Modeling of zoning patterns in garnet: Thermodynamic and kinetic aspects

ISSN 1075�7015, Geology of Ore Deposits, 2013, Vol. 55, No. 8, pp. 625–636. © Pleiades Publishing, Ltd., 2013.Original Russian Text © Yu.L. Gulbin, 2013, published in Zapiski Rossiiskogo Mineralogicheskogo Obshchestva, 2013, No. 1, pp. 3–22.

625

1 INTRODUCTION

Zoning of garnet is an important source of petroge�netic information. The computation of mineral equi�libria is one of the most efficient approaches to itsinterpretation (Spear, 1988; Powell et al., 1998; DeCapitani and Patrakakis, 2010). This method makes itpossible to plot phase diagrams of mineral systems andto reconstruct the sequence of phase transformations.The next step is to provide insights into the mechanismof metamorphic crystallization by simulation of theconcentration profiles in garnet (Spears et al., 1991;Okudaira, 1996; Konrad�Schmolke et al., 2005; Inui,2006; Gaidies et al., 2008).

This paper is concerned with the modeling of zon�ing in garnet from metapelites occurring in the north�ern Ladoga region (Gulbin, 2012, 2013), using pseu�dosections by application of THERMOCALC soft�ware (Powell et al., 1998) with a special emphasis onthe kinetics of porphyroblast growth. In addition, thepossiblity of a description of zoning on the basis offractional crystallization model is considered. Atten�tion is also paid to the P–T–t paths and the geody�namic implications of the revealed trend.

1 Corresponding author: Yu.L. Gulbin. E�mail: [email protected]

RESEARCH METHODS

A sample for studying has been taken from the stau�rolite–andalusite zone of the Paleoproterozoic meta�morphic sequence in the northern Ladoga region(Gulbin, 2012). The sample contains euhedral garnetcrystals up to 4 mm in size with distinct concentriczoning. The temperature interval of their formationwas estimated at 515–570°C using a garnet–biotitegeothermometer. A similar interval estimated with theisopleth technique is shifted by 20–30°C to the regionof higher temperatures. The peak conditions of meta�morphism are estimated at 595–605°C and 4.0–4.2 kbar (Gulbin, 2012).

The studied sample displays a correlation betweenthe chemical composition in the centers of garnetcrystals and their size, indicating progressive nucle�ation in the course of porphyroblast growth. Withallowance for similar composition of rims in crystals ofvariable size, this correlation is used as evidence forthe achievement of local equilibrium in the system andmakes it possible to use mole fractions of componentsas temporal markers for studying phase transformationkinetics (Gulbin, 2013).

The graphs of Mn isopleths were plotted for thepurpose of simulating garnet zoning. The main prob�lem arising in plotting of such graphs is related to thefractionation of chemical elements in the process ofporphyroblast growth (Evans, 2004; Zeh, 2006). The

Modeling of Zoning Patterns in Garnet:Thermodynamic and Kinetic Aspects

Yu. L. Gulbin1

St. Petersburg State Mining University, 21�aya liniya 2, St. Petersburg, 199106 St. PetersburgReceived October 17, 2012

Abstract—Results from the modeling of compositional zoning patterns in garnet porphyroblasts from themedium�grade metapelitic schist of northern Ladoga area are considered. The P–T pseudosections in themodel KMnFMASH system were calculated for this purpose using THERMOCALC software (Powell et al.,1998). Particular emphasis is placed upon the effect of garnet growth kinetics on the model zoning profilesfor Mn (Gulbin, 2013). They fit the observed profiles if intergranular diffusion�controlled growth is assumedfor porphyroblasts. Additionally, a model of metamorphic fractional crystallization is used to characterize theoscillations in both the garnet core and rim. Starting from the assumption that a reservoir, where garnet grows,consists of chlorite, and that this mineral is intensely replaced with biotite and staurolite at the onset of crys�tallization, a partial release of Mn from the chlorite structure and the concentration of this component inintergranular space is inferred. In terms of the model under consideration, the coefficient of the Mn partitionbetween garnet and reservoir temporarily increases at the early stage of garnet growth, giving rise to theenrichment of the intermediate zone of porphyroblasts in Mn. In addition to the modeling of garnet growthzoning, its subsequent diffusion modification is estimated on the basis of intracrystalline diffusion profile sim�ulation. The reverse zoned, Mn�rich and Mg�poor garnet rims are related to retrograde growth of garnet atthe late stage of porphyroblast formation. The data obtained are used to constrain metamorphic evolutionand the P–T–t path of staurolite�bearing rocks in the northern domain of the studied area.

DOI: 10.1134/S1075701513080035

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following technique was used to solve this problem.Based on petrographic examination (Gulbin, 2012)and the involvement of stereological reconstruction(Gulbin, 2008), a number of garnets were found per�taining to various grain�size classes incorporated into1 cm3 of rock. Graphs of zoning for each grain�sizeclass were based on microprobe profiles goingthrough crystal centers and expressed in (1) wt %oxides and (2) mole fractions, under the conditionthat XFe + XMg + XMn = 1. The points with contentsdivisible by 0.01 were fixed in each XMn profile. Thesepoints divided profiles into zones, which were num�bered so that synchronous zones in the crystals differ�entiated by size would have similar numbers. To deter�mine the position of zones within crystals, their radiiwere measured.

