not have been active. Suites of such rocks from two areas were studied. Five thin sections of permeation gneiss from the Colla Firth permeation belt of Shetland (Flinn 1954, p. 187) were selected from several dozen such sections and were studied first. Later, a group of 35 thin sections cut from 16 Siberian granulites were studied. The granulites were selected after hand-specimen and thin section examination of a large number of specimens from the Archean crystalline complex in the Aldanski shield in southern Yakutia. The Shetland rocks were composed of assemblages made up from the following minerals: Quartz, plagioclase (oligoclase), biotite, and muscovite. Rocks containing K-felspar were not studied as this felspar shows a late replacement appearance. The Siberian rocks contained assemblages of the following minerals: plagioclase, K-felspar, biotite, apatite, opaque ores, hornblende, quartz, augitic clino-pyroxene. Neither the hand-specimens nor the thin sections cut from these rocks showed any signs of inhomogeneity or of tectonite fabric. The hand-specimens carried no orientation marks so that the thin sections can be considered as having been cut at random. Tests of homogeneity No statistical tests of homogeneity were carried out, partly because the specimens showed no obvious signs of inhomogeneity and partly because
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inhomogeneity does not seem to affect the hypothesis. A sharp junction between areas of very different composition might give rise to abnormal grain relation within the junction, but no such junctions were present. T e s t s o f grain relations The null hypothesis tested was that grains occur in a random arrangement in the rock. That is to say, the numbers of grains of two given types that occur in contact depend only on the total numbers of grains of each of the taro types present. There are two alternative hypotheses. One is that grains tend to occur in a regular pattern, like sites in a crystal lattice. This arrangement would lead to less contacts between like phases than the null hypothesis predicts. The other is that like grains tend to occur in groups, giving rise to more contacts between like phases than the null hypothesis predicts. The possible effects of these alternatives on the frequencies of contacts between non-like phases are complex and difficult to predict. The test selected as most useful was designed to distinguish between the null hypothesis and the alternatives without distinguishing between the alternatives. This test was subsequently used without modification as a test of AVA (Flinn 1965) and later by Kretz (1969, p. 58), who named it the 'linetransect' method and obtained it from Vistelius (1966). There is no need to describe the test again, except to note that for each thin section a sequence of not less than 1000 grains were noted for testing. All of the 39 sections gave sequences of grain types which differed from random at a very high level of significance when this test was applied. The degrees of freedom used in the tests varied from 1 to 9, since the assemblages varied from 2 to 4, after grouping the less common grain types until no cells had an expected frequency of less than 5. The smallest value found was 7.2=16 and was associated with one degree of freedom. Thus the pattern of grain distribution is most unlikely to be a random one. Kretz carried out the same test in the same way on his Fig. 1 (Kretz 1969, p. 40) and found that the null hypothesis of randomness was just acceptable at the 95 per cent level of significance. However, it should be noted that the observed number of contacts between like phases is less than the expected number in all three cells in his Table 14 and in all six in his Table 12 (Kretz 1969, p. 60). This arrangement would be expected for Table 14 once in every eight trials if the grains were randomly arranged and for Table 12, once in every 64 trials. Taken alone, these facts are not enough to prove that Kretz's Fig. 1 is a non-random distribution of grains. However, most of the different types of contacts between like phases considered in the 21 rocks I investigated occurred less frequently than expected (see Table 1), and this appears to be a characteristic feature of recrystallized rocks. Thus Kretz's Fig. 1 is probably a non-random distribution but the test he used was not powerful enough to recognize this fact. It is an unfortunate feature of this test that the degrees of freedom are in no way dependent on the
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DrREKFLINN
Table 1. F r e q u e n c y greater than one Type of contaetratio PP KK BB Ap Ap OO HH XX QQ MM
o f o c c u r r e n c e o f g r a i n c o n t a c t r a t i o s less a n d
>1.0 4 4 6 0 0 0 5 0 -
Siberia <1.0 100xProb. 31 11 20 26 33 20 26 13 -
1.5 x 4.2 0.3 1.5 x 1.2x 9.5 x 7.9 X 1.2x
1 0 -~
10 10 10 10 10
-6 -s -s -3 -z
-
>1.0
Shetland <1.0 100xProb.
0 1 0 0 _ _ 1
5 4 3 4 _ _ 4
4
0
3.1 15.6 12.5 6.3 _ _ 15.6 6.3
Probability of observed frequeneies=nCr(½) n n=total number of events r=number of events of one type F o r c o n t a c t r a t i o s y m b o l s see T a b l e 2a.
amount of data collected, so that it is difficult to determine how much data to collect. It is possible that the collection of more data (I noted four times as many grain contacts per section as Kretz did) would have led Kretz to the rejection of the null hypothesis.
