Structural Geology (The Journal of the Tectonic Research Group of Japan), No. .., .1ῌ/1, ,***

Di#usion-controlled garnet growth during Sambagawa metamorphism
Mitsuhiro Toriumi* and Akiko Nomizo**
Abstract : Garnet from higher grade zones of the Sambagawa metamorphic belt commonly shows reverse zoning of MnO in the ﬁnal stages of growth. The reverse zoning
formed during growth process rather than by resorption. The magnitude of reverse
zoning increases with increasing grain size. Given that grain size is proportional tolinear growth velocity, a model of reverse zoning and growth velocity requires averagecompositions of chlorite di#erent from equilibrium compositions. The model presented here is derived from competition of di#usive mass transfer of component ions ofgarnet in the matrix and interface migration due to garnet growth. This can well
explain the occurrence of garnet cluster and relationships between the magnitude ofreverse zoning and grain size.
Key words : garnet, grain growth, di#usion, Sambagawa

Introduction
Compositional zoning of garnet in medium tohigh grade metamorphic rocks can be used toinfer P-T-t paths of metamorphism (e.g. Spear &Peacock, +323 ; Vance & O’Nions, +33,). Garnet
zoning in pelitic schists of the greenschist toamphibolite facies grade has been interpreted toresult from continuous growth accompanied byan increase of temperature, or by Rayleigh typefractionation during a constant temperature(Hollister, +300). Fractionation, however, isunable to produce the systematic change in the
Fe-Mg partitioning between garnet and chlorite
from early to late stage growth of garnet. The
former model, on the other hand, allows us tointerpret P-T paths by means of the metamorphic inversion method (Spear & Peacock, +323).
Tracy (+32,), St-Onge (+321) and Spear & Peacock (+323) developed modeling of garnet
growth together with the sliding equilibrium
and proposed a metamorphic inversion method
which provides a quantitative and continuous
P-T record, of metamorphism in a single rock.
Received +* October ,*** ; accepted +/ November ,**** Department of Complexity Science and Engineering,
Graduate School of Frontier Sciences, University ofTokyo, Hongo, Bunkyo-ku, Tokyo** Medical School of Juntenndo University Hospital, Hongo,
Tokyo

Their method for inversion is based on surface
equilibrium and growth zoning due to P-Tchange. During the period of garnet growth,
di#usion of component ions is very fast in thematrix relative to the growth velocity and thenenrichment of Mn in the matrix at the growth
front never occurs. In the above model, compositional zoning explicitly represents a recordof P-T-t.
Hollister (+300), Kretz (+31-) and Banno & Chii(+312) proposed models of garnet growth interpreting the Bell-shape zoning of MnO. In theirmodels, reverse zoning of MnO is due to growthduring retrograde metamorphism. Sakai et al.
(+320) clariﬁed growth zoning of garnet bymeans of back scattered electron images, andshow two images which indicate growth andresorption. The electron images showing theresorption of garnet display growth outlines
cross-cutting the inner contour lines of MnO andFeO contents.
On the contrary, crystal size distributions ofgarnet from low to high grade regional metamorphic rocks of the Appalachian belt in Maine
were measured by Cashman & Ferry (+322) andfrom high pressure type regional metamorphic
rocks of Kyushu, Japan by Miyazaki (+33+).
They attributed coarsening mechanism ofgarnet to Ostwald ripening during metamor-

῍ 47 ῍Mitsuhiro Toriumi and Akiko Nomizo

Fig. +
Index and zonal maps of the Sambagawa metamorphic belt in Shikoku, SW Japan. The study area is shown by theclosed square in the western part of the map. MTL ; Median tectonic Line, SMB ; Sambagawa metamorphic belt, OBZ ;oligoclase-biotite zone, ABZ ; albite-biotite zone, GAZ ; garnet zone, CHZ ; chlorite zone.

phism because the cyrstal size distributions are
similar to an LSW type pattern. However, Ostwald ripening requires reduction in size of small
grains rather than growth of large grains anddoes not depend on the di#usive mass transfer
of component ions in the matrix. Neverthelessthere is no evidence for the preferential resorption of small grains of garnet in metamorphic
rocks (Toriumi, +31/), suggesting a growth process of garnet other than Ostwald ripening.
Kretz (+31.) and Banno & Chii (+312) proposedthat reverse zoning (MnO enrichment in the rim)
is formed by the resorption of garnet. However, reversely zoned garnet grains formed by continuous growth are common in the Sambagawa
metamorphic belt (Nomizo, +323). Furthermore,
Nomizo clariﬁed that the compositional range ofreverse zoning depend strongly on the grain sizeof garnet. Such reverse zoning requires a chemical gradient from the equilibrium interface tothe matrix and consequenctly the di#usive masstransfer. In this paper, we propose a model ofgarnet growth that involves an advancing gar-

net-matrix interface competing with di#usion of
constituent ions in the matrix. The ratio ofgrowth velocity and di#usion rate of ions through chlorite in the matrix controls the degreeof reverse zoning garnet growth.
Outline of the Sambagawa metamorphic belt
The samples have been collected from the
Sambagawa metamorphic belt in central Shikoku, Japan. Sambagawa metamorphism occurred from 1* to ++* Ma, by means of K-Ar (Itaya& Takasugi +322) and Ar-Ar method (Takasu &Dallemeyer, +33*). The peak metamorphism at
about 0**῏ and +/ kb occured at ++* Ma andretrograde metamorphism through -**῏ and /ῌ
1 kb at 1* Ma. Thus the cooling rate is estimated to be ῎+*῏/million years.
The Sambagawa metamorphism is a high pressure intermediate type regional metamorphismof pumpellyite-actinolite facies, glaucophane
schist facies, and epidote amphibolite facies (e.g.
Toriumi, +33*). The detailed spatial distribu-

῍ 48 ῍Structural Geology No. ..

Di#usion-controlled garnet growthtion of metamorphic zones has been clariﬁed by
the distribution of garnet, biotite and oligoclasebearing assemblages in chemically homogeneous pelitic schists (Kurata & Banno, +31. ; Higashino, +33*). The isothermal surfaces deﬁned by
the garnet, biotite and oligoclase-biotite isograds are penetratively folded with a southward
vergence. Toriumi (+33*) and Wallis (+33+) postulated that the large penetrative fold of isothermal surfaces is attributed to a sheath fold of /km NS width and with an EW kinematic ﬂow
direction.
Recently, Nomizo (+33,) suggested that thehighest grade of the Sambagawa metamorphic
belt reaches eclogite facies conditions. The
physical conditions of Sambagawa metamorphism in the eclogite facies are thus estimated at
0**ῌ1** and +/ῌ,* kb, and the epidote amphibolite facies conditions are inferred to be /**ῌ
0** and +*ῌ+/ kb (Takasu, +323). The P-T
paths of high grade rocks were traced near the
adiabatic depression, suggesting rapid uplift
process relative to thermal di#usion. Retrograde metamorphism is widely recognized in
the Sambagawa metamorphic belt as chlorite
formation after garnet in pelitic schists and actinolite formation in pull-apart structure within
hornblende.
Intense plastic deformation during metamorphism occurred with an E-W transport direction(Faure, +32- ; Toriumi, +32,, +320, +33*). Faure
pointed out that synmetamorphic ductile deformation of the Sambagawa belt is represented by
top-to-east shear sense, although Wallis (+33+)
documented top-to-west sense of the ductile
shear in central Shikoku. Further, Mori (+320)
and Toriumi (+323) suggested that the metamorphic belt is divided into many domains having
top-to-the east and top-to-the west shear sense,and concluded inhomogeneous ductile shear
during Sambagawa metamorphism. Late stage
deformation during retrograde metamorphism
resulted in the displacement of the thermal structure deﬁned by the surface of the isograd, andis thought to be represented by the NS vergent
pile nappe structure of thrust sheets (Hara et al.,+33*). However, many retrograde cracks sealed
by quartz and calcite record E-W extension andour data suggest that metamorphic isograds are
not cut by thrusts as proposed by Hara et al.

(+33*). This indicates that no large NS displacement by thrusts can be found.
Mineral assemblages of pelitic schists of thechlorite zone, the garnet zone, the albite-biotite
zone and the oligoclase-biotite zone are chlorite,
chlorite-garnet, chlorite-garnet-biotite, and chlorite-garnet-biotite-oligoclase together withalbite and quartz, respectively. During Sambagawa metamorphism, the divariant reaction producing garnet across the garnet zone is as follows,
chloritequartzgarnetH, O
Thus, the isograd deﬁned by this equation requires constant XFe in chlorite (and the bulkrock). With increasing metamorphic grade, thevolume fraction of chlorite decreases and that ofgarnet increases in zones higher than the garnetzone. The Mg/Fe ratio in chlorite increases
with increasing metamorphic grade (Higashino,+33*). In the biotite zone, the reaction proposed
above involves muscovite in the left and biotite
in the right hand side, because of the appearenceof biotite. However, Kurata & Banno (+31.) andBanno & Sakai (+33*) obtained gradual increaseof the partition coe$cients of FeO and MgO
between garnet and chlorite and gradual increase of MgO and decrease of MnO content ingarnet from the garnet zone to the oligoclasebiotite zone. Thus it is concluded that the metamorphic temperature increases gradually fromthe garnet zone to the oligoclase-biotite zone
and that the change from the divariant to theunivariant reaction does not control the reverse
zoning of MnO content in garnet growth. Basicrocks show a systematic change of mineral assemblages with increasing metamorphic temperature from the pumpellyite zone, pumpellyite-chlorite zone, pumpellyite-actinolite zone,
barroisite zone, and the hornblende zone (Otsuki& Banno, +33*).
Experimental procedures
The analysed samples were obtained from thealbite-biotite zone in the Saijo district in central
Shikoku, as shown in Fig. +. Detailed sampling
points are shown in Fig. ,. The mineral assemblages of pelitic schists and basic schists are
garnet-muscovite- albite- biotite-ilmenite-quartzcalcite and hornblende-chlorite-garnet-albite- 49 

Mitsuhiro Toriumi and Akiko Nomizo

Grain size distribution of garnet

Fig. ,
Relative sample location of studied pelitic andbasic schists in the locality shown in Fig. +., samples P +to
B , are the same as those denoted by q -0 in Table +. The
distance between B , and T -* b is - meters. The mineralcompositions of the samples are shown in (b). Qz ;quartz, Ga ; garnet, Ms ; muscovite, Ab ; albite, Bt ; biotite,
Hb ; hornblende, Cc ; calcite.

quartz- muscovite, respectively. Volume fractions of these minerals are shown in Fig. ,. The
size of garnet varies largely from *.+ to +* mm.
Volume fractions of garnet are ranged from / to+* percents.
Crystal size frequency distributions (CSD) ofgarnet were measured using an optical microscope. The total number of grains in schists
having large grain of garnet is +30 from two +*cm῎+* cm wide thin sections. CSDs of micaceous and quartz-albite layers were also measured to clarify the e#ect of mineral compositions. Local means of grain spacing and grain
size were calculated as the mean of / grains.
The microprobe analyses and Xray composition mappings of garnet and chlorite were carried out by a JEOL XMA type 20** the University of Tokyo, and partial analyses of bulk compositions of pelitic and basic schists were performed by XRF at the Geological Institute, University of Tokyo.

Garnet displays a commonly euhedral outline.
Two types of microstructures are apparent porphyroblastic grains containing abundant mineral inclusions, and small grains containing small
numbers of inclusions. Although there is agreat di#erence of the mean grain size, the pattern of the crystal size frequency distribution ofgarnet is very similar in both porphyroblastic
and small garnet bearing pelitic schists (Fig. -).
The CSD function is approximated to be a LSWtype except for large grain size region but notlognormal type as shown by Cashman & Ferry(+322), Miyazaki (+33+) and Toriumi (+31/), andthey attributed the CSD pattern to Ostwald ripening during coarsening of garnet grains. The
ﬁgure , indicates that CSDs have a relative
narrow distribution pattern with no long term
tail in both large garnet and small garnet bearing schists indicating LSW type distributionfunction. Nevertheless, the discrepancy of
CSDs from LSW function can be found in large
grain sizes from +./ Rm to -.* Rm. This discrepancy is also shown in CSDs obtained by Cashman & Ferry (+322) and Miyazaki (+33+).
The density of grains in large-garnet bearing
schists is very small compared with that insmall-garnet bearing schists. The number density gradually decreases in the boundary zone
between large- and small-garnet bearing schists.
The modal compositions of muscovite, hornblende, albite (oligoclase), quartz and biotite aresimilar in these rock types. Furthermore, theFe/Mg ratio in both rock types is similar. It is
concluded from this that the chemical composition of rocks does not control the grain size ofgarnet. The CSDs of basic schists are similar to
those of pelitic schists as mentioned above, suggesting similar growth process in large andsmall grains of garnet studied here.
Representative chemical compositions ofgarnet are listed in Table +. As described in
Banno et al. (+320), chemical compositions of thecore of large grains of garnet are the same as
those of small grains, and the compositions of
the rim are also the same in grains of various
sizes. Thus average growth velocity is largely
di#erent even in homogeneous pelitic schists

῍ 50 ῍Structural Geology No. ..

Di#usion-controlled garnet growthFig. Crystal size distributions of garnet in pelitic (A)and basic (B) schists. The CSDs of garnet in micaceous
layer and in quartz albite layers are shown in (A). The
ﬁgure (C) indicated the normalized CSD pattern,suggesting a similarity of CSD of garnet from basic (large
grains ; square) and pelitic schist CSDs (small grains ;circle).

Table +

Representative analyses of garnet cores, mid-regions and rims.

within a ,./ m distance : large grains formed bylarge growth velocities, and small garnet grains
represent small growth velocities.
The apparent increase of garnet growth velocity with increasing grain size has been describedby Sakai (+32/) and Banno et al. (+320) in the
Sambagawa metamorphic belt from single sections based on zonal structures. All grains ofgarnet in single rock samples have a similar
compositional trend from the core to the rim asshown in Fig. .. This fact indicates that small
grains did not experience any resorption. Considering the presence of chlorite in studied
garnet bearing schists, the LSW type CSDs are
not attributed to post-growth annealing. Therefore, CSD patterns in such samples may bederived from growth processes rather than Ostwald type post growth annealing.
Mean sizes of garnet grains are locally variable even in a single hand specimen. The CSDs

῍ 51 ῍Mitsuhiro Toriumi and Akiko Nomizo

Fig. .
Compositional zoning of garnet in q -0B, indicating a gradual increases of MgO toward the rim andreverse zoning of MnO in the outer layer of garnet.

depend on local mineral compositions. In micaceous layers large grains are dominant, although small and large grains are present inquartz-albite layers. Grain sizes are apparently
dependent on the mean spacing between garnet
grains : the distance between small grains iscommonly small and that of large grains islarge. The relations between local spacingsand grain sizes of garnet are shown in Fig. /,suggesting a linear relation between them. It
follows that small garnet grains tend to form ina cluster and large grains tend to be more isolated.
Grain size and reverse zoning of garnet
Chemical zoning of representative garnet
grains Sambagawa samples are shown in Fig. 0.
The MnO content decreases, and the FeO content increases from the core to the rim of garnetin pelitic and basic schists. The CaO content inpelitic schists increases from the core to theintermediate region and decreases again to therim. The core composition of small grains are
similar to those of relatively large grains, as
described by Sakai et al. (+320). Sakai et al.
(+320) attributed this to growth velocity being
proportional to the grain radius.

Fig. / Relationship between average grain size andmean spacing to neighbouring ﬁve grains. This suggeststhat grain spacing is proportional to grain size.

Zoning proﬁles of analysed garnet grains arecharacterized by MnO enrichment at the core ofgrains although MgO increases gradually fromthe core to the rim. The pattern of MnO enrichment appears that the MnO content increases
rapidly and again decreases gradually outward
as shown in Fig. .. The outline of iso-MnO content at the beginning of MnO enrichment (reverse zoning) can be found by Xray mapping

῍ 52 ῍Structural Geology No. ..

Di#usion-controlled garnet growthFig. 0Compositional paths of garnet in sample q-0 P + in Mn-Fe-Mg diagram. The initiation andcessation points of the reverse zoning of grains areshown by I +-I - and E +-E -, respectively. Thisshows that the large grain indicates markedly
reverse zoning. The intensity of reverse zoning isdeﬁned as the di#erence of MnO content from I to
E, and is indicated by Cbs MnO.

large grains of garnet, but not in small grains.
In addition, the di#erence of MnO content between the beginning of the reverse zoning andthe middle part of the reverse zone in a single
garnet increases with increasing grain size.
The relationship between the compositional
range of the MnO content (hereafter called asbackstep composition of MnO) of the reverse
zoning and grain size is shown in Fig. 1. The
MnO-backstep content shows an asymptotic relationship to grain size. The relationship between the MnO-backstep and the grain size inbasic schists is similar to that in pelitic schists.
Fig. 1
Relationship between compositional di#erence
Cbs in MnO from the initiation to the cessation points of
the reverse zoning and grain size of garnet.

Discussion
Growth of garnet in clusters
Cashman & Ferry (+322) and Miyazaki (+33+)
interpreted garnet growth in metamorphic
rocks and resultant LSW type pattern of crystal
size distribution (CSD) in terms of Ostwald ripening of garnet. Ostwald ripening requiresthat the growth velocity depends on the interfacial energy between garnet and surrounding
matrix minerals. In this process, the growthvelocity of small grains is negative and that oflarge grains is positive. However, continuous
zonings of MnO, FeO and MgO suggests that+.

(CMA mapping) using automatic XMA. The
representative CMA image of the reverse-zoned
garnet is shown in Fig. 2. It is apparent thatthe outline of the beginning of the reverse
zoning is consistent with the growth surface
outlined by ++* Therefore, the reverse
zoning is not due to resorption of garnet assuggested by Kretz (+31-) and Banno & Chii(+312), but rather resulted from growth processes.
The reverse zoning of MnO content is found in 53 

Mitsuhiro Toriumi and Akiko Nomizo

Fig. 2Mapping of MnO content in thereversely zoned garnet from q -0 PB sample
in Fig. ,. the band of the reverse zoning ismarked by R.

selective resorption of small garnet cannot befound in either pelitic or basic schists. Consequently, the Ostwald ripening might not operatestrongly on the growth of garnet due to consumption of chlorite and quartz.
The kinetics of di#usion of the component
ions of garnet in matrix chlorite and reaction at
the growth surface of garnet is considered here.
The system has multigrains in the core region of
the di#usional ﬁeld concerning component ionsof garnet. In the di#usional ﬁeld, where thenumber of garnet grains is n and the averagegrain size is R, we have the following massbalance equation ;. npR, dR/dt.pr, Ddc/dr(+)
in which c and D are the concentration of component ions at the distance r from the cluster
center and di#usion constant of the slowest
component ion of garnet and chlorite in thematrix, respectively. Integrating this equation
at the boundary conditions ofcco at rinﬁnite(,)
andccoo at rR (growth surface)(-)
we havedCn R V/D(.)
where VdR/dt and dCcocoo. Then , RdR/
dt becomes dR,/dt. Therefore, we obtain the
average growth velocity Vm
Vm(+/t)V(t)dt
R/t
(, dC D/t)+/, n+/,.
(/)

Furthermore, according to eqn. (/), the relationship between the average spacing and the grain
size of garnet can be inferred as follows ;RdC+/, (l/L+./)d+./(0)
in which l is the di#usional length of component
ions during growth of garnet (l,, Dt) and L isthe scale of a garnet cluster as deﬁned byn d-L-.
(1)
As shown in Fig. /, the mean spacing of
garnet grains is proportional to the grain size ofgarnet and the ratio of d+.//R is about -.*ῌ..*.
Considering dC*.*+, l reaches about ,ῌ/ timesof L, indicating that the scale of the di#usional
ﬁeld is larger than that of the cluster.
,. A model of reverse zoning
Kretz (+31-) and Banno & Chii (+312) attributed
the reverse zoning of garnet to the retrograde
growth of garnet and the di#erence between theequilibrium composition of chlorite at thegrowth surface and the average composition far
from the growth surface. Unless the equilibrium composition exists at the growing surface
both in reactant chlorite and product garnet, thegrowth of garnet during decreasing temperature and/or pressure requires excess Mn chlorite relative to the equilibrium composition atpeak pressure (Fig. 3). Reverse zoning resulting from the retrograde growth of garnet should
then continue until the average composition ofchlorite is equal to the equlibrium one (at pointr in Fig. 3). Then, the MnO backstep magnitude is governed by the average composition of 54 

Structural Geology No. ..

Di#usion-controlled garnet growthFig. 3
Proposed growth model of reversely
zoned garnet in the system garnet-chloritebiotite-muscovite-quartz-H , O at the highpressure conditions. The equilibrium compositions of garnet and chlorite at their interfaces
are assumed, and thus the reverse zoning of
garnet requires a decrease in pressure because
continuous enrichment of MgO in garnet inreverse zoning indicates pressure increase at
this stage. G ; garnet, c-p-r ; pathe of bulk
chlorite composition in the case of fast growthvelocity, c-q-s ; pathe in the case of slow growthvelocity of garnet (see text).

matrix chlorite at the turning pressure to theretrograde metamorphism.
Consider a simple model in which averagecompositions of the matrix chlorite are represented by a simple average of chemical composition in the di#usional ﬁeld and its exterior
region. The di#usional length can be obtainedby di#usion equation with moving boundary
between chlorite and growing garnet in the spherical coordinates. Assuming the velocity ofboundary migration due to garnet growth Vm,
we have the following di#erential equation,
D d, c/dr,Vm dc/dr*(2)
In solving this equation, we have a speciﬁc solutions near the growth front as follows ;c(r)exp(Vm r/D).
(3)
Thus, the di#usional length L around a single
garnet can be deﬁned byc(L)coo e+.
Therefore, we obtainkD/Vm(+*)
The di#usional ﬁeld has a volume surrounding asingle garnet grain of
vr./- pk-.
(++)
On the other hand, the matrix volume surrounding a single grain of garnet represents
vN+
(+,)

in which N is the population density of thegarnet grain (cm-). Consequently, the averagecompositions, x, of di#usional ﬁeld and its exterior region surrounding a garnet grain becomes(+-)
x(C+(v-vr)C, vr)/v
in which C+ is the initial MnO content in chlorite
and C, is the equilibrium MnO content in chlorite (see Fig. 3) at the beginning of the retrograde metamorphism.
The magnitude of the MnO backstep is considered to be proportional to the di#erence between the equilibrium and the average MnOcontents of chlorite as illustrated in Fig. 3.
Thus, we obtainCbsg(x-C,)
g(C+C,)(+vr/v)
gC(+. pNDt-/- R-)(+.)
in which C is the di#erence between C+ and C,,
g is a constant, and t is the time of growth,which is constant in a single hand specimen.
The backstep compositions Cbs should be zero if
the value in the blanket of the above equation issmaller than zero. Thus, we obtain
vr/v+(+/)
Then
DVm d,(+0)
Further, if we observe Rc at Cbs* as shown in 55 

Mitsuhiro Toriumi and Akiko Nomizo

Fig. 2, we obtain the growth time of garnet, tg tobetgRc d/D(+1)
where d is the mean spacing of garnet grainsdeﬁned earlier. And, the average growth velocity of garnet becomes,
VRc/tgD/d.
(+2)
These relations between grain size and backstep
magnitude are shown in Fig. 2. The binary
loops of the Mn-Fe garnet system are approximately box type because enthalpy change accompanied with this reaction is very large(Banno et al., +320). Then, C, is very small sothat C is approximated to be C+. As shown in
Fig. 3, the MnO backstep contents seem to converge to +.- wt ῌ which will be g C+. If g isabout +*, then C+ becomes *.+ wt ῌ which is near
the average MnO content in chlorite zone chlorite (Higashino et al., +32+). From eqn. (+1),
using Rc*.*+ cm, d+ cm and D+*+0 cm,/sec,
we obtain tg+*+. sec, suggesting that the modelis reliable. Further, average growth velocity is
about +*+0 cm/sec.

zoning formed during decrease of pressure
under isothermal conditions. The growthmodel presented here is consistent with the relationship between grain size and population densities of grains, and the relationship between themagnitude of reverse zoning of MnO and grainsize.
Acknowledgements
The authors thank Tadao Nishiyama andAaron Stallard for critical reviews.
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Grain sizes of garnet in higher grade pelitic
and basic schists of the Sambagawa high pressure metamorphic belt vary from *.*+ to +* mmwithin a single outcrop, and the CSD is similar
to the LSW pattern. The local mean spacing of
garnet grains increases with increasing local average of grain size, suggesting that grains inclusters create a competitive environment forthe component ions of garnet from matrix chlorite around the cluster. Furthermore, large
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