At the next step, the profiles expressed in wt %oxides were divided into zones on the basis of mea�sured radii. The mass of oxides contained within eachzone was calculated. The obtained values were sub�tracted from the mass of oxides contained in 1 cm3 ofrock; the latter figures were found from chemical com�position of rock. For each component, the calculationwas carried out according to the following algorithm:

where M is the mass of oxide contained in 1 cm3 ofrock; Mf, the apparent mass of oxide in the same vol�ume as a function of the fractionation degree f; mij, themass of oxide contained in zone j of the crystal fromgrain�size class i; pi, the number of crystals from grain�

Mf M pimij, fi 1=

k

∑j 1=

f

∑– 1 2…q,,= =

size i in 1 cm3 of rock; i, the grain�size class number;k, the number of grain�size classes; j, zone number; f,the number of zones taken into account in the calcu�lation of apparent mass. The found apparent masseswere recalculated to mol %. Some calculation resultsare shown in Table 1.

As follows from the mode of calculation, an value (equilibrium Mn content in rims of garnet nucleiat stage f of their crystallization) corresponds to eachcalculated Mf value. This implies that the calculatedMf values can be used for the plotting of the corre�sponding Mn isopleths in the P–T pseudosection.This problem was solved with the THERMOCALCv.3.33 program on the basis of internally consistentthermodynamic dataset and activity�compositionrelationships of the solid solutions (Holland and Pow�ell, 1998) for the KMnFMASH system. In the processof computation, it was assumed that quartz and H2Ooccur in excess. The plotted diagram is shown in Fig.1a.

To simulate concentration profiles, a P–T paththat reflects the possible evolution of the mineral sys�tem in the course of porphyroblast growth was plottedin the pseudosection. Abscissas of intersection pointsof this path and isopleths were read from the plot and,further in assumption of uniform heating and dimen�sionless time, the relationship XMn versus t* was plot�ted, where t* = (T – To)/(Te – To); To and Te are tem�peratures of the onset (o) and end (e) of garnet crystal�lization (Fig. 2). To plot the concentration profile,variable t* was replaced with variable r*(τi), i.e., thenormalized radius of nucleus i as a function (generally

XMnf

Table 1. Chemical composition of the metapelite used for modeling (sample H4/1, northern Ladoga region)

ComponentBulk

composition, wt %

Model composition, mol %

KMnFMASH (without

fractionation)

KMnFMASH (with effect of fractionation)

f = 10 f = 15 f = 20 f = 26

SiO2 54.9 65.17 65.26 65.45 66.01 66.70

TiO2 0.62

A12O3 18.4 12.87 12.84 12.80 12.69 12.52

FeO* 12.3 12.21 12.16 12.05 11.73 11.22

MnO 0.32 0.32 0.29 0.26 0.19 0.13

MgO 3.89 6.88 6.90 6.87 6.78 6.78

CaO 2.43

Na2O 2.71

K2O 3.35 2.54 2.55 2.56 2.60 2.65

P2O5 0.24

LOI 0.84

Total 100.00 100.00 100.00 100.00 100.00 100.00

The contents of the components were determined with XRF at the Central Laboratory of the All�Russia Geological Research Institute,analyst B.A. Tsimoshenko. * Total iron.


MODELING OF ZONING PATTERNS IN GARNET 627

nonlinear) of time τi = t* – ti* elapsed from themoment of nucleus origination ti*. The kinetic modelsas the basis of such replacement have been consideredearlier (Gulbin, 2013).

As seen from the results of modeling (Fig. 3), theshape of the concentration profiles observed in garnetcrystals variable in size is best described by the equa�tion ri* = [ki(t* –ti*)]1/2, where ki is a kinetic factor,which remains constant during growth of the nucleusarising at moment ti* and decreases when passing tothe nuclei arising at subsequent moments (Fig. 3,model 3). A similar relationship was earlier establishedfor the studied sample with Kretz diagrams (Gulbin,2013).

Along with Mn isopleths in garnet, the calculatedapparent compositions Mf were used for plotting isop�leths of mineral contents in P–T pseudosection (Fig. 1b).The graph of the evolution of the mineral compositionof the model system along the P–T path was plottedon this basis (Fig. 4).

ZONING AND FRACTIONAL CRYSTALLIZATION

Comparison of Figs. 2–4 shows that the shape ofconcentration profiles is closely related to thesequence of phase reactions that accompany thegrowth of porphyroblasts. In the central zones of thelargest porphyroblasts, the concentration profiles arecharacterized by oscillations caused by a symmetricannulus surrounding the Mn�rich core and enrichedin Mn to the greatest extent (Fig, 3, grain 2A). Such

annuli are observed from time to time in garnets frommetapelites with flat�top concentration profiles. Theirorigin is explained by breaks in mineral formation andthe partial dissolution of nuclei (Stallard and Hickey,2002); the breakdown of auxiliary Mn�bearing phases,which attends garnet growth (Yang and Rivers, 2002);pressure release (Toriumi and Nomizo, 2000); orrelaxation of the system after nonequilibrium nucle�ation (Meth and Carlson, 2005).

3

500 550 600

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grt stbt chl

grt sill

crd btcrd

btst

grt

grt and crd bt

st16

st17

grt 2 bt

35.5

bt36

grt 3

XMn

P, kbar P, kbar

T, °C T, °C

(a) (b)

Fig. 1. (a) P–T pseudosection with isopleths of XMn in garnet and (b) contour lines of garnet, biotite, and staurolite contents.Sample H4/1, metapelitic schist, northern Ladoga region. The arrow indicates P–T path based on zoning of garnet. Here andhereafter, abbreviations of minerals are given after Kretz (1983).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.05

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540530 550 560 570 580 590 600

XM

n

T, °C

t*

1 2 3 4 5 6

Fig. 2. The equilibrium Mn content in garnet (XMn) versusthe time (t*) elapsed from onset of crystallization. 1–6 aremineral assemblage numbers.

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0.05

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12345

2A 15A

6A 2�15A

14A 2�8A

k = 0.57

k = 0.53k = 0.95

k = 0.63 k = 0.54

XM

nX

Mn

XM

n

r* r*

Fig. 3. Mn concentration profiles in garnet plotted on the basis of (1) experimental data and (2–5) results of calculations with

THERMOCALC with the assumption of the fulfillment of the following kinetic conditions: (2) = τi; (3) = (kiτi)1/2; (4) =

[exp(γτ*) ⋅ ]1/2, γ = 1, and (5) = . Numerals at graphs correspond to grain numbers. Values of kinetic factor k estimated

in terms of model (3) are shown as well. To compare profiles related to one crystal, their lengths are normalized.

ri* ri

* ri*

τi* ri

* τi1/3

10

0530 540

20

30

40

550 560 570 580 590 600

1

0530 540 550 560 570 580 590 600

2

3

4

51 2 3 4 5 6 1 2 3 4 5 6Qtz

Ms

Bt

St

Chl Grt

Crd

Sill

Grtm

Grtn

T, °CT, °C

No

rmal

ized

mo

l %

No

rmal

ized

mo

l %/v

ol %

(a) (b)

Fig. 4. Evolution of the rock’s mineral composition along P–T path: (a) variation of mineral contents from the results of model�ing; (b) garnet contents calculated using the method of isopleths (Grtm) and estimated from data of granulometric analysis takinginto account stereological reconstruction (Gulbin, 2008) and zoning of crystals variable in size (Grtn). Mole fractions of mineralsare normalized to the total of mole fractions of oxides which make them up.



In popular opinion, disequilibrium plays an impor�tant role at the early stage of nucleus crystallizationand is caused by a low rate of intergranular diffusion ofchemical elements (primarily, Mn and Ca) at a lowertemperature, which prevents the leveling of compo�nent concentrations in the intergranular medium on ascale comparable with nucleus size (a few millime�ters). Because of a low diffusion rate, garnets withoverprint zoning are formed, consuming Mn and Cafrom the replaced phases (Hirsch et al., 2003). In theabsence of such phases near nuclei, the low�Mn andlow�Ca garnets are formed. Afterward, owing to a risein temperature, the diffusion rate increases and theconcentrations of components in growing facesachieve the equilibrium level (Meth and Carlson,2005).

The latter mechanism is questionable, because itsupposes garnet nucleation in the medium depleted inMn. At the same time, it is well known that the incor�poration of Mn into garnet expands its stability field atlow temperature and pressure (Mahar et al., 1997). Onthe contrary, a deficiency in Mn is a serious obstaclefor the growth of a new phase under these conditions.This implies that it is reasonable to return to the ideasthat treat the observed zoning in a more conventionalmanner.

As has been shown by previous investigation,nucleation of garnet in the studied sample started atthermal overstepping reaching 40–50°C (Gulbin,2012). Owing to this, the first garnet nuclei appear at atemperature of 530°C in the stability field of the Grt–St–Chl–Ms assemblage. As follows from model calcula�tions, the mineral composition of rocks markedlychanges at 530–550°C. Owing to the divariant reaction

Chl + Ms St + Bt + Qtz + H2O, (1)

muscovite disappears and the concentration of chlo�rite rapidly decreases, along with fast growth of biotiteand staurolite (Fig. 4a; 4b). While the temperaturerises further, first chlorite and then staurolite disappearat 584 and 604°C, respectively. The decomposition ofthe latter mineral occurs at the upper temperatureboundary of staurolite subfacies below the triple pointof Al2SiO5 polymorphs due to the well�known reaction

St + Qtz Grt + Crd + Sill + H2O (2)

(Korikovsky, 1979).

