J. metamorphic Geol., 2009, 27, 83–91
doi:10.1111/j.1525-1314.2008.00805.x
Matrix rheology effects on reaction rim growth II: coupled
diffusion and creep model
D. W. SCHMID,1 R. ABART,2 Y. Y. PODLADCHIKOV1 AND R. MILKE2
1
Physics of Geological Processes, University Oslo, Pb 1048 Blindern, 0316 Oslo, Norway (schmid@fys.uio.no)
2
Institute of Geological Sciences, Free University Berlin, Malteserstrasse 74-100, 12249 Berlin, Germany
ABSTRACT
Chemical reactions and phase changes generally involve volume changes. In confined settings this will
cause a mechanical deformation of the matrix that surrounds the reaction sites where the volume
change takes place. Consequently, mineral reactions and the mechanical response of the rock matrix
are coupled. A companion paper in this issue illustrates this coupling with experiments where quartz
and olivine react to form enstatite reaction rims under ambient conditions of 1 GPa and 1000 C. It
has been demonstrated that for identical run conditions, the thickness of the reaction rims depends on
whether quartz grains are embedded in an olivine matrix or olivine grains are included in a quartz
matrix. The experimental conditions, the nature of the results, and the large volume change of the
reaction ()6%) leave only viscous creep as a viable matrix response to the reaction progress. A model
is developed for this reaction, which combines diffusion of chemical components through the growing
rim and viscous creep of the matrix. The resulting rate law for reaction rim growth in spherical
geometry shows that the progress rate is proportional to the reaction overstepping and controlled by
the slower of the two competing processes; either diffusion
pffiffi or creep. If diffusion is rate limiting the
usual linear proportionality between rim growth and t results. However, if viscous creep is rate
limiting, then the reaction rates are reduced and may become effectively creep controlled. With respect
to the experiments in the companion paper it is inferred that the effective viscosity of the two matrix
materials, i.e. polycrystalline quartz and olivine, differ by approximately one order of magnitude with
the quartz being the stronger one. The absolute values of the inferred viscosities correspond well to
published flow laws. The rheological properties of natural rocks are well within the parameter range
for which significant mechanical control on reaction rim growth is expected. This implies that for the
interpretation of natural reaction rims and corona structures both diffusion and mechanical control
need to be considered. In addition the mechanical effect also needs to be considered when
interdiffusion coefficients are retrieved from rim growth experiments. This should also be considered
for geospeedometry analyses. Furthermore, the control on reaction rate because of slow creep of the
matrix is expected to be even more important, compared to the experiments, under colder crustal
conditions and may contribute substantially to the frequent observation of only partially completed
reactions. We suggest that this phenomenon is referred to as Ômechanical closureÕ, which may be an
important mechanism in the kinetic displacement of the boundaries between the stability fields of
phase assemblages.
Key words: diffusion; mechanical effects; reaction; rheology; rim growth.
INTRODUCTION
Much growth and dissolution of minerals takes place
under conditions of macroscopically non-hydrostatic
stress. It has long been recognized that mineral reactions and non-hydrostatic stress may interfere. Grain
scale stress heterogeneity and anisotropy may be
internally produced through volume changes associated with phase transformation and mineral reactions
(Fletcher & Merino, 2001; Lenze et al., 2005). Reaction-induced stress is expected to produce initially
elastic and later ductile deformation (Rubie &
Thompson, 1985). Reaction-induced stress acts against
2008 Blackwell Publishing Ltd
the reaction by which it is generated, and it may slow
down and eventually quench the respective reaction
(Hacker & Peacock, 1995; Paterson, 1995; Mosenfelder
& Bohlen, 1997).
A number of models have been developed that
analyse the mechanical effects of a matrix (elastic or
elasto-plastic) on an inclusion that undergoes a phase
transformation (e.g. Lee et al., 1980; Morris, 1992,
2002; Fischer et al., 1994; Zhang, 1998; Barron, 2003;
Guiraud & Powell, 2006). Yet, the interplay between
net transfer mineral reactions, i.e. reactions that
involve chemical transport, and reaction-induced stress
is poorly understood. Transport-controlled mineral
83
84 D. W. SCHMID ET AL.
reactions typically produce spatially organized microstructures such as reaction rims or corona structures
(Joesten, 1977), which form between two mutually
incompatible phases or mineral assemblages (e.g.
Ashworth et al., 1998; Markl et al., 1998; Abart et al.,
2001; Milke et al., 2001). Rim growth experiments in
simple systems such as MgO-SiO2 (Brady & McCallister, 1983; Fisler et al., 1997; Yund, 1997; Milke et al.,
2001), CaCO3 – SiO2 (Milke & Heinrich, 2002), or
MgO – Al2O3 (Watson & Price, 2002) were used
to extract effective component diffusivity from
measured rim growth rates. So far, all experimental
rim growth studies rely on the assumption of isothermal-isobaric conditions. However, such a scenario is
not compatible with the notion that minerals are
mechanically strong, i.e. can sustain differential stress.
As a consequence, in a mineral aggregate, the local
volume change will act against the confinement; related
to this is the Ôforce of crystallizationÕ concept that goes
back a century or more (Taber, 1916; see also Vernon,
2004). Carmichael (1987), Wheeler (1987), and
Fletcher & Merino (2001) argued for the existence of
reaction-induced variations in local stress in the context of reaction rim growth. The potential effects of
reaction-induced non-hydrostatic stress on rim growth
rates have, however, not been addressed in a systematic
manner.
