Diffusion Kinetics in Minerals

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Diffusion kinetics in minerals:Principles and applications to tectono-metamorphic

processes

JIBAMITRA GANGULY

Bayerisches Geoinstitut, University of Bayreuth

D-95440, Bayreuth, Germany;

Permanent address: Department of Geosciences, University of ArizonaTucson, AZ 85721, USA;

e-mail: [email protected]

Introduction

Diffusion is the process by which atoms or ions or ionic species migrate within a mediumin the absence of a bulk flow. Diffusion in solids has been a subject of interest in thefields of solid state science (physics, chemistry and metallurgy) for nearly a century,starting with the work of Einstein on the relationship between random atomic movement

and diffusion process. The phenomenological study of diffusion began even 50 yearsearlier, with the empirical formulation of Fick on the relationship between the diffusionflux of a component and its concentration gradient.

The subject of diffusion in the solids may be subdivided into volume, grain boundary and surface diffusion. The last topic has, however, received very little attentionin the study of geological processes. The study of grain boundary diffusion is importantto the understanding of many metamorphic processes including the problems of masstransport, fluid/rock interactions, thermal history and crystal growth. The interestedreader is referred to Joestein (1991) for an excellent review of the subject of grain boundary diffusion and its applications to geological problems.

Diffusion controlled processes within a mineral preserve important records of thethermal and physico-chemical history of the host rocks. Volume diffusion, that isdiffusion through the crystal lattices, affects development of compositional zoning inminerals, ordering of atoms in nonequivalent crystallographic sites of a mineral,formation and coarsening of exsolution lamellae, and retention of isotopic characteristicsin minerals that can serve as quantitative chronometers in their thermal and growthhistory. In this chapter, I present a brief overview of the fundamental principles of volume diffusion kinetics, primarily at a phenomenological level, and discuss thevarious factors that affect diffusion kinetics in minerals. Finally, I discuss someapplications of the diffusion kinetic studies in minerals to the understanding of tectono-metamorphic processes in major continent–continent collisional environments. Itshould, however, be emphasized that the scope of applications of volume diffusion

EMU Notes in Mineralogy, Vol. 4 (2002), Chapter 10, 271–309



kinetics to geological and planetary problems is much wider than what I have attemptedto cover in this chapter. Some incidental references to additional applications have beenmade in the appropriate places.

List of symbols

Di

*, Di

+ and Di Tracer, self- and chemical diffusion coefficients, respectively, of thecomponent i

D(i – j) Chemical interdiffusion coefficient of the components i and j

Di(EB) Effective binary diffusion coefficient of the component i in a multi-component system

D Matrix of diffusion coefficients Dij An element of the D matrix

τ Diagonal matrix of the eigenvalues of a D matrixτ i An eigenvalue of the D matrixB A matrix composed of the eigenvectors of D matrixL Matrix of kinetic coefficients (Onsager matrix)G A thermodynamic matrix that relates D and L matrices

f Isotopic correlation factor Q and ∆V

+ Activation energy and activation volume of diffusion, respectively J i Flux of a component i

J A column vector of the fluxes of n –1 independent components in an n-component system

C i Concentration of the component i, expressed in atomic units per unitvolume

C A column vector of the concentrations of n –1 independent components inan n-component system

X i, ai and γ i Atomic fraction, activity and activity coefficient, respectively, of thecomponent i

µi Chemical potential of the component i

W Non-ideal interaction parameter k B Boltzmann constant

η A cooling time constant ( K –1

t –1

)

Γ

Phenomenological theory of diffusion

Fick’s laws

Let us consider a planar section that has a fixed position in an isotropic medium with

respect to a coordinate system measured normal to the section. According to Fick’s law,which was formulated by analogy with Fourier’s law of heat conduction, the flux (i.e.

∫′t

dt t D0

)(

J. Ganguly272



the rate of transfer per unit area) of a component, J i, through this planar section is proportional to its local concentration gradient. Thus,

(1)

where C i is the concentration of i, in atomic units per unit volume, which decreases inthe direction of increasing x, and Di is the diffusion coefficient of i (with dimension of

L2/t ). (The negative sign in the above expression is introduced to make the flux positivein the direction of decreasing C i.) Implicit in the above statement is the assumption thatthere is no external force (such as electrical and gravitational forces) acting on thediffusing species. The modifications for the expression of flux to incorporate the effectsof an external force and the movement of the planar section, with respect to a fixed

coordinate system, are discussed below.From the continuity relation that stems simply from the principle of conservationof matter, it follows (e.g. Crank, 1983, p. 2–4) that if diffusion is one-dimensional (i.e.

there is a concentration gradient only along the x axis), then

(2)

so that, if the flux is given by Equation 1, then

(3a)

If the diffusion coefficient is independent of position, then

(3b)

The dependence of D on x arises from its compositional dependence and the variation of composition as a function of position.

For three dimensional diffusion in an isotropic medium, the equationscorresponding to Equations 3a and 3b are written by simply replacing ∂ by the gradient

operator (i.e. i(δ /δ x) + j(δ /δ y) + k (δ /δ z)) and ∂2 by the Laplacian operator ∇2 (i.e. δ 2/δ x2

+ δ 2/δ y2 + δ 2/δ z2) in the right-hand side of the equations. The equation expressing thetime dependence of concentration, that is Equation 3b or its three-dimensional form, isknown as the diffusion equation in Cartesian coordinates. Other forms of the diffusionequation in different coordinate systems follow simply from the appropriatetransformation of coordinates. The solutions of the diffusion equation, either analyticalor numerical, for the appropriate initial and boundary conditions, permit us to modeldiffusion controlled properties to retrieve quantitative information about geological, planetary and other processes.

.2

2

x

C D

t

C ii

i

∂∂

=∂

∂

.

∂

∂

∂

∂=

∂

∂

x

C D

xt

C iii

),( ii J

xt

C

∂∂

−=∂

∂

, xC D= J i

ii ∂∂−

Diffusion kinetics in minerals: Principles and applications 273



Irreversible thermodynamic formulation

While Fick’s law is an empirical law, the flux equation can be formulated rigorously

from the principles of irreversible thermodynamics. It follows from the latter that, in theabsence of interference from diffusion of other species, the appropriate driving force of diffusion of a species i in an isotropic medium is – ∂(µi/T )/∂ x, where µi is the chemical potential of the component i. Thus, the flux of a component is given by

(4)

where Li is a phenomenological coefficient. This relation, however, assumes that thehigher order terms of the driving force has negligible effect on the flux (i.e. Eqn. 4 holdswithin the domain of validity of linear irreversible thermodynamics).

Since at a constant p – T condition ∂µi = RT ∂lnai = RT ∂ln(C iγ i), where ai and γ i arethe activity and activity coefficient of the component i, respectively, it is easy to see that

. (5)

Comparing Equations 1 and 5, we have

, (6a)

where

. (6b)

The quantity within the parentheses of Equation 6a is usually referred to as thethermodynamic factor. We will henceforth refer to it as D(thermo). From definitionC i = ni / NV = X i /V , where X i is the mole fraction of the component i, V is the molar

volume and N is the total number of moles in the system. Thus, if the molar volume of the substance remains constant, then d lnC i = d ln X i.

Diffusion coefficients: Terminology and definitions

Before proceeding further, it is important to discuss certain properties of diffusioncoefficient and the related terminology. In the literature, one encounters terms liketracer, self-, inter-, and chemical diffusion coefficient, the meaning of which is notusually clear to the reader. In addition, there is a lack of uniformity in the usage of these terms, which is a source of confusion in the literature on diffusion kinetics. It

is, thus, important to define these terms clearly in the sense these are used in anywork.

i

ii

C

RL D =+

∂

∂+= +

i

iii

C D D

ln

ln1

γ

∂

∂

∂

∂+−=

x

C

C C

RL J i

i

i

i

ii

ln

ln1

γ

,

/

∂

∂

∂

∂−=

∂

∂−=

∂

∂−=

x

C

C T

L

xT

L

x

T L J

i

i

iiii

iii

µµ

µ

J. Ganguly274



Tracer, self-, and chemical diffusion coefficients

In this paper, the diffusion coefficient of an isotope of an element that describes its flux solely in response to the isotopic concentration gradient in a chemically homogeneousmedium will be called the tracer diffusion coefficient of the element i, and be denoted by the symbol D*

i( I ) or simply D*i, where ( I ) stands for the specific isotope. By chemically

homogeneous we mean homogeneity with respect to the concentration of chemicalelements. The term self-diffusion coefficient will be used to define the diffusioncoefficient that describes the flux of an element solely in response to its ownconcentration gradient, and under the condition that its thermodynamic interaction withthe solvent matrix is independent of its concentration so that the “thermodynamic factor”(Eqn. 6a) is unity. Thus, the self-diffusion coefficient of an element is the quantitydefined by Di

+ in Equation 6b. The diffusion coefficient Di, which is a product of the self-

diffusion coefficient and the thermodynamic factor (Equation 6a), will be referred to asthe chemical diffusion coefficient of the component i. The self- and tracer diffusioncoefficients are equivalent when all isotopes of the element have the same diffusivities.Although this is strictly not the case, the terms self- and tracer diffusion coefficients have been used interchangeably. (In the literature, the term self-diffusion coefficient of anelement has also been applied to the limiting case of tracer diffusion, in the sense definedabove, when the diffusing tracer isotope and the non-tracer solvent belong to the sameelement, e.g. diffusion of 26Mg in Mg2SiO4.)

Chemical interdiffusion coefficient in binary metallic and ionic systems

When two or more components diffuse simultaneously in a given medium, the flux of the components becomes coupled. We first consider the case of diffusion of two neutralspecies ( A and B) across a welded plane (Fig. 1), as in a binary metallic alloy. If thecomponent A diffuses faster than B, then the right-hand side of the couple will swellwhile the left-hand part will shrink. If, however, the specimen is held at a fixed position,then the interface will move leftwards. This phenomenon was first noticed bySmigelskas & Kirkendall (1947) for the interdiffusion of Cu and Zn by placing finemolybdenum markers at the interface, and is usually referred to as Kirkendall effect(instead of Smigelskas effect or Smigelskas–Kirkendall effect!). In analysing this resultof Smigelskas and Kirkendall, Darken (1948) raised the important question “what isdiffusion?” and introduced the concept of frame of reference in diffusion studies.

In the above example, one can describe the diffusion process with reference to twoalternative coordinate systems, x and x′, as follows. In the first system, x′ = 0 is fixed tothe interface (which may be located by some inert markers, as in the experiment of Smigelskas & Kirkendall, 1947), and increases to the right. In the second system, x = 0is located at one end of the diffusion couple, say the left end (this frame of reference isusually called the “laboratory frame of reference”). In the x′ coordinate system, the fluxof the component A across a plane located at a fixed distance, say x′ = 0, is simply given by Equation 1. However, in the x coordinate system, the flux of A across a plane at

x = k (which we may take as the same plane as at x′ = 0) must involve an additional term,vC A( x = k ), in order to account for the effect of movement of the plane, where v is thevelocity of the plane. Thus, in the x coordinate system,




. (7a)

So that, assuming D to be independent of x

. (7b)

In the above example, the velocity v is obviously a consequence of the difference between the individual diffusivities of the two components A and B. Thus, assuming that

the volume of the system has remained constant (and also ignoring any effect due tovacancy flow and interactions of the atoms with the vacancies), solution for v in termsof these individual chemical diffusivities, DA and DB, in the x coordinate system, andrearrangement of terms yields (Darken, 1948)

, (8a)

where D(A–B) is given by

D( A – B) = [ X A DB + X B DA], (8b)

and is called a chemical interdiffusion coefficient (in the above example, for a metallicsystem). The flux of the component A in the x coordinate system is also given by

x

C B – A D J B

B ∂

∂−= )(

∂

∂+

∂

∂=

∂

∂−=

∂

∂

x

C v

x

C D J

xt

C B

2

B2

BBB )(

k xvC x

C D J =+

∂

∂−= )( B

BBB

J. Ganguly276

Fig. 1. Schematic illustration of the Kirkendall effect for the inter-diffusion of two neutral species, A and B,with D+

A > D+B. The interface, which is shown by a dashed line, moves to the left if the specimen is held at a

fixed position. (a) is the initial configuration of the diffusion couple, and (c) is the configuration after anelapsed time. In the x′ coordinate system, the moving interface is located at x′ = 0, whereas in the x coordinate

system, the interface is located at x = k (modified from Haasen, 1978).

– –



Equation 8a upon simply replacing C B by C A. Note that both fluxes are described by the same diffusion coefficient, D( A – B).

It may be noted incidentally that Equation 7 is a general equation that describesdiffusion under the influence of a “driving force”. The latter is defined to be a force, suchas those arising from an electrical field or non-ideal mixing property, which causes anatomic jump in one direction across a potential energy barrier to be more probable thanthat in the reverse direction across the same barrier (Manning, 1968). Equation 7 is alsoapplied to the solution of diffusion-reaction problem where the interface between twocrystals moves, with respect to a stationary coordinate system, due to reaction along withdiffusion (see later for an application).

Instead of neutral atoms, let us now consider the problem of diffusion of ions of thesame charge, z. In this case, if A diffuses faster than B, then there would be an accumulation

of excess positive charges on the right hand side of the couple, and a correspondingaccumulation of positive charge vacancies on the left. This would create an electrical field E d (which implies a driving force zE d, where z is the charge of the diffusing ion) that wouldaffect the diffusion of ions so that local electrical neutrality of the sample is preserved. Thelatter requirement implies that, in the absence of any other mechanism of chargecompensation, the net flux of ions at the interface must be zero (i.e. there must be equalnumber of A and B crossing the interface per unit area per unit time). As a result, the interfacemust remain fixed with respect to a stationary coordinate system if the molar volumeremains constant. Expressing the mean drift velocity of the ionic species in terms of E d (i.e.

v = ( Di) zE d/k BT ) in Equation 7, and also in the analogous expression for J A, and solving E d

in terms of DA and DB under the constraint that J A + J B = 0, yields (e.g. Manning, 1968)

. (9)

We now write

D( A z – B z) = (10)

Equation 10 defines the chemical interdiffusion coefficient in a binary system of equally charged species. By the requirement of mass balance, ∂C B/∂ x = – ∂C A/∂ x. As

emphasised by Lasaga (1979), the chemical interdiffusion coefficient in an ionic systemwill, in general, be overestimated, especially around X = 0.5, if one uses the Darkenrelation, i.e. Equation 8b instead of Equation 10. The Darken relation has been appliedto mineralogical systems, but from a theoretical point of view, it is not applicable to suchsystems since the diffusing species are ions instead of neutral atoms.

For the interdiffusion of unequally charged species, the chemical interdiffusion

coefficient in a volume fixed reference frame is given by (Barrer et al., 1963; Brady, 1975)

D( A Z A – B Z B) = (11)

where Z A and Z B represent the charges on the specified ionic species. Equation 11reduces to Equation 10 when Z A = Z B.

( ),

BB2BAA

2A

BBAABA

D X Z D X Z

X Z X Z D D

+

+

.BBAA

BA

D X D X

D D

+

B

BBAA

BAAB

∂

∂+

−=−= X

C

D X D X

D D J J




Thermodynamic effect on interdiffusion coefficient

Substitution of Equation 6a in the expressions of interdiffusion coefficient in either a

binary metallic system (Eqn. 8b) or a binary ionic system (Eqns. 10 and 11), andapplication of the Gibbs–Duhem relation (i.e. n1d µ1 + n2d µ2 = 0 at constant p – T

condition) yield, for a system with constant molar volume,

(12)

where D+(i – j) is an appropriate (metallic or ionic) interdiffusion coefficient had the twocomponents mixed ideally. It is expressed according to the forms of Equations 8b, 10 and11, as appropriate, by substituting the self-diffusion coefficient, Di

+, for Di. Thus, for

example, if the interdiffusion is between two equally charged ions, then

(13)

Note that in Equations 12 and 13, component i can be either A or B, since, according tothe Gibbs–Duhem relation and stoichiometric constraint for a binary solution (i.e.

dX A = – dX B) with a constant molar volume, d ln aA/d ln C A = d ln aB/d ln C B. Also recallthat if the molar volume of the material is constant, then d ln C i = d ln X i. It should benoted that, in general, the self-diffusion coefficients are also functions of composition.

Thus, Di

+

values should be for the same composition for which one wishes to compute D( A – B). However, usually we do not have enough data for geologically importantsystems to treat self-diffusion coefficients as function of composition.

If the binary solution has a sub-regular thermodynamic mixing property, i.e. theexcess Gibbs energy of mixing can be expressed according to ∆Gxs = X 1 X 2(W 12 X 2 +W 21 X 1), where W ij represents the Margules parameters (see, for example, Ganguly &Saxena, 1987), then

(14)

For the special case of “Simple Mixture” or “Regular Solution”, W 12 = W 21 = W , so that

. (15)

As an illustration of thermodynamic effect, we show in Figure 2 thethermodynamic factor calculated by Brady & McCallister (1983) at 1150 °C for thequasi-binary Ca–Mg(+Fe) interdiffusion between pigeonite lamellae and sub-calcicdiopside host. The sample is a diopside megacryst (Fe/(Fe + Mg) = 0.14) from Mabukikimberlite, Tanzania. The experimentally determined critical mixing temperature (T c)

between the Ca and Mg (+ Fe) components is 1132 °C, and the critical mixingcomposition is ~ 20 mol% diopside. Also shown in the figure is the thermodynamicfactor (dashed curve) calculated from the mixing energy data of Lindsley et al. (1981) in

RT

X WX

X

i 21

1

2

ln

ln−=

∂∂ γ

( ) ( )[ ].4242ln

ln12212112

21

1

X X W X X W RT

X X

X

i −+−=∂∂ γ

.ln

ln1

ln

ln1)()(

BBAA

AA

∂∂

+

+=

∂∂

+−=−++

+++

i

i

i

i

X D X D X

D D

X B A D B A D

γ γ

,ln

ln1)(

ln

ln)()(

∂∂

+−=

∂∂

−=− ++

i

i

i

i

X ji D

C

a ji D ji D

γ

J. Ganguly278

D+A D

+B

γ 1

γ 1



the diopside–enstatite join at temperatures that have the same relative position withrespect to T c (1500 °C) in this join as the inferred temperature of the natural sample haswith respect to its own T c. It is evident from Figure 2 that the thermodynamic effect onthe diffusion coefficient is very pronounced near the critical mixing temperature (alsosee Christoffersen et al., 1983).

It should be noted from Equation 13, and the equivalent expression for the metallicsystem that follows from Equation 8b, that as X A → 0, D(thermo) = 1 (since γ i =constant). Consequently,

Thus, we arrive at the rather counter-intuitive conclusion that the interdiffusioncoefficient in a binary system approaches the self-diffusion coefficient of the dilute

component (instead of the major component). This conclusion can be shown to be validalso for multi-component solutions by examining the limiting behaviour of the extensionof Equation 13 for the ternary solution (see below, Eqn. 24).

Diffusion, atomic motions and correlation effect

Diffusion takes place via atomic jumps. The correlation effect arises from the non-randomness of the atomic jumps. To illustrate this point, let us consider the common caseof a vacancy mediated diffusion. In this case, an atom moves by interchanging positionwith a vacancy after it arrives (or diffuses) into one of the neighbouring lattice sites. But

.)(lim 0+

→ =− i X D ji Di


Fig. 2. Thermodynamic factor (TF) for the inter-diffusion of Ca–Mg(+Fe) as a function of diopside content of clinopyroxene, as calculated by Brady & McCallister (1983). The dashed line is for the quasi-binary system

Ca–Mg(+Fe), with Fe/(Fe + Mg) = 0.14, corresponding to the composition of a natural diopside megacryst from

Mabuki kimberlite, Tanzania, at 1150 °C. The critical mixing temperature in this join is 1132 °C. The solid line isfor Fe-free system at a temperature that has the same ratio with the T c in this join, which is 1500 °C, as the

chosen temperature of the Fe-bearing sample has with its own T c.



after making the first jump, there is a greater probability for the atom to return to itsoriginal position (because it now finds a vacancy in that position) when it executes thenext jump, than moving into any of the other neighbouring lattice sites that are occupied by atoms. Thus, a certain fraction of the atomic jumps are “wasted”. The correlationfactor, f , accounts for this problem by expressing Di as a product of f i and Di(random).The latter is the value of D that should be obtained under conditions of completelyrandom atomic jumps. By definition, f i ≤ 1.

It can be shown that for one dimensional diffusion, D(random) = ⟨ X 2⟩/2t , where ⟨ X 2⟩is the mean square displacement of the atoms after time t (for random atomic jumps of equal length, we obviously have ⟨ X ⟩ = 0, but ⟨ X 2⟩ ≠ 0). This relation between the meansquare displacement and diffusion coefficient for random atomic motion is often referredto as the Einstein relation, and provides the basis for the determination of D by

molecular dynamics simulation. In this approach, ⟨ X 2

⟩ is determined at several differenttime steps, and the slope of the linear relation between ⟨ X 2⟩ vs. t yields the diffusion

coefficient (e.g . Tirone, 2002).Hermeling & Schmalzried (1984) determined the correlation factors for the

diffusion of Fe2+ and Mg in olivine. So far these constitute the only measurements of f ifor diffusion in rock forming minerals. Their results are illustrated in Figure 3.Diffusivities of the different isotopes of an element would differ from each other becauseof the differences in their masses, which affect their jump frequencies, and thecorrelation coefficient. If D*

i(α) and D*

i(β ) are the tracer diffusion coefficients of two

J. Ganguly280

Fig. 3. Correlation factors of Fe2+ and Mg as a function of X Mg in binary Fe–Mg olivine at 1 bar, 1130 °C, and

log f O2 = –10.67 (modified from Hermeling & Schmalzried, 1984).



indicate the extent of hydrodynamic coupling in the diffusion process, i.e. the extent towhich the flux of a given component is influenced by the concentration gradients of theother independent components. If these off-diagonal terms are significant, then onecould get a positive flux of a component in the direction of increasing concentration,leading to what is known as uphill diffusion. Indeed, because of the effect of cross terms,a component could diffuse even in the direction of its increasing chemical potential .Uphill diffusion has not been documented in any mineralogically important system, buthas been found in several silicate melts (e.g. Chakraborty et al., 1995a, 1995b) that areof interest in the understanding of magmatic processes.

D matrix

Using the principle of matrix multiplication, we can re-write Equation 17 as

, (18a)

or, in matrix notation

(18b)

so that

, (18c)

where J and C are (n – 1) column vectors and D is an (n – 1)×(n – 1) matrix of diffusioncoefficients, which is usually referred to as the D matrix. For n = 2, Equation 17 reducesto the expression of flux in binary diffusion, in which case D11 is a binary interdiffusioncoefficient.

In general, the D matrix is not symmetrical. However, from the principles of irreversible thermodynamics, it can be related to two symmetric (n – 1) × (n – 1)

matrices, L and G, as

D = LG. (19)

The matrix L is the Onsager matrix or the matrix of kinetic, or phenomenological,coefficients, and G is the thermodynamic matrix. (Recall from Eqn. 6 that thechemical diffusion coefficient of a species consists of a product of an L coefficientand a thermodynamic factor.) The L and G matrices are also positive definite, thatis, they have real and positive eigenvalues. Consequently, the D matrix must alsohave real and positive eigenvalues. This requirement provides important constraintson the values of the elements of the D matrix (see Lasaga, 1998, for further discussion). Furthermore, Equation 19 permits one to test the mutual compatibilityof the data on the diffusion kinetic and thermodynamic mixing properties of species

∂

∂

∂

∂=

∂

∂ x xt

CD

C

x∂

∂−=

CDJ

∂∂

∂∂∂∂

⋅

−=

−−−−

−

−

− xC

xC

xC

D D

D D D

D D D

J

J

J

nnnn

n

n

n /

.............

/

/

.......

...............................

..........

..........

....

1

2

1

)1)(1(1)1(

)1(22221

)1(11211

1

2

1

J. Ganguly282



in a solution. One can also extract the unknown value of a diffusion kinetic or thermodynamic mixing property, if other values are well constrained, by insertingguessed values of these parameters into the L and G matrices until the product of Dand G –1 yields a symmetric and positive definite matrix (Chakraborty & Ganguly,1994; Chakraborty, 1994).

In terms of the L matrix, we can write an expression for flux analogous to Equation18b as

. (20)

The symmetry of the L matrix is a consequence of Onsager reciprocity principle(Onsager, 1931a, 1931b). In a system of constant molar volume, which is a goodapproximation for most diffusion process in minerals, the elements of G matrix can becalculated as follows (e.g. Loomis, 1978).

, (21)

where the nth component has been chosen to be the dependent component.The diffusion matrix, as defined above, has the property that it can always be

diagonalised (Toor, 1964; Cullinan, 1965). This enables reduction of multicomponentdiffusion to the mathematical forms of binary diffusion (Toor, 1964; Cullinan, 1965), asfollows. If τ is a diagonal matrix of the eigenvalues of the D matrix, and B is a matrix for

which the columns are composed of the corresponding eigenvectors, then B –1

DB = τ (or B

–1D = τ B

–1). Therefore, on pre-multiplying both sides of Equation 18c by B –1, the

multi-component diffusion equation can be expressed in the following form

, (22)

where C′ = B –1

C. Thus, instead of the coupled diffusion equations of the originalcomponents, we obtain uncoupled or independent diffusion equations of the transformedcomponents, C i′, as

, (23)

where the eigenvalue τ i is the diffusion coefficient of the transformed component C i′.These equations can be solved for C i′( x, t ) using solutions of diffusion equationscharacterised by a single concentration gradient (e.g. Crank, 1983). The solution for C ′( x, t ) can then be converted to that of the real component C ( x, t ) using the relationship between the two variables.

The elements of the D matrix can be calculated from the self-diffusion andthermodynamic mixing property data of the species from the extensions of Equations 8b

and 11 to multi-component systems. These extensions are due to Hartley & Crank (1949)for the metallic system and Lasaga (1979) for the ionic systems. The equation derived

∂′∂

∂∂=

∂′∂

x

C

xt

C ii

i τ

∂

′∂∂∂

=∂

′∂

x xt

CCτ

i

niij

X G

∂−∂

=)( µµ

x

T

∂

∂−=

)( ì/ LJ


(∂µ/T )



by Lasaga (1979), which is appropriate for the mineralogical problems, is as follows if the activity coefficients of the diffusing components (γ i) are constant within the domainof compositional variation

, (24)

where δ ij is the Kronecker delta (δ ij = 1 when i = j, and δ ij = 0, when i ≠ j). Full treatmentto incorporate the effects of variation of γ i can be found in Lasaga (1979).

Because of the paucity of experimental data on diffusion in multicomponentmineralogical systems, the off-diagonal terms of the D matrix are usually neglected.Such approximations, however, are not always justified. Chakraborty & Ganguly (1991,1992) have discussed examples of D matrix in garnets that show significant off-diagonalterms. An example from their study is given below, where the matrix elements are inunits of cm2/s, and Ca was treated as the dependent component.

Mn Mg FeMn 8.38(10 –20) –9.91(10 –23) –4.68(10 –21)Mg –2.78(10 –21) 7.26(10 –21) –8.81(10 –23)Fe –7.16(10 –20) –4.81(10 –23) 1.19(10 –20)

This D matrix was calculated for a fixed garnet composition (Alm0.79Prp0.06Sps0.10Grs0.05)in the Barrovian zone rocks (Dempster, 1985) at the inferred peak metamorphiccondition of 600 °C, 5 kbar, f O2 = graphite–O2 equilibrium (see below for discussionabout the dependence of D on f O2). It is evident that the magnitude of some of the off-diagonal terms are comparable to the on-diagonal ones (compare DFeFe with DFeMn and

DMgMg with DMgMn) so that neglect of these cross effects on diffusion could lead tosignificant errors in the modelling of diffusion modification of compositional zoning inmulti-component garnets.

To illustrate the importance of accounting for the multi-component interactions, Ishow in Figure 4 the diffusion profiles for Fe, Mg, Mn and Ca that would be generated

in a semi-infinite garnet/garnet diffusion couple according to the D matrix given above.Although the D matrix is a function of composition, and hence of position, a constant D

matrix has been assumed for the sake of simplicity. The profiles were calculated usingthe program PROFILER, which is discussed in detail in Glicksman (2000). The initialcomposition of Mn on the two sides of the couple were chosen to be the same, and Cawas treated as the dependent component. The simulation is for 40 Myr, and each divisionon the distance axis of the main figure equals 1 µm. It is interesting to note that Mnshows uphill diffusion and develops a wavy pattern near the interface of the couple dueto the influence of the other diffusing species.

It should be noted that uphill diffusion of a component in a semi-infinite diffusioncouple not only depends on the magnitude of the off-diagonal terms of the D matrix, butalso on the nature of the compositional difference of the components on two sides of the

( **

1

*2

*

*nink

k

k k k

i jii

ijiij D D

D X Z

X Z Z D D D −

−=

∑=

=

δ

J. Ganguly284

( Di

* – Dn

*) D j

*



couple (see, for example, Chakraborty et al ., 1995a; Glicksman, 2000). Because of themulti-component interaction, one or more components in a multi-component diffusioncan have either stationary or moving zero flux planes (ZFP). A ZFP defines a planewhere the flux of a component vanishes. In other words, the component with a ZFP

diffuses on both sides of the plane, but not across the plane. The dynamics of ZFP haverecently been explored in detail by Glicksman & Lupulescu (in press). This property hasinteresting industrial and potential geological applications.