The above�described scheme of metamorphic evo�lution suggests that, except for garnet nucleation, allprocesses responsible for the formation and change ofmineral assemblages are controlled by thermody�namic factors. The absence of cordierite in the studiedrock shows that this suggestion is fulfilled here onlypartly. As in the case of garnet, this circumstance canbe caused by the kinetics of phase transformation, inparticular, by only slight overstepping (a few degrees)of the system, which is insufficient for formation ofporphyroblasts (Fig. 4a, assemblage 5).

To describe garnet zoning at the early stage of por�phyroblast growth, let us dwell on the Rayleigh frac�tionation model. In its classical version (Hollister,1966), the growth of garnet is regarded as an isother�mal process proceeding in a closed system and withmaintenance of equilibrium between the environment(reservoir) and the rim of the growing nucleus in theabsence of diffusion within the crystal. Under suchconditions, the concentration of trace elements in therim of crystal CS depends on the relative amount of thefeeding medium F as

CS =λC0F(λ – 1), (3)

where λ = CS/CR is the partition coefficient of elementbetween crystal and reservoir (Nernst coefficient), CR

is the mass concentration of an element in the reser�voir; C0 is the initial concentration of element in thereservoir; F= MS/M0; MR is the mass of the reservoir;M0, mass of the system as a whole. Although λ hasbeen determined only for trace elements, Cygan andLasaga (1982) showed that Eq. (3) is a good approxi�mation for the description of bell�shaped Mn profilesin garnet.

In Hollister’s model, the partition coefficient isaccepted as a constant; however, λ is actually a func�tion of temperature and of the proportions of femic sil�icates in rock, which vary during crystallization. Anexpression like (3), which takes into account the vari�ability of the partition coefficient, was introduced byGreenland (1970) for the linear relationship between λand the mass fraction of the reservoir: λ = a + bF. Thisexpression looks like

(4)

where a and b are certain constants. Using theapproach of Greenland, we write a similar expressionfor nonlinear relationships of λ versus F. If this rela�tionship is approximated by polynom λ = b0 + b1F +b2F

2 + … + bkFk, where b, b1, and …bk are certain con�stants, k is order of polynom, formula (4) takes theform of

(5)

Let chlorite play the role of the reservoir, wheregarnet crystallizes. This suggestion seems evident,because, according to model calculations, chlorite is asole femic mineral at the onset of porphyroblastgrowth. Taking into consideration the lagging nucle�ation of garnet, it may be assumed that the composi�tion of chlorite corresponds to that at the lowerboundary of the Grt–Chl–Ms stability field (Gulbin,2012, Fig. 3). As follows from computation (Table 2),at 478°C and 2 kbar, this mineral belongs to the daph�

CR

C0

��ln a 1–( ) F b F 1–( ),+ln=

CR

C0

��ln a 1–( ) F b F 1–( )b2

2�� F2 1–( )++ln=

+ …bk

k�� Fk 1–( ).+

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nite–amesite series and contains 0.075 apfu or 0.67 wt% Mn. Such chlorites are typical of metapelites fromthe chlorite zone of regional metamorphism (Inui andToriumi, 2004).

As proceeds from comparison of chlorite composi�tion with the composition of the Mn�bearing core ofthe largest garnet crystals, the initial partition coeffi�cient of Mn between garnet and chlorite is close to 12.At the early stage of crystallization, when chlorite isintensely replaced with biotite and staurolite contain�ing less than 0.14 and 0.24 wt % Mn, respectively(Table 2), a part of manganese is released from thechlorite and concentrates in the intergranular space.Being highly mobile, this Mn is absorbed by garnetmore readily than structural manganese. In terms ofthe model under consideration, this leads to anincrease in λ in the process of phase transformation.

Let us estimate the mass of the reservoir at themoment of initial porphyroblast growth. With allow�ance for the simultaneous crystallization of garnet,biotite, and staurolite, we simplify the situation andassume that each mineral recovers Mn from a separateportion of chlorite.

The balance of Mn during phase transformation(Table 3) shows that 64% of the initial mass of chlorite

(78 g in 100 cm3 of rock) falls on garnet. We use thisfigure to estimate a difference 1 – F (the ratio of massof crystallized garnet to the initial mass of the reser�voir) at the moment corresponding to the temperatureof Mn�bearing annulus formation within the innerzones of large garnet porphyroblasts (550°C) and tothe temperature of chlorite disappearance (584°C).According to crystal size distruibution and microprobedata, the masses of garnet arising at the correspondingstages of porphyroblast growth are 0.26 and 8.89 g,respectively. Thus, in the first case, 1 – F is 0.003 and0.11 in the second case.

The calculated estimates can be input parametersfor plotting XMn – (1 – F) graphs using Eq. (5). Theexamples of such graphs, which are given in Figs. 5aand 5b, show the possibility of episodic growth of XMn

at the early stages of fraction crystallization due to aninsignificant increase in λ. When combining themodel under discussion with experimental data onkinetics of garnet growth (Fig. 5c), a profile of Mncontentration in garnet can be plotted. Except for theinner Mn�rich core, its shape satisfactorily coincideswith that of the microprobe profile.

Table 2. Compositional parameters of the minerals used for estimation of equilibrium Mn contents in garnet from Eq. (6).Sample H 4/1, Northern Ladoga region

Mineral x y m AlVI, apfuMn

apfu wt %

Chlorite 0.63 0.66 0.015 1.32 0.075 0.67Biotite 0.60 0.48 0.004 0.48 0.012 0.136Staurolite 0.90 – 0.018 – 0.072 0.236

Compositional parameters were determined using the THRMOCALC for the lower boundary of the Grt–Chl–Ms assemblage stabilityfield at 478°C and 2 kbar (chlorite) and the upper boundary of the Grt–St–Bt–Chl–Ms assemblage stability field (biotite, staurolite) at550°C and 3.75 kbar. The following structural formulae for phyllosilicates were used in calculations: chlorite[M2]2[M3]2[M1][M4]{[T1]2[T2]2}O10(OH)8; biotite [A][M2]2[M1]{[T]4}O10(OH)2; staurolite formula was written as (Fe,Mg,Mn)4Al18Si7.5O48H4; x = Fe/(Fe + Mg), y(chl) = {A1[M1] + A1[M4]}/2, y(bt) = A1[M1], m(chl) = Mn[M2] = Mn[M3] = Mn[M1],m(bt) = Mn[M1] = Mn[M2], m(st) = Mn/(Fe + Mg + Mn).

Table 3. Balance of Mn in garnet crystallization

Mineral Content of mineral, vol %

Mass of mineral in 100 cm3 of rock, g

Mn content in mineral, wt %

Mass of Mn in mineral, g

Mn fraction of total content

in rock, %

Initial moment of crystallization

Chlorite 43 122 0.67 0.82 100

Final stages of crystallization

Biotite 36 112 0.136 0.15 18

Staurolite 16 60 0.236 0.14 17

Garnet 3.2 11 7.89–1.74 0.53 64

Masses of minerals are estimated with the assumption that chlorite, biotite, and staurolite solid solutions correspond to theoretical for�mulae Ames45Daph55, Phl40Ann60, Mg�St10Fe�St90, respectively. The data of crystal size distribution studies and microprobe analyseswere used to calculate the mass of garnet and Mn mass in garnet.



EFFECT OF RETROGRADE PROCESSES

Along with oscillations in the cores of the largestcrystals, the normal zoning of garnet is complicated byreverse zoning at grain rims. Such rims are enriched inMn and depleted in Mg as compared with the outerzones of crystals (Fig. 6). The width of rims dependson grain size. If in the crystals 4–2 mm in size the widthdoes not exceed 50–60 μm, then in smaller grains (1.5–0.5 mm) it reaches 70–90 μm for Mn and 150–200 μmfor Mg. Because of this, many small garnets are char�acterized by reverse zoning (Gulbin, 2012).

The appearance of reverse zoned rims in garnetcrystals is commonly regarded as being related to ret�rograde processes. It is suggested that their formationis controlled by the diffusion of Mn released in theprocess of resorption conversely into garnet. This pro�cess is facilitated by distortion of the structure inresorbed rims with increased oxygen activity (DeBéthune et al., 1975; Woodswoorth, 1977). The for�mation of rims depleted in Mg and enriched in Fe innonresorbed garnet is also related to diffusion. Theirappearance is regarded as a result of cation exchangebetween garnet and biotite at the retrograde stage. Inaddition to secondary alteration, it is assumed thatreverse zoned rims are formed during the growth ofgarnet. Ilmenite (Hollister, 1969), Mn�bearing epi�dote (Yang and Rivers, 2002), and chlorite (Banno and

Chii, 1978) are hypothetical sources of Mn. In the lat�ter case, the garnet grows under retrograde conditions.

Simulation of diffusion profiles is an efficient toolfor the study of intracrystalline diffusion as a mecha�nism responsible for zoning in garnet (Spear 1991;Okudaira, 1996; Ganguly, 2010). In a general case,implementation of this approach requires the solutionof a system of differential equations, which describediffusion in multicomponent systems. In this work,the THERMAL HISTORY program (Robl et al.,2007) has been used. Simulation is based on simplifieddescription of intracrystalline diffusion as a binaryprocess proceeding in an ideal solid solution in Fe–Mg garnet which forms a spherical crystal, whose sur�face is in equilibrium with biotite. To take into accounta temperature�dependent equilibrium constant ofexchange reaction between femic silicates, the Ferryand Spear (1978) calibration of garnet�biotite geo�thermometer was used. The composition of biotite wasaccepted as XMg = 0.44 and the contents of minerals as93 and 7 vol % of biotite and garnet, respectively.Parameters of diffusion (pre�exponential factor D0 andactivation energy Q) were taken from Ganguly et al.(1998). The diffusion profiles have been calculated fortwo values of peak temperature of metamorphism (570and 600°C) in the assumption that garnet crystals ini�tially were homogeneous and the cooling rate was 0.5,1, 3 and 10°C/Ma (Fig. 7). To take into account the