T e s t s o f t h e n a t u r e o f the n o n - r a n d 0 m grain d i s t r i b u t i o n Once the non-random nature of the grain distributions has been established, it becomes necessary to find out the way in which the distribution departs from random. In particular it is necessary to decide whether the different grain types are more regularly distributed or more grouped than would be expected in a random pattern. This is most easily done by a study of the like against like contacts. For this purpose my data, plotted in tables similar to Kretz's (1969, p. 60) Table 12, were treated in the following way. It was observed that in all the tables the number of observed unlike contacts of type say XY were ahvays almost the same as the number of contacts of type YX. Out of about 1000 such compared frequencies only two showed a significant difference at 95% level of significance using the Z2 test. This result was expected since the data was collected by traversing backwards and forwards across the slide (see Kretz 1969, Fig. 6). Therefore all such pairs were summed. Tables were then made for each of the 39 sections showing the ratio of observed to expected contacts of each type. Ratios greater than one indicate more contacts than would be expected in a random arrangement, while ratios less than one indicate less contacts. (a) Like against like contacts. It was pointed out above that if the different grain types are distributed in a regular manner through the rock-like sites in a crystal lattice, then the frequency of contacts between like phases should
GRAIN CONTACTS
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be reduced below that to be expected in a random arrangement. On the other hand grouping of grains of like type should lead to an increase in observed contacts betaveen like phases. The tables for the 39 rocks were examined and it was found that in general there were far more cases of frequencies being less than expected on the random hypothesis than of the reverse (Table 1). The probabilities that the observed frequencies could arise by chance from a random pattern of grains were calculated and are given in Table 1. It is clear from Table 1 that the distribution pattern of the grain types in the thin sections examined is most unlikely to be random and it is also clear that grains of any given type tend to avoid contact with themselves (except for muscovite). The same test could be applied to all the other observed types of contacts. However, a quantitative test has advantages so the following one was carried out instead. (b) Tests of the contact ratios (observed[expected). As mentioned above, for each slide examined a table of contact ratios was constructed. A weighted mean ratio and a weighted standard deviation of the mean were calculated for each type of contact. The weighting factor used was the number of observations; each individual ratio was multiplied by the number of observations on which it was based. Tables 2a and b show the weighted mean ratio and the weighted standard deviation of the mean. If the grain distribution were random then the expected mean ratio would be 1.0. Using this expected mean the t-test was applied for each cell in Table 2 and the results of the test are shown. Where the test proved to be not significant at the 95% level this does not signify that the grains were randomly distributed but merely that an), departure from randomness was not sufficiently marked to be detected by the test when using the amount of data employed. For instance, the BB ratio does not differ significantly from 1.0 according to the test, yet the test used in Table 1 indicates that the distribution is most unlikely to be random. Duplicate and even triplicate sections were cut from most of the Siberian specimens. This made possible a 'one way model II weighted analysis of variance' (Hald 1952, p. 437) to be carried out on the contact ratios. The object of this test was to determine whether the variations of the ratios determined from duplicate slides cut from the same hand-specimen were sufficiently large to explain the variations between the ratios determined from different hand-specimens. If the within hand-specimen variation was significantly less than the between hand-specimen variation, then the pattern of distribution or at least its strength of development must vary from rock to rock.
The results of the analysis of variance were that: PP, PO; PK, BO, BH; KK, BB, BX, OH, all showed a significantly greater between rocks variation, than within rocks variation (symbols explained in Table 2a). The first two differed at 99.9%, the next three at 99%, and the last four at 95%. The rest
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DEREKFLINN
Table 2a. S i b e r i a n Rocks. x,Veighted m e a n a n d w e i g h t e d s t a n d a r d de vi a t i on o f the m e a n of contact ra t i os (ob-
P
P 0.7784-0.021 V=34 ***
K
K 1.0194-0.051 V=14 NS 0.8794-0.050 V=10 *
B
B 1.1564-0.021 V=23 *** 1.2234-0.084 V=ll * 0.8754-0.14 V=20 NS
Ap
Ap 1.2074-0.055 V=22 *** 1.0174-0.059 V=6 NS 0.9194-0.172 V=10 NS i ns uffi c i e nt data
O
O 1.102-4-0.054 V=31 NS 0.8494-0.046 V=9 ** 1.4854-0.197 V=24 * 2.4254-0.432 V=8 * insufficient data
H X
Q
t-test
V - d e g r e e s of f r e e d o m N S - n o t significant at 95.0% * significant at 95.0% ** * * 99.0% * ** * ~ 99.9%
P K B Ap -
plagioclase K-felspar biotite apatite
O H X Q
-
o p a q u e ores hornblende clino-pyroxene quartz
showed no significant difference at the 95% level between these sources of variation. These results mean that there is a tendency for the distribution patterns of the grains to vary from rock to rock. More tests would be needed to determine whether the variation was in type or in strength of development of the pattern, and also to determine whether it depended on rock composition or position in the field.