Diffusion in anisotropic crystals: Diffusion tensor

In anisotropic crystals, diffusion properties are, in principle, different along differentdirections. In such medium, Equation 1 holds only for the special case that there isconcentration gradient only along the x direction. If there are concentration gradientsalong the other directions that are orthogonal to x, then the flux along any direction is

linearly related (within the domain of validity of linear irreversible thermodynamics) tothe concentration gradients along all three orthogonal directions. Thus, in the absence of a driving force, the flux of a component along the x direction is given by

. (25)

Similar relation holds for the flux of the component along the other directions. Thus,using the principle of matrix multiplication, as in Equations 17 and 18, we have

J = – D∇C (26)

where J and ∇C are column vectors of the directional fluxes and concentration gradients,respectively, and D is a symmetric matrix of the diffusion coefficients, which is known

z

C D

y

C D

x

C D J xz xy xx x ∂

∂−

∂

∂−

∂

∂−=


Distance

C o n c e n t r a t i o n s

(

)

a t %

–3 –2 –1 0 1 2 3

Mg

Fe

Mn

Ca0

10

20

30

40

50

60

70

(a)

Fig. 4. (a) Calculated diffusion profiles of Fe, Mg, Mn and Ca in a semi-infinite garnet/garnet diffusion coupleusing the D matrix given in the text. The initial concentration of all components was homogeneous in eachgarnet crystal, and that of Mn was the same on both sides of the couple. Multi-component interaction producesthe uphill diffusion and wavy pattern of Mn profile near the interface, which is magnified in (b). The simulationis for 40 Myr. Each division on the distance axis in (a) equals 1 µm.



as the diffusion tensor. It is, however, always possible to find three orthogonaldirections ξ1, ξ2, ξ3 in an anisotropic medium such that the flux along any of thesedirections depends only on the concentration gradient along the specific direction, i.e.

J ξi = – Dξi(∂C /∂ξi). These directions and the corresponding diffusion coefficients areknown as the principal diffusion axes and principal diffusion coefficients, respectively.Diffusion along any arbitrary direction κ , which makes angles θ 1, θ 2, θ 3 with the principal diffusion axes (1, 2 and 3), is given by

, (27)

where Dκ = Dξ 1 cos2θ ξ 1 + Dξ 2 cos2θ ξ 2 + Dξ 3 cos2θ ξ 3, and ∂C /∂κ is the concentrationgradient along the direction κ .

The direction of a crystallographic symmetry axis coincides with that of a principaldiffusion axis. Thus, the a, b and c crystallographic directions in cubic, tetragonal,orthorhombic and hexagonal systems constitute the directions of principal diffusionaxes. For monoclinic system, the b axial direction constitutes the direction of one of the principal diffusion axes. The other two, which must lie in the a – c plane, can bedetermined by three measurements of diffusion coefficients in that plane. For triclinicsystem, one needs measurements of diffusion coefficients in six different directions todetermine the directions of the three principal diffusion axes (see Nye, 1957 for further discussions).

Anisotropic diffusion was measured in several non-cubic minerals such as olivine

(orthorhombic: Buening & Buseck, 1973; Misener, 1974; Jurewicz & Watson, 1988;Chakraborty et al., 1994), orthopyroxene (orthorhombic: Schwandt et al., 1988),clinopyroxene (Sneeringer et al., 1984; Tirone, 2002) and feldspar (triclinic: e.g.

Christoffersen et al., 1983). Because of the relaxation of structure with increasingtemperature, diffusion anisotropy should be expected to decrease with increasingtemperature. The anisotropic diffusion data for olivine were summarised and discussed by Morioka & Nagasawa (1991). The tracer diffusion coefficients of Ni, Co, Ca, as wellas Fe–Mg interdiffusion coefficient, were found to be fastest parallel to the c axis, andslowest parallel to the b axis in olivine. The observed anisotropy is consistent with the

arrangement of divalent cation sites and the energetics of defect formation in the olivinestructure in that the M 1 sites form a closely spaced chain parallel to the c axis, and theenergy of formation of cation vacancies in the M 1 site is significantly less than that inthe M 2 site (e.g. Ottonello, 1997). Niemeier et al. (1996) observed by high temperatureMössbauer spectroscopy that cation diffusion in olivine takes place predominantly via

M 1– M 1 jumps along the c direction. In contrast to the above results, Jurewicz & Watson(1988) found that at f O2 < 10 –8 bar, Mn and Fe show highest diffusion rates parallel tothe a axis. These authors tried to explain the unexpected anisotropic behaviour in termsof different diffusion mechanisms along a and b/c crystallographic axes. For alkalifeldspars, Na and K self-diffusion and Na–K interdiffusion were found to be faster

within (010) plane than normal to it. As discussed by Christoffersen et al. (1983), theobserved diffusion anisotropy is consistent with the feldspar structure in that the alkalisites are much closer to each other within the (010) plane than normal to it. The above

κ κ κ ∂

∂−=

C D J

J. Ganguly286



examples suggest that consideration of the packing density of the host structural sitescould provide useful, although not unfailing, guidelines about the expected diffusionanisotropy in minerals.

Factors affecting diffusion coefficient

Diffusion can take place through a number of different mechanisms such as throughexchange of position between atoms and lattice vacancies (vacancy mechanism),migration of atoms through interstitial sites (interstitial mechanism) etc. These have beendiscussed in detail in many standard books on diffusion in solids (e.g. Shewmon, 1963;Bokshtein et al., 1985; Borg & Dienes, 1988). Vacancy mechanism is by far the mostimportant of all diffusion mechanisms. All cases of substitutional diffusion seem tooperate through a vacancy mechanism (Borg & Dienes, 1988). Thus, any physico-

chemical factor that affects the vacancy concentrations in a crystal significantly also hasa significant effect on its diffusion properties.

Since diffusion involves climbing an energy barrier by atoms between two stablestates, temperature has the strongest effect on diffusion coefficient as it provides theenergy (k BT ) to elevate the atoms over the energy barrier. The other factors affecting thevolume diffusion coefficient of a species are pressure, volatile species ( f O2 and thefugacities of the “water” related species), dislocations, bulk composition of the phase,

and radioactive damage. The last topic is not of any interest in the context of the presentchapter, and, thus, will not be discussed any further. The interested readers are referredto Borg & Dienes (1988) for a general discussion about the theory, and to Cherniak (1993) for discussions in the context of geological systems.

Effect of temperature and pressure

The temperature and pressure dependencies of a diffusion coefficient are given by

(28)

, (29)

where Q( p) and ∆V +(T ) are known as the activation energy (at pressure p) and activationvolume (at temperature T ) of diffusion, respectively. Note that the above equations areformally similar to those governing the temperature and pressure dependencies of theequilibrium constant , K . In general, any kinetic constant or coefficient has the sameformal dependencies on temperature and pressure. Assuming that Q( p) is independent of temperature, integration of Equation 28 yields

, (30)

where D0 is D(T = ∞). Assuming that ∆V + is independent of pressure, we haveQ( p) = Q( p′) + ∆V +( p – p′).

The activated state is an energetically higher transient state that a system passes

RT

pQ

e D D

)(

0

−=

RT

T V

p

D )(ln +∆−=

∂

∂

R

pQ

T

D )(

)/1(

ln−=

∂

∂




through in any kinetic process. For diffusion in solids, the activation energy is the sumof the potential energy (or enthalpy) barrier, ∆ H +m, that an atom must climb over in order to move from one lattice position to the next (i.e. enthalpy barrier to migration), and theenthalpy of formation of vacancies, ∆ H +v, if the diffusion process is controlled by intrinsicvacancies. The activation volume has analogous definition, i.e. ∆V

+ = ∆V +m + ∆V

+v, where

∆V +m is the transient volume change of the crystal during the process of atomic migration

and ∆V +v is the volume change associated with the formation of intrinsic vacancies when

the diffusion process is in an intrinsic domain. The pre-exponential factor, D0, is proportional to e

∆S +/ R, where ∆S+ is the activation entropy of diffusion, and hascomponents associated with both migration and vacancy formation, as in the other activation terms.

Available experimental data on diffusion in minerals show that at pressures up to

several tens of kilobars, ∆V +

> 0, so that increasing pressure within this range will reducethe diffusion coefficient. Review of the experimental data on divalent cation diffusion inolivine, garnet and spinel (Chakraborty & Ganguly, 1992; Chakraborty et al., 1994;Chakraborty & Rubie, 1996; Misener, 1974; Liermann & Ganguly, 2002) show ∆V + to be less than 10 cm3/mol, and probably around half as much. Thus, on the basis of theavailable experimental data, one would expect a relatively small pressure effect on D for divalent cation diffusion in minerals. For example, at 1000 K, the expected effect of achange of 5 kbar pressure would be to reduce log D by no more than 0.26, and probably by half as much. In principle, ∆V

+ is a function of pressure. However, withinexperimental error, no pressure dependence of ∆V + could be detected in garnet up to ~

85 kbar (Chakraborty & Rubie, 1996). On the other hand, diffusion kinetics of asubstance at a given temperature seems to bear a relation with the degree of proximityof the temperature to the melting temperature, T m (e.g. Borg & Dienes, 1988). Meltingtemperature maximum has been found in olivine, and probably all minerals have thesame property. Thus, after the pressure exceeds that of T m(max), increasing pressure at

a fixed temperature may enhance the diffusion kinetics in a mineral since it would bring

the mineral progressively closer to T m.

Effect of f O2

In solids containing elements that can have variable oxidation states, such as iron, f O2 isexpected to influence the diffusion property by changing the vacancy concentrationthrough the change of the oxidation state of the element. For example, if thehomogeneous redox equilibrium of iron in a solid is governed by the reaction

3Fe2+(l ) + ½ O2( g ) ↔ 2Fe3+(l ) + FeO(surface) + V Fe , (a)

where l , g and V Fe stand for lattice site, gas and vacancy in Fe lattice site, respectively,then it is easy to show that V Fe, and hence the diffusion coefficient, would varyapproximately as ( f O2)

1/6. This relation follows by combining the expression of equilibrium constant of the above reaction with the relation 2(Fe3+) = V Fe (which follows

from the requirement of charge conservation if the vacancies are neutral), and assumingthat X Fe2+ is not significantly altered by the oxidation. It is further assumed, although rarely

J. Ganguly288



stated explicitly, that the activities of both Fe2+ and Fe3+ are proportional to their molefractions (in the spirit of the laws of dilute solutions). Experimental data on the diffusioncoefficients in ferromagnesium olivine (Buening & Buseck, 1973; Nakamura &Schmalzried, 1983) show that D varies approximately as ( f O2)

1/6. It should be noted that f O2 affects the diffusion coefficient not just of Fe, but also that of other cations ( i.e. Fedoes not have any exclusive right to utilise the vacancies created by its oxidation).

The dominant diffusion mechanism could change as a function of Fe concentrationand f O2. For example, the experimental data of Chakraborty et al. (1994) suggest thatthe above defect forming reaction is of primary importance in the diffusion behaviour of olivine only when the Fe content exceeds a threshold value of ~ 150 ppm. Dieckmann &Schmalzried (1975, 1977) showed that at a fixed temperature, log D*

Fe in Fe3O4 vs. log f O2

has a minimum, which shifts to higher f O2 with increasing temperature. The

experimental data can be matched very well by a theoretical relation between D*Fe and f O2

that they derived by invoking that Fe diffuses through both vacancy and interstitialmechanisms. The vacancy diffusion plays the dominant role at f O2 above the minimumwhereas the interstitial diffusion plays the dominant role at lower f O2.

Effect of hydrous condition

Diffusion in the presence of water has been investigated by a number of workers. Thespecies affecting diffusion may be H+, (OH) – or H2O. We will simply refer to these as“water”. A summary of the available experimental data on the effect of “water” on

diffusion in mineralogically important systems may be found in Cherniak (1993). Onreviewing these data, she concluded that “water” may not play a significant role in theinterdiffusion process that involves only a simple exchange between cations of the samecharge. On the other hand, “water” (the actual species could be proton) has been shownto have a significant enhancement effect on the interdiffusion or ordering process infeldspars that involves Al–Si exchange (e.g. Yund & Snow, 1989; Goldsmith, 1991;Graham & Elphick, 1991). Also, experimental study on the effect of “water” at 300 barsand f O2 defined by the Ni–NiO buffer on the Fe2+ –Mg interdiffusion in olivine alsoshows an increase of D(Fe2+ –Mg) by a factor of ~ 10 relative to the dry diffusion data of Chakraborty (1997) (Kohlstedt, pers. commun.). The experimental data for olivine seem

contrary to the conclusion of Cherniak (1993). The effect of “water” on diffusionkinetics needs to be carefully investigated so the experimental data can be applied tonatural systems in a meaningful way. Our understanding of the problem at the presentstage is sketchy at best.

Effect of dislocations

Diffusion along dislocations, commonly referred to as pipe diffusion, is much faster thandiffusion through crystal lattice. In the presence of distributed dislocations, the apparentvolumetric diffusion would reflect a combination of the diffusion through normal crystal

lattice and that through the dislocations. Yund et al. (1989) investigated the effect of dislocations on the apparent volumetric diffusion in albite–adularia diffusion couples.Comparing the result of experiments at hydrostatic condition with that in which the




diffusion couple was strained at a rate of 10 –6 s –1 during the process of diffusion at thesame p – T condition (1.5 kbar, 1000 °C), they concluded that distributed dislocations areunlikely to have any significant effect on the bulk volumetric diffusion in alkali feldsparsat all metamorphic conditions.

The above result, however, does not guarantee that dislocations have, in general,negligible effect on the bulk volumetric diffusion in minerals during metamorphism,especially when these are in motion. Nonetheless, the effect of distributed dislocationson the bulk diffusion process in minerals at the strain rate of metamorphic rocks may not be a matter of major concern. The effect of localised dislocations should be apparent inthe extended nature of the diffusion profile in a mineral as compared to those in other parts of the same mineral in a rock. These anomalous profiles should obviously beavoided in modelling compositional profiles that are aimed at retrieving time scales of

metamorphic processes (see below).

Change of diffusion mechanism: Applicability of laboratory data to

geological problems

A point of critical importance in the application of laboratory experimental data tonatural systems is the possible change of diffusion mechanism in moving from thelaboratory to the natural conditions. There are two important issues in this respect,namely, (a) the change of mechanism as a function of temperature and other physicalvariables such as f O2 and pressure, and (b) change of mechanism due to the “purity” of

mineral composition that are sometimes used in the laboratory experiments. These problems are discussed below.

The lattice vacancies originate for two different reasons. First, there are always anequilibrium number of lattice vacancies in a crystal, which varies as a function of temperature. These are known as intrinsic lattice vacancies, and result from the effect of entropy of mixing between the vacancies and ions in lowering the Gibbs energy (G) of the system. The Gibbs energy of a solution must decrease in the terminal compositionalsegments of any solution (for a proof of this statement, see, for example, Ganguly &Saxena, 1987, p. 37). Inasmuch as the vacancy can also be treated as a component in thesolid solution, G of a crystal must also decrease as function of X v (atomic fraction of

vacancies) in the terminal region of X v = 0. Second, vacancies are created by thereplacement of an ion in a lattice site by an ion of different charge (e.g. replacement of Na+ by Cd2+), or by the oxidation of an ion in a lattice site (e.g. oxidation of Fe2+ to Fe3+).These are known as extrinsic vacancies since their formation results from interactionwith an external source.

As discussed by Chakraborty (1997), one should distinguish between the extrinsicvacancies created by an impurity substitution, in which case there is no defect formationenergy, and those created by the redox reaction of a transition metal element. In the latter case, there is a defect formation energy, which equals the enthalpy change of the redox

reaction, and the defect concentration changes as a function of temperature. Chakraborty(1997) called the first case as pure extrinsic diffusion (PED) and the second case astransition metal extrinsic diffusion (TaMED).

J. Ganguly290



The atomic fraction of the intrinsic lattice vacancies have an exponential dependence

on temperature according to X v ∝ exp(– ∆ H 0v/ RT ), where ∆ H 0v is the enthalpy of formation per mole of the particular type of vacancy†. Consequently, at high temperature, X v(intrinsic) >>

X v(pure extrinsic), so that diffusion takes place essentially through the intrinsic vacancies.On the other hand, at low temperature, X v(intrinsic) << X v(pure extrinsic) so that diffusiontakes place dominantly through the extrinsic defects created by heterovalent substitution.The temperature for transition from a dominantly intrinsic to a dominantly extrinsicmechanism depends on the system. An example of this transition for the self-diffusion of Na+

in Cd2+ doped NaCl is shown in Figure 5. As discussed above, the activation energy in theintrinsic domain is the sum of the energies of defect formation and atomic migration, whereas


† Note that if the vacancies are of Schottky type, that is created in pairs of cation (c) and anion (a) vacancies,and H 0v is the enthalpy of formation of the Schottky pair, then from the expression of equilibrium constantof the vacancy forming reaction, we have X v(c) X v(a) ∝ exp (– ∆ H 0v/ RT ), or X v(c) ∝ exp (– ∆ H 0v/2 RT ).

Fig. 5. Change of self-diffusion mechanism of Na+ in Cd2+ doped NaCl as a function of temperature. At high

temperature, the diffusion is controlled dominantly by the equilibrium or intrinsic point defects whereas at lowtemperature, it is controlled by the extrinsic point defects created by the substitution of Cd2+ for Na+ according to

2 Na+ → Cd2+ + V (Na+) , where V (Na+) stands for a vacancy in the sodium site (modified from Mapother et al., 1950).



that in the pure extrinsic domain is only due to atomic migration. Thus, the difference between these two activation energies yields the energy of (intrinsic) defect formation.

Similar to the case of intrinsic diffusion, the defect concentration for TaMEDdiffusion also vary as a function of temperature as exp(– ∆ H r

0/ RT ), where – ∆ H r 0 is the

enthalpy change of the appropriate redox reaction. The latter is, however, much less thanthe enthalpy of formation of intrinsic defects. Thus, the log D vs. 1/T slope in the TaMEDdomain is less than that in the intrinsic domain.

A qualitatively similar behaviour to the extrinsic–intrinsic transition for volumediffusion is also shown by the transition from grain boundary to volume diffusion in polycrystalline aggregates. At low temperature, the pathways offered by the grain boundariesdominate those due to intrinsic defects, but the situation reverses at high temperature.

In mineralogical systems, extrinsic–intrinsic transition was first suggested for cation

diffusion in olivine at ~ 1100 °C by Buening & Buseck (1973). These data served as anillustration of the change of volume diffusion mechanism in many mineralogical publications. Also, these diffusion data have been used extensively to model thermalhistories of terrestrial and planetary samples. However, Buening & Buseck (1973) used adiffusion couple consisting of Mg-rich olivine single crystal and powdered syntheticfayalite. They noted that the observed change of mechanism could have also been due to achange of grain boundary to volume diffusion – a point that seems to have been ignored infavour of an interpretation of extrinsic–intrinsic transition. Recently Chakraborty et al.

(1994), Meissner et al. (1998) and Chakraborty (1997) determined both tracer diffusion of Mg ( D*

Mg) and interdiffusion of Fe–Mg ( D(Fe–Mg)) in olivine single crystals over the

temperature range of 980–1300 °C. Their data (Fig. 6) show no change of volume diffusionmechanism in olivine within this temperature range. Chakraborty & Ganguly (1991) andGanguly et al. (1998) also showed that there is no change of self- or tracer diffusionmechanism of Mg in garnet within the temperature range of ~ 750–1475 °C (Fig. 6).