Milke et al. (2009) performed rim growth experiments in the system MgO-FeO-SiO2. They used
olivine and quartz starting materials to grow orthopyroxene reaction rims at quartz-olivine interfaces.
To test the potential effect of matrix rheology on rim
growth rate, they used different matrix-inclusion
arrangements. In one set of experiments, about 10
vol.% of olivine grains with an average grain size of
150 lm were immersed in a matrix of fine grained
quartz (grain size 20 lm), and in the other set,
about 10 vol.% of quartz grains with an average
grain size of 150 lm were immersed in a matrix of
fine grained olivine (grain size 20 lm). The samples
were loaded into the capsules without drying so that
the total water content was about 0.1 wt%. To ensure
similar run conditions for both matrix-inclusion
arrangements, both mixtures were placed into a
single capsule. FTIR analyses of the run products
showed that within 80 h run duration olivine equilibrated with an aqueous fluid throughout the capsule.
In contrast, quartz largely retained its original low
water content.
Despite similar run conditions, the width of the
orthopyroxene rims are significantly different between
the two compartments of the assembly: orthopyroxene
rims have an average thickness of 10.3 lm in the
compartment with quartz inclusion in the olivine
matrix and they are on average 6.1 lm thick in the
compartment where olivine inclusions are immersed in
the quartz matrix.
The experimental setup rules out difference in the
effective component diffusivity between the two
experimental configurations as an explanation for the
difference in reaction rim thickness. The volume
change of the reaction is too large to be accommodated
by purely elastic response of the matrix. Furthermore,
the two matrix materials creep under the given conditions and strain rates (see detailed discussion below
and Milke et al., 2009). Hence the bulk of the volume
change will be accommodated by creep and we therefore develop a coupled reaction-deformation model
where the chemical mass transport associated with the
rim growth reaction is combined with the viscous creep
response of the matrix material. The importance of
viscous creep has, in the context of phase change
experiments, previously been pointed out by a number
of authors (Liu et al., 1998; Morris, 1992; Morris,
2002; Mosenfelder et al., 2000; Rubie & Thompson,
1985).
MODEL DERIVATION
Our model deals with an isolated (dilute assumption),
perfectly spherical inclusion of reactant mineral A
in a matrix that consists of the reactant mineral B.
The product mineral C is assumed to form a continuous
rim between A and B. It is assumed furthermore that
the interfaces between the reactant and product phases
are perfectly mobile so that the chemical potentials of
the mobile components are continuous functions of the
radial distance across the interfaces. The setting of rim
growth associated with a reaction of the type
A + B = C is illustrated schematically in Fig. 1.
B
C
A
rB
rA
rC
φ
r
θ
Fig. 1. Cross-sectional sketch of the analysed three-dimensional
spherical system. The inclusion A and the matrix B represent
forsterite or quartz depending on the configuration; C represents
the enstatite rim that is produced by the reaction. rA and rC are
the radial distances of the inner and outer limit of the reaction
rim. rB is the hypothetical inner interface of matrix B if there was
no mechanical adjustment to the volume change caused by the
reactions. The observed mechanical response is indicated by the
arrows. B is assumed to extend to infinity where a constant
pressure boundary condition is applied; the temperature is
constant throughout the entire domain.
2008 Blackwell Publishing Ltd
COUPLED DIFFUSION AND CREEP MODEL 85
Mass and volume balance
At the run conditions of 1000 C and 1 GPa, olivine
and quartz cannot coexist stably and an orthopyroxene
rim grows from the original quartz-olivine interface
into both directions, replacing quartz on one side and
olivine on the other. The bulk reaction may be written
as:
olivine ((Mg, Fe)2 SiO4 Þ þ quartzðSiO2 Þ
¼ orthopyroxeneððMg, FeÞ2 Si2 O6 Þ
ð1Þ
For the purpose of this derivation the complexities
that may arise from the partitioning of Fe–Mg
between olivine and orthopyroxene are neglected, and
both are considered as stoichiometric phases in
the MgO–SiO2 model system (i.e. En = enstatite
(Mg2Si2O6), Fo = forsterite (Mg2SiO4), and Qtz =
quartz (SiO2)). In this case, rim growth requires
transfer of SiO2 and ⁄ or MgO components between the
two reaction interfaces where orthopyroxene replaces
quartz and olivine, respectively, see Fig. 2. Note that
molar quantities are used throughout this paper unless
stated otherwise.
The bulk reaction may be split into two half reactions (Abart et al., 2004). At the enstatite ⁄ forsterite
interface:
1 1
þ c En þ ð1 cÞMgO
ð2Þ
1Fo þ cSiO2 ¼
2 2
and at the enstatite ⁄ quartz interface as
1 1
c En þ cSiO2
ð1 cÞMgO þ 1Qtz ¼
2 2
ð3Þ
The coefficient c expresses the relative fluxes of SiO2
and MgO in molar units. For feasible mass balance
scenarios 0 £ c £ 1. Using the thermodynamic data of
Holland & Powell (1998), a negative volume change
of )6% is calculated for the reaction at the conditions
of the rim growth experiments. Microstructural and
textural evidence suggests that, irrespective of the
matrix-inclusion arrangement, rim growth occurs at
close to constant volume replacement at the enstatiteforsterite interface (Abart et al., 2004; Milke et al.,
2009) and the volume change is largely accommodated
by creep at the enstatite-quartz reaction front. In terms
of mass balance this implies that c 0.39.
Reaction affinity and the driving force for component
diffusion
The orthopyroxene reaction rims typically have sharp
interfaces with the reactant minerals indicating
attainment of local equilibrium during transportcontrolled rim growth (Thