EMU Notes in Mineralogy, Vol. 4 (2002), Chapter 10, 271–309
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 Arizona
Tucson, AZ 85721, USA;
e-mail: ganguly@geo.arizona.edu
Introduction
Diffusion is the process by which atoms or ions or ionic species migrate within a medium
in the absence of a bulk flow. Diffusion in solids has been a subject of interest in the
fields 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 years
earlier, with the empirical formulation of Fick on the relationship between the diffusion
flux 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 attention
in the study of geological processes. The study of grain boundary diffusion is important
to the understanding of many metamorphic processes including the problems of mass
transport, fluid/rock interactions, thermal history and crystal growth. The interested
reader 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 the
thermal and physico-chemical history of the host rocks. Volume diffusion, that is
diffusion through the crystal lattices, affects development of compositional zoning in
minerals, ordering of atoms in nonequivalent crystallographic sites of a mineral,
formation and coarsening of exsolution lamellae, and retention of isotopic characteristics
in minerals that can serve as quantitative chronometers in their thermal and growth
history. In this chapter, I present a brief overview of the fundamental principles of
volume diffusion kinetics, primarily at a phenomenological level, and discuss the
various factors that affect diffusion kinetics in minerals. Finally, I discuss some
applications of the diffusion kinetic studies in minerals to the understanding of tectonometamorphic processes in major continent–continent collisional environments. It
should, however, be emphasized that the scope of applications of volume diffusion
272
J. Ganguly
kinetics to geological and planetary problems is much wider than what I have attempted
to cover in this chapter. Some incidental references to additional applications have been
made in the appropriate places.
List of symbols
D*i , Di+ and Di Tracer, self- and chemical diffusion coefficients, respectively, of the
component 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 multicomponent system
D
Matrix of diffusion coefficients
Dij
An element of the D matrix
τ
Diagonal matrix of the eigenvalues of a D matrix
τi
An eigenvalue of the D matrix
B
A matrix composed of the eigenvectors of D matrix
L
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
Ji
Flux of a component i
J
A column vector of the fluxes of n–1 independent components in an ncomponent system
Ci
Concentration of the component i, expressed in atomic units per unit
volume
C
A column vector of the concentrations of n–1 independent components in
an n-component system
Xi, ai and γi Atomic fraction, activity and activity coefficient, respectively, of the
component i
µi
Chemical potential of the component i
W
Non-ideal interaction parameter
kB
Boltzmann constant
η
A cooling time constant (K–1t–1)
t′
Γ
∫ D(t )dt
0
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.
273
Diffusion kinetics in minerals: Principles and applications
the rate of transfer per unit area) of a component, Ji, through this planar section is
proportional to its local concentration gradient. Thus,
J i = − Di
∂Ci
,
∂x
(1)
where Ci is the concentration of i, in atomic units per unit volume, which decreases in
the 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 positive
in the direction of decreasing Ci.) Implicit in the above statement is the assumption that
there is no external force (such as electrical and gravitational forces) acting on the
diffusing species. The modifications for the expression of flux to incorporate the effects
of 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 conservation
of 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
∂Ci
∂
= − ( J i ),
∂t
∂x
(2)
so that, if the flux is given by Equation 1, then
∂Ci
∂  D ∂C 
=  i i .
∂t
∂x  ∂x 
(3a)
If the diffusion coefficient is independent of position, then
∂C i
∂ 2 Ci
.
= Di
∂t
∂x 2
(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 equations
corresponding to Equations 3a and 3b are written by simply replacing ∂ by the gradient
operator (i.e. i(δ/δx) + j(δ/δy) + k(δ/δz)) and ∂2 by the Laplacian operator ∇2 (i.e. δ 2/δx2
+ δ 2/δy2 + δ 2/δz2) in the right-hand side of the equations. The equation expressing the
time dependence of concentration, that is Equation 3b or its three-dimensional form, is
known as the diffusion equation in Cartesian coordinates. Other forms of the diffusion
equation in different coordinate systems follow simply from the appropriate
transformation of coordinates. The solutions of the diffusion equation, either analytical
or numerical, for the appropriate initial and boundary conditions, permit us to model
diffusion controlled properties to retrieve quantitative information about geological,
planetary and other processes.
274
J. Ganguly
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 the
absence 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
 ∂µ / T 
J i = − Li  i

 ∂x 
L  ∂µ 
L
=− i  i =− i
T  ∂x 
T
 ∂µ i

 ∂C
 i
  ∂Ci 

 ∂x  ,


(4)
where Li is a phenomenological coefficient. This relation, however, assumes that the
higher order terms of the driving force has negligible effect on the flux (i.e. Eqn. 4 holds
within the domain of validity of linear irreversible thermodynamics).
Since at a constant p–T condition ∂µi = RT∂lnai = RT∂ln(Ciγi), where ai and γi are
the activity and activity coefficient of the component i, respectively, it is easy to see that
Ji = −
RLi
Ci
 ∂ ln γ i
1 +
 ∂ ln C
i

  ∂Ci 

  ∂x  .

(5)
Comparing Equations 1 and 5, we have

∂ ln γ i
Di = Di+ 1 +
 ∂ ln Ci
where
Di+ =

,


RLi .
Ci
(6a)
(6b)
The quantity within the parentheses of Equation 6a is usually referred to as the
thermodynamic factor. We will henceforth refer to it as D(thermo). From definition
Ci = ni /NV = Xi /V, where Xi 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 dlnCi = dlnXi.
Diffusion coefficients: Terminology and definitions
Before proceeding further, it is important to discuss certain properties of diffusion
coefficient and the related terminology. In the literature, one encounters terms like
tracer, self-, inter-, and chemical diffusion coefficient, the meaning of which is not
usually 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 any
work.
Diffusion kinetics in minerals: Principles and applications
275
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 homogeneous
medium 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 chemical
elements. The term self-diffusion coefficient will be used to define the diffusion
coefficient that describes the flux of an element solely in response to its own
concentration gradient, and under the condition that its thermodynamic interaction with
the 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 quantity
defined by D+i in Equation 6b. The diffusion coefficient Di, which is a product of the selfdiffusion coefficient and the thermodynamic factor (Equation 6a), will be referred to as
the chemical diffusion coefficient of the component i. The self- and tracer diffusion
coefficients 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 an
element has also been applied to the limiting case of tracer diffusion, in the sense defined
above, when the diffusing tracer isotope and the non-tracer solvent belong to the same
element, 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 neutral
species (A and B) across a welded plane (Fig. 1), as in a binary metallic alloy. If the
component A diffuses faster than B, then the right-hand side of the couple will swell
while 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 by
Smigelskas & Kirkendall (1947) for the interdiffusion of Cu and Zn by placing fine
molybdenum markers at the interface, and is usually referred to as Kirkendall effect
(instead of Smigelskas effect or Smigelskas–Kirkendall effect!). In analysing this result
of Smigelskas and Kirkendall, Darken (1948) raised the important question “what is
diffusion?” and introduced the concept of frame of reference in diffusion studies.
In the above example, one can describe the diffusion process with reference to two
alternative coordinate systems, x and x′, as follows. In the first system, x′ = 0 is fixed to
the 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 = 0
is located at one end of the diffusion couple, say the left end (this frame of reference is
usually called the “laboratory frame of reference”). In the x′ coordinate system, the flux
of 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,
vCA(x = k), in order to account for the effect of movement of the plane, where v is the
velocity of the plane. Thus, in the x coordinate system,
276
J. Ganguly
Fig. 1. Schematic illustration of the Kirkendall effect for the inter-diffusion of two neutral species, A and B,
with DA+ > 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 an
elapsed 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).
∂CB
+ (vCB ) x = k .
∂x
So that, assuming D to be independent of x
J B = − DB
(7a)
∂ 2CB
∂C
∂
∂CB
(7b)
–+ v B .
=–
− ( J B ) = DB
2
∂x
∂t
∂x
 ∂x 
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 to
vacancy flow and interactions of the atoms with the vacancies), solution for v in terms
of these individual chemical diffusivities, DA and DB, in the x coordinate system, and
rearrangement of terms yields (Darken, 1948)
J B = − D( A–B )
∂CB
,
∂x
(8a)
where D(A–B) is given by
D(A–B) = [XADB + XBDA],
(8b)
and is called a chemical interdiffusion coefficient (in the above example, for a metallic
system). The flux of the component A in the x coordinate system is also given by
277
Diffusion kinetics in minerals: Principles and applications
Equation 8a upon simply replacing CB by CA. 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 describes
diffusion under the influence of a “driving force”. The latter is defined to be a force, such
as those arising from an electrical field or non-ideal mixing property, which causes an
atomic jump in one direction across a potential energy barrier to be more probable than
that in the reverse direction across the same barrier (Manning, 1968). Equation 7 is also
applied to the solution of diffusion-reaction problem where the interface between two
crystals moves, with respect to a stationary coordinate system, due to reaction along with
diffusion (see later for an application).
Instead of neutral atoms, let us now consider the problem of diffusion of ions of the
same 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 corresponding
accumulation of positive charge vacancies on the left. This would create an electrical field
Ed (which implies a driving force zEd, where z is the charge of the diffusing ion) that would
affect the diffusion of ions so that local electrical neutrality of the sample is preserved. The
latter requirement implies that, in the absence of any other mechanism of charge
compensation, the net flux of ions at the interface must be zero (i.e. there must be equal
number of A and B crossing the interface per unit area per unit time). As a result, the interface
must remain fixed with respect to a stationary coordinate system if the molar volume
remains constant. Expressing the mean drift velocity of the ionic species in terms of Ed (i.e.
v = (Di)zEd/kBT) in Equation 7, and also in the analogous expression for JA, and solving Ed
in terms of DA and DB under the constraint that JA + JB = 0, yields (e.g. Manning, 1968)
J B = −J A = −
We now write
D(Az – Bz) =
=
DA DB
 ∂C B  .


X A DA + X B DB  ∂X 
(9)
DA DB
.
X A DA + X B DB
(10)
Equation 10 defines the chemical interdiffusion coefficient in a binary system of
equally charged species. By the requirement of mass balance, ∂CB/∂x = –∂CA/∂x. As
emphasised by Lasaga (1979), the chemical interdiffusion coefficient in an ionic system
will, in general, be overestimated, especially around X = 0.5, if one uses the Darken
relation, i.e. Equation 8b instead of Equation 10. The Darken relation has been applied
to mineralogical systems, but from a theoretical point of view, it is not applicable to such
systems 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 (AZA – BZB) =
DA DB (Z A X A + Z B X B )2
Z A2 X A DA + Z B2 X B DB
,
(11)
where ZA and ZB represent the charges on the specified ionic species. Equation 11
reduces to Equation 10 when ZA = ZB.
278
J. Ganguly
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), and
application of the Gibbs–Duhem relation (i.e. n1dµ1 + n2dµ2 = 0 at constant p–T
condition) yield, for a system with constant molar volume,
 ∂ ln ai
D (i − j ) = D + (i − j )
 ∂ ln Ci


∂ ln γ i
 = D + (i − j )1 +

 ∂ ln X i

,

(12)
where D+(i–j) is an appropriate (metallic or ionic) interdiffusion coefficient had the two
components mixed ideally. It is expressed according to the forms of Equations 8b, 10 and
11, as appropriate, by substituting the self-diffusion coefficient, D+i , for Di. Thus, for
example, if the interdiffusion is between two equally charged ions, then

∂ ln γ i
D ( A − B ) = D + ( A − B )1 +
 ∂ ln X i
+
DA++D
 
DBA+
 = 
+
+
  X A DA + X B DB

∂ ln γ i
1 +
 ∂ ln X i

.

(13)
Note that in Equations 12 and 13, component i can be either A or B, since, according to
the Gibbs–Duhem relation and stoichiometric constraint for a binary solution (i.e.
dXA = –dXB) with a constant molar volume, d ln aA/d ln CA = d ln aB/d ln CB. Also recall
that if the molar volume of the material is constant, then d ln Ci = d ln Xi. It should be
noted that, in general, the self-diffusion coefficients are also functions of composition.
Thus, D+i 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 important
systems to treat self-diffusion coefficients as function of composition.
If the binary solution has a sub-regular thermodynamic mixing property, i.e. the
excess Gibbs energy of mixing can be expressed according to ∆Gxs = X1X2(W12X2 +
W21X1), where Wij represents the Margules parameters (see, for example, Ganguly &
Saxena, 1987), then
∂ ln γ1i
X X
= 1 2 [W12 (2 X 1 − 4 X 2 )+ W21 (2 X 2 − 4 X 1 )].
∂ ln X 1
RT
(14)
For the special case of “Simple Mixture” or “Regular Solution”, W12 = W21 = W, so that
∂ ln γ1i
2WX 1 X 2 .
=−
∂ ln X 1
RT
(15)
As an illustration of thermodynamic effect, we show in Figure 2 the
thermodynamic factor calculated by Brady & McCallister (1983) at 1150 °C for the
quasi-binary Ca–Mg(+Fe) interdiffusion between pigeonite lamellae and sub-calcic
diopside host. The sample is a diopside megacryst (Fe/(Fe + Mg) = 0.14) from Mabuki
kimberlite, Tanzania. The experimentally determined critical mixing temperature (Tc)
between the Ca and Mg (+ Fe) components is 1132 °C, and the critical mixing
composition is ~ 20 mol% diopside. Also shown in the figure is the thermodynamic
factor (dashed curve) calculated from the mixing energy data of Lindsley et al. (1981) in
Diffusion kinetics in minerals: Principles and applications
279
the diopside–enstatite join at temperatures that have the same relative position with
respect to Tc (1500 °C) in this join as the inferred temperature of the natural sample has
with respect to its own Tc. It is evident from Figure 2 that the thermodynamic effect on
the diffusion coefficient is very pronounced near the critical mixing temperature (also
see Christoffersen et al., 1983).
It should be noted from Equation 13, and the equivalent expression for the metallic
system that follows from Equation 8b, that as XA → 0, D(thermo) = 1 (since γi =
constant). Consequently,
lim X i →0 D (i − j ) = Di+ .
Thus, we arrive at the rather counter-intuitive conclusion that the interdiffusion
coefficient 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 valid
also for multi-component solutions by examining the limiting behaviour of the extension
of 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 nonrandomness of the atomic jumps. To illustrate this point, let us consider the common case
of a vacancy mediated diffusion. In this case, an atom moves by interchanging position
with a vacancy after it arrives (or diffuses) into one of the neighbouring lattice sites. But
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 is
for Fe-free system at a temperature that has the same ratio with the Tc in this join, which is 1500 °C, as the
chosen temperature of the Fe-bearing sample has with its own Tc.
280
J. Ganguly
after making the first jump, there is a greater probability for the atom to return to its
original position (because it now finds a vacancy in that position) when it executes the
next 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 correlation
factor, f, accounts for this problem by expressing Di as a product of fi and Di(random).
The latter is the value of D that should be obtained under conditions of completely
random atomic jumps. By definition, fi ≤ 1.
It can be shown that for one dimensional diffusion, D(random) = ⟨X2⟩/2t, where ⟨X2⟩
is the mean square displacement of the atoms after time t (for random atomic jumps of
equal length, we obviously have ⟨X⟩ = 0, but ⟨X2⟩ ≠ 0). This relation between the mean
square displacement and diffusion coefficient for random atomic motion is often referred
to as the Einstein relation, and provides the basis for the determination of D by
molecular dynamics simulation. In this approach, ⟨X2⟩ is determined at several different
time steps, and the slope of the linear relation between ⟨X2⟩ 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 fi
for 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 because
of the differences in their masses, which affect their jump frequencies, and the
correlation coefficient. If Di(*α) and Di(*β) are the tracer diffusion coefficients of two
Fig. 3. Correlation factors of Fe2+ and Mg as a function of XMg in binary Fe–Mg olivine at 1 bar, 1130 °C, and
log fO2 = –10.67 (modified from Hermeling & Schmalzried, 1984).
281
Diffusion kinetics in minerals: Principles and applications
isotopes α and β of an element i, then with certain simplifying assumptions (see
Bokshtein et al., 1985 for further details), it can be shown that

 Mβ
∆D
≈ fα 
*
M
Di ( β )
 α
1

2 
 − 1 ,



(16)
where ∆D = Di(*α) – Di(*β) and M stands for the masses of the specified isotopes. There is only
one correlation factor in the above equation since the correlation factors of two isotopes of
an element are interrelated (e.g. Bokshtein et al., 1985; Borg & Dienes, 1988). In the
treatment of multi-component diffusion, it is usually assumed that the self-diffusion
coefficient of an element is the same as the diffusion coefficient of a tracer isotope of the
element. While this is not strictly correct, the error introduced by this assumption is usually
small compared to other sources of error, especially in complex geological problems.
Multi-component diffusion
Extension of Fick’s laws
During geological processes, diffusion in many minerals, e.g. garnet, is often multicomponent in nature in that it involves simultaneous flow of more than two components.
Understanding of multi-component diffusion processes is, therefore, important to the
interpretation of diffusion induced compositional modifications of minerals during
geological processes. In this section, I would try to provide a brief overview of some of
the important phenomenological concepts of multi-component diffusion.
In a multi-component system, the diffusion flux of any component does not
depend only on its own concentration or chemical potential gradient, but also on those
of all other diffusing components. Assuming that only the first spatial derivatives of
concentrations are important for the flux of any component, Fick’s law can be
extended to a system of n components in a volume fixed reference frame as follows
(Onsager, 1945):
 ∂C 
 ∂C 
 ∂C 
J 1 = − D11  1  − D12  2  ........................... − D1( n−1)  n−1 
∂
∂
x
x




 ∂x 
 ∂C 
 ∂C 
 ∂C 
J 2 = − D21  1  − D22  2  ......................... − D2( n−1)  n−1 
 ∂x 
 ∂x 
 ∂x  .
.............................................................................................
(17)
 ∂C 
 ∂C 
 ∂C 
J n−1 = − D( n−1)1  1  − D( n−1) 2  2 ........... − D( n−1)( n−1)  n−1 
 ∂x 
 ∂x 
 ∂x 
In the above system, there are n – 1 flux equations since in an n-component system of
fixed mass and volume, the concentration of one component is fixed by those of others
at any given point.
An on-diagonal term in Equation 17 shows the extent by which the flux of a
component is affected by its own concentration gradient, whereas the off-diagonal terms
282
J. Ganguly
indicate the extent of hydrodynamic coupling in the diffusion process, i.e. the extent to
which the flux of a given component is influenced by the concentration gradients of the
other independent components. If these off-diagonal terms are significant, then one
could 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, but
has been found in several silicate melts (e.g. Chakraborty et al., 1995a, 1995b) that are
of interest in the understanding of magmatic processes.
D matrix
Using the principle of matrix multiplication, we can re-write Equation 17 as
 D11 D12 ..........D1( n −1)  ∂C1 / ∂x 
 J1 




D D ..........D2( n −1)  ∂C2 / ∂x 


 J 2  = −  21 22
...............................  ⋅ .............  ,
.... 
 




 D( n −1)1 ....... D( n −1)( n −1)  ∂Cn −1 / ∂x 
 J n −1 
(18a)
or, in matrix notation
∂C
∂x
(18b)
∂C ∂  ∂C
= D
,
∂t ∂x  ∂x 
(18c)
J = −D
so that
where J and C are (n – 1) column vectors and D is an (n – 1)×(n – 1) matrix of diffusion
coefficients, which is usually referred to as the D matrix. For n = 2, Equation 17 reduces
to the expression of flux in binary diffusion, in which case D11 is a binary interdiffusion
coefficient.
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 the
chemical diffusion coefficient of a species consists of a product of an L coefficient
and a thermodynamic factor.) The L and G matrices are also positive definite, that
is, they have real and positive eigenvalues. Consequently, the D matrix must also
have real and positive eigenvalues. This requirement provides important constraints
on the values of the elements of the D matrix (see Lasaga, 1998, for further
discussion). Furthermore, Equation 19 permits one to test the mutual compatibility
of the data on the diffusion kinetic and thermodynamic mixing properties of species
283
Diffusion kinetics in minerals: Principles and applications
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 inserting
guessed values of these parameters into the L and G matrices until the product of D
and 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 Equation
18b as
((∂µ/T)
∂ ì/ T)
J = −L
.
(20)
∂x
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 good
approximation for most diffusion process in minerals, the elements of G matrix can be
calculated as follows (e.g. Loomis, 1978).
Gij =
∂(µi − µ n )
,
∂X i
(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 multicomponent
diffusion to the mathematical forms of binary diffusion (Toor, 1964; Cullinan, 1965), as
follows. 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–1DB = 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
∂ C′ ∂  ∂ C′ 
= τ
,
∂x  ∂x 
∂t
(22)
where C′ = B–1C. Thus, instead of the coupled diffusion equations of the original
components, we obtain uncoupled or independent diffusion equations of the transformed
components, Ci′, as
∂Ci′ ∂  ∂Ci′ 
= τ i

∂t
∂x  ∂x  ,
(23)
where the eigenvalue τi is the diffusion coefficient of the transformed component Ci′.
These equations can be solved for Ci′(x, t) using solutions of diffusion equations
characterised 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 and
thermodynamic 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
284
J. Ganguly
by Lasaga (1979), which is appropriate for the mineralogical problems, is as follows if
the activity coefficients of the diffusing components (γi) are constant within the domain
of compositional variation


 D*Z Z X 
ij i j i
 (D
Di*i* –− D
Dij = Di*δ ij −  k = n
Dn*n*) ,

2
*
 ∑ Z k X k Dk 
 k =1

(
)
(24)
where δij is the Kronecker delta (δij = 1 when i = j, and δij = 0, when i ≠ j). Full treatment
to incorporate the effects of variation of γi can be found in Lasaga (1979).
Because of the paucity of experimental data on diffusion in multicomponent
mineralogical 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-diagonal
terms. An example from their study is given below, where the matrix elements are in
units of cm2/s, and Ca was treated as the dependent component.
Mn
Mg
Fe
Mn
8.38(10–20)
–2.78(10–21)
–7.16(10–20)
Mg
–9.91(10–23)
7.26(10–21)
–4.81(10–23)
Fe
–4.68(10–21)
–8.81(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 metamorphic
condition of 600 °C, 5 kbar, fO2 = graphite–O2 equilibrium (see below for discussion
about the dependence of D on fO2). It is evident that the magnitude of some of the offdiagonal 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 to
significant errors in the modelling of diffusion modification of compositional zoning in
multi-component garnets.
To illustrate the importance of accounting for the multi-component interactions, I
show 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 using
the program PROFILER, which is discussed in detail in Glicksman (2000). The initial
composition of Mn on the two sides of the couple were chosen to be the same, and Ca
was treated as the dependent component. The simulation is for 40 Myr, and each division
on the distance axis of the main figure equals 1 µm. It is interesting to note that Mn
shows uphill diffusion and develops a wavy pattern near the interface of the couple due
to the influence of the other diffusing species.
It should be noted that uphill diffusion of a component in a semi-infinite diffusion
couple not only depends on the magnitude of the off-diagonal terms of the D matrix, but
also on the nature of the compositional difference of the components on two sides of the
285
Diffusion kinetics in minerals: Principles and applications
70
Concentrations (at%)
60
50
40
Mg
30
Fe
20
Mn
10
0
–3
(a)
Ca
–2
0
–1
1
2
3
Distance
Fig. 4. (a) Calculated diffusion profiles of Fe, Mg, Mn and Ca in a semi-infinite garnet/garnet diffusion couple
using the D matrix given in the text. The initial concentration of all components was homogeneous in each
garnet crystal, and that of Mn was the same on both sides of the couple. Multi-component interaction produces
the uphill diffusion and wavy pattern of Mn profile near the interface, which is magnified in (b). The simulation
is for 40 Myr. Each division on the distance axis in (a) equals 1 µm.
couple (see, for example, Chakraborty et al., 1995a; Glicksman, 2000). Because of the
multi-component interaction, one or more components in a multi-component diffusion
can have either stationary or moving zero flux planes (ZFP). A ZFP defines a plane
where 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 have
recently been explored in detail by Glicksman & Lupulescu (in press). This property has
interesting industrial and potential geological applications.
Diffusion in anisotropic crystals: Diffusion tensor
In anisotropic crystals, diffusion properties are, in principle, different along different
directions. In such medium, Equation 1 holds only for the special case that there is
concentration gradient only along the x direction. If there are concentration gradients
along 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) to
the 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
J x = − Dxx
∂C
∂C
∂C
.
− Dxy
− Dxz
∂x
∂y
∂z
(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
286
J. Ganguly
as the diffusion tensor. It is, however, always possible to find three orthogonal
directions ξ1, ξ2, ξ3 in an anisotropic medium such that the flux along any of these
directions depends only on the concentration gradient along the specific direction, i.e.
Jξi = – Dξi(∂C/∂ξi). These directions and the corresponding diffusion coefficients are
known as the principal diffusion axes and principal diffusion coefficients, respectively.
Diffusion along any arbitrary direction κ, which makes angles θ1, θ2, θ3 with the
principal diffusion axes (1, 2 and 3), is given by
J κ = − Dκ
∂C
∂κ ,
(27)
where Dκ = Dξ1 cos2θξ1 + Dξ2 cos2θξ2 + Dξ3 cos2θξ3, and ∂C/∂κ is the concentration
gradient along the direction κ.
The direction of a crystallographic symmetry axis coincides with that of a principal
diffusion axis. Thus, the a, b and c crystallographic directions in cubic, tetragonal,
orthorhombic and hexagonal systems constitute the directions of principal diffusion
axes. 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 be
determined by three measurements of diffusion coefficients in that plane. For triclinic
system, one needs measurements of diffusion coefficients in six different directions to
determine 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 increasing
temperature, diffusion anisotropy should be expected to decrease with increasing
temperature. The anisotropic diffusion data for olivine were summarised and discussed
by Morioka & Nagasawa (1991). The tracer diffusion coefficients of Ni, Co, Ca, as well
as Fe–Mg interdiffusion coefficient, were found to be fastest parallel to the c axis, and
slowest 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 olivine
structure in that the M1 sites form a closely spaced chain parallel to the c axis, and the
energy of formation of cation vacancies in the M1 site is significantly less than that in
the M2 site (e.g. Ottonello, 1997). Niemeier et al. (1996) observed by high temperature
Mössbauer spectroscopy that cation diffusion in olivine takes place predominantly via
M1–M1 jumps along the c direction. In contrast to the above results, Jurewicz & Watson
(1988) found that at fO2 < 10–8 bar, Mn and Fe show highest diffusion rates parallel to
the a axis. These authors tried to explain the unexpected anisotropic behaviour in terms
of different diffusion mechanisms along a and b/c crystallographic axes. For alkali
feldspars, 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), the
observed diffusion anisotropy is consistent with the feldspar structure in that the alkali
sites are much closer to each other within the (010) plane than normal to it. The above
287
Diffusion kinetics in minerals: Principles and applications
examples suggest that consideration of the packing density of the host structural sites
could provide useful, although not unfailing, guidelines about the expected diffusion
anisotropy in minerals.
Factors affecting diffusion coefficient
Diffusion can take place through a number of different mechanisms such as through
exchange of position between atoms and lattice vacancies (vacancy mechanism),
migration of atoms through interstitial sites (interstitial mechanism) etc. These have been
discussed 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 most
important of all diffusion mechanisms. All cases of substitutional diffusion seem to
operate through a vacancy mechanism (Borg & Dienes, 1988). Thus, any physicochemical factor that affects the vacancy concentrations in a crystal significantly also has
a significant effect on its diffusion properties.
Since diffusion involves climbing an energy barrier by atoms between two stable
states, temperature has the strongest effect on diffusion coefficient as it provides the
energy (kBT) to elevate the atoms over the energy barrier. The other factors affecting the
volume diffusion coefficient of a species are pressure, volatile species (fO2 and the
fugacities 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 present
chapter, and, thus, will not be discussed any further. The interested readers are referred
to 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
∂ ln D
Q( p)
=−
∂ (1 / T )
R
(28)
∂ ln D
∆V + (T )
,
=−
∂p
RT
(29)
where Q(p) and ∆V +(T) are known as the activation energy (at pressure p) and activation
volume (at temperature T) of diffusion, respectively. Note that the above equations are
formally similar to those governing the temperature and pressure dependencies of the
equilibrium constant, K. In general, any kinetic constant or coefficient has the same
formal dependencies on temperature and pressure. Assuming that Q(p) is independent of
temperature, integration of Equation 28 yields
D = D0 e
−
Q( p)
RT
,
(30)
where D0 is D(T = ∞). Assuming that ∆V + is independent of pressure, we have
Q(p) = Q(p′) + ∆V +(p – p′).
The activated state is an energetically higher transient state that a system passes
288
J. Ganguly
through in any kinetic process. For diffusion in solids, the activation energy is the sum
of 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 the
enthalpy of formation of vacancies, ∆H+v, if the diffusion process is controlled by intrinsic
vacancies. 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 has
components 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 reduce
the diffusion coefficient. Review of the experimental data on divalent cation diffusion in
olivine, 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 the
available 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 a
change of 5 kbar pressure would be to reduce logD by no more than 0.26, and probably
by half as much. In principle, ∆V + is a function of pressure. However, within
experimental 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 a
substance at a given temperature seems to bear a relation with the degree of proximity
of the temperature to the melting temperature, Tm (e.g. Borg & Dienes, 1988). Melting
temperature maximum has been found in olivine, and probably all minerals have the
same property. Thus, after the pressure exceeds that of Tm(max), increasing pressure at
a fixed temperature may enhance the diffusion kinetics in a mineral since it would bring
the mineral progressively closer to Tm.
Effect of fO2
In solids containing elements that can have variable oxidation states, such as iron, fO2 is
expected to influence the diffusion property by changing the vacancy concentration
through the change of the oxidation state of the element. For example, if the
homogeneous redox equilibrium of iron in a solid is governed by the reaction
3Fe2+(l) + ½ O2(g) ↔ 2Fe3+(l) + FeO(surface) + VFe ,
(a)
where l, g and VFe stand for lattice site, gas and vacancy in Fe lattice site, respectively,
then it is easy to show that VFe, and hence the diffusion coefficient, would vary
approximately as (fO2)1/6. This relation follows by combining the expression of
equilibrium constant of the above reaction with the relation 2(Fe3+) = VFe (which follows
from the requirement of charge conservation if the vacancies are neutral), and assuming
that XFe is not significantly altered by the oxidation. It is further assumed, although rarely
2+
Diffusion kinetics in minerals: Principles and applications
289
stated explicitly, that the activities of both Fe2+ and Fe3+ are proportional to their mole
fractions (in the spirit of the laws of dilute solutions). Experimental data on the diffusion
coefficients in ferromagnesium olivine (Buening & Buseck, 1973; Nakamura &
Schmalzried, 1983) show that D varies approximately as (fO2)1/6. It should be noted that
fO2 affects the diffusion coefficient not just of Fe, but also that of other cations (i.e. Fe
does not have any exclusive right to utilise the vacancies created by its oxidation).
The dominant diffusion mechanism could change as a function of Fe concentration
and fO2. For example, the experimental data of Chakraborty et al. (1994) suggest that
the 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, logD*Fe in Fe3O4 vs. log fO2
has a minimum, which shifts to higher fO2 with increasing temperature. The
experimental data can be matched very well by a theoretical relation between D*Fe and fO2
that they derived by invoking that Fe diffuses through both vacancy and interstitial
mechanisms. The vacancy diffusion plays the dominant role at fO2 above the minimum
whereas the interstitial diffusion plays the dominant role at lower fO2.
Effect of hydrous condition
Diffusion in the presence of water has been investigated by a number of workers. The
species 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). On
reviewing these data, she concluded that “water” may not play a significant role in the
interdiffusion process that involves only a simple exchange between cations of the same
charge. On the other hand, “water” (the actual species could be proton) has been shown
to have a significant enhancement effect on the interdiffusion or ordering process in
feldspars 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 bars
and fO2 defined by the Ni–NiO buffer on the Fe2+–Mg interdiffusion in olivine also
shows 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 diffusion
kinetics needs to be carefully investigated so the experimental data can be applied to
natural systems in a meaningful way. Our understanding of the problem at the present
stage is sketchy at best.
Effect of dislocations
Diffusion along dislocations, commonly referred to as pipe diffusion, is much faster than
diffusion through crystal lattice. In the presence of distributed dislocations, the apparent
volumetric 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
290
J. Ganguly
diffusion couple was strained at a rate of 10–6 s–1 during the process of diffusion at the
same p–T condition (1.5 kbar, 1000 °C), they concluded that distributed dislocations are
unlikely to have any significant effect on the bulk volumetric diffusion in alkali feldspars
at 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 dislocations
on 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 in
the 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 be
avoided 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 to
natural systems is the possible change of diffusion mechanism in moving from the
laboratory 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 physical
variables such as fO2 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 an
equilibrium 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 compositional
segments 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 the
solid solution, G of a crystal must also decrease as function of Xv (atomic fraction of
vacancies) in the terminal region of Xv = 0. Second, vacancies are created by the
replacement 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 interaction
with an external source.
As discussed by Chakraborty (1997), one should distinguish between the extrinsic
vacancies created by an impurity substitution, in which case there is no defect formation
energy, 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 as
transition metal extrinsic diffusion (TaMED).
Diffusion kinetics in minerals: Principles and applications
291
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 low
temperature, 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).
The atomic fraction of the intrinsic lattice vacancies have an exponential dependence
on temperature according to Xv ∝ exp(–∆H0v/RT), where ∆H0v is the enthalpy of formation per
mole of the particular type of vacancy†. Consequently, at high temperature, Xv(intrinsic) >>
Xv(pure extrinsic), so that diffusion takes place essentially through the intrinsic vacancies.
On the other hand, at low temperature, Xv(intrinsic) << Xv(pure extrinsic) so that diffusion
takes place dominantly through the extrinsic defects created by heterovalent substitution.
The temperature for transition from a dominantly intrinsic to a dominantly extrinsic
mechanism 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 the
intrinsic 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 H0v is the enthalpy of formation of the Schottky pair, then from the expression of equilibrium constant
of the vacancy forming reaction, we have Xv(c)Xv(a) ∝ exp (–∆H0v /RT), or Xv(c) ∝ exp (–∆H0v /2RT).
292
J. Ganguly
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 TaMED
diffusion also vary as a function of temperature as exp(–∆Hr0/RT), where –∆Hr0 is the
enthalpy change of the appropriate redox reaction. The latter is, however, much less than
the enthalpy of formation of intrinsic defects. Thus, the log D vs. 1/T slope in the TaMED
domain is less than that in the intrinsic domain.
A qualitatively similar behaviour to the extrinsic–intrinsic transition for volume
diffusion is also shown by the transition from grain boundary to volume diffusion in
polycrystalline aggregates. At low temperature, the pathways offered by the grain boundaries
dominate 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 an
illustration of the change of volume diffusion mechanism in many mineralogical
publications. Also, these diffusion data have been used extensively to model thermal
histories of terrestrial and planetary samples. However, Buening & Buseck (1973) used a
diffusion couple consisting of Mg-rich olivine single crystal and powdered synthetic
fayalite. They noted that the observed change of mechanism could have also been due to a
change of grain boundary to volume diffusion – a point that seems to have been ignored in
favour 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 diffusion
mechanism in olivine within this temperature range. Chakraborty & Ganguly (1991) and
Ganguly et al. (1998) also showed that there is no change of self- or tracer diffusion
mechanism 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 that
volume diffusion in olivine and garnet does not show any change of mechanism within the
temperature range of geological interest. From comparison of the intrinsic defect formation
energy (~ 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 in
olivine is in the extrinsic domain. Chakraborty et al. (1994) argued, from consideration of
the intrinsic vacancy content that follows from conservative estimate of defect formation
energy, that intrinsic diffusion is unlikely to be observed in silicates at temperatures below
1300 °C. Wuensch (1982) came to similar conclusion for refractory oxides.
The above conclusion about the extrinsic nature of diffusion in silicates and
refractory 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 interest
since the experimental data are usually collected at temperatures that are higher than
those at which these processes take place in nature. The Arrhenian relation could be
extrapolated linearly to lower temperatures with, of course, due consideration for the
statistical uncertainties associated with the regression of the experimental data. On the
other hand, once the diffusion is in the extrinsic domain, it becomes vulnerable to
Diffusion kinetics in minerals: Principles and applications
293
Fig. 6. Summary of Fe–Mg diffusion data in (a) garnet (open symbols: Ganguly et al., 1998; filled circles:
Chakraborty & Rubie, 1996; circles with inscribed crosses: Cygan & Lasaga, 1985) and (b) olivine of
composition Fo86 (Chakraborty et al., 1994; Meissner et al., 1998) as function of temperature. For garnet (a),
all data have been normalised by Ganguly et al. (1998) to p = 10 kbar, and f O2 corresponding to those defined
by graphite in the system C–O. For olivine (b) the filled symbols represent D(Fe–Mg) data derived from
composition profile determined by analytical transmission electron microscope, whereas the open symbols
represent those derived from compositional profiles determined by an electron microprobe.
294
J. Ganguly
substitutions of ions that have different charge than the host species. Experimental
studies on iron-bearing garnets, however, did not show any significant dependence of the
diffusion 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 by
Chakraborty et al. (1994), that the equilibrium vacancies controlled by the ferrous–ferric
equilibrium in an iron-bearing mineral greatly dominates its vacancy content.
Chakraborty et al. (1994) found that the activation energy for Mg self-diffusion in
nominally pure synthetic forsterite (Fo100) at 1000–1300 °C is 400 (±60) kJ/mol, which
is in contrast to that of 275 (±25) kJ/mol in San Carlos olivine (Fo92) within the same
temperature range. Both sets of experiments were carried out by them following the
same experimental technique. Also the D(Mg) in the San Carlos olivine had a much
stronger fO2 dependence than that in Fo100 (the latter was essentially independent of fO2).
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 from
diffusion-couple experiments at 20 kbar, 950–1325 °C. The retrieved activation energy
is much lower than that determined by Sheng et al. (1992) for Mg tracer diffusion in
spinel at 1 bar, 1261–1553 °C (202±8 kJ/mol vs. 343±8 kJ/mol). The latter workers used
a 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 in
the studies of Sheng et al. (1992) relative to that in their work is due to the extremely
small, probably below the detection limit in microprobe analysis, amount of FeO content
in 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 observed
difference in the activation energies in the two sets of experiments. In summary, one
must 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 amenable
to analytical treatment. These problems have to be dealt with numerically, and
considerable 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.
295
Diffusion kinetics in minerals: Principles and applications
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 dΓ = D(t) dt so that the diffusion equation
(Equation 3b) transforms to
∂C ∂ 2C
=
.
∂Γ ∂x 2
.
(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 in
the 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
t′
Γ = ∫ D(t )dt .
(32)
0
This relation provides an important constraint on the thermal history of the sample in that
the integral of D(t) dt over the postulated T–t path must equal the value of Γ derived from
modelling the compositional zoning. Examples of the application of this concept to
geological problems are discussed in the section on tectono-metamorphic processes.
Let us consider a case in which the observed compositional zoning had developed
during cooling, and suppose that the system had cooled according to an “asymptotic”
relation
1 1
(33)
= + ηt ,
T T0
where T0 is the initial temperature at the onset of cooling, and η is a cooling time constant
with the dimension of K–1t–1. The Arrhenian relation of diffusion coefficient
(Eqn. 30) then transforms to
(34)

Q 1
D = D0 e
or
−
Q
RT
= D0 e
−
 + ηt 

R  T0

,
D(t ) = D(T0 )e −η′t ,
where η′ = Qη/R, and D(T0) is the diffusion coefficient at T0. In general, the time
dependence of D can be expressed as D(t) = D(T0)f(t).
Substituting Equation 34 in 32, we obtain (Ganguly et al., 1994)
t′
Γ ≡ ∫ D(t )dt = −
o
[
]
D (T0 ) −η′t′
[e − 1]..
η′
(35)
296
J. Ganguly
Since the time scale of geological processes is very large (at least for those for which
cooling rates are of any interest), the above equation simplifies to
t′
Γ ≡ ∫ D(t )dt =
0
D(T0 ) D (T0 ) R
.
=
η′
Qη
(36)
Using an exponential cooling model, i.e. T = T0e–αt, Kaiser & Wasserburg (1983)
obtained the following expression for the integral quantity:
t′
Γ ≡ ∫ D (t )dt =
0
D (T0 ) RT0
αQ
(37)
Equations 36 and 37 were used to retrieve the cooling rates of meteorites from the values
of Γ obtained from modelling the observed compositional zoning in minerals according
to 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 “characteristic
temperature”, Tch, and the related diffusion coefficient D(Tch). In a non-isothermal
process, it is always possible to find a temperature, Tch, such that
t′
D(Tch )∆t = ∫ D(t )dt ≡ Γ .
(38)
0
Chakraborty & Ganguly (1992) explored the characteristic temperature that satisfies the
above relation in T–t cycles of metamorphic rocks, which are characterised by a single
thermal peak. They found that Tch ≈ 0.97Tpeak, where Tpeak is the peak metamorphic
temperature 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 mathematical
analysis 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 diffusing
solute and a solvent matrix. By applying chain rule to the expression of flux in a multicomponent system in the linear domain (Eqn. 17), we obtain

∂C
∂C
∂C  ∂C
J1 = −  D11 + D12 2 + D13 3 + ... + D1( n −1) n−1  1
∂C1
∂C1
∂C1  ∂x

or
J1 = − D1 (EB)
∂C1 ,
∂x
(39)
(40)
Diffusion kinetics in minerals: Principles and applications
297
where D1(EB) is the effective binary diffusion coefficient (EBDC) of component 1, and
equals 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 that
Equation 40 has the property of Fickian diffusion, that is, flux ∝ force. Specifically, the
concept of effective binary diffusion holds for diffusion in a semi-infinite diffusion
couple in a multi-component system that does not show any inflection of the diffusion
profile. Chakraborty & Ganguly (1992) discussed applications of this approach in
modelling diffusion profiles in multi-component diffusion couple experiments. An
example 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 metamorphic
rocks. It participates in a large number of cation exchange and discontinuous reactions
that are used to calculate the p–T conditions of rocks from the compositions of the
coexisting mineral phases. It is amenable to geochronological studies using a number of
decay systems, and often shows compositional zoning that preserve records of its
tectono-metamorphic and exhumation history. In addition, since garnet is isotropic,
diffusion in garnet has no directional dependence – a property that offers a practical
advantage in the modelling of garnet compositional zoning. I will discuss here several
types of compositional zoning in garnet in metamorphic rocks from collisional
environments, and retrieval of the time scales of the attendant metamorphic and
exhumation 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 many
terrestrial and planetary samples. When the overgrowth and core segment have
significantly different compositions, as in polymetamorphic rocks, the composite sample
forms a natural diffusion couple with continuous concentration profiles of components
across 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 on
temperature 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 compositional
zoning of this type, I summarise below the analysis of a natural garnet–garnet diffusion
couple by Ganguly et al. (1996a).
Figure 7a shows a backscattered electron image of a composite garnet collected
from the biotite grade rock from eastern Vermont. The couple consists of a
grossular–spessartine garnet that had formed during regional metamorphism on an
almandine core, which had crystallised during an earlier period of contact
metamorphism at 411±5 Ma. The regional metamorphism took place during the Acadian
298
J. Ganguly
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 divalent
cations 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).
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 diffusion
induced compositional zoning between the two segments was too narrow to be clearly
resolved by electron microprobe analyses because of convolution or spatial averaging
effect (Ganguly et al., 1988). The compositional zoning was, thus, determined by an
analytical transmission electron microscope (ATEM), which had negligible convolution
effect because of the very small size of the excited analytical volume resulting from the
small beam size and thinness of the sample to electron transparency. The results are
shown in Figure 7b. Mg profile is not shown since the XMg is between 0.001 (overgrowth)
and 0.06 (core).
Since in this problem we are dealing with a semi-infinite diffusion couple, and
there is no inflection in the diffusion profile, the diffusion problem may be treated in
terms of an effective binary diffusion coefficient, as discussed above. Assuming that the
299
Diffusion kinetics in minerals: Principles and applications
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 couple
is semi-infinite so that the initial concentrations are preserved at sufficiently large
distances from the interface, which is located at x = 0, and that there is no initial
concentration gradient on either side of the interface. For an isothermal diffusion
process, the solution is (Crank, 1983; Equation 2.14)
Ci (t , x) = Ci (0) +

∆C 0 
x
1 − erf

2 
2 Di ( EB)t 

(41)
where ∆C0 represents the initial difference between the concentrations of the
components on the two sides of the couple, and Ci(0) is the lower of the two initial values
of Ci. If diffusion had taken place under condition of variable temperature, then, as
discussed above, Dt in the above equation is to be replaced by Γ (Eqn. 32). Using
Equation 41, a single value of Γ = 7.5×10–12 cm2 was found to match well both Fe and
Ca diffusion profiles (Fig. 7c). No attempt was made to fit the Mn profile because of the
irregularity in the measured data points.
On the basis of the Fe–Mn fractionation data between the overgrowth garnet and
ilmenites, which are present as inclusions within this garnet, the peak metamorphic
temperature 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 Ganguly
et 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 time
scale of the metamorphic process:
∆t =
7.5 ×10 −12 cm 2
,
Di ( EB) (Tch )
(42)
where the term in the denominator represents the effective binary diffusion coefficient
of either Ca or Fe at the characteristic temperature, which is ~ 0.97 × Tpeak.
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 spessartine
crystals, 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 and
Fe 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 inferred
Tch = 343 °C. Ganguly et al. (1996a) discussed the potential uncertainties in the above
calculation 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 the
core–overgrowth interface of garnet.
300
J. Ganguly
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 garnet
according to Grt + Qtz → Opx + Plag at ~ 1000 Ma, while the fracture cleavage within
the 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 be
associated with a regional Pan-African tectono-metamorphic event that has been
interpreted 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 to
the fracture cleavage (Fig. 9) is a consequence of the fact that the composition of garnet
in equilibrium with the symplectitic orthopyroxenes at the p–T condition of M2 is
different from its initial composition, which is preserved in the core, and that the duration
of M2 was too short to homogenise the garnet by volume diffusion. Using the Fe–Mg
distribution coefficient between the garnet rim and adjacent orthopyroxenes and the
thermometric formulation of Ganguly et al. (1996b) for Fe–Mg exchange between
garnet 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 the
compositional zoning in garnet developed and froze before the rock experienced
sufficient cooling following peak M2, so that the reaction diffusion process may be
approximated by an isothermal process.
With the above framework, and assuming that the D(Fe–Mg) is not significantly
affected by compositional change within the diffusion zone, the appropriate diffusion
equation 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 boundary
conditions are C = C0 at x > 0, t = 0 and C = Cr 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
C ( x, t ) = C0 +

 x + vt
1
(Cr − C0 ) erfc
2
 2 Dt

 x − vt

 − vx 
 + exp
erfc
D


 2 Dt


 .