Conclusions The conclusion to be drawn from the results of the tests reported above is that the mineral grains in the thin sections studied are not distributed at random. Since the thin sections were cut in a random manner from apparently homogeneous hand specimens, all the conclusions drawn from the study of the thin sections may be extended to the whole hand-specimen. Thus the grains are not distributed at random in the rock. The tests vary in their ability to detect departure from randomness. The most powerful test seems to be that used in Table 1 if at least seven determinations of the ratio are made. In general the other tests are more informative. One of the tests used by Kretz (1969, p. 58) was the same as one of the tests used above. For him it did not lead to rejection of the hypothesis that
GRAIN CONTACTS
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served/expected) H 1.3144-0.021 V=19 *'* 1.2874-0.038 V=4 ** 0.8724-0.175 V=16 NS 1.3574-0.330 V=13 NS 1.2974-0.189 V=18 NS 0.6404-0.023 V=16 ***
x 1.179--1-0.043 V=28 *** 1.0894-0.234 V=9 NS 1.0454-0.081 V=14 NS 1.4544-0.183 V=8 * 1.3834-0.077 V=25 *** 1.1944-0.055 V=19 '** 0.729-4-0.053 V=24 ***
Q 1.3194-0.020 V=12 *** 0.9604-0.081 V=3 NS 0.955 4-0.064 V=2 NS 1.5834-0.849 V=3 NS 0.9974-0.119 V=9 NS 0.5604-0.057 V=3 ** 0.9484-0.020 V=12 * 0.6124-0.035 V=10 ***
P K B Ap O H X Q
the grains in his thin section were randomly distributed, but a study of the data he tested shows that they were very similar to the data used for this paper in that contacts between like phases occurred less frequently than expected. If this fact is taken into account Kretz's thin section can be recognized as probably having a weakly non-random distribution of the mineral grains. Rogers & Bogy (1958) and Mahan & Rogers (1968) have also worked on these lines. For each type of grain contact they calculate the ratio 'percentage of contact length of mineral A occupied by mineral B divided by modal percent of mineral B'. If for 'modal percent of mineral B' is substituted 'grain percent of mineral B' then their ratio would be algebraically identical to the contact ratio used above in this paper. The use of grain percent instead of modal percent is an obvious improvement in a study of grain contacts, and also avoids the necessity of having to determine the mode. Use of modal percent implies that all grain types occur with the same grain size distribution. This is not necessarily or even usually true. Rogers & Bogy (1958) determined the ratio for 31 granitic rocks and Mahan & Rogers (1968) for 18 metamorphic rocks. They averaged the ratios for each contact type and applied the t-test to detect significant differences between the observed ratios and 1.0, the ratio expected if the grain distribution is random. They used geometric means instead of arithmetrie means as used above in the similar test. This is correct but does not make much difference for means close to 1.0. They apparently did not use weighting which could make a considerable difference if their rocks varied in composition. The averaged ratios resemble those of Kretz (1969, Table 14) and of this paper (Table 2) in that contacts between like phases occur less frequently than would be expected if the grain types were randomly distributed
368
DEREK FLINN
Table 2b. Shetland granitic gneisses.
P B
P 0.7194-0.025 V=5 ***
B 1.1154-0.012 V=5 *** 0.9084-0.020 V=5 "*
Q AI
Q 1.2234-0.022 V=5 "** 1.0364-0.011 V=5 * 0.9304-0.041 V=5 NS
AI 1.0134-0.055 V=5 NS 1.2294-0.385 V=5 NS 0.6064-0.010 V=4 *** 2.4254-1.017 V=4 NS
See Table 2a for meaning of symbols.
(except for biotite). Their results differ in that they show considerable differences within tables between AB and BA type contacts. These differences are so large that they are probably significant, in which case they could imply a polar pattern of distribution of grains within the rock, if these authors always traversed in the same direction in all their rocks when collecting data. This seems to be most unlikely. The cause of these differences is probably the use of modal instead of grain percentages in calculating the ratios. All the data reviewed in this paper indicate that in crystalline rocks grain types are not distributed at random. The authors of the earlier papers discuss their results in terms of the nucleation of the grains without discussing the mechanism by which the nucleation is controlled. However, some authors, in particular DeVore (1956) and Laffitte (1957) predicted nonrandom grain arrangements from a consideration of the role of interracial energy in rocks. Laffitte (1957, p. 328) predicted that because cases of metamorphic differentiation and segregation occur, this must be due to the redistribution of phases in such a way that the total interfacial energy is minimized. From this he concluded that contacts between like phases have less interfacial energy than contacts between unlike phases. However, it is well known that 'interface energies of boundaries between different phases are invariably lower than the energies of boundaries between two crystals of the same phase' (Sinnott 1958, p. 273). This agrees with the observation that in recrystallized rocks contacts between like phases are reduced in number. Thus metamorphic differentiation and segregation involve an increase in interfacial energy and these processes must be driven by some other source of energy. DeVore (1956, p. 54) came to the conclusion that 'it is the interface energy which largely controls the sites of nucleation, crystal g r o ~ h . . .'. However, he assumed that this control resulted not in certain preferred patterns of grain distribution in rocks but in certain ratios of the minerals present. Thus he concluded that rocks owe their compositions to interfacial energy control. Material is brought into or driven out of the rock so that the 'resulting ratio of minerals present can be considered as an equilibrium condition in which
GRAIN CONTACTS
369
the total energy of the mineral assemblage is lower than any other mineral ratio'. From the context, 'total energy' seems to mean total interracial energy. He cites the fact that granites tend to contain 30°,/0 quartz. Apart from the alternative explanation provided by igneous petrology, there is also the fact that the rocks concerned were only called granite and therefore taken into consideration by him because they happened to contain about 30% quartz. The conclusion I draw from the data is that during strong recrystallization a state approaching minimum interfacial energy is attained by reducing the relatively high energy contacts between like phases relative to the lower energy contacts between unlike phases. This is done by a redistribution of the phases already present in the rock and may involve changes in grain size and therefore frequency of occurrence of grains. There is no evidence that changes in bulk composition are caused. The redistribution of grains may take place by diffusion, nucleation and growth of unlike phases in boundaries between like phases. From the interfacial energy point of view, a phase could nucleate more easily in a high energy boundary between grains of another phase than in a boundary against its own phase. However, it seems likely that much of the redistribution of phases takes place during recrystallization as a result of grain boundary migration arising in part from grain growth and in part from the attempt of the boundaries to establish an equilibrium 'triple point network' pattern. Since contacts bet~veen like phases have a relatively high energy there will always be a tendency for grain boundaries to migrate in such a way that unlike phases are inserted between like phases wherever possible so as to produce lower energy boundaries. It seems unlikely that in a thoroughly recrystallized rock any conclusions can be drawn about the nature of the original interface within which any given grain was nucleated. This type of study has been used in an attempt to distinguish igneous from metamorphic granites (Mahan & Rogers 1968, Vistelius 1966). These works are based on the assumption that the granite as seen now in thin section shows the grain contacts produced by crystallization from the melt. However, granites characteristically have grain boundaries which seem to indicate much reaction and recrystallization so that even if individual phases can be said to have crystallized early or late it cannot be assumed that their grains are now occurring in contact with the grains they were in contact with when the rock first crystallized. Thus the grain relationships described by Rogers & Bogy (1958) for their granites may resemble the grain relationships found by Mahan & Rogers (1968) for metamorphic rocks, because both result from recrystallization in the solid state.
ACKNOWLEDGEMENTS. T h e author is grateful to the University of Chicago and Prof. H. Ramberg for financial support and facilities in 1957-8 when this work was started, and to the Royal Society, the Academy of Sciences of the U.S.S.R., I.G.E.M., Moscow, and Acade-
370
DEREKFLINN
mician D.C. Korzhinskii for financial support and facilities in 1960 when the work was continued. ~I.A. Litzarev kindly supplied the specimens of Siberian granulites. April 1969
Jane tterdman Laboratories of Geology, University of Liverpool, England
REFERENCES
DEVORE, G.~V. 1956: Surface chemistry as a chemical control on mineral associations. flour. GeoL 64, 31-56. FLINN, D. 1954: On the time relations between regional metamorphism and permeation in Delting, Shetland. Quart. flour. Geol. Soc. Lond. 110, 177-201. FL1NN, D. 1965: On the statistical analysis of axial distribution diagrams. NeuesJb. l~liner. Mh. 2, 54-64. HALD, A. 1952: Statistical Theory with Engbteerbtg Applications. Wiley, New York. Krm'TZ, R. 1969: On the spaclal distribution of crystals in rocks. Lithos 2, 39-65. LAFFIa'rE, 1957: hztroduction a l'dtude des roches mltamorphiques et des gltes m~tallifbres. Paris. MAHAN, S.M. & ROCERS, J.J.W. 1968: A study of grain contacts in some high grade metamorphic rocks. Am. llllneral. 53, 323-27. ROCERS, J.J.W. & BooY, D.B. 1958: A study of grain contacts in granitic rocks. Science 127, 470-71. SINNOTT, M.J. 1958: Solid State for Enghteers. Wiley, New York. VlSTEUUS, A.B. 1966: Genesis of the Mt. Belaya granodlorite, Kamchatka (an experiment in stochastic modeling). Doklady Acad. ScL USSR ; Earth Science section, 167, 48-50
CONTACTS
IN
CRYSTALLINE
ROCKS
FLINN
FLINN, D. 1969: Grain contacts in crystalline rocks. Lithos 3, 361-370. The distributions of mineral grains in gneisses from Shetland and Siberia are studied. Statistical tests show that the grains, instead of being distributed at random, are specially arranged so that grains of the same phase tend not to occur in contact with each other. It is concluded that this arrangement arises from grain boundary migration leading to the insertion of grains of one phase between pairs of like grains of other phases. This process arises from the fact that the interfacial energies of contacts between like phases are greater than those between unlike phases.
Some years ago it was suggested that recrystallization of a rock might give rise to a distribution within the rock of the mineral grains of different type in a pattern minimizing the interfacial (grain boundary) energy, and further that to minimize the total interfacial energy of the rock would bring about not only an ideal arrangement of the mineral grains but also an ideal proportional relationship between the grains, which would result in rocks of certain compositions (De Vore 1956, p. 54; Laffitte 1957, p. 328). The second process does not necessarily follow from the first, and indeed seems unlikely from the field occurrence and appearance of rocks. However, the first process, the rearrangement of mineral grains into a pattern tending to minimize the interfacial energy, seems sufficiently possible to be worth seeking the results in natural rocks. The writers mentioned above did not test their hypotheses in this way but Rogers & Bogy (1958) did investigate grain contacts. In 1957-8 I had time and opportunity to investigate this problem, and working on granitic gneisses from Shetland, developed statistical tests of randomness of mineral grain distribution. In 1960 another opportunity arose and I spent much time testing a suite of granulites from Siberia, but I was unable to finish the project because of the multiplicity of tests that may be employed. However, in a recent issue of this journal there appeared a study of grain contacts in a metamorphic rock employing a test identical to one used by me (Kretz 1969, pp. 39-65). The very detailed paper by Kretz makes it possible for me to publish my results in a much briefer form than I could have otherwise employed. Furthermore, my results are made more interesting by the fact that although they are very similar to those of Kretz, they led me to diametrically opposite conclusions.
