Journal of Petrology 2004 45(7):1369-1392; doi:10.1093/petrology/egh016
Journal of Petrology 45(7) © Oxford University Press 2004; all rights reserved
A Theoretical Study on the Formation of Growth Zoning in Garnet Consuming Chlorite
M. INUI1,* and
M. TORIUMI2
1 DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING, COLLEGE OF ENGINEERING, KOKUSHIKAN UNIVERSITY, 4-28-1 SETAGAYA, SETAGAYA-KU, TOKYO 154-8515, JAPAN
2 INSTITUTE OF FRONTIER RESEARCHES OF EARTH EVOLUTION, JAPAN MARINE SCIENCE AND TECHNOLOGY CENTER, YOKOSUKA 237-0061, JAPAN
RECEIVED
SEPTEMBER 10, 2002;
ACCEPTED
JANUARY 1, 2003
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ABSTRACT
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Chemical zoning of garnet is often used to deduce
P–
T paths of rocks by inverse calculation. To validate the derived
P–
T paths, it is desired to establish a method to predict
the chemical compositions of garnet theoretically. This study
proposes a new forward calculation of the formation of Mg–Fe–Mn
garnet from chlorite, which solves the non-linear simultaneous
equations using nested iterative calculations. Growth of garnet
consuming chlorite and quartz was modelled in a MnO–FeO–MgO–Al
2O
3–SiO
2–H
2O
system, using the most recent thermodynamic data for the minerals.
The prograde
P–
T history of the Sambagawa metamorphic
belt, SW Japan, was modelled. To reproduce growth zoning, crystallized
garnet was removed step by step from the system; perfect diffusion
was assumed for chlorite. The proposed model derived the evolution
of molar amounts and chemical compositions of Mg–Fe–Mn
chlorite and garnet. It successfully reproduced the shape of
the observed chemical profile of garnet, although the temperature
condition was higher than general observations. The Mn content
of the garnet core was generally high, and Mg/Fe ratio always
started rising rapidly after Mn was depleted. Thermodynamic
properties of minerals, initial chlorite composition,
P–
T path, H
2O partial pressure, and Ca content in garnet were varied
to test the behaviour of the system. The properties of Mn phases
influenced only the chemical composition of the garnet core.
The temperature range in which garnet grew depended on the H
2O
partial pressure or the Ca content in garnet.
KEY WORDS: chemical equilibrium; chemical zoning; garnet; forward modelling; Sambagawa metamorphic belt
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INTRODUCTION
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Chemical zoning retained in metamorphic minerals is one of the
most important clues for the
P–
T and reaction history
of the rock. It is widely accepted that the surface of growing
garnet is always in thermodynamic equilibrium with matrix minerals
during prograde metamorphism, gradually changing its composition
in response to changing conditions. As a result of the limited
chemical diffusion rate in garnet, the inner part of the grain
is isolated as it is covered by the new growth surface. Thus,
every chemical composition retained in garnet represents the
equilibrium composition at different times, provided these compositions
are not homogenized later through intracrystalline diffusion.
The relationship between the mineral compositions in metamorphic rocks is expressed by equations of thermodynamic equilibrium. If the mineral assemblage is known, it is possible to obtain the P–T trajectory. To estimate the P–T paths inversely, the differential thermodynamic method (Gibbs' method) is often applied (Spear & Selverstone, 1983
; Spear 1989, 1993). The method linearizes the non-linear simultaneous equations by differentiation. Given the composition changes of garnet, the changes of all other conditions including the intensive variables are calculated.
However, there are several weaknesses in the differential thermodynamic method. First, it requires the exact starting conditions. These are the absolute pressure, temperature, and the equilibrium mineral compositions under the P–T, which are not always clear. Second, it is difficult to estimate whether garnet grows or decomposes along the calculated P–T path. This information is desirable as it confirms the calculation result. It is possible to include volume changes of minerals as additional constraints. In this case, however, it is even more difficult to define the starting conditions concerning the molar amount of each mineral. Third, equations are solved using only the linear term in a Taylor's series expansion. The ignored terms might cause differences to the results. One way to solve the problem is to use a petrogenetic grid, which shows the validity of the assumed mineral assemblage. For instance, the effect of Mn on the stability of garnet-bearing assemblages has been investigated (Mahar et al., 1997). However, it is difficult to treat continuous reactions with this method, as mineral compositions must be assumed to obtain a univariant line. To analyse the behaviour of garnet more quantitatively, forward models have been proposed by several workers (Hollister, 1966; Banno et al., 1986). Thermodynamic properties of minerals have been significantly refined since these publications.
The aim of this study is to construct a new forward growth model of Mg–Fe–Mn garnet using the latest thermodynamic property data for the minerals. With the advancement of computers it has become possible to solve the total system without linearization. The initial chemical conditions and P–T path of the Sambagawa metamorphic rocks in central Shikoku, SW Japan, have been considered as a case study. Parameter studies have been performed with regard to the thermodynamic properties of minerals, P–T conditions, bulk-rock chemistry, H2O partial pressure, and Ca content in garnet. The sensitivity of the resulting garnet chemical trend is discussed.
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METHODS
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Basic equations
Formation of Mg–Fe–Mn garnet from Mg–Fe–Mn
chlorite was modelled in the MnO–FeO–MgO–Al
2O
3–SiO
2–H
2O
system. Quartz and water are involved to balance the stoichiometry.
A closed system was assumed, except for SiO
2 and H
2O. With fixed
bulk chemical composition the thermodynamic system is divariant,
as specified in Duhem's theorem. This means that the molar amounts
of all minerals (except quartz and water) and their chemical
compositions are uniquely determined at any pressure and temperature
of interest. Applying a series of
P–
T conditions, it is
possible to trace the progress of the metamorphic reaction.
Phases and end-members involved in the model in this study are listed in Table 1. Garnet and chlorite are treated as solid solutions consisting of three and four end-members, respectively. Grossular content was ignored, but will be discussed further in the discussion section below. The fourth end-member of chlorite (amesite) was necessary to take Tschermak's substitution into account.
The unit formula of garnet is generally written as
. M1 is the dodecahedral site, M2 is the octahedral site, and
T is the tetrahedral site. Ignoring the Ca end-member, garnet
is expressed as a solid solution of three end-members listed
in
Table 1. The mole fraction of each end-member is defined
as follows:
where
represents the mole fraction of end-member b in phase A, C
M1 the fraction of cation C in
M1 site, and C represents the number of cation C per unit formula.
The chlorite unit formula for normal metamorphic chlorite is generally written as 
. M2, M1, and M4 are the octahedral sites, and T1 and T2 are the tetrahedral sites. Mole fractions of end-members are defined as follows, according to the model of Holland et al. (1998):
Tschermak's substitution (MgSi–AlAl) was represented only by Mg end-members (clinochlore–amesite). The mole fraction of amesite is known by the amount of excess Al cations that preferentially fill the M1 site. No preferential distribution was assumed with regard to Fe and Mn, because of limited availability of the thermodynamic data.
In total, there are 11 variables in the system: T, P, MGrt, MChl, 
, 
, 
, 
, 
, 
, and 
. T is the temperature, P is the pressure, MA is the molar amount of phase A, and represents the mole fraction of end-member b in phase A. To constrain these variables, nine constraints are active in the system as required by Duhem's theorem. These include chemical equilibrium, constraints of stoichiometry, and constraints of mass balance. Each constraint is described below.
Three independent reactions exist in this system to make three non-linear constraints of chemical equilibrium:
 | (1) |
 | (2) |
 | (3) |
Reaction (1) is the garnet-forming
reaction that consumes chlorite. Exchange
reactions (2) and
(3) control the behaviour of Fe and Mn end-members. When the
system is in equilibrium, the following equation of chemical
equilibrium is true for each of the above reactions:
G is the Gibbs' free energy (J),
H is the enthalpy (J),
S is the entropy (J/K),
CP is the heat
capacity (J/K),
V is the volume [J/bar = J/(0·1 MPa)],
and
R is the gas constant.
Ar represents the change of
A as
a result of the reaction.
T and
P are the temperature (K) and
pressure (bar = 0·1 MPa) of interest, respectively.
Vr concerns only solid phases. Volume dependence on pressure is
ignored in this study. The contribution of water is involved
in the fugacity term mentioned below.
H,
S,
CP, and
V for each
mineral end-member were taken from the internally consistent
thermodynamic dataset of Holland & Powell (1998)
.
P–
T correction terms appear as the data are standardized to 25°C
and 0·1 MPa.
K is the equilibrium constant, through which
mineral compositions are involved in the equations:
where
a is the activity and
is the
stoichiometric coefficient.
The ideal parts of the activities of garnet are
The mixing between pyrope and almandine was modelled
as a regular solution, with the Margules parameter (
WPrp-Alm = 0·8 kJ/mol) as required by Holland & Powell (1998)
.
The mixing of other binary joins in garnet was calculated using
the asymmetric mixing parameters of Ganguly
et al. (1996)
.
The activity–composition relationships of chlorite were formulated as follows using the mole fractions of end-members. This is mathematically equivalent to the ideal mixing on site model (Holland et al., 1998; Inui & Toriumi, 2002):
The
mixing of chlorite was modelled as a regular solution using
Margules parameters from Holland & Powell (1998)
, and the
revised version from their web site (
http://www.esc.cam.ac.uk/astaff/holland/ds5/HP98_index.html):
WCln-Ame = 18,
WCln-Dph = 2·5,
WAme-Dph = 13·5
(kJ/mol). Ideal mixing was assumed between Mn-chlorite and other
end-members, as was required for the dataset (Holland &
Powell, 1998
).
The polynomial for water fugacity, fH2O, given by Holland & Powell (1990) was used to introduce the activity of water. For example, the activity term for reaction (1) is as follows:
Constraints of stoichiometry are the simple rules as follows:
 | (4) |
 | (5) |
Constraints of mass balance are the expression of a closed system. As quartz and water are in excess, there are four constraints of mass balance:
 | (6) |
 | (7) |
 | (8) |
 | (9) |
BC is the total molar amount of
the C cation. Using
constraints (1)–
(9), the molar amounts
and chemical compositions of garnet and chlorite were uniquely
determined at the
P and
T of interest.
Algorithm
Seven of the nine constraints are non-linear equations. Simultaneous non-linear equations had to be solved using an iterative algorithm described in the Appendix. The model was coded using Microsoft® Visual Basic® for Applications.
The calculation is valid only when both garnet and chlorite are thermodynamically stable, i.e. MGrt and MChl are both positive. It is necessary to delimit the P–T conditions before starting the calculation. It is generally known that garnet forms by consuming chlorite in response to temperature rise. Formation of garnet from chlorite provides the lower temperature boundary of the P–T range, in which both garnet and chlorite are stable. The boundary serves as the starting condition of the calculation in this study. The boundary, however, shifts in accordance with mineral compositions, as the degree of freedom of the garnet formation reaction is four. In this study, the low-temperature boundary for fixed chlorite composition and P was sought by gradually raising T. The resulting MGrt value is usually negative at first and turns positive at a certain T. The temperature was raised 0·5 or 1°C at a time and the boundary was identified when 1 mole of garnet first formed (MGrt > 1) from 10 000 moles of material chlorite.
Forward calculations can be performed giving serial P–T conditions. In addition, when simulating the formation of growth zoning in garnet, diffusion was assumed to be zero in garnet and very fast in chlorite. Garnet with a new chemical composition always formed enclosing the old crystal, whereas chlorite remained homogeneous. Chlorite chemistry (practically being the bulk-rock chemistry) was gradually changed as crystallized garnet was removed from the system (Fig. 1). The calculation proceeded as follows.
- Input initial temperature, pressure, and composition of chlorite. Derive the amounts and chemical compositions of garnet and chlorite that are in equilibrium.
- Subtract the amount of each cation newly incorporated into garnet from the previous bulk-rock chemical composition.
- Input the next temperature, pressure, and chlorite chemical composition, and obtain the amounts and chemical compositions of garnet and chlorite that are in equilibrium.
- Iterate steps (2) and (3).
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Fig. 1. Calculation algorithm of the proposed forward model. Chemical composition of chlorite is always recalculated after new garnet is formed and removed from the system.
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CASE STUDY
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The Sambagawa metamorphic belt
Garnet growth in the Sambagawa metamorphism (Inui & Toriumi,
2002
) was modelled as a case study. The Sambagawa metamorphic
belt is a subduction-related metamorphic belt in SW Japan (Wallis
& Banno, 1990
). It originates mainly from Mesozoic trench
sediments accreted to the eastern margin of the Eurasian continent.
The Sambagawa sequence was subject to regional metamorphism
up to the epidote–amphibolite facies, probably during
the Late Cretaceous (116 ± 10 Ma), followed by exhumation
and cooling during the interval 110–50 Ma (Isozaki &
Itaya, 1990
). It consists mainly of pelitic schists with smaller
amounts of mafic and quartz schists. Their foliations generally
have gentle northward dips and mineral lineations are east–west
and subhorizontal, slightly oblique compared with the Sambagawa
belt itself. The lithological layering and the metamorphic zonation
are roughly parallel to the foliation. Metamorphic zones are
defined using mineral assemblages in pelitic rocks: the chlorite
zone, the garnet zone, the albite–biotite zone, and the
oligoclase–biotite zone, in order of increasing metamorphic
grade (Higashino, 1990
). Major minerals in the pelitic schists
of the chlorite zone are muscovite, chlorite, albite, and quartz.
The garnet zone is defined by the appearance of garnet; biotite
and hornblende begin to occur in the albite–biotite zone;
oligoclase is found at the rim of albite porphyroblasts in the
oligoclase–biotite zone. The case seems suitable for this
study because garnet is considered to have grown mainly consuming
chlorite.
Observed zoning in garnet
Garnet in the Sambagawa metamorphic rocks is known to exhibit conspicuous chemical zoning, which is usually regarded as the record of prograde metamorphism. Inui & Toriumi (2002) estimated the prograde P–T path of the Sambagawa metamorphism using garnet in the pelitic schists from the albite–biotite zone in central Shikoku. The pelitic schists in this area mainly contain garnet, muscovite, chlorite, plagioclase, and quartz (Fig. 2a). The schistosity is dominantly defined by muscovite. Roughly half of the samples contain epidote. Biotite and hornblende are found in a few samples. Titanite and calcite are common as accessory phases. Graphite is abundant as inclusions in plagioclase porphyroblasts in some samples. Garnet in pelitic schists occurs mainly as subhedral to rounded porphyroblasts with a grain size of 0·7–4 mm (Fig. 2b). Sigmoidal quartz inclusion trails in garnet grains are common. Muscovite, paragonite, and epidote inclusions are often present. Muscovite often wraps around large garnet porphyroblasts, indicating that the formation of the dominant schistosity postdated garnet growth. Pressure shadows consisting of quartz are associated with garnet porphyroblasts. Typical chemical trends of garnet grains are shown in Fig. 3. Mn content in garnet decreases from core to rim. Mg/(Fe + Mg) usually increases from around 0·05 in cores to 0·10–0·13 at rims. Ca content initially increases from the core and then in some samples decreases toward the rim. The maximum XCa [= Ca/(Fe + Mg + Mn + Ca)] is generally over 0·30. The number of Al cations is always close to 2 p.f.u. (per formula unit), indicating that the Fe3+ content in garnet is insignificant. All chemical zonation maps and quantitative chemical analyses of garnet and other minerals were performed using a JEOL electron probe microanalyzer JXA-8900L at the University of Tokyo. The acceleration voltage was 15 kV, the beam current was set to 12 nA, and the beam diameter was focused (2–3 µm). Correction procedures follow the methods of Bence & Albee (1968).
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Fig. 2. Photomicrographs of the pelitic schists from the albite–biotite zone of the Sambagawa metamorphic belt. (a) Sample F0702, plane-polarized light. Muscovite, chlorite, and epidote form the lineation. (b) Sample F1403, plane-polarized light. Anastomosing muscovite and pressure shadows of quartz are found around a garnet porphyroblast. Epidote and muscovite inclusions occur in the garnet.
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Fig. 3. Typical chemical trends of garnet in the albite–biotite zone of the Sambagawa metamorphic belt. (a) Compositional profiles of grains from sample F0702 (Fig. 2a) and F1403 (Fig. 2b). (b) and (c) Mg–Fe–Mn triangular diagrams showing the chemical trends of garnet from the core to the rim.
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Conditions
The prograde
P–
T path estimated by Inui & Toriumi
(2002)
was used in this case study (see
Fig. 4). An Fe
2O
3–MnO–Na
2O–CaO–K
2O–FeO–MgO–Al
2O
3–SiO
2–H
2O
system was modelled by Inui & Toriumi (2002)
with the phase
assemblage garnet + muscovite + paragonite + chlorite + albite
+ quartz + epidote + water. Samples of pelitic schists with
the mineral assemblage garnet (+ biotite) + muscovite + chlorite
+ albite + quartz + epidote were selected. Application of the
model to biotite-bearing samples was possible because biotite
was not identified as inclusions in garnet. The fluid phase
was assumed as pure water. Inui & Toriumi (2002)
first estimated
the
P–
T condition at the garnet rim, which was confirmed
to be consistent with previous studies (e.g. Enami
et al., 1994
).
Then, using the condition as well as chemical analyses of garnet
and matrix minerals, the
P–
T trajectories shown in
Fig. 4 were derived by the differential thermodynamic method (Gibbs'
method; e.g. Spear, 1993
). Inui & Toriumi (2002)
concluded
that the studied garnet–muscovite–chlorite schists
experienced a pressure increase of
c. 0·3 GPa along with
heating of
c. 50–70°C, to achieve the peak metamorphic
conditions of around 520°C and 0·9 GPa at the rim.
A
P–
T path similar to the estimated path was applied in
the case study.
The average chemical composition of chlorite from the chlorite
zone of the Sambagawa metamorphic belt was used as the initial
chlorite composition in the case study. This corresponds to
the typical chlorite chemical composition in chlorite zone rocks
compiled from Higashino (1975)
and is considered to reflect
the chlorite composition before garnet formation in other areas
in the Sambagawa metamorphic belt (
Table 2). Total cation number
is 9·78 p.f.u., suggesting that the assumed cation distribution
in chlorite is imperfect. From charge balance, however, it is
clear that ferric iron can be ignored in this chlorite, as ferric
iron acts to further decrease the cation number per formula
unit. The deficiency of cation number might be due to di/tri-octahedral
exchange [(Al
VI)
2–Mg
3]. Thermodynamic data considering
the di/tri-octahedral exchange of Mg–Fe chlorite have
been published recently (Vidal
et al., 2001
). However, the model
does not consider Mn-chlorite, which is of profound importance
in the formation of garnet. Therefore, di/tri-octahedral exchange
was not considered in this study and the chlorite model of Holland
et al. (1998)
was used instead.
The major Mg–Fe–Mn minerals in the rocks are chlorite
and garnet. Other phases are ignored in this study. Biotite
is not considered to have existed in equilibrium with growing
garnet. Ca-bearing phases and grossular are ignored but will
be discussed later. Even if other phases had contributed to
garnet formation, the modelled system must have been valid as
a partial system. In that case, mass balance constraints may
be affected.
PH2O = Prock was assumed. Titanite is stable; rutile occurs in higher-grade zones, suggesting buffered CO2 content in the fluid phase (Goto, 2002). The influence of lower H2O partial pressure will be tested in the discussion below.
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RESULTS AND DISCUSSION
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Before starting the calculation, a valid
P–
T range must
be identified, as the calculation is possible only in the
P–
T field where both garnet and chlorite are stable. Through preliminary
calculations, a boundary line can be drawn where
MGrt turns
positive (dashed curve in
Fig. 5). The garnet-forming isopleth
of mode
0 has high d
P/d
T, suggesting that the beginning of garnet
formation largely depends on temperature. The initial
P–
T condition was sought along the model
P–
T path. About 2·1
moles of garnet formed from 10 000 moles of material chlorite
at around 490°C and 0·71 GPa, which was used as the
standard initial condition in this study. After that, a uniform
increase in
P–
T was used to model growth. The increments
of temperature rise and pressure increase were around 3·2°C
and 0·02 GPa, respectively. This path will be referred
to as the standard
P–
T path hereafter (
Fig. 5). The calculations
were generally performed up to 1·3 GPa.
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Fig. 5. Calculated stability range of garnet and the standard P–T path applied to the forward model in this study. Dashed line is the isopleth of garnet mode 0 (‘garnet in’) derived for the assumed initial chlorite composition in this study (Table 2). The calculation is applicable only to the high-temperature side of the dashed line, as garnet is unstable at the low-temperature side.