(43)
Ganguly et al. (2001) assumed that the breakdown of garnet within the fracture
cleavage proceeded symmetrically in both directions so that v = RZ/t, where RZ is the
observed half-width of the reaction zone. Substitution of this relation in Equation 43
eliminates 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 Mg
in garnet (Ganguly et al., 1998) at the median composition of the diffusion zone
according to Equation 10.
Fig. 8. Photomicrograph showing reaction texture and fracture cleavage of a
large garnet grain in a granulite sample from the Søstrene Island, Antarctica. The
coarse outer grain symplectite consists primarily of ortho-pyroxene (Opx), and
plagioclase (Plag), which were interpreted to have formed by the breakdown of
garnet during decompression following an earlier metamorphism (M1) at ca.
1000 Ma. The finer grained symplectite mantling the garnet and within the
fracture zones were interpreted to have formed by the breakdown of garnet
according to Grt → Opx + Plag + Spl during a subsequent metamorphism (M2)
accompanying Pan-African collision event at ca. 500 Ma. Scale is 1.7 mm. Cpx:
clinopyroxene, Hbl: hornblende, Ilm: ilmenite, Mag: magnetite
Diffusion kinetics in minerals: Principles and applications
301
302
J. Ganguly
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).
Modelling of the measured compositional zoning in garnet was carried out
according to Equation 43 by linking it with a non-linear optimisation program (see
Ganguly et al., 2001 for details). The initial composition, C0, and time, t, were the
floating variables. The optimisation program finds values of the floating variables that
lead 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 in
Prydz Bay, Fitzsimons (pers. commun.) suggested 17±13 Myr for the duration of peak
Pan-African metamorphism. Comparing the time scale inferred from modelling the
compositional 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 to
smoothen the compositional gradients that developed during the growth of a few mm
size garnets at the lower grade conditions (Chakraborty & Ganguly, 1991). During the
exhumation, however, the rim composition of garnet re-equilibrates with the matrix in
response to the changing p–T condition, but the interior of the garnet does not come to
equilibrium with the matrix because of the slow volume diffusion kinetics. Thus, the
garnet crystals develop compositional gradients between the core and rim segments
during 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
Diffusion kinetics in minerals: Principles and applications
303
composition “freezes” at ~ 500–550 °C for cooling rates experienced during the
exhumation of regionally metamorphosed rocks.
Lasaga (1983) developed the theoretical groundwork for the retrieval of cooling
rate 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) have
since followed this basic idea to retrieve cooling rates of metamorphic rocks and
planetary samples from the retrograde compositional zoning of minerals, especially
garnet. Spear & Parrish (1996) and Weyer et al. (1999) compared the cooling rates
obtained from modelling the retrograde compositional zoning of garnet in metamorphic
rocks with those constrained by the closure age and closure temperatures (TC) of multiple
geochronological systems. They found that the cation diffusion data in garnet by
Chakraborty & Ganguly (1992) yield cooling rates that are in good agreement with those
obtained 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 core
composition. The sample is from northern Sikkim in the eastern Himalayas, where the
metamophic isograds are disposed in an arcuate regional fold pattern. The thermal
evolution and exhumation of the Himalayan metamorphic rocks have been subjects of
extraordinary interest among earth scientists as these record the thermo-tectonic
evolution of rocks in a major collisional and continental subduction environment, and
also 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 cooling
and exhumation rates of rocks from the upper part of the High Himalayan Crystalline
Complex (HHC) in the Sikkim–Darjeeling section on the basis of the retrograde
compositional zoning of garnet crystals along with phase equilibrium constraints. The
HHC is bounded on the south and north by the Main Central Thrust (MCT) zone and the
South Tibetan Detachment System (STDS), respectively. The MCT is a southerly
directed thrust (20–23 Ma) whereas the STDS is a northerly directed system of normal
faults (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 in
Figure 10. The compositions of biotite grains in contact with garnet were homogeneous
and were essentially the same as those of other biotite grains within the same thin
section. 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 behaved
essentially as a homogeneous infinite reservoir of the exchanging components, Fe2+ and
Mg.
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 cooling
rate smaller) than that due to diffusion normal to the plane. The most definitive way to
ensure normalcy of the traverse with respect to the interfacial plane is to obtain three-
304
J. Ganguly
0.30
(a)
0.26
X(Mg)
(b)
X(m) = X(t)/cosQ
0.22
Q
Interface for (a)
46
Interface for (b)
relative to (a)
0.18
0
40
80
120
Distance (µm)
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 diffusion
equation, 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 the
orientation 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 with
respect 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.)
dimensional image of the rock by computer aided X-ray tomography, and cut the thin
section normal to the interfacial plane (Carlson & Denison, 1992). As a practical
substitute 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 cooling
rate, and chose a profile that did not seem to have been affected significantly by the
rotation of the interface from the vertical (see Ganguly et al., 2000, for further details
about this procedure).
Multiplying both sides of Equation 31 by a2, we obtain
dC d 2C
,
=
dΓr
dx r2
(44)
Diffusion kinetics in minerals: Principles and applications
305
where Γr and xr are
Γ
x
and xr = .
2
a
a
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’ relation
given by Equation 33, (b) the interface composition of garnet was governed by exchange
equilibrium between garnet and biotite, (c) biotite behaved as a homogeneous infinite
reservoir of the exchanging components, as justified above, and (d) the garnet
composition was homogeneous at the peak metamorphic condition (T0). Equation 44 was
solved numerically subject to the boundary condition that dC/dx = 0 at x = a. The
normalising distance a may be any distance from the interface where dC/dx = 0 (i.e. it
need not be the distance to the core of the crystal). As above, the numerical program was
linked to an optimisation program that was allowed to choose the best value of X(T0), but
it returned the observed core composition of garnet as the best choice implying that it was
not 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, which
enables 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. The
retrieved value is 8.18×10–5 cm2. As a next step, Ganguly et al. (2000) calculated the
exhumation velocity (Vz) of the rock by calculating the T–t path for a given Vz,
integrating D(t)dt over this path, and repeating the process until the above value of
∫D(t)dt was obtained. This procedure yields Vz = 2 mm/year. However, the phase
equilibrium constraints that are imposed by the retrograde reaction relations in the rock,
which could be deduced from petrographic observations, require a nearly isothermal
exhumation, corresponding to Vz ≈ 15 mm/year from a depth of ~ 34 km to ~ 15 km. The
potential geodynamic implications of this change of exhumation velocity have been
discussed by Ganguly et al. (2000, 2001).
Γr =
Diffusion modification of growth zoning in garnet
A mineral develops compositional zoning during the growth process if it fractionates
components with the matrix, and the diffusion within both the matrix and the mineral is
slow compared to the growth velocity of the crystal. This may be understood by
considering crystal growth in incremental steps. Owing to the fractionation of components
with 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, and
enriched in those that have the reverse behaviour. Consequently, during the next growth
step, the adjacent matrix would have a bulk composition that is effectively different from
that in the previous step. Thus, the segment of the crystal that had grown in the second step
would 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. Such
growth zoning is very common in the garnet zone rocks of regionally metamorphosed
306
J. Ganguly
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 and
zoned rim, as in the Himalayan garnet sample discussed above. On the other hand, if the
initial growth zoning were partly or completely preserved, then the crystal would have
complicated zoning profiles in which some of the components could show extrema in
their profiles from the centre to the rim of a grain. Modelling of these partially modified
growth zoning profiles hold great promise in the retrieval of the full thermal history of the
rock. Discussion of this modelling procedure is, however, beyond the scope of this review,
as it requires consideration of crystal growth and resorption processes simultaneously
with volume and intergranular diffusion kinetics. The interested readers are referred to
Florence & Spear (1993), Okudaira (1996) and Carlson (2002) for an appreciation of the
topic 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 using
the program PROFILER, and Alexander von Humboldt Foundation for support through
a research award (Forschungspreis) during the preparation of this manuscript at the
Bayerisches Geoinstitut, Bayreuth, Germany.
References
Barrer, R.M., Bartholomew, R.F. & Rees, L.V.C. (1963): Ion exchange in porous crystals. Part II. The
relationship between self and exchange-diffusion coefficients. J. Phys. Chem. Solids, 24:309–317.
Bokshtein, B.S., Bokshtein, S.Z. & Zhukhovitskii, A.A. (1985): Thermodynamics and kinetics of diffusion in
solids. New Delhi: Oxonian Press.
Borg, R.J. & Dienes, G.J. (1988): An introduction to solid state diffusion. Boston (Mass.): Academic Press.
Brady, J.B. (1975): Reference frames and diffusion coefficients. Am. J. Sci., 275:954–983.
Brady, J.B. & McCallister, R.H. (1983): Diffusion data for clinopyroxenes from homogenization and selfdiffusion experiments. Am. Mineral., 68:95–105.
Buening, D.K. & Buseck, P.R. (1973): Fe-Mg lattice diffusion in olivine. J. Geophys. Res., 78:6852–6862.
Carlson, W.D. (2002): Scales of disequilibrium and rates of equilibration during metamorphism. Am. Mineral.,
87:185–204.
Carlson, W.D. & Denison, C. (1992): Mechanism of porphyroblast crystallization: Results from highresolution computed X-ray tomography. Science, 257:1236–1239.
Carslaw, H.C. & Jaeger, J.C. (1959): Conduction of heat in solids. Oxford: Clarendon Press.
Chakraborty, S. (1994): Relationships between thermodynamic mixing and diffusive transport in multicomponent solutions: Some constraints and potential applications. J. Phys. Chem., 98:4923–4926.
Chakraborty, S. (1997): Rates and mechanisms of Fe-Mg interdiffusion in olivine at 980–1300 °C. J. Geophys.
Res., 102:12,317–12,331.
Chakraborty, S. & Ganguly, J. (1991): Compositional zoning and cation diffusion in aluminosilicate garnets.
In Ganguly, J. (ed.): Diffusion, atomic ordering and mass transport /Adv. Phys. Geochem., 8/. Berlin:
Springer-Verlag, 120–175.
Diffusion kinetics in minerals: Principles and applications
307
Chakraborty, S. & Ganguly, J. (1992): Cation diffusion in aluminosilicate garnets: experimental determination
in spessartine-almandine diffusion couples, evaluation of effective binary diffusion coefficients, and
applications. Contrib. Mineral. Petrol., 111:74–86.
Chakraborty, S. & Ganguly, J. (1994): A method to constrain thermodynamic mixing properties and diffusion
data in multicomponent solutions. Mater. Sci. Forum, 155-156:279–283.