362
DEREKFLINN
The problem investigated Is the position of a mineral grain of a given type dependent to any extent on the positions of the same or other types of mineral grains ? If so, is this dependence due to interfacial energy effects ? Clearly, if a rock is metasomatised by the migration of material along grain boundaries a special arrangement of mineral grains could arise by chemical reaction. A special arrangement of mineral grains might also arise during penetrative deformation, due to mechanical differences in the different grains. An attempt was made to eliminate mineral grain patterns of the last tnvo types by special selection of the rocks studied. Selection of material Regional metamorphic rocks with the following characteristics were studied: uniform grain size, equidimensional grains, simple pattern of grain boundaries (triple point network), lack of signs of mechanical strain in grains, no obvious signs of preferred orientation, and no signs of reaction betnveen the grains. Such rocks are likely to have resulted from prolonged recrystallization under static conditions. If surface energy influences the relative positions of mineral grains of different type then that influence should be most apparent in such rocks, and the other processes mentioned above which tend to produce special arrangements of grains should not have been active. Suites of such rocks from two areas were studied. Five thin sections of permeation gneiss from the Colla Firth permeation belt of Shetland (Flinn 1954, p. 187) were selected from several dozen such sections and were studied first. Later, a group of 35 thin sections cut from 16 Siberian granulites were studied. The granulites were selected after hand-specimen and thin section examination of a large number of specimens from the Archean crystalline complex in the Aldanski shield in southern Yakutia. The Shetland rocks were composed of assemblages made up from the following minerals: Quartz, plagioclase (oligoclase), biotite, and muscovite. Rocks containing K-felspar were not studied as this felspar shows a late replacement appearance. The Siberian rocks contained assemblages of the following minerals: plagioclase, K-felspar, biotite, apatite, opaque ores, hornblende, quartz, augitic clino-pyroxene. Neither the hand-specimens nor the thin sections cut from these rocks showed any signs of inhomogeneity or of tectonite fabric. The hand-specimens carried no orientation marks so that the thin sections can be considered as having been cut at random. Tests of homogeneity No statistical tests of homogeneity were carried out, partly because the specimens showed no obvious signs of inhomogeneity and partly because
GRAIN CONTACTS
363
inhomogeneity does not seem to affect the hypothesis. A sharp junction between areas of very different composition might give rise to abnormal grain relation within the junction, but no such junctions were present. T e s t s o f grain relations The null hypothesis tested was that grains occur in a random arrangement in the rock. That is to say, the numbers of grains of two given types that occur in contact depend only on the total numbers of grains of each of the taro types present. There are two alternative hypotheses. One is that grains tend to occur in a regular pattern, like sites in a crystal lattice. This arrangement would lead to less contacts between like phases than the null hypothesis predicts. The other is that like grains tend to occur in groups, giving rise to more contacts between like phases than the null hypothesis predicts. The possible effects of these alternatives on the frequencies of contacts between non-like phases are complex and difficult to predict. The test selected as most useful was designed to distinguish between the null hypothesis and the alternatives without distinguishing between the alternatives. This test was subsequently used without modification as a test of AVA (Flinn 1965) and later by Kretz (1969, p. 58), who named it the 'linetransect' method and obtained it from Vistelius (1966). There is no need to describe the test again, except to note that for each thin section a sequence of not less than 1000 grains were noted for testing. All of the 39 sections gave sequences of grain types which differed from random at a very high level of significance when this test was applied. The degrees of freedom used in the tests varied from 1 to 9, since the assemblages varied from 2 to 4, after grouping the less common grain types until no cells had an expected frequency of less than 5. The smallest value found was 7.2=16 and was associated with one degree of freedom. Thus the pattern of grain distribution is most unlikely to be a random one. Kretz carried out the same test in the same way on his Fig. 1 (Kretz 1969, p. 40) and found that the null hypothesis of randomness was just acceptable at the 95 per cent level of significance. However, it should be noted that the observed number of contacts between like phases is less than the expected number in all three cells in his Table 14 and in all six in his Table 12 (Kretz 1969, p. 60). This arrangement would be expected for Table 14 once in every eight trials if the grains were randomly arranged and for Table 12, once in every 64 trials. Taken alone, these facts are not enough to prove that Kretz's Fig. 1 is a non-random distribution of grains. However, most of the different types of contacts between like phases considered in the 21 rocks I investigated occurred less frequently than expected (see Table 1), and this appears to be a characteristic feature of recrystallized rocks. Thus Kretz's Fig. 1 is probably a non-random distribution but the test he used was not powerful enough to recognize this fact. It is an unfortunate feature of this test that the degrees of freedom are in no way dependent on the
364
DrREKFLINN
Table 1. F r e q u e n c y greater than one Type of contaetratio PP KK BB Ap Ap OO HH XX QQ MM
o f o c c u r r e n c e o f g r a i n c o n t a c t r a t i o s less a n d
>1.0 4 4 6 0 0 0 5 0 -
Siberia <1.0 100xProb. 31 11 20 26 33 20 26 13 -
1.5 x 4.2 0.3 1.5 x 1.2x 9.5 x 7.9 X 1.2x
1 0 -~
10 10 10 10 10
-6 -s -s -3 -z
-
>1.0
Shetland <1.0 100xProb.