0.050.00200.050

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r*

XM

n

r*X

Mn

1 2λ

F F

λ

1 – F 1 – F

Mass fraction of garnet

(a) (b)

(c) (d)

Fig. 5. Modeling of zoning in garnet using equations of fractional crystallization applied to Mn: (a, b) equilibrium Mn content ingarnet crystals versus mass fraction of crystals in the system; relationships between λ versus degree of crystallization approximatedby polynoms of 4–6 orders are shown as insets; (c) normalized nucleus size versus mass fraction of garnet in the system; (d) Mnconcentration profiles in garnet: (1) experimental and (2) calculated.

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effect of uncertainty in diffusion coefficients at lowtemperature (Carlson, 2006), the curves of diffusionzoning based on D0 and Q values published by Carlson(2006) and Perchuk et al. (2009) have been addition�ally calculated (Fig. 8).

Comparison of simulation results with microprobedata shows a substantial difference between observedand calculated diffusion profiles (Fig. 6). Although amaximum width of diffusion rims under the given con�

ditions can reach 100–150 μm, with constraintsimposed by the duration of retrograde stage (seebelow), it actually hardly exceeds 40–50 μm. Thisvalue approximately corresponds to the width ofreverse zoned rims in larger crystals but obviouslyyields to that in small grains. It is even more importantthat the diffusion model is not able to explain the causeof such variations.

Let us consider the possibility of forming reversezoned rims in the process of garnet growth as an alter�native to the diffusion model. According to modelingwith the use of pseudosections, the final stages of por�phyroblast growth proceeded under conditions corre�sponding to the stability field of the Grt–Sill–Crd–Btassemblage. The equilibrium crystallization of silli�manite and cordierite, if it actually took place, wasaccompanied by the dissolution of staurolite, whichfacilitated redistribution of Mn from staurolite intogarnet, at least, hypothetically. By several reasons, thisscenario hardly is probable. The main point is that it ispoorly reconciled with the absence of sillimanite andcordierite in the studied sample.

It cannot be ruled out that the latter circumstancepoints to a lower temperature of metamorphism incomparison with that estimated using the isoplethsmethod. For reference, the temperature calculated bygarnet�biotite geothermometer is close to 570°C. TheGrt–St–Bt–Chl assemblage is stable at this tempera�ture. As follows from petrographic examination, mus�covite is added to this assemblage, and thus the rocksare substantially overstepped at the final stage of por�phyroblast growth. The most probable cause of ther�mal overstepping is related to the lagged crystallizationof staurolite. It may be suggested that under these con�ditions dehydration giving rise to garnet formation willproceed not only at increasing but also at decreasingtemperature, owing to an excess of the thermal energynecessary for the destruction of crystal lattices of phyl�losilicates. If this is actually the case, the appearance

0.06

0.040

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0.10

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100 200 300 400

0.06

0.08

0.10

0.07

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0.040

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10050 150 200 250 300

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10050 150 200

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0.30.20.10

0.160.24

0

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0.040.50.3

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0.40.20.10

0.12

0,040.3

565

542

544

580

501

452

541530

568576

r, mm

r, mm

r, mm

2A

2�8A

13A

XM

nX

Mn

XM

nX

Mg

XM

gX

Mg

Fig. 6. XMn and XMg profiles in the outer zones of garnetcrystals variable in size based on the data of microprobeanalysis (Camscan MV2300, analyst V.F. Sapega, CentralLaboratory of the All�Russia Geological Research Insti�tute) and calculated from modeling results. Open circlesand boxes are analytical points; the dashed lines are theaveraging curves; solid lines are diffusion profiles; dottedlines are profiles calculated using equations of fractionalcrystallization. In the latter case, normalization was used:width of reverse zoned rims in modeling profiles (Figs. 5d,9c) was equated with width of microprobe profiles (60 μmfor grain 2A and 100–200 μm for grains 2�8A, 13A). Thediffusion profiles for grain 2A were calculated at Tmax =600°C on the basis of diffusion parameters taken fromGanguly et al. (1998); for grains 2�8A and 13�A, at Tmax =600°C, ΔT/Δt = 1°C/Ma on the basis of diffusion parame�ters taken from Carlson (2006). For convenience of com�parison with microprobe data, diffusion profiles have beenshifted along vertical axis. Numerals near analytical pointscorrespond to crystallization temperature, °C estimated withgarnet–biotite geothermometer (Gulbin, 2011, Eq. 12).



of reverse zoned rims in garnet porphyroblasts receivesa simple explanation.