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Perfect diffusion in garnet
First, perfect diffusion was assumed both in garnet and chlorite
applying the standard
P–
T path: the calculation algorithm
is different from that shown in
Fig. 1 in that the garnet formed
at each
P–
T step was not removed from the system. Perfect
diffusion implies homogeneous garnet and chlorite at any one
moment, so that no chemical zoning results. Calculated chemical
composition changes in this case (
Fig. 6) represent only the
temporal changes of the mineral compositions while the system
follows the model
P–
T path.
Figure 6a is plotted against
temperature for convenience. The molar amount of garnet,
MGrt,
formed at each temperature increment (
MGrt/
T) is also shown.
A small amount of Mn-rich garnet formed initially, gradually
changing into Fe-rich garnet, and then into Fe–Mg garnet
at high temperatures. The chlorite composition changes in response
to mass balance, increasing in Mg (represented by
) and decreasing in Al, Fe, and Mn (represented
by
,
, and
, respectively). In
Fig. 6b,
it is observed that the Mn decrease in garnet predominates
at first and after that the Mg/Fe ratio increases rapidly (the
trend curves towards the Mg-apex). The latter stage will be
referred to as the ‘rapid Mg/Fe rise’ hereafter.
Figure 7 shows the compositional ratios of garnet and chlorite
plotted against temperature. These are the so-called pseudobinary
loops for Fe–Mn and Mg–Fe systems, which partly
explain the behaviour of the system. The Fe–Mn loop system
was located at lower temperature conditions compared with that
of the Mg–Fe system. Therefore, garnet was mainly produced
by the Fe–Mn system reaction at lower temperatures, and
the system shifted into the Mg–Fe system at higher temperatures.
Because Mn is basically a trace element,
MGrt/
T was small when
the Fe–Mn system was active. The production rate increased
rapidly at temperatures at which Mg–Fe garnet became stable.
Figure 8 shows the changing volume ratio of the minerals. If
the bulk rock consisted of 10 000 moles of chlorite and 50 000
moles of quartz, the volume of the rock will change as shown
in the figure. The garnet–chlorite ratio is uniquely determined,
whereas the amount of quartz is arbitrary because it is an excess
phase. The amount of water evolved is proportional to the amount
of garnet formed.
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Fig. 6. Results assuming perfect diffusion both in garnet and chlorite. (a) Chemical composition change of garnet and chlorite plotted against temperature. Chlorite composition effectively represents the bulk chemistry. The molar amount of garnet formed at each temperature rise is also shown. The amount of garnet is calculated assuming 10 000 moles of material chlorite. (b) Triangular diagram showing the chemical composition change along the P–T path. It should be noted that this figure represents only the temporal changes of the mineral compositions while the system follows the standard P–T path. The resultant garnet is homogeneous and does not have any zoning within the crystal in this case.
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Fig. 7. Compositional ratios plotted against temperature assuming perfect diffusion in garnet. The amount of garnet formed per temperature increment is also shown. These diagrams help to explain the behaviour of the ternary system by dividing it into two pseudobinary diagrams. It should be noted that the peak of garnet formation rate is located at the temperature where the Mg–Fe binary loop starts to function.
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Fig. 8. Changing volume ratio of minerals in the whole rock along the standard P–T path, assuming perfect diffusion in garnet and chlorite. It is assumed that the initial rock consists of 10 000 moles of chlorite and 50 000 moles of quartz. Quartz is given as excess. The amount of water evolved is proportional to the amount of garnet formed. The following molar volumes are used: 11·5 [J/bar = J/(0·1 MPa)] for garnet, 21 (J/bar) for chlorite, 2·25 (J/bar) for quartz.
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Mg–Fe–Mn growth zoning in garnet
Standard case
To simulate garnet growth accompanied by growth zoning, no diffusion
was assumed in garnet. The calculation was performed following
the algorithm described in
Fig. 1. The predicted chemical change
of garnet and chlorite along the standard Sambagawa
P–
T path is shown in
Fig. 9. In this case, the chlorite composition
is the bulk-rock chemistry in effect. This result will be referred
to as the standard case hereafter.
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Fig. 9. ‘Standard’ results of garnet growth calculation assuming no diffusion in garnet and perfect diffusion in chlorite. (a) Changing chemical compositions of garnet and chlorite plotted against temperature. The molar amount of garnet formed at each temperature rise is shown. The amount of garnet is calculated assuming 10 000 moles of material chlorite. The result obtained with larger calculation increments is shown by open diamonds, all of which overlap the filled diamonds of the standard result. (b) Triangular diagram showing the chemical composition change of garnet along the P–T path, which will be preserved as the chemical profile of garnet from the core to the rim. (c) Predicted chemical profile from the core to the rim of the garnet. Radius (horizontal axis) is the garnet volume raised to the one-third power. The bell-shaped profile of XSps should be noted.
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To check the influence of the step width of calculation, the
result calculated with a larger step is also shown in
Fig. 9a.
It is seen that there is little calculation artifact as a result
of the step width. The overall chemical trend was similar to
that with perfect diffusion, except that the d
X/d
T changed more
abruptly as Mg–Fe garnet started forming. Also, it is
observed in
Fig. 9b that the rapid Mg/Fe rise did not occur
before Mn was depleted from the system. This difference results
from the change of bulk chemical composition owing to the removal
of crystallized garnet from the system. Most of the Mn was fractionated
into garnet and depleted from chlorite before Fe–Mg garnet
became stable. Some of the Fe was also removed from the system.
The smaller amount of available Mn and Fe raised the starting
temperature of Mg–Fe garnet, and the onset became more
abrupt. It is shown in
Fig. 9a that the rate of garnet formation
at this stage increased more rapidly in the standard case. The
peak is sharper when zoned garnet is formed, as a result of
Mn depletion. Pseudobinary diagrams in
Fig. 10 illustrate that
the peak in garnet growth is located around the temperature
at which the Mg–Fe binary system begins to be effective.
The calculated volume ratio change of minerals in the bulk system
is shown in
Fig. 11. The chemical profile that will be observed
in the resultant garnet grain is shown in
Fig. 9c. The bell-shaped
profile of Mn that is commonly observed in natural garnet is
reproduced because of the marked variation in the garnet formation
rate.
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Fig. 10. Compositional ratios of the standard result plotted against temperature. Amount of garnet formed per temperature increment is also shown. The peak of garnet formation rate is sharper and is located at the temperature where the Mg–Fe binary loop starts to function.
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Fig. 11. Changing volume ratio of minerals calculated along the standard P–T path. Conditions are the same as in Fig. 8, except that no diffusion is considered in garnet.
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Although the general chemical trend is reconstructed in this
standard case, the calculated garnet zoning differs from natural
samples in several respects. Calculated Mn content is very high
at the core and Mn is almost completely depleted in the calculation
result before Mg–Fe garnet starts growing. To test the
sensitivity of these features to mineral property data, parameter
studies are described below. The calculated garnet formation
also occurs at higher temperatures compared with the estimated
conditions (
Fig. 4). One of the reasons for this is the lack
of consideration of grossular content in the model. This fourth
end-member obviously stabilizes the Mg–Fe–Mn garnet
to lower temperatures, which may be closer to reality. Calculations
involving the Ca component will be discussed later.
Garnet formation is regarded as one of the dominant dehydration reactions in metamorphic rocks. In subduction-related metamorphism such as the Sambagawa metamorphism, a consistent change in P–T conditions is usually expected. In the standard case, the garnet formation rate (= water evolution) doubles during the first 40°C of temperature rise, and increases ten-fold during the following 30°C of heating. After the peak, it takes less than 30°C before the rate halves. It is suggested that the dehydration from subducting sedimentary rocks during garnet formation might occur intensively over a restricted depth, in spite of the fact that the reaction is continuous.
Sensitivity analyses
Thermodynamic properties of mineral end-members were varied to examine the behaviour of the theoretical ternary system. Figure 12 shows the results using varied enthalpy, entropy, and molar volume values; magnitudes of variance are shown in the figures. Values for 1 SD on the enthalpy of formation (
H) listed in the dataset of Holland & Powell (1998) are: H-Prp = 1260 (J), H-Alm = 1320 (J), H-Sps = 3180 (J), H-Cln = 1850 (J), H-Ame = 2020 (J), H-Dph = 3330 (J), H-Mnc = 8810 (J). The applied fluctuation of ±10 000 (J) was chosen to amplify the difference of behaviour, and should be regarded as being unrealistically large except for Mn-chlorite. Fluctuations of 1 (J/K) on the entropy and 0·1 [J/bar = J/(0·1 MPa)] on the molar volume were applied according to previous thermodynamic studies (e.g. Kohn, 1993). Decreased enthalpy of formation (increased absolute value) of phase A simulates the case when phase A is more stable. Likewise, increased entropy and decreased molar volume of phase A illustrates the case when phase A becomes more stable along the P–T path. Naturally, when one of the garnet end-members is stabilized, formation of garnet is accelerated. As seen in Fig. 12a, stabilizing pyrope resulted in larger amount of pyrope incorporated in garnet throughout its growth, whereas the starting condition shifted little. Stabilizing almandine (Fig. 12b) results in low-temperature onset of garnet growth, with higher almandine and lower spessartine content. As for the chlorite properties (Fig. 12d–g), it is observed that the influence is generally the contrary. Clinochlore and amesite properties are both relevant to Mg end-member and also involve substitution of Al. The initial chlorite chemical composition used in this study contains so much Al that most Mg is assigned to amesite. Destabilizing amesite results in lower-temperature onset of garnet growth with higher pyrope content, which is similar to the result with stabilized pyrope. On the other hand, is 0·3% at the beginning and increases in response to garnet growth. Destabilizing clinochlore does not accelerate garnet formation, but garnet with higher pyrope content is formed as a result. Destabilizing daphnite produces similar results to that of stabilizing almandine. Triangular diagrams in Fig. 13 summarize how the Mg and Fe end-member properties affect garnet zoning. The changing stability of Mg-phases resulted in different Mg/Fe ratios throughout garnet growth. Changing stability of Fe-phases led to different Mn contents in garnet cores and different Mg/Fe ratios throughout garnet growth.
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Fig. 13. Triangular diagrams showing the predicted chemical trends of garnet with varied thermodynamic properties. (a) and (b) Chemical trends with varied garnet properties. (c) and (d) Chemical trends with varied chlorite properties. It should be noted that varying properties of pyrope (a) and clinochlore (c) act contrary to each other. Varying properties of almandine (b) and daphnite (d) also act contrary to each other.
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It must be noted that the fluctuation of ±10 000 (J)
is far larger than 1 SD on enthalpy, and the true uncertainties
related to the property data are about 1/5 to 1/8 of those shown
in
Fig. 13. Considering that
H-Prp is about 1/8 of the given
variance, 1 SD on
at 580°C caused by the uncertainty in the enthalpy of formation of pyrope
is only around 0·009 units. Likewise, 1 SD on
in garnet cores caused by the uncertainty
in the enthalpy of amesite and clinochlore is around 0·0004
and 0·001 units, respectively. The uncertainty introduced
in the nucleation temperature of garnet is around 4°C and
3°C, respectively. One standard deviation on
and
of garnet cores caused by the uncertainty in the enthalpy of almandine
is around 0·02 units for each. The uncertainty introduced
in the nucleation temperature of garnet is
c. 3°C. One standard
deviation on
and
in garnet cores caused by the uncertainty
in the enthalpy of daphnite is around 0·03 units for
each. One standard deviation on the nucleation temperature of
garnet is around 5°C. These uncertainties do not have much
precision because the increment of calculation is around 3°C.
In contrast to changing Fe or Mg end-member properties, stabilizing spessartine (Fig. 12c) or destabilizing Mn-chlorite have little influence on the chemical composition trajectory of garnet except at the core: the growth starts from much lower temperature and high spessartine content. The insensitivity is due to the very small amount of Mn in the bulk-rock chemistry. The amount of spessartine formed is constrained by the Fe–Mn mass balance relations, to allow very small amounts of garnet at the lower temperatures. Mn is not much depleted from the bulk system to affect garnet chemical compositions. Considering the estimated H-Sps, 1 SD on in garnet cores caused by the uncertainty in the enthalpy of spessartine and Mn-chlorite is around 0·06 and 0·10 units, respectively. One standard deviation on the nucleation temperature of garnet is around 11°C and 17°C.
Thermodynamic properties of garnet and chlorite end-members have been of much interest, so that they are generally very well constrained. The largest possible error caused by the thermodynamic data is due to the Mn-chlorite data, and thus could change the spessartine content by up to 10 mol %. It is remarkable, however, that fluctuation of Mn-chlorite properties contributes only to the starting temperature condition and does not greatly affect the resulting chemical composition trajectory of garnet zoning during growth. In addition, it should be noted that in all results, the rapid Mg/Fe rise occurred only after Mn was significantly depleted.
The influence of the chemical composition of initial chlorite has also been explored (Table 3). Generally, uncertainties in the mineral compositions are divided into the analytical and the geological uncertainties. Random analytical uncertainties owing to X-ray counting statistics are usually around 0·5% or less for major cations. Geological uncertainties in mineral compositions as a result of invalid assumptions of equilibrium are generally considered to be ±5% (e.g. Kohn, 1993). Geological uncertainties in this study also include variations of bulk chlorite chemistry in the rocks, which are far larger than those of the analytical errors. Therefore, in this study, proper chlorite compositions have been applied to test the sensitivity of the model to them, instead of performing accurate uncertainty analysis for the analytical error. Mn content most powerfully affects the starting temperature of garnet formation (Fig. 14). Increase of 0·005 on (symbolized as XMnc+) causes the garnet formation temperature to fall by 20°C or more. When is increased by 0·1 with no change in (XDph+), garnet starts forming at temperatures 10°C lower than the standard case. When is decreased by 0·1 with no change in and (XAme–), the lower temperature stability range of garnet changes only slightly. Garnet growths from different chlorite compositions are shown in Fig. 15. Figure 15a shows that, although Mn-richer chlorite starts producing garnet earlier, the resulting chemical trend and the temperature of garnet growth are not much different from the standard case, as was the case with changing thermodynamic properties of Mn end-members. On the other hand, when initial chlorite has higher by 10 mol %, both Fe–Mn and Mg–Fe garnet growth occurs at lower temperatures than the standard case (Fig. 15b). In this case, garnet is generally lower in Mn, and the formation rate of Mg–Fe garnet increases rapidly toward a sharp peak. This chemical trend is shown on a ternary diagram in Fig. 15c to illustrate that this case is similar to the results with stabilized almandine (Fig. 13b or d). Values of 1 SD on analysed chlorite compositions compiled from Higashino (1975) are Xame = 0·046, Xdph = 0·014 and Xmnc = 0·006. Accordingly, 1 SD on and nucleation temperature as a result of initial Mn content in chlorite is around 0·13 and 23°C, respectively. These values are slightly larger than the error due to the uncertainty of the thermodynamic data for Mn-chlorite. One standard deviation on and nucleation temperature owing to the initial Fe content in chlorite is around 0·006 and 1°C, respectively. These values are far smaller than the error predicted from the uncertainty of the thermodynamic data of daphnite.
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Fig. 14. Isopleths of garnet mode 0 (‘garnet in’) influenced by the initial chlorite chemical composition. Garnet is unstable at the low-temperature side of the lines, but becomes stable at the high-temperature side. The content of Mn-chlorite is increased by 0·005 in ‘XMnc+’. Daphnite content is increased by 0·1 in ‘XDph+’, without varying Mn-chlorite content. Amesite content is decreased by 0·1 in ‘XAme–’, without varying Mn-chlorite and daphnite contents. Detailed input compositions are shown in Table 3.
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Fig. 15. Sensitivity of garnet growth to different initial chlorite compositions. The amount of garnet is calculated assuming 10 000 moles of reactant chlorite. (a) Chemical composition changes and amount of garnet produced from chlorite with higher Mn content. The trend is scarcely influenced except for the cores. (b) Chemical composition changes and amount of garnet produced from chlorite with lower Mg/Fe ratio. Garnet with higher Fe content starts growing from lower temperature. (c) Triangular diagram showing chemical trend of garnet grown from lower Mg/Fe ratio. (d) Chemical composition changes and amount of garnet produced from chlorite with lower Al content. The trend is scarcely influenced.
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Garnet growth along different
P–
T paths has also been
calculated.
Figure 16a illustrates the results for isothermal
paths in which pressure increases from 0·5 to 1·5
GPa, and
Fig. 16b shows the results for isobaric heating paths
from 460 to 600°C.
Figure 16a demonstrates that small amounts
of garnet grow under constant temperature conditions. Because
the amount of garnet removed from the system is small, more
Mn remains in the bulk system, and garnet still contains significant
spessartine at a pressure of 1·5 GPa. The chemical trend
is similar to the earliest core part of the standard result.
With rising temperature at constant pressure, the amount of
garnet formed is comparable with the result with a standard
P–
T path (
Fig. 16b). It is suggested that temperature
rise is essential in Mg–Fe–Mn garnet growth, when
the reacting mineral is mainly chlorite and the reaction involves
dehydration. This is consistent with the garnet formation isopleth
(
Fig. 5), which showed little dependence on pressure. Compared
with constant pressure cases, garnet that grows at higher pressure
has higher Mn content in the cores, as it begins growing at
a lower temperature.
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Fig. 16. Calculated garnet growth in the Mg–Fe–Mn system along different P–T paths. (a) Changing garnet chemical composition and formation rate at constant temperatures of 450°C and 500°C, plotted against pressure. (b) Results at constant pressures of 0·5 GPa and 1·0 GPa, plotted against temperature.
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The influence of lower H
2O fugacity has been tested for the
Mg–Fe–Mn system. The metamorphic fluid phase associated
with the Sambagawa prograde metamorphism has often been assumed
as pure H
2O, because most of the protoliths were sediments on
the ocean floor. However, judging from the occurrence of calcite
and titaniferous accessory minerals, it is likely that minor
amounts of CO
2 were present at least during the later stage
of prograde metamorphism (e.g. Goto, 2002
). Lower H
2O fugacity
has been incorporated into the forward model as equal to 0·9
Prock and the simple system recalculated. As seen in
Fig. 17, lower
H
2O fugacity accelerates the rate of garnet growth through
reaction (1),
and offsets the whole growth process by 20°C or more
to a lower temperature range. As a result of the lower starting
temperature, spessartine is higher in the core. Pseudobinary
diagrams in
Fig. 18 show that the resulting chemical trend is
similar to the standard result with pure H
2O.
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Fig. 17. Calculated garnet growth in the Mg–Fe–Mn system at lower H2O fugacity. PH2O = 0·9Prock. Standard P–T path is applied.
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Fig. 18. Compositional ratios of the result with lower H2O partial pressure plotted against temperature. Amount of garnet formed per temperature increment is shown. The temperature range of Mg–Fe–Mn garnet growth is lowered by around 20°C. The peak temperature for garnet formation rate is also lowered but is still located at the temperature at which the Mg–Fe binary loop starts to function.
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Introduction of Ca end-members
So far, the calculated garnet growth patterns are consistent
with natural samples as regards chemical composition trends
from the core to the rim; however, the temperature range over
which the growth occurs is generally higher than that estimated
in the Sambagawa metamorphic belt. One reason for this is the
lack of consideration of the Ca end-member component. Garnet
in the Sambagawa metamorphic rocks generally contains 15–35%
of grossular (Grs) (e.g. Banno & Kurata, 1972
; Enami, 1998
;
Inui & Toriumi, 2002
). As this reduces the activities of
the Mg–Fe–Mn components in garnet, Mg–Fe–Mn–Ca
garnet becomes stable at lower temperatures compared with free
garnet.
For the simulation of garnet growth considering the grossular content, epidote was chosen as the reactant mineral in this study. Epidote is known to occur in 90% of the Sambagawa pelitic metamorphic rocks (Goto et al., 2002), and is often considered as the source for Ca incorporated into garnet. To introduce epidote, it is only necessary to add epidote as a new phase. Calcite and titanite also are possible sources of Ca (Goto et al., 2002). To involve these phases, however, it is required to introduce more than one new phase. The simpler system was explored in this study, as the objective was to examine the potential effect of Ca on the temperature interval of garnet growth and epidote is observed to occur as inclusions in garnet.
The epidote-bearing model
To introduce epidote (Epi), the mineral was treated as a binary solid solution, the end-members being clinozoisite [Czo: Ca2Al3Si3O12(OH)] and pistacite [Ptc: Ca2Fe3+Al2Si3O12(OH)]. All Fe in epidote was assumed as Fe3+, which preferentially occupies one of the M sites. Activity–composition relationships were defined as ideal solutions according to Holland & Powell (1998):
Four
variables (
,
MEpi,
,
) were added, which means that four more constraints came into
effect in addition to
constraints (1)–
(9) described previously.
One is the constraint of chemical equilibrium of the following
reaction:
 | (10) |
The univariant
line for this reaction is highly sensitive to temperature. Another
constraint is the stoichiometry of epidote:
 | (11) |
Two further constraints on mass balance are
relevant to Ca and Fe
3+:
 | (12) |
 | (13) |
Pistacite is not involved in any
chemical reaction within the system. It is considered mainly
to simulate the change of activity of clinozoisite. In addition,
constraints (4) and
(9) must be revised as follows:
 | (4') |
 | (9') |
The calculation algorithm is illustrated in Fig. 19. The initial chemical composition of epidote was set to 
, as epidote inclusions in garnet cores in the Sambagawa pelitic rocks generally have around 0·7–0·8 of clinozoisite (Inui & Toriumi, 2002). The initial molar amount of epidote was set to about 700 moles, whereas chlorite was set to 10 000 moles as in the standard case. The model was again coded using Microsoft® Visual Basic® for Applications. The actual calculation was performed using two different codes: one is the simple Mg–Fe–Mn system proposed in this study; the other is the code to solve the chemical equilibrium of reaction (10). To solve the whole system at a certain P–T condition, the two codes were alternatively run at the P–T until the two solutions converged (differences less than 0·00001).
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Fig. 19. Calculation algorithm of the epidote-bearing model. Chemical equilibrium between chlorite, epidote, and newly formed garnet is always satisfied. The chemical compositions of chlorite and epidote are always recalculated after new garnet is formed and removed from the system.
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Results
Figure 20 shows the low-temperature boundary of garnet stability.
Garnet starts forming at temperatures only around 5°C lower
than in the Ca-free system. The calculated chemical composition
changes of garnet, chlorite, and epidote along the standard
P–
T path are shown in
Fig. 21. The figure shows the results
only for the temperature range over which they converged: the
solution diverged at higher temperatures. Predicted grossular
contents are about 5–9 mol %, generally increasing from
cores toward the rims.
MGrt/
T achieves a maximum at a temperature
more than 10°C lower than the standard case. It is natural
that the presence of another end-member expands the stability
field of Mg–Fe–Mn garnet to lower temperatures.
The ternary diagram (
Fig. 21b) shows that the behaviour of the
Mg–Fe–Mn system is similar to that of the Ca-free
standard system. The predicted chemical profile of garnet (
Fig. 21c)
exhibits bell-shaped Mn zoning. A slight decrease of grossular
content observed at the last stage of garnet growth is a result
of the rapid increase in garnet formation rate, which consumes
much clinozoisite and, as a result, decreases the clinozoisite
content in epidote. This calculated decrease of clinozoisite
is rather large compared with natural samples. Observed clinozoisite
content in epidote in the Sambagawa metamorphic belt is generally
around 0·5 or higher. The calculated volume change of
minerals is illustrated in
Fig. 22. It is seen that the given
volume of epidote was small and decreased rapidly in response
to garnet formation. The occurrence of the decrease of clinozoisite
content in epidote largely depends on the initial amount of
clinozoisite assumed in the calculations.
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Fig. 20. Isopleth of garnet mode 0 (‘garnet in’) considering epidote and grossular. Garnet is stable only at the high-temperature side of the dashed curve. The boundary is about 5°C lower than in the Ca-free system (continuous curve).
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Fig. 22. Changing volume ratio of minerals in the epidote-bearing system calculated along the standard P–T path. Conditions are the same as in Fig. 11. A molar volume of 13·5 [J/bar = J/(0·1 MPa)] for epidote is used.
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Garnet growth along different
P–
T paths in the Ca-bearing
system is illustrated in
Fig. 23.
Figure 23a and b shows that
temperature rise is essential to garnet growth, as was the case
in the Ca-free model. A simple isothermal increase in pressure
(
Fig. 23a) produces very small amounts of garnet compared with
the cases accompanied by temperature rise (
Fig. 23b). As a result
of the small quantity of garnet formed under isothermal conditions,
the removal of Mn from the system is minor and garnet still
contains significant Mn even at 1·5 GPa. The difference
in growth zoning along different
P–
T paths is similar
to that seen in Mg–Fe–Mn garnet. Calculation of
Mg–Fe–Mn–Ca garnet growth under lower H
2O
partial pressures is shown in
Fig. 23c. Garnet is stabilized
generally, and the peak temperature of the garnet formation
rate is more than 20°C lower than in the case with pure
H
2O fluid.
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Fig. 23. Calculated garnet growth in the epidote-bearing system under different conditions. (a) Changing garnet chemical composition and formation rate at a constant temperature of 490°C plotted against pressure. (b) Result at a constant pressure of 1·0 GPa plotted against temperature. (c) Result at lower H2O fugacity along the standard P–T path. PH2O = 0·9Prock.
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Uncertainties in the epidote-bearing model
Predicted grossular contents in the forward model are much smaller
than those observed in Sambagawa metamorphic rocks. Because
any possible reaction should be in equilibrium, it is not logical
to assign the discrepancy to the incorrect choice of grossular-forming
reaction. It might suggest, rather, that the initial Al content
in chlorite was not realistic. In
reaction (10), the contents
of clinochlore and amesite strongly control the clinozoisite–grossular
equilibrium. The initial chlorite in this study (
Table 2) is
very much enriched in Al (amesite-rich), and little clinochlore
is left to lower the equilibrium grossular content. If
was decreased by 0·1, as attempted
in
Fig. 15d in the Mn–Fe–Mg system, about 12 mol
% grossular is incorporated in garnet cores and increases during
growth (
Fig. 24). This shows that the amount of Al in the bulk
chemistry can significantly affect the incorporation of grossular
in garnet. The remaining chlorite in the present Sambagawa metamorphic
rocks may not retain its original chemical composition, even
where garnet has not formed, so that it is difficult to estimate
the exact Al content in chlorite. The simple Mg–Fe–Mn
standard results can still be consistent, as the Mg–Fe–Mn
system was not much affected by the amesite content of the initial
chlorite, as described above.
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Fig. 24. Calculated garnet growth in the epidote-bearing system assuming lower Al content in initial chlorite. Standard P–T path is applied.
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Constraints on the bulk chemistry can also be one of the reasons
for the insensitivity of the garnet growth zoning to the
P–
T path (
Fig. 23). As described above, calcite or titanite could
have contributed to the formation of grossular in the Sambagawa
metamorphism. In that case, the assumption in the present model
that the system is closed for Al and Ca would not be valid,
as both chlorite and epidote can change their chemical composition
through reactions involving other minerals. Furthermore, observed
epidote retains growth zoning, which violates the assumption
of perfect diffusion in the matrix. The inner parts of epidote
grains might always have been insulated from the system. However,
it is known that clinozoisite content increases from the core
to the rim in its growth zoning. As garnet grew consuming epidote,
the clinozoisite content in the surface of epidote must have
decreased gradually, even if no diffusion occurred in epidote.
The clinozoisite content also decreases during progress in the
present model. Accordingly, it is possible that the simulation
is not very far from what is intended.
Other uncertainties
One of the remaining uncertainties in the forward calculation is the solution model of chlorite. Definitions of molar fractions of end-members can change by introducing new end-members such as Fe-amesite or sudoite. The cation number ratio suggests that the di/tri-octahedral substitution might have affected the Sambagawa chlorite. Clinochlore/amesite and Mg/Fe ratios are affected by such changes and may have an influence on the equilibrium calculation. The proposed model will have to be revised in response to revisions of chlorite models.
Another uncertainty arises from the assumption that the system was closed with regard to Mg, Fe, and Mn. In particular, Mn is a trace element and therefore can be significantly affected by the presence of other sources such as magnetite or ilmenite. To involve these minerals in the model is very difficult, as it is then essential to determine the initial amount of those source oxides. None the less, their contributions may be one of the reasons, in natural samples, for Mn sometimes not being much depleted before the rapid rise in Mg/Fe in garnet occurs.
Kinetic problems have not been considered in this study, but can also cause the rapid Mg/Fe rise before Mn depletion. If diffusion occurred to some degree in garnet cores, it may lessen the Mn contents of those cores. Garnet generally retains a euhedral growth zoning pattern, which argues against chemical diffusion. However, diffusion affects the core parts more than rims and it may not always be evident because the diffused area may be too small to confirm if it is polygonal or round. On the other hand, if diffusion were not perfect in chlorite, the ‘effective’ bulk-rock chemical composition might be different. More Mn will remain in the matrix, which stabilizes the Mg–Fe–Mn garnet and makes the rapid Mg/Fe rise occur before Mn is completely depleted from the system. The result will approach the case of perfect diffusion in garnet. Further parameter studies will be necessary in the future to consider non-equilibrium factors, such as imperfect homogenization in chlorite.
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CONCLUSION
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Forward modelling using a straightforward iterative algorithm
has been proposed to simulate the growth of garnet consuming
chlorite. The chemical composition and formation rate of garnet
have been calculated quantitatively assuming equilibrium between
chlorite and the surface of garnet, and using the most recent
thermodynamic properties of minerals and activity–composition
relationships. The
P–
T path and chemical composition of
chlorite from the Sambagawa metamorphic belt, Japan, have been
used for a case study.
The overall shape and trend of chemical zoning of garnet in the Sambagawa case study have been reproduced, with decreasing Mn and increasing Mg, Fe, and Mg/Fe ratio from the core to the rim. Calculated Mn contents of garnet cores are generally higher than those observed, but uncertainties of 10 mol % or more in spessartine are expected as a result of thermodynamic property data or initial chlorite composition. The uncertainty in the Mg/Fe ratio of garnet as a result of the choice of initial chlorite composition is smaller than that related to the thermodynamic property data, which is a few mole percent. In all calculated growth zoning profiles, the rapid Mg/Fe rise in garnet occurs only after Mn is practically depleted from the system. In natural samples, however, the rapid Mg/Fe rise sometimes occurs while is a few mole percent. This may suggest either that the assumption of no diffusion in garnet and perfect diffusion in chlorite is not strictly true, or that Mn was supplied from some other source in the natural samples.
The amount of garnet formed during each step of temperature rise or pressure increase (garnet formation rate) is small when the Fe–Mn binary system is active, but increases rapidly to form a peak at temperatures at which Mg–Fe garnet becomes stable. It is suggested that under conditions of continuously rising temperature during subduction, dehydration of metamorphic rocks is focused into a narrow temperature interval because of this variation in garnet growth rate. It is confirmed that the characteristic bell-shaped zoning profile of Mn results from the rapid increase and decrease of garnet formation rate.
The temperature range predicted for garnet growth is much higher (by around 50°C) than general observations. However, it is confirmed that higher Fe content in chlorite or lower H2O partial pressure will promote Mg–Fe–Mn garnet growth at lower temperatures. The simplest modelled Ca-bearing system involving clinozoisite in epidote only partially explains the grossular content in garnet, which probably needs to be considered in a more complex system involving additional Ca- bearing minerals.
|
APPENDIX: ITERATIVE ALGORITHM
|
|---|
Figure A1 shows the flow chart of the iterative algorithm to
obtain the amounts and chemical compositions of garnet and chlorite
that are in equilibrium at a given temperature, pressure, and
bulk chemistry. The algorithm consists of three nested calculation
loops. (Abbreviations are defined in the main text.) First,
temporary values must be given to three of the nine variables
(
MGrt,
, and
, in this study).
Equations (4)–
(9) can be
solved for the remaining six variables by a substitution method,
as only two of the variables are multiplied with each other.
As the result, a set of values for the nine variables is obtained
that satisfies
equations (4)–
(9). At this stage, it is
possible to rule out unrealistic sets of values in which any
of the
values is less than zero or exceeds one. Next, equations of
equilibrium, (1)–
(3),
are calculated and the changes of Gibbs' free energy as a result
of each reaction [
Gr(
n) for reaction (
n);
n = 1, 2, 3] are obtained.
If
Gr(
n) is zero, the reaction (
n) is in equilibrium with the
given
P,
T, and the compositions of minerals. If otherwise,
the reaction is not in equilibrium with the temporary values
and therefore the temporary values must be reconsidered. To
increase the speed of calculation, iteration was performed systematically
to seek the solution.
is incrementally changed, separately for each reaction. As a result, three sets
of the nine variables are obtained so that each set satisfies
one of the
equations (1),
(2), or
(3). The equation was judged
satisfied when
Gr(
n) < 1 (J). At this stage, again, temporary
values that do not yield valid sets of data can be ruled out.
Finally, the remaining temporary values (
MGrt and
) are incrementally changed until all three equations
yield identical values of
. Iteration was performed until the differences in
for
reactions (1)–
(3) were less than 0·001.
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Fig. A1. Flow chart showing the iterative algorithm to solve the non-linear simultaneous equations of chemical equilibrium.
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ACKNOWLEDGEMENTS
|
|---|
We would like to express our sincere thanks to Simon Harley
for his encouragement and guidance. Critical reading and advice
by J. Hermann and an anonymous reviewer greatly enhanced our
discussions and we are very grateful to them. We owe our gratitude
to T. Okudaira for his advice and insightful discussions. We
are grateful to H. Yoshida for his thorough care concerning
the chemical analyses. Discussions with members of the structural
seminar of the Geological Institute, University of Tokyo, were
greatly beneficial.
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FOOTNOTES
|
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* Corresponding author. Telephone and fax: +81-3-5481-3263. E-mail: inui{at}kokushikan.ac.jp
|
REFERENCES
|
|---|
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