On the basis of the evidence presented above, it seems reasonable to conclude thatvolume diffusion in olivine and garnet does not show any change of mechanism within thetemperature range of geological interest. From comparison of the intrinsic defect formationenergy (~ 800 kJ/mol) and the experimental activation energy of diffusion in olivine(~ 275 kJ/mol), Chakraborty et al. (1994) pointed out that the latter cannot represent a

combination of defect formation and migration energies. Thus, the observed diffusion inolivine is in the extrinsic domain. Chakraborty et al. (1994) argued, from consideration of the intrinsic vacancy content that follows from conservative estimate of defect formationenergy, that intrinsic diffusion is unlikely to be observed in silicates at temperatures below1300 °C. Wuensch (1982) came to similar conclusion for refractory oxides.

The above conclusion about the extrinsic nature of diffusion in silicates andrefractory oxides at the laboratory conditions is very important from the point of view of extrapolation of the experimental data to temperatures of geological processes of interestsince the experimental data are usually collected at temperatures that are higher thanthose at which these processes take place in nature. The Arrhenian relation could be

extrapolated linearly to lower temperatures with, of course, due consideration for thestatistical uncertainties associated with the regression of the experimental data. On theother hand, once the diffusion is in the extrinsic domain, it becomes vulnerable to

J. Ganguly292



substitutions of ions that have different charge than the host species. Experimentalstudies on iron-bearing garnets, however, did not show any significant dependence of thediffusion coefficient on garnets from different sources (Chakraborty & Ganguly, 1992;Ganguly et al., 1998). This is fortunate, and is probably due to the fact, as discussed byChakraborty et al. (1994), that the equilibrium vacancies controlled by the ferrous–ferricequilibrium in an iron-bearing mineral greatly dominates its vacancy content.

Chakraborty et al. (1994) found that the activation energy for Mg self-diffusion innominally pure synthetic forsterite (Fo100) at 1000–1300 °C is 400 (±60) kJ/mol, whichis in contrast to that of 275 (±25) kJ/mol in San Carlos olivine (Fo92) within the sametemperature range. Both sets of experiments were carried out by them following thesame experimental technique. Also the D(Mg) in the San Carlos olivine had a muchstronger f O2 dependence than that in Fo100 (the latter was essentially independent of f O2).

Chakraborty et al. (1994), thus suggested that the diffusion mechanism in the nominally pure forsterite is different from that in olivine, which contains significant amount of Fe2+.When the FeO content falls below a critical value, the vacancies created by the Fe2+ –Fe3+

equilibria have relatively minor role in the diffusion process.Liermann & Ganguly (2002) determined the Fe and Mg self-diffusion coefficients

in spinel, (Fe2+,Mg)Al2O4, from modelling the Fe–Mg interdiffusion data obtained fromdiffusion-couple experiments at 20 kbar, 950–1325 °C. The retrieved activation energyis much lower than that determined by Sheng et al. (1992) for Mg tracer diffusion inspinel at 1 bar, 1261–1553 °C (202±8 kJ/mol vs. 343±8 kJ/mol). The latter workers useda tracer isotope of Mg on an essentially pure end member natural MgAl2O4 spinel (they

did not report any FeO in their microprobe analysis of the sample, for which the wt%MgO, Al2O3, SiO2 and CaO add up to 100.88). Thus, it seems very likely, as discussed by Liermann & Ganguly (2002), that the high activation energy of Mg self-diffusion inthe studies of Sheng et al. (1992) relative to that in their work is due to the extremelysmall, probably below the detection limit in microprobe analysis, amount of FeO contentin the sample used by Sheng et al. (1992). It seems highly unlikely that the change of extrinsic–intrinsic transition of diffusion mechanism is responsible for the observeddifference in the activation energies in the two sets of experiments. In summary, onemust exercise great caution in the application of diffusion data obtained from pure

crystals to natural samples that contain iron since there could be major difference between the defect formation energies in the two types of materials.

Some modelling simplifications for geological problems

Complex diffusion processes in geological and planetary systems are not often amenableto analytical treatment. These problems have to be dealt with numerically, andconsiderable progress has indeed been made along these directions over the past decade(e.g. Florence & Spear, 1993; Okudaira, 1996; Carlson, 2002; Tirone, 2002). However,some simplifications may be made to the problems of diffusion in natural processes,which in many cases make it possible to treat the problems analytically to gain useful

insights about the behaviour of the system without requiring extensive computations.Some of these simplifications are discussed below.

J. Ganguly294



Time dependence of diffusion coefficient

Diffusion during geological and planetary processes usually takes place over a range of

temperature, which changes as a function of time, t . Thus, the diffusion coefficient becomes a function of time. Problem with the time dependent diffusion coefficient can be handled in a simple way as follows.

Let us define a new variable Γ as d Γ = D(t ) dt so that the diffusion equation(Equation 3b) transforms to

..

(31)

This equation may be viewed as a diffusion equation in which C is a function of a new

variable Γ, and D = 1 (note, however, that Γ does not have the dimension of time).Solutions of the diffusion equation under isothermal condition can often be expressed inthe form C ( x, t ) = f [ x/( Dt )½]. One such solution is given in a later section (Eqn. 41).Solution for Equation 31 is the same as that for the standard diffusion Equation 3b, with

Dt replaced by Γ. Now from the definition of Γ, we also have the relation

. (32)

This relation provides an important constraint on the thermal history of the sample in thatthe integral of D(t ) dt over the postulated T – t path must equal the value of Γ derived frommodelling the compositional zoning. Examples of the application of this concept togeological problems are discussed in the section on tectono-metamorphic processes.

Let us consider a case in which the observed compositional zoning had developedduring cooling, and suppose that the system had cooled according to an “asymptotic”relation

, (33)

where T 0 is the initial temperature at the onset of cooling, and η is a cooling time constant

with the dimension of K

–1

t

–1

. The Arrhenian relation of diffusion coefficient(Eqn. 30) then transforms to

(34)

or

where η′ = Qη/ R, and D(T 0) is the diffusion coefficient at T 0. In general, the timedependence of D can be expressed as D(t ) = D(T 0) f (t ).

Substituting Equation 34 in 32, we obtain (Ganguly et al., 1994)

. (35)[ 1)(

)( 0 −′

−=≡Γ ′′−′

∫ t t

o

eT D

dt t D η

η

,)()( 0t eT Dt D η′−=

,

1

000

t T R

Q

RT

Q

e De D Dη

+−−

==

t T T

η+=0

11

∫′

=Γt

dt t D0

)(

2

2

x

C C

∂

∂=

Γ∂

∂


[e ].



Since the time scale of geological processes is very large (at least for those for whichcooling rates are of any interest), the above equation simplifies to

. (36)

Using an exponential cooling model, i.e. T = T 0e – αt , Kaiser & Wasserburg (1983)

obtained the following expression for the integral quantity:

(37)

Equations 36 and 37 were used to retrieve the cooling rates of meteorites from the valuesof Γ obtained from modelling the observed compositional zoning in minerals accordingto the appropriate solutions of the diffusion equation (Kaiser & Wasserburg, 1983;Ganguly et al., 1994).

Characteristic diffusion coefficient

Another useful simplifying concept in the treatment of natural processes in which D

changes as a function of time due to the change of temperature is that of “characteristictemperature”, T ch, and the related diffusion coefficient D(T ch). In a non-isothermal process, it is always possible to find a temperature, T ch, such that

(38)

Chakraborty & Ganguly (1992) explored the characteristic temperature that satisfies theabove relation in T – t cycles of metamorphic rocks, which are characterised by a singlethermal peak. They found that T ch ≈ 0.97T peak , where T peak is the peak metamorphictemperature in K.

Concept of effective binary diffusion in multi-component systems

Although the phenomenological theory and mathematical treatment of multi-component

diffusion is well developed in the linear domain, one can simplify the mathematicalanalysis of multi-component diffusion in some cases by using the concept of effective binary diffusion coefficient (EBDC), as if there are only two components, the diffusingsolute and a solvent matrix. By applying chain rule to the expression of flux in a multi-component system in the linear domain (Eqn. 17), we obtain

(39)

or

, (40) x

C D J ∂

∂−= 111 )EB(

x

C

C

C D

C

C D

C

C D D J n

n ∂∂

∂∂

++∂∂

+∂∂

+−= −−

1

1

1)1(1

1

313

1

212111 ...

.)()(0

Γ≡=∆ ∫′t

ch dt t Dt T D

Q

RT T Ddt t D

t

α

00

0

)()(∫

′

=≡Γ

ηη Q RT DT Ddt t D

t

)()()( 0

0

0 =′

=≡Γ ∫′

J. Ganguly296



where D1(EB) is the effective binary diffusion coefficient (EBDC) of component 1, andequals the quantity within the square brackets in Equation 39. It is important to note that,unlike true binary diffusion, Di(EB) for each component is different.

Cooper (1968) showed that Di(EB) must be a single valued function of composition, and hence independent of the spatial concentration gradient, in order thatEquation 40 has the property of Fickian diffusion, that is, flux ∝ force. Specifically, theconcept of effective binary diffusion holds for diffusion in a semi-infinite diffusioncouple in a multi-component system that does not show any inflection of the diffusion profile. Chakraborty & Ganguly (1992) discussed applications of this approach inmodelling diffusion profiles in multi-component diffusion couple experiments. Anexample of application to a natural diffusion couple is discussed below.

Time scales of tectono-metamorphic processes in collisional

environments: Records in garnet zoning

Garnet is the single most important mineral in the study of p – T – t history of metamorphicrocks. It participates in a large number of cation exchange and discontinuous reactionsthat are used to calculate the p – T conditions of rocks from the compositions of thecoexisting mineral phases. It is amenable to geochronological studies using a number of decay systems, and often shows compositional zoning that preserve records of itstectono-metamorphic and exhumation history. In addition, since garnet is isotropic,

diffusion in garnet has no directional dependence – a property that offers a practicaladvantage in the modelling of garnet compositional zoning. I will discuss here severaltypes of compositional zoning in garnet in metamorphic rocks from collisionalenvironments, and retrieval of the time scales of the attendant metamorphic andexhumation processes from modelling of the observed compositional profiles.

Compositional zoning in a natural garnet–garnet diffusion couple

Overgrowth of a mineral on itself is a well-documented petrographic feature in manyterrestrial and planetary samples. When the overgrowth and core segment havesignificantly different compositions, as in polymetamorphic rocks, the composite sampleforms a natural diffusion couple with continuous concentration profiles of componentsacross the interface as a result of diffusion driven by the initial compositional contrasts between the core and overgrowth. The extent of these diffusion profiles depends ontemperature and the time scale over which diffusion was effective. As an illustration of the retrieval of the time scale of a geological process from modelling compositionalzoning of this type, I summarise below the analysis of a natural garnet–garnet diffusioncouple by Ganguly et al. (1996a).

Figure 7a shows a backscattered electron image of a composite garnet collectedfrom the biotite grade rock from eastern Vermont. The couple consists of a

grossular–spessartine garnet that had formed during regional metamorphism on analmandine core, which had crystallised during an earlier period of contactmetamorphism at 411±5 Ma. The regional metamorphism took place during the Acadian




orogeny, which is believed to be a short-lived tectonic event that involved the collision

between two plates and the closing of an ocean basin (Naylor, 1971). The diffusioninduced compositional zoning between the two segments was too narrow to be clearlyresolved by electron microprobe analyses because of convolution or spatial averagingeffect (Ganguly et al., 1988). The compositional zoning was, thus, determined by ananalytical transmission electron microscope (ATEM), which had negligible convolutioneffect because of the very small size of the excited analytical volume resulting from thesmall beam size and thinness of the sample to electron transparency. The results areshown in Figure 7b. Mg profile is not shown since the X Mg is between 0.001 (overgrowth)and 0.06 (core).

Since in this problem we are dealing with a semi-infinite diffusion couple, andthere is no inflection in the diffusion profile, the diffusion problem may be treated interms of an effective binary diffusion coefficient, as discussed above. Assuming that the

J. Ganguly298

Fig. 7. (a) Backscattered electron (BSE) image of the overgrowth of spessartine–grossular garnet on an

almandine core during the Acadian orogeny, eastern Vermont, USA; (b) compositional profiles of the divalentcations across the core–overgrowth interface, as determined in an analytical transmission electron microscope

(ATEM); (c) fits to the measured ATEM profiles in (b) according to the solution of Equation 41. The fits yield

a value of ∫ D(t )dt = 7.5 × 10 –12 cm2 (modified from Ganguly et al., 1994).



EBDC of a component is independent of distance within the diffusion zone, we seek solution of the diffusion equation (Eqn. 3b) for the conditions that the diffusion coupleis semi-infinite so that the initial concentrations are preserved at sufficiently largedistances from the interface, which is located at x = 0, and that there is no initialconcentration gradient on either side of the interface. For an isothermal diffusion process, the solution is (Crank, 1983; Equation 2.14)

(41)

where ∆C0 represents the initial difference between the concentrations of thecomponents on the two sides of the couple, and C i(0) is the lower of the two initial values

of C i. If diffusion had taken place under condition of variable temperature, then, asdiscussed above, Dt in the above equation is to be replaced by Γ (Eqn. 32). UsingEquation 41, a single value of Γ = 7.5×10 –12 cm2 was found to match well both Fe andCa diffusion profiles (Fig. 7c). No attempt was made to fit the Mn profile because of theirregularity in the measured data points.

On the basis of the Fe–Mn fractionation data between the overgrowth garnet andilmenites, which are present as inclusions within this garnet, the peak metamorphictemperature for the biotite grade regional metamorphism was estimated to be 353±15 °C(the thermometric formulation is due to Pownceby et al., 1991, as corrected in Gangulyet al., 1996a). Using now the concept of characteristic temperature (Eqn. 38) and the

above value of Γ, Ganguly et al. (1996a) obtained the following relation for the timescale of the metamorphic process:

, (42)

where the term in the denominator represents the effective binary diffusion coefficientof either Ca or Fe at the characteristic temperature, which is ~ 0.97 × T peak .

The calculation of EBDC of Ca and Mn requires data for the self-diffusion of Fe,Mg, Mn and Ca. Chakraborty & Ganguly (1992) determined the self-diffusion of the first

three elements using diffusion couple made from natural almandine and spessartinecrystals, but well constrained data for the diffusion of Ca was not available. However,one can simultaneously solve for both ∆t and DCa by calculating EBDCs of both Ca andFe for guessed values of DCa according to Equation 24, and satisfying the condition that∆t calculated from the DCa(EB) and DFe(EB) according to Equation 42 must be the same. This procedure yields ∆t = 47 Myr and DCa = 9.7 × 10 –27 cm2/s at the inferredT ch = 343 °C. Ganguly et al. (1996a) discussed the potential uncertainties in the abovecalculation of time scale, and suggested ∆t ≈ 40–50 Ma as the probable time scale of biotite grade regional metamorphism reflected by the diffusion zoning across thecore–overgrowth interface of garnet.

)(

cm105.7

ch)EB(

212

T Dt

i

−×=∆

−

∆+=

t D

xerf

C C xt C

i

ii

)EB(

0

21

2)0(),(




Reaction-diffusion zoning in garnet: Pan-African tectono-metamorphic event

Ganguly et al. (2001) have carried out modelling of compositional zoning in a garnet

from granulite facies rocks in Søstrene Island, which is located in Prydz Bay, Antarctica.The garnet (Fig. 8) shows reaction textures corresponding to two metamorphic episodes,M1 and M2. The outer reaction texture formed during M1 by the breakdown of garnetaccording to Grt + Qtz → Opx + Plag at ~ 1000 Ma, while the fracture cleavage withinthe garnet and the included fine-grained symplectites, which formed by the reaction Grt→ Opx + Plag + Spl, developed during M2 at ~ 500 Ma. The latter is believed to beassociated with a regional Pan-African tectono-metamorphic event that has beeninterpreted to represent a continent–continent collision, followed by extensional collapse(Fitzsimons, 1996, 2000).

The garnet shows Fe–Mg zoning parallel and normal to the fracture cleavage. The

latter developed during M2 by a combined process of reaction and diffusion. The zoning parallel to fracture cleavage developed during both M1 and M2. The zoning normal tothe fracture cleavage (Fig. 9) is a consequence of the fact that the composition of garnetin equilibrium with the symplectitic orthopyroxenes at the p – T condition of M2 isdifferent from its initial composition, which is preserved in the core, and that the durationof M2 was too short to homogenise the garnet by volume diffusion. Using the Fe–Mgdistribution coefficient between the garnet rim and adjacent orthopyroxenes and thethermometric formulation of Ganguly et al. (1996b) for Fe–Mg exchange betweengarnet and orthopyroxene yield T ≈ 730±20 °C at p = 6 kbar. This temperature estimate

is in good agreement with that inferred by Thost et al. (1991) and Hensen et al. (1995)as the peak temperature of M2 at the same pressure. This agreement suggests that thecompositional zoning in garnet developed and froze before the rock experiencedsufficient cooling following peak M2, so that the reaction diffusion process may beapproximated by an isothermal process.

With the above framework, and assuming that the D(Fe–Mg) is not significantlyaffected by compositional change within the diffusion zone, the appropriate diffusionequation to be solved is Equation 7b, where v is the velocity of the garnet–matrix interface,which is set at x = 0, towards a fixed marker point at x > 0. The initial and boundaryconditions are C = C 0 at x > 0, t = 0 and C = C r at x = 0, t > 0. The solution of the diffusion

equation can then be easily obtained as a special case of that derived by Carslaw & Jaeger (1959, p. 388, Equation 7) for heat conduction in a moving body. The solution is

+ (43)

Ganguly et al. (2001) assumed that the breakdown of garnet within the fracturecleavage proceeded symmetrically in both directions so that v = RZ/t , where RZ is theobserved half-width of the reaction zone. Substitution of this relation in Equation 43eliminates one variable. C ( x, t ) can then be solved in terms of time (t ) if the value of

D(Fe–Mg) is known. The latter was calculated from the self-diffusion data of Fe and Mgin garnet (Ganguly et al., 1998) at the median composition of the diffusion zoneaccording to Equation 10.

.2

exp

−

−

Dt

vt xerfc

D

vx

2)(

2

1),( 0r 0

+−+=

Dt

vt xerfcC C C t xC

J. Ganguly300



Modelling of the measured compositional zoning in garnet was carried outaccording to Equation 43 by linking it with a non-linear optimisation program (see

Ganguly et al., 2001 for details). The initial composition, C 0, and time, t , were thefloating variables. The optimisation program finds values of the floating variables thatlead to the best match between the calculated and the observed zoning data. This procedure yields t ≈ 5–16 Myr for the duration of peak M2 at the inferred temperature of 750–710 °C. The model fit to the measured compositional profile in garnet is illustrated by a solid line in Figure 9. From geochronological constraints in an adjacent area inPrydz Bay, Fitzsimons (pers. commun.) suggested 17±13 Myr for the duration of peak Pan-African metamorphism. Comparing the time scale inferred from modelling thecompositional zoning of garnet with the geochronological constraints, Ganguly et al.

(2001) suggested that the duration of peak M2 during the Pan-African event was

probably not significantly in excess of 16 Myr.