The relationship between λ and F during fractionalcrystallization is shown by the curve displayed in Fig. 5a(inset). As was mentioned above, the Mn partitioncoefficient is a function of temperature and decreaseswith rise of T. At the same time, the partition coeffi�cient depends on the chlorite/biotite ratio in rock andincreases in the course of porphyroblast growth.Owing to the effects of differently directed factors, λfirst retains its constant value and then rapidlyincreases with the growth of F, which is treated as aresult of a fall in temperature by several dozen degreesat the final stage of garnet growth. The maximum λvalues (40–45) approximately correspond to the Mnpartition coefficient between garnet and chlorite at thelate stage of crystallization. The plots of zoning basedon the λ–F relationship reproduce well the shape ofMn concentration profiles for garnet crystals variablein size (Fig. 5d, 6).

In contrast to Mn, the Mg partition coefficientbetween garnet and femic silicates (chlorite, biotite)decreases with a fall in temperature. Therefore retro�grade growth of garnet is accompanied by the forma�tion of rims depleted in Mg. The temperature�depen�dent variation of λ is described by equation (Cyganand Lasaga, 1982)

(6)

where λ0 is the partition coefficient at a certain initialtemperature T0; ΔH, the enthalpy of garnet–femic sil�icate exchange reaction; R, gas constant; and ΔT= T–T0. Let us use formula (6) to simulate the Mg concen�tration profile in garnet. Because of the relatively closeMg mole fraction in chlorite and biotite, it may beassumed that Mg is supplied into garnet in the courseof exchange reaction GrtFe + BtMg = GrtMg + BtFe. Letus accept λ0 = 0.12, ΔH = 6.15 kJ, and T0 = 515°C. Atthe first step of calculation, we postulate the relation�

λ λ0ΔHR

�� ΔTTT0

��⎝ ⎠⎛ ⎞ ,exp=

ship T – (1 – F) (Fig. 9a, inset). Assume that propor�tionality takes place between two variables at the pro�grade stage of garnet growth and nonuniform (acceler�ating with an increase in the crystallization degree)drop in temperature at the retrograde stage. Such arelationship takes place, for example, when the rate ofphase transition gradually decreases at the final stage.At the second step, we calculate λ for a series of pre�scribed temperatures and approximate relationship (1–F) – λ by polynom (Fig. 9a). On the basis of this rela�tionship, the (1 –F) – XMg diagram will be plotted(Fig. 9b). As in the case of Mn, the problem of simu�lating the concentration profile with be solved withinvolvement of the relationship (1 –F) – r* (Fig. 5c).

The plotted graph (Fig. 9c) reflects in idealizedform the zonal Mg distribution in the largest garnetporphyroblasts. Besides this, it is used for a moredetailed reproduction of reverse zoned rims in crystalsvarying in size (Fig. 6).

0.11

0.100 10 20 30 40 50

0.12

0.11

0.100 20 40 60

0.12

0.13

0.14

80

XM

g

XM

g

10 31

10310.5 Tmax = 570°C Tmax = 600°C

Distance, μm Distance, μm

(a) (b)

Fig. 7. Mg diffusion profiles in garnet at its contact with biotite, calculated using diffusion parameters taken from Ganguly et al.(1998). Numerals in the figure correspond to the chosen values of peak temperature of metamorphism and rates of cooling(°C/Ma). The dashed lines indicate the most probable boundaries of diffusion rims.

0.11

0.101000 50

0.12

0.13

0.14

150 200Distance, μm

10

1

101

Tmax = 600°C

XM

g

Fig. 8. Mg diffusion profiles in garnet at its contact withbiotite calculated using diffusion parameters taken fromPerchuk et al. (1998) and Carlson (2000), dot�and�dashand two�dots�and�dash lines, respectively.

634


GULBIN

CONCLUSIONS

Compare our results on the evolution of mineralformation conditions in the process of porphyroblastgrowth with published data on regional geological history.As is known, the northern Ladoga region occupies thesoutheastern end of the Svecofennian Belt as a part of theaccretionary orogen formed 1.89–1.78 Ga ago (Balty�baev, 2005). The considered territory is heterogeneousin geology. The northern domain, from where thestudied sample has been taken, is a margin of theArchean craton overlapped by Paleoproterozoic

metaterrigenous rocks, deformed and metamor�phosed under conditions of greenschist to amphibolitefacies. This domain contacts along the near�latitudi�nal thrust fault zone with granulites of the southern(Svecofennian proper) domain. According to isotopicgeochronological data, the peak of metamorphism inboth domains is dated at 1.88–1.87 Ga and thus cor�responds to the peak of the Early Svecofennian Orog�eny (Baltybaev et al., 2008). At the postcollision stage(1.86–1.85 Ga), the Svecofennides were involved inrising, accompanied by gradual cooling, of metamor�phic sequences.

The P–T–t path plotted on the basis of model zon�ing in garnet and garnet–biotite geothermometrycharacterizes the direction and duration of metamor�phism in the zone of staurolite�bearing rocks. Thispath is clockwise progressively demonstrating (1) anincrease in temperature and pressure from 515°C and3.1 kbar up to a baric peak (570°C, 4.3 kbar), (2) theongoing heating of rocks up to a thermal peak (585°C,4.1 kbar) against the background of decompression,and (3) a fall in temperature by a few dozen degrees(Fig. 10). If the ascending branch of the trend (stages1, 2) obviously reflects prograde nucleation andgrowth of garnet grains, then the descending branch(stage 3) corresponds to conditions of retrogradegrowth or the diffusion modification of zoning in gar�net.