Chakraborty, S. & Rubie, D. (1996): Mg tracer diffusion in aluminosilicate garnets at 750–850 °C. 1 atm. and
1300 °C, 8.5 GPa. Contrib. Mineral. Petrol., 122:406–414.
Chakraborty, S., Farver, J.R., Yund, R.A. & Rubie, D.C. (1994): Mg tracer diffusion in synthetic forsterite and
San Carlos olivine as a function of P, T and fO2. Phys. Chem. Miner., 21:489–500.
Chakraborty, S., Dingwell, D.B. & Rubie, D.C. (1995a): Multi-component diffusion in ternary silicate melts
in the system K2O–Al2O3–SiO2: I. Experimental measurements. Geochim. Cosmochim. Acta,
59:225–264.
Chakraborty, S., Dingwell, D.B. & Rubie, D.C. (1995b): Multi-component diffusion in ternary silicate melts
in the system K2O–Al2O3–SiO2: II. Mechanisms, systematics, and geological applications. Geochim.
Cosmochim. Acta, 59:265–277.
Cherniak, D.J. (1993): Lead diffusion in titanite and preliminary results on the effects of radiation damage on
Pb transport. Chem. Geol., 110:177–194.
Christoffersen, R., Yund, R.A. & Tullis, J. (1983): Inter-diffusion of potassium and sodium in alkali feldspars:
diffusion couple experiments. Am. Mineral., 68:1126–1133.
Cooper, A.R. (1968): The use and limitation of the concept of an effective binary diffusion coefficient for
multicomponent diffusion. In Wachtman, J. B., Jr., & Franklin, A.D. (eds.): Mass transport in oxides.
Proc. Symp. held at Gaithersburg, Maryland, October 22–25, 1967 /U.S. Nat. Bur. Stand. Spec. Publ.,
296/.Washington (D.C.): U.S. Gov. Print. Off., 79–84.
Crank, J. (1983): Mathematics of diffusion. Oxford: Oxford Univ. Press.
Cullinan, H.T., Jr. (1965): Analysis of flux equation of multicomponent diffusion. Ind. Eng. Chem. Fundam.,
4:133–139.
Cygan, R.T. & Lasaga, A.C. (1985): Self-diffusion ot magnesium in garnet at 750 to 900 °C. Am. J. Sci.,
285:328–350.
Darken, L.S. (1948): Diffusion, mobility and their interrelation through free energy in binary metallic systems.
Am. Inst. Min. Metall. Eng. Trans., 175:184–201.
Dempster, T.J. (1985): Garnet zoning and metamorphism of the Barrovian type area, Scotland. Contrib.
Mineral. Petrol., 89:30–38.
Dieckmann, R. & Schmalzried, H. (1975): Point defects and cation diffusion in magnetite. Z. Phys. Chem.,
Neue Folge, 96:331–333.
Dieckmann, R. & Schmalzried, H. (1977): Neue Ergebnisse zum Ionentransport in Magnetit und ihre
Bedeutung für die Eisenoxidation. Arch. Eisenhüttenw., 48:611–616.
Fitzsimons, I.C.W. (1996): Metapelitic migmatites from Brattstrand Bluffs, East Antarctica - metamorphism,
melting and exhumation of the mid crust. J. Petrol., 37:395–414.
Fitzsimons, I.C.W. (2000): A review of tectonic events in the East Antarctica Shield, and their implications for
Gondwana and earlier supercontinents. J. African Earth Sci., 31:3–23.
Florence, F.P. & Spear, F.S. (1993): Influences of reaction history and chemical diffusion on P-T calculations
for staurolite schists from the Littleton Formation, north-western New Hampshire. Am. Mineral.,
78:345–359.
Ganguly, J. & Saxena, S.K. (1987): Mixtures and mineral reactions. Berlin: Springer-Verlag.
Ganguly, J., Bhattacharya, R. N. & Chakraborty, S. (1988): Convolution effect in the determination of
compositional zoning by microprobe step scans. Am. Mineral., 73: 901–909.
Ganguly, J., Yang, H. & Ghose, S. (1994): Thermal history of mesosiderites: Quantitative constraints from
compositional zoning and Fe-Mg ordering in orthopyroxenes. Geochim. Cosmochim. Acta,
58:2711–2723.
Ganguly, J., Chakraborty, S., Sharp, T. & Rumble, D. (1996a): Constraint on the time scale of biotite-grade
metamorphism during Acadian orogeny from a natural garnet-garnet diffusion couple. Am. Mineral.,
81:1208–1216.
308
J. Ganguly
Ganguly, J., Cheng, W. & Tirone, M. (1996b): Thermodynamics of aluminosilicate garnet solid solution: new
experimental data, an optimized model, and thermometric applications. Contrib. Mineral. Petrol.,
126:137–151.
Ganguly, J., Cheng, W. & Chakraborty, S. (1998): Cation diffusion in aluminosilicate garnett: Exprerimental
determination in pyrope–almandine diffusion couples. Contrib. Mineral. Petrol., 131:171–180.
Ganguly, J., Dasgupta, S., Cheng, W. & Neogi, S. (2000): Exhumation history of a section of the Sikkim
Himalayas, India: records in the metamorphic mineral equilibria and compositional zoning. Earth.
Planet. Sci. Lett., 183:471–486.
Ganguly, J., Hensen, B. & Cheng, W. (2001): Reaction texture and Fe-Mg zoning in granulite garnet from
Søstrene Island, Antarctica: Modeling and constrain on the time scale of metamorphism during PanAfrican collisional event. Proc. Indian Acad. Sci., Earth Planet. Sci., 110:305–312.
Glicksman, M.E. (2000): Diffusion in solids: Field theory, solid state principles and applications. New York
(N.Y.): Wiley.
Glicksman, M.E. & Lupulescu, A.O.: Dynamics of multicomponent diffusion with zero flux planes. Acta Mater.,
in press.
Goldsmith, J.R. (1991): Pressure enhanced Al-Si diffusion and oxygen isotope exchange. In Ganguly, J. (ed.):
Diffusion, atomic ordering and mass transport /Adv. Phys. Geochem., 8/. Berlin: Springer-Verlag, 221–247.
Graham, C. & Elphick, S.C. (1991): Some experimental constraints on the role of oxygen and hydrogen
diffusion and Al–Si interdiffusion in silicates. In Ganguly, J. (ed.): Diffusion, atomic ordering and mass
transport /Adv. Phys. Geochem., 8/. Berlin: Springer-Verlag, 248–285.
Haasen, P. (1978): Physical metallurgy. Cambridge: Cambridge Univ. Press.
Hartley, G.S. & Crank, J. (1949): Some fundamental definitions and concepts in diffusion processes. Faraday
Soc. Trans., 45:801–818.
Hensen, B.J., Zhou, B. & Thost, D.E. (1995): Are reaction textures reliable guides to metamorphic histories?
Timing constraints from Sm-Nd chronology for “decompression” textures in granulites from Søstrene
Island, Prydz Bay, Antarctica. Geol. J., 30: 261–271.
Hermeling, J. & Schmalzried, H. (1984): Tracer diffusion of the Fe-cations in olivine (FexMg1–x)2SiO4 (III).
Phys. Chem. Miner., 11:161–166.
Joestein, R. (1991): Grain boundary diffusion kinetics in silicate and oxide minerals. In Ganguly, J. (ed.):
Diffusion, atomic ordering and mass transport /Adv. Phys. Geochem., 8/. Berlin: Springer-Verlag,
345–395.
Jurewicz, A.J.G. & Watson, B. (1988): Cations in olivine, Part 2: Diffusion in olivine xenocrysts, with
applications to petrology and mineral physics. Contrib. Mineral. Petrol., 99:186–201.
Kaiser, T. & Wasserburg, G. J. (1983): The isotopic composition and concentration of Ag in iron meteorites
and the origin of exotic silver. Geochim. Cosmochim. Acta, 47:43–58.
Lasaga, A.C. (1979): Multicomponent exchange and diffusion in silicates. Geochim. Cosmochim. Acta,
43:455–468.
Lasaga, A.C. (1983): Geospeedometry: an extension of geothermometry. In Saxena, S.K. (ed.): Kinetics and
equilibrium in mineral reactions. /Adv. Phys. Geochem., 2/. Berlin: Springer-Verlag, 81–114.
Lasaga, A.C. (1998): Kinetic theory in the earth sciences. Princeton (N.J.): Princeton Univ. Press.
Liermann, H-P. & Ganguly, J. (2001): Compositional properties of coexisting orthopyroxene and spinel in
some Antarctic diogenites: Implications for thermal history. Meteor. Planet. Sci., 36:155–166.
Liermann, H-P. & Ganguly, J. (2002): Diffusion kinetics of Fe2+ and Mg in aluminous spinel: experimental
determination and applications. Geochim. Cosmochim. Acta, 66:2903–2913.
Lindsley, D.H., Grover, J.E. & Davidson, P.M. (1981): The thermodynamics of the pyroxene
(Mg2Si2O6–CaMgSi2O6) join: a review and an improved model. In Saxena, S.K. (ed.): Thermodynamics
of minerals and melts /Adv. Phys. Geochem., 1/. Berlin: Springer-Verlag, 149–175.
Lindstrom, R., Viitanen, M., Juhanoja, J. & Höltä, P. (1991): Geospeedometry of metamorphic rocks: examples
in the Rantasalami-Sulkava and Kiuruvesi areas, eastern Finland. Biotite-garnet diffusion couples. J.
Metamorph. Geol., 9:181–190.
Loomis, T.P. (1978): Multicomponent diffusion in garnet: I. Formulation of isothermal models. Am. J. Sci.,
278:1099–1078.
Diffusion kinetics in minerals: Principles and applications
309
Manning, J.R. (1968): Diffusion kinetics for atoms in crystals. Princeton (N.J.): D. Van Nostrand.
Mapother, D.E., Crooks, H.N. & Mautter, J.R. (1950): Self-diffusion of sodium in sodium chloride and sodium
bromide. J. Chem. Phys., 18:1231–1236.
Meissner, E., Sharp, T. & Chakraborty, S. (1998): Quantitative measurement of short compositional profiles
using analytical transmission microscopy. Am. Mineral., 83:546–552.
Misener, D.J. (1974): Cation diffusion in olivine to 1400 °C and 35 kbar. In Hofmann, A.W., Giletti, B.J.,
Yoder, H.S., Jr. & Yund, R.A. (eds.): Geochemical transport and kinetics /Carnegie Inst. Wash. Publ.,
634/. Washington (D.C.): Carnegie Inst., 117–129.
Morioka, M. & Nagasawa, H. (1991): Ionic diffusion in olivine. In Ganguly, J. (ed.): Diffusion, atomic
ordering and mass transport /Adv. Phys. Geochem., 8/. Berlin: Springer-Verlag, 176–197.
Nakamura, A. & Schmalzried, H. (1983): On the nonstoichiometry and point defects of olivine. Phys. Chem.
Miner., 10:27–37.
Naylor, R. S. (1971): Acadian orogeny: abrupt and brief event. Science, 172:558–560.
Niemeier, D., Chakraborty, S. & Becker, K.D. (1996): A high temperature Mössbauer study of the directional
geometry of diffusion in fayalite, Fe2SiO4. Phys. Chem. Miner., 23:284.
Nye, J. F. (1957): Physical properties of crystals. Oxford: Clarendon Press.
Okudaira, T. (1996): Temperature-time path for the low-pressure Ryoke metamorphism, Japan, based on
chemical zoning in garnet. J. Metamorph. Geol., 14: 427–440.
Onsager, L. (1931a): Reciprocal relations in irreversible processes-I. Phys. Rev., 37:405–426.
Onsager, L. (1931b): Reciprocal relations in irreversible processes-II. Phys. Rev., 38:2265–2279.
Onsager, L. (1945): Theories and problems of liquid diffusion. Ann. N.Y. Acad. Sci., 46:241–265.
Ottonello, G. 1997): Principles of geochemistry. New York (N.Y.): Columbia Univ. Press.
Pownceby, M.I., Wall, V.J., & O’Neill, H.St.C. (1991): An experimental study of the effect of Ca on garnetilmenite exchange equilibria. Geochim. Cosmochim. Acta, 76:1582–1590.
Schwandt, C.S., Cygan, R.T. & Westrich, H.R. (1998): Magnesium self-diffusion in orthoenstatite. Contrib.
Mineral. Petrol., 130:390–396.
Sheng, Y. J., Wasserburg, G. J. & Hutcheon, I. D. (1992): Self-diffusion of magnesium in spinel and in
equilibrium melts: Constraints on flash heating of silicates. Geochim. Cosmochim. Acta, 56:2535–2546.
Shewmon, P.G. (1963): Diffusion in solids. New York (N.Y.): McGraw-Hill.
Smigelskas, A.D. & Kirkendall, E.D. (1947): Zinc diffusion in alpha brass. Trans. Am. Inst. Mining Metal.
Eng., 171:130–142.
Sneeringer, M., Hart, S.R. & Shimizu, N. (1984): Strontium and samarium diffusion in diopside. Geochim.
Cosmochim. Acta, 48:1589–1608.
Spear, F. & Parrish, R. (1996): Petrology and cooling rates of the Valhalla Complex, British Columbia, Canada.
J. Petrol., 37:733–765.
Thost, D.E., Hensen, B.J. & Motoyoshi, Y. (1991): Two stage decompression in garnet-bearing mafic
granulites from Søstrene Island, Prydz Bay, East Antactica. J. Metamorph. Petrol., 9:245–256.
Tirone, M. (2002): Diffusion of rare earth elements in garnets and pyroxenes: experiment, theory, and
applications. Ph.D. Thesis, Univ. of Arizona, U.S.A.
Toor, H.L. (1964): Solution of the linearized equations of multicomponent mass transfer: II. Matrix methods.
J. Am. Inst. Chem. Eng., 8:460–465.
Weyer, S., Jarick, J. & Mezger, K. (1999): Quantitative temperature-time information from retrograde
diffusion zoning in garnet: Constraints for the P–T–t history of the Central Black Forest, Germany. J.
Metamorph. Geol., 17: 449–461.
Wuensch, B.J. (1982): Diffusion in stoichiometric close-packed oxides. In Beniere, F. & Catlow, C.R.A. (eds.):
Mass transport in oxides /NATO ASI Ser. B, 97/. New York (N.Y.): Plenum Publ. Co., 353–376.
Yund, R.A. & Snow, E. (1989): Effects of hydrogen fugacity and confining pressure on the inter-diffusion rate
of NaSi-CaAl in plagioclase. J. Geophys. Res., 94:10,662–10,668.
Yund, R.A., Quigley, J. & Tullis, J. (1989): The effect of dislocations on bulk diffusion in feldspars during
metamorphism. J. Metamorph. Geol., 7:337–341.