Cooling and exhumation of metamorphic rocks

Diffusion in garnets becomes sufficiently rapid at the temperature of sillimanite grade tosmoothen the compositional gradients that developed during the growth of a few mmsize garnets at the lower grade conditions (Chakraborty & Ganguly, 1991). During theexhumation, however, the rim composition of garnet re-equilibrates with the matrix inresponse to the changing p – T condition, but the interior of the garnet does not come toequilibrium with the matrix because of the slow volume diffusion kinetics. Thus, the

garnet crystals develop compositional gradients between the core and rim segmentsduring the exhumation process, the extent of which depends on the cooling rate defined by the exhumation velocity, peak temperature and grain size. Typically, the rim

J. Ganguly302

Fig. 9. Mg/(Mg + Fe) profile in garnet from Søstrene Island, Antarctica, normal to a fracture cleavage (see Fig.8), and model fit to the data according to the solution of Equation 43. Reproduced from Ganguly et al. (2001).



composition “freezes” at ~ 500–550 °C for cooling rates experienced during theexhumation of regionally metamorphosed rocks.

Lasaga (1983) developed the theoretical groundwork for the retrieval of coolingrate from the forward modelling of retrograde compositional zoning. Several workers(Lindstrom et al., 1991; Chakraborty & Ganguly, 1992; Spear & Parrish, 1996; Weyer et

al., 1999; Ganguly et al., 2000; Liermann & Ganguly, 2001; Ganguly et al., 2001) havesince followed this basic idea to retrieve cooling rates of metamorphic rocks and planetary samples from the retrograde compositional zoning of minerals, especiallygarnet. Spear & Parrish (1996) and Weyer et al. (1999) compared the cooling ratesobtained from modelling the retrograde compositional zoning of garnet in metamorphicrocks with those constrained by the closure age and closure temperatures (T C) of multiplegeochronological systems. They found that the cation diffusion data in garnet by

Chakraborty & Ganguly (1992) yield cooling rates that are in good agreement with thoseobtained from the geochronological data.I discuss below an example on the retrieval of cooling and exhumation rates from

modelling the retrograde compositional zoning in garnets with homogeneous corecomposition. The sample is from northern Sikkim in the eastern Himalayas, where themetamophic isograds are disposed in an arcuate regional fold pattern. The thermalevolution and exhumation of the Himalayan metamorphic rocks have been subjects of extraordinary interest among earth scientists as these record the thermo-tectonicevolution of rocks in a major collisional and continental subduction environment, andalso because of the presence of an inverted (Barrovian) metamorphic sequence almost

along the entire length of the mountain belt. Ganguly et al. (2000) calculated the coolingand exhumation rates of rocks from the upper part of the High Himalayan CrystallineComplex (HHC) in the Sikkim–Darjeeling section on the basis of the retrogradecompositional zoning of garnet crystals along with phase equilibrium constraints. TheHHC is bounded on the south and north by the Main Central Thrust (MCT) zone and theSouth Tibetan Detachment System (STDS), respectively. The MCT is a southerlydirected thrust (20–23 Ma) whereas the STDS is a northerly directed system of normalfaults (16–23 Ma).

Ganguly et al. (2000) determined the compositions of a number of garnet crystals

along traverses that are normal to the traces of the interface between garnet and biotite.The zoning profiles for Mg along two traverses in one of the samples are illustrated inFigure 10. The compositions of biotite grains in contact with garnet were homogeneousand were essentially the same as those of other biotite grains within the same thinsection. Because of its compositional homogeneity and large mass compared to that of thin garnet rims affected by cation exchange, it was assumed that the biotite behavedessentially as a homogeneous infinite reservoir of the exchanging components, Fe2+ andMg.

Ideally, the compositional zoning in a crystal used for retrieving cooling rate should be measured along a direction that is normal to the interface; otherwise, the length of the

measured concentration profile would be longer (and consequently the retrieved coolingrate smaller) than that due to diffusion normal to the plane. The most definitive way toensure normalcy of the traverse with respect to the interfacial plane is to obtain three-




dimensional image of the rock by computer aided X-ray tomography, and cut the thinsection normal to the interfacial plane (Carlson & Denison, 1992). As a practicalsubstitute of this laborious procedure, which is rarely used, Ganguly et al. (2000)analysed the effect of geometric distortion of the zoning profile on the retrieved coolingrate, and chose a profile that did not seem to have been affected significantly by therotation of the interface from the vertical (see Ganguly et al., 2000, for further detailsabout this procedure).

Multiplying both sides of Equation 31 by a2, we obtain

, (44)2r

2

r dx

C d

d

dC =

Γ

J. Ganguly304

Q

X X t Q( ) = (m )/cos

46

(b)

(a)

Interface for (a)Interface for (b)relative to (a)

Distan ( m)ce µ

12080400

0.18

0.22

X ( M g )

0.26

0.30

Fig. 10. Mg zoning in garnet in contact with a large mass of biotite in a granulite sample from the

Sikkim–Darjeeling section of the eastern Himalayas, ~ 10 km south of the South Tibetan Detachment System.

X (Mg) = Mg/(Mg + Fe). Line (a) is a fit to the filled squares according to the numerical solution of the diffusionequation, as discussed in the text. Line (b) is derived from (a) by 46° counterclockwise rotation of the

garnet–biotite interface (see inset). X (m) in the inset stands for the measured length of the profile when theorientation of the interface deviates from the vertical by an angle Q, whereas X (t ) is the length of the diffusion

profile normal to the interface. Line (b) fits the data (circles) along another traverse in a garnet in the same thin

section. The garnet–biotite interface for (b) is, thus, interpreted to have been rotated 46° counterclockwise withrespect to that for (a). If it is assumed that the interface for (b) is not rotated from the vertical by more than 75°,

then Q < 30°, which has insignificant effect on the retrieved value of cooling rate (CR) from the profile in (a).

(Modified from Ganguly et al., 2000.)



where Γr and xr are

Both Γr and xr are dimensionless quantities.The following assumptions were made to model the retrograde compositional

zoning in garnet in the Himalayas: (a) the cooling rate followed the ‘asymptotic’ relationgiven by Equation 33, (b) the interface composition of garnet was governed by exchangeequilibrium between garnet and biotite, (c) biotite behaved as a homogeneous infinitereservoir of the exchanging components, as justified above, and (d) the garnetcomposition was homogeneous at the peak metamorphic condition (T 0). Equation 44 wassolved numerically subject to the boundary condition that dC /dx = 0 at x = a. Thenormalising distance a may be any distance from the interface where dC /dx = 0 (i.e. itneed not be the distance to the core of the crystal). As above, the numerical program waslinked to an optimisation program that was allowed to choose the best value of X (T 0), butit returned the observed core composition of garnet as the best choice implying that it wasnot affected by diffusion. The same conclusion was arrived at from other observations.

Figure 10 shows the model fit to the measured compositional data of garnet, whichenables one to retrieve the quantity Γ, that is the integral of D(t )dt from the peak metamorphic condition to the “freezing” of the compositional profile in garnet. Theretrieved value is 8.18×10 –5 cm2. As a next step, Ganguly et al. (2000) calculated theexhumation velocity (V z) of the rock by calculating the T – t path for a given V z,

integrating D(t )dt over this path, and repeating the process until the above value of ∫ D(t )dt was obtained. This procedure yields V z = 2 mm/year. However, the phaseequilibrium constraints that are imposed by the retrograde reaction relations in the rock,which could be deduced from petrographic observations, require a nearly isothermalexhumation, corresponding to V z ≈ 15 mm/year from a depth of ~ 34 km to ~ 15 km. The potential geodynamic implications of this change of exhumation velocity have beendiscussed by Ganguly et al. (2000, 2001).

Diffusion modification of growth zoning in garnet

A mineral develops compositional zoning during the growth process if it fractionatescomponents with the matrix, and the diffusion within both the matrix and the mineral isslow compared to the growth velocity of the crystal. This may be understood byconsidering crystal growth in incremental steps. Owing to the fractionation of componentswith the crystal, the matrix in the immediate vicinity of the crystal (or “adjacent” matrix)would become depleted in the components that partition preferentially into the crystal, andenriched in those that have the reverse behaviour. Consequently, during the next growthstep, the adjacent matrix would have a bulk composition that is effectively different fromthat in the previous step. Thus, the segment of the crystal that had grown in the second stepwould have a different composition than that in the previous step (because it had

partitioned components with two different matrix compositions in the two steps).Continuation of the process would lead to the development of a zoned crystal. Suchgrowth zoning is very common in the garnet zone rocks of regionally metamorphosed

.and r 2r

a

x x

a

=Γ

=Γ




Barrovian sequence, in which the garnets typically show a bell shaped Mn profile and bowl shaped Fe, Mg and Ca profiles, with the extrema near the centre of the crystals.

The retrograde compositional adjustment of the rim leads to two types of zoning profiles in garnet. If the crystal had homogenised earlier, then it would have a flat core andzoned rim, as in the Himalayan garnet sample discussed above. On the other hand, if theinitial growth zoning were partly or completely preserved, then the crystal would havecomplicated zoning profiles in which some of the components could show extrema intheir profiles from the centre to the rim of a grain. Modelling of these partially modifiedgrowth zoning profiles hold great promise in the retrieval of the full thermal history of therock. Discussion of this modelling procedure is, however, beyond the scope of this review,as it requires consideration of crystal growth and resorption processes simultaneouslywith volume and intergranular diffusion kinetics. The interested readers are referred to

Florence & Spear (1993), Okudaira (1996) and Carlson (2002) for an appreciation of thetopic and interesting examples of applications to metamorphic processes.

Acknowledgements

Thanks are due to Prof. Sumit Chakraborty for an insightful review in a very short notice,Prof. Carlo Maria Gramaccioli for inviting me to the short course in Budapest, Prof.Tamás Weiszburg and his supporting staff and family for their hospitality, and Prof.Martin Glicksman and Dr. Afina Lupulescu for their help in calculating Figure 4 usingthe program PROFILER, and Alexander von Humboldt Foundation for support through

a research award (Forschungspreis) during the preparation of this manuscript at theBayerisches Geoinstitut, Bayreuth, Germany.

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