0.100.85 0.90

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0.20

0.25

0.95 1.00

0.02–0.25 0.50

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0.020 0.05

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590

550

510

0.15

1 – F1 – F

1 – F

T, °C

XM

g

r*

λ

XM

g

12

(a) (b)

(c)

Fig. 9. Simulation of zoned Mg distribution in garnet using equations of fractional crystallization.

P,

kbar

3

500

4

5

550 600

19

17

15

13

11

KySill

And

0.05 Δ t

H,

km

T, °C

Fig. 10. P–T–t path.



To decide which of these alternatives is preferred,compare the obtained data with available estimates ofretrograde stage duration and cooling rate of meta�morphic sequences in the studied and adjacentregions. The U–Pb monazite age and Sm–Nd garnetage of granulites in the West Uusimaa area pertainingto the Late Svecofennian granite–migmatize zone ofsouthern Finland (Mouri et al., 2005) show that theinterval between the moment of crystallization at 800–850°C and the time of isotopic system closure at600°C does not exceed 15–20 Ma. This implies thatthe rate of retrograde cooling is 10–15°C/Ma. Similarestimates (15–20°C/Ma) characterize the rate ofcooling during multistage migmatization in the north�ern Ladoga region (Baltybaev et al., 2006). It isassumed that after a peak of thermal activity, the cool�ing rate in the southern domain could have reached30–40°C/Ma (Baltybaev, 2005). Such a relatively slowretrograde cooling rate is inherent to the ancientaccretionary orogens, where erosion was the maindriving force of exhumation, in contrast to theyounger subduction�related orogens exhumed underthe action of tectonic forces (Gapais et al., 2009). Atthe temperature gradient of 40–50°C/km, this ratefacilitates passage of the P–T path near the geothermand achievement of thermal equilibrium in the system.As is shown by comparison of U–Pb zircon and K–Arbiotite ages of migmatites from southern Finland(Lindstrom et al., 1991), the cooling rate slows downto 3–15°C/Ma, indicating deceleration of tectonicprocesses in the low�temperature (300°C) region.

The constraints imposed upon the duration of theretrograde stage (if they are applicable to the northerndomain) markedly diminish the role of intracrystallinediffusion in the formation of zoning in garnet. First,this concerns a width of diffusion rims. As follows fromsimulation results at the peak of metamorphism reach�ing 600°C and the rate of retrograde cooling of10°C/Ma and higher, the width does not exceed 40–50 μm (for Mg). As was mentioned above, this estimateappreciably yields to the actual width of the rims depletedin Mg in the studied garnets, where they are 150–200 μmwide. Taking into account the formerly stated suggestionregarding possible garnet crystallization at the retrogradestage, this discrepancy is evidence in favor of the growthorigin of reverse zoned rims.

To apply the drawn conclusion to the estimation ofthe temporal component of the P–T–t trend, let usimpose restrictions on the total duration of porphyro�blast growth Δt, issuing from possible mechanisms ofmetamorphic crystallization. If the garnet growth waslimited by intergranular diffusion, Δt falls, with a highprobability, into the time interval 0.4–3.8 Ma (Gulbin,2013). Further, using the parabolic relationship r2 =kΔt, let us estimate the duration of the retrogradestage. For this purpose, we calculate the duration ofthe formation of a rim 50 μm wide in a porphyroblast4 mm in diameter. The duration will be 0.05Δt, i.e.,0.19 Ma, if the upper limit of Δt is used in calculation.

This value can be used to calculate heating andcooling rates ΔT/Δt of the studied sample in the pro�cess of metamorphism. Since the prograde crystalliza�tion of garnet is accompanied by a rise in temperatureestimated at 50–70°C, the heating rate falls within aninterval of 15–20°C/ Ma. The rock temperature grewat such a rate, if the duration of the prograde stage was3.6 Ma. It is evident that this estimate corresponds tothe lower limit of the heating rate. At shorter Δt, it willbe measured by many dozen C/Ma. The cooling rate atΔT = 20–25°C (Fig. 7, grain 2a) is close to 100–130°C/Ma. This value is obviously overestimated rela�tive to the aforementioned estimates for the Svecofen�nian domains but is comparable with the rate of retro�grade cooling estimated for some other regions ofHT–LP metamorphism (Kalt et al., 2000). On theone hand, this result is consistent with the elevatedtemperature gradient in the considered regions, atwhich a high rate of retrograde cooling can be reacheddue to a moderate rate of exhumation. On the otherhand, it is also probable that the retrograde growth rateof garnet is overestimated, owing to a decrease in theintergranular diffusion coefficient induced by lowtemperature or by a drop in H2O activity in the mineralsystem at the retrograde stage.

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Translated by V. Popov

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Modeling of zoning patterns in garnet: Thermodynamic and kinetic aspects