0 1 0 0 _ _ 1
5 4 3 4 _ _ 4
4
0
3.1 15.6 12.5 6.3 _ _ 15.6 6.3
Probability of observed frequeneies=nCr(½) n n=total number of events r=number of events of one type F o r c o n t a c t r a t i o s y m b o l s see T a b l e 2a.
amount of data collected, so that it is difficult to determine how much data to collect. It is possible that the collection of more data (I noted four times as many grain contacts per section as Kretz did) would have led Kretz to the rejection of the null hypothesis.
T e s t s o f t h e n a t u r e o f the n o n - r a n d 0 m grain d i s t r i b u t i o n Once the non-random nature of the grain distributions has been established, it becomes necessary to find out the way in which the distribution departs from random. In particular it is necessary to decide whether the different grain types are more regularly distributed or more grouped than would be expected in a random pattern. This is most easily done by a study of the like against like contacts. For this purpose my data, plotted in tables similar to Kretz's (1969, p. 60) Table 12, were treated in the following way. It was observed that in all the tables the number of observed unlike contacts of type say XY were ahvays almost the same as the number of contacts of type YX. Out of about 1000 such compared frequencies only two showed a significant difference at 95% level of significance using the Z2 test. This result was expected since the data was collected by traversing backwards and forwards across the slide (see Kretz 1969, Fig. 6). Therefore all such pairs were summed. Tables were then made for each of the 39 sections showing the ratio of observed to expected contacts of each type. Ratios greater than one indicate more contacts than would be expected in a random arrangement, while ratios less than one indicate less contacts. (a) Like against like contacts. It was pointed out above that if the different grain types are distributed in a regular manner through the rock-like sites in a crystal lattice, then the frequency of contacts between like phases should
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be reduced below that to be expected in a random arrangement. On the other hand grouping of grains of like type should lead to an increase in observed contacts betaveen like phases. The tables for the 39 rocks were examined and it was found that in general there were far more cases of frequencies being less than expected on the random hypothesis than of the reverse (Table 1). The probabilities that the observed frequencies could arise by chance from a random pattern of grains were calculated and are given in Table 1. It is clear from Table 1 that the distribution pattern of the grain types in the thin sections examined is most unlikely to be random and it is also clear that grains of any given type tend to avoid contact with themselves (except for muscovite). The same test could be applied to all the other observed types of contacts. However, a quantitative test has advantages so the following one was carried out instead. (b) Tests of the contact ratios (observed[expected). As mentioned above, for each slide examined a table of contact ratios was constructed. A weighted mean ratio and a weighted standard deviation of the mean were calculated for each type of contact. The weighting factor used was the number of observations; each individual ratio was multiplied by the number of observations on which it was based. Tables 2a and b show the weighted mean ratio and the weighted standard deviation of the mean. If the grain distribution were random then the expected mean ratio would be 1.0. Using this expected mean the t-test was applied for each cell in Table 2 and the results of the test are shown. Where the test proved to be not significant at the 95% level this does not signify that the grains were randomly distributed but merely that an), departure from randomness was not sufficiently marked to be detected by the test when using the amount of data employed. For instance, the BB ratio does not differ significantly from 1.0 according to the test, yet the test used in Table 1 indicates that the distribution is most unlikely to be random. Duplicate and even triplicate sections were cut from most of the Siberian specimens. This made possible a 'one way model II weighted analysis of variance' (Hald 1952, p. 437) to be carried out on the contact ratios. The object of this test was to determine whether the variations of the ratios determined from duplicate slides cut from the same hand-specimen were sufficiently large to explain the variations between the ratios determined from different hand-specimens. If the within hand-specimen variation was significantly less than the between hand-specimen variation, then the pattern of distribution or at least its strength of development must vary from rock to rock.
The results of the analysis of variance were that: PP, PO; PK, BO, BH; KK, BB, BX, OH, all showed a significantly greater between rocks variation, than within rocks variation (symbols explained in Table 2a). The first two differed at 99.9%, the next three at 99%, and the last four at 95%. The rest
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Table 2a. S i b e r i a n Rocks. x,Veighted m e a n a n d w e i g h t e d s t a n d a r d de vi a t i on o f the m e a n of contact ra t i os (ob-
P
P 0.7784-0.021 V=34 ***
K
K 1.0194-0.051 V=14 NS 0.8794-0.050 V=10 *
B
B 1.1564-0.021 V=23 *** 1.2234-0.084 V=ll * 0.8754-0.14 V=20 NS
Ap
Ap 1.2074-0.055 V=22 *** 1.0174-0.059 V=6 NS 0.9194-0.172 V=10 NS i ns uffi c i e nt data
O
O 1.102-4-0.054 V=31 NS 0.8494-0.046 V=9 ** 1.4854-0.197 V=24 * 2.4254-0.432 V=8 * insufficient data
H X
Q
t-test
V - d e g r e e s of f r e e d o m N S - n o t significant at 95.0% * significant at 95.0% ** * * 99.0% * ** * ~ 99.9%
P K B Ap -
plagioclase K-felspar biotite apatite
O H X Q
-
o p a q u e ores hornblende clino-pyroxene quartz
showed no significant difference at the 95% level between these sources of variation. These results mean that there is a tendency for the distribution patterns of the grains to vary from rock to rock. More tests would be needed to determine whether the variation was in type or in strength of development of the pattern, and also to determine whether it depended on rock composition or position in the field.
Conclusions The conclusion to be drawn from the results of the tests reported above is that the mineral grains in the thin sections studied are not distributed at random. Since the thin sections were cut in a random manner from apparently homogeneous hand specimens, all the conclusions drawn from the study of the thin sections may be extended to the whole hand-specimen. Thus the grains are not distributed at random in the rock. The tests vary in their ability to detect departure from randomness. The most powerful test seems to be that used in Table 1 if at least seven determinations of the ratio are made. In general the other tests are more informative. One of the tests used by Kretz (1969, p. 58) was the same as one of the tests used above. For him it did not lead to rejection of the hypothesis that
GRAIN CONTACTS
367
served/expected) H 1.3144-0.021 V=19 *'* 1.2874-0.038 V=4 ** 0.8724-0.175 V=16 NS 1.3574-0.330 V=13 NS 1.2974-0.189 V=18 NS 0.6404-0.023 V=16 ***
x 1.179--1-0.043 V=28 *** 1.0894-0.234 V=9 NS 1.0454-0.081 V=14 NS 1.4544-0.183 V=8 * 1.3834-0.077 V=25 *** 1.1944-0.055 V=19 '** 0.729-4-0.053 V=24 ***
Q 1.3194-0.020 V=12 *** 0.9604-0.081 V=3 NS 0.955 4-0.064 V=2 NS 1.5834-0.849 V=3 NS 0.9974-0.119 V=9 NS 0.5604-0.057 V=3 ** 0.9484-0.020 V=12 * 0.6124-0.035 V=10 ***
P K B Ap O H X Q
the grains in his thin section were randomly distributed, but a study of the data he tested shows that they were very similar to the data used for this paper in that contacts between like phases occurred less frequently than expected. If this fact is taken into account Kretz's thin section can be recognized as probably having a weakly non-random distribution of the mineral grains. Rogers & Bogy (1958) and Mahan & Rogers (1968) have also worked on these lines. For each type of grain contact they calculate the ratio 'percentage of contact length of mineral A occupied by mineral B divided by modal percent of mineral B'. If for 'modal percent of mineral B' is substituted 'grain percent of mineral B' then their ratio would be algebraically identical to the contact ratio used above in this paper. The use of grain percent instead of modal percent is an obvious improvement in a study of grain contacts, and also avoids the necessity of having to determine the mode. Use of modal percent implies that all grain types occur with the same grain size distribution. This is not necessarily or even usually true. Rogers & Bogy (1958) determined the ratio for 31 granitic rocks and Mahan & Rogers (1968) for 18 metamorphic rocks. They averaged the ratios for each contact type and applied the t-test to detect significant differences between the observed ratios and 1.0, the ratio expected if the grain distribution is random. They used geometric means instead of arithmetrie means as used above in the similar test. This is correct but does not make much difference for means close to 1.0. They apparently did not use weighting which could make a considerable difference if their rocks varied in composition. The averaged ratios resemble those of Kretz (1969, Table 14) and of this paper (Table 2) in that contacts between like phases occur less frequently than would be expected if the grain types were randomly distributed
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DEREK FLINN
Table 2b. Shetland granitic gneisses.
P B
P 0.7194-0.025 V=5 ***
B 1.1154-0.012 V=5 *** 0.9084-0.020 V=5 "*
Q AI
Q 1.2234-0.022 V=5 "** 1.0364-0.011 V=5 * 0.9304-0.041 V=5 NS
AI 1.0134-0.055 V=5 NS 1.2294-0.385 V=5 NS 0.6064-0.010 V=4 *** 2.4254-1.017 V=4 NS
See Table 2a for meaning of symbols.
(except for biotite). Their results differ in that they show considerable differences within tables between AB and BA type contacts. These differences are so large that they are probably significant, in which case they could imply a polar pattern of distribution of grains within the rock, if these authors always traversed in the same direction in all their rocks when collecting data. This seems to be most unlikely. The cause of these differences is probably the use of modal instead of grain percentages in calculating the ratios. All the data reviewed in this paper indicate that in crystalline rocks grain types are not distributed at random. The authors of the earlier papers discuss their results in terms of the nucleation of the grains without discussing the mechanism by which the nucleation is controlled. However, some authors, in particular DeVore (1956) and Laffitte (1957) predicted nonrandom grain arrangements from a consideration of the role of interracial energy in rocks. Laffitte (1957, p. 328) predicted that because cases of metamorphic differentiation and segregation occur, this must be due to the redistribution of phases in such a way that the total interfacial energy is minimized. From this he concluded that contacts between like phases have less interfacial energy than contacts between unlike phases. However, it is well known that 'interface energies of boundaries between different phases are invariably lower than the energies of boundaries between two crystals of the same phase' (Sinnott 1958, p. 273). This agrees with the observation that in recrystallized rocks contacts between like phases are reduced in number. Thus metamorphic differentiation and segregation involve an increase in interfacial energy and these processes must be driven by some other source of energy. DeVore (1956, p. 54) came to the conclusion that 'it is the interface energy which largely controls the sites of nucleation, crystal g r o ~ h . . .'. However, he assumed that this control resulted not in certain preferred patterns of grain distribution in rocks but in certain ratios of the minerals present. Thus he concluded that rocks owe their compositions to interfacial energy control. Material is brought into or driven out of the rock so that the 'resulting ratio of minerals present can be considered as an equilibrium condition in which
GRAIN CONTACTS
369
the total energy of the mineral assemblage is lower than any other mineral ratio'. From the context, 'total energy' seems to mean total interracial energy. He cites the fact that granites tend to contain 30°,/0 quartz. Apart from the alternative explanation provided by igneous petrology, there is also the fact that the rocks concerned were only called granite and therefore taken into consideration by him because they happened to contain about 30% quartz. The conclusion I draw from the data is that during strong recrystallization a state approaching minimum interfacial energy is attained by reducing the relatively high energy contacts between like phases relative to the lower energy contacts between unlike phases. This is done by a redistribution of the phases already present in the rock and may involve changes in grain size and therefore frequency of occurrence of grains. There is no evidence that changes in bulk composition are caused. The redistribution of grains may take place by diffusion, nucleation and growth of unlike phases in boundaries between like phases. From the interfacial energy point of view, a phase could nucleate more easily in a high energy boundary between grains of another phase than in a boundary against its own phase. However, it seems likely that much of the redistribution of phases takes place during recrystallization as a result of grain boundary migration arising in part from grain growth and in part from the attempt of the boundaries to establish an equilibrium 'triple point network' pattern. Since contacts bet~veen like phases have a relatively high energy there will always be a tendency for grain boundaries to migrate in such a way that unlike phases are inserted between like phases wherever possible so as to produce lower energy boundaries. It seems unlikely that in a thoroughly recrystallized rock any conclusions can be drawn about the nature of the original interface within which any given grain was nucleated. This type of study has been used in an attempt to distinguish igneous from metamorphic granites (Mahan & Rogers 1968, Vistelius 1966). These works are based on the assumption that the granite as seen now in thin section shows the grain contacts produced by crystallization from the melt. However, granites characteristically have grain boundaries which seem to indicate much reaction and recrystallization so that even if individual phases can be said to have crystallized early or late it cannot be assumed that their grains are now occurring in contact with the grains they were in contact with when the rock first crystallized. Thus the grain relationships described by Rogers & Bogy (1958) for their granites may resemble the grain relationships found by Mahan & Rogers (1968) for metamorphic rocks, because both result from recrystallization in the solid state.
ACKNOWLEDGEMENTS. T h e author is grateful to the University of Chicago and Prof. H. Ramberg for financial support and facilities in 1957-8 when this work was started, and to the Royal Society, the Academy of Sciences of the U.S.S.R., I.G.E.M., Moscow, and Acade-
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DEREKFLINN
mician D.C. Korzhinskii for financial support and facilities in 1960 when the work was continued. ~I.A. Litzarev kindly supplied the specimens of Siberian granulites. April 1969
Jane tterdman Laboratories of Geology, University of Liverpool, England
REFERENCES
DEVORE, G.~V. 1956: Surface chemistry as a chemical control on mineral associations. flour. GeoL 64, 31-56. FLINN, D. 1954: On the time relations between regional metamorphism and permeation in Delting, Shetland. Quart. flour. Geol. Soc. Lond. 110, 177-201. FL1NN, D. 1965: On the statistical analysis of axial distribution diagrams. NeuesJb. l~liner. Mh. 2, 54-64. HALD, A. 1952: Statistical Theory with Engbteerbtg Applications. Wiley, New York. Krm'TZ, R. 1969: On the spaclal distribution of crystals in rocks. Lithos 2, 39-65. LAFFIa'rE, 1957: hztroduction a l'dtude des roches mltamorphiques et des gltes m~tallifbres. Paris. MAHAN, S.M. & ROCERS, J.J.W. 1968: A study of grain contacts in some high grade metamorphic rocks. Am. llllneral. 53, 323-27. ROCERS, J.J.W. & BooY, D.B. 1958: A study of grain contacts in granitic rocks. Science 127, 470-71. SINNOTT, M.J. 1958: Solid State for Enghteers. Wiley, New York. VlSTEUUS, A.B. 1966: Genesis of the Mt. Belaya granodlorite, Kamchatka (an experiment in stochastic modeling). Doklady Acad. ScL USSR ; Earth Science section, 167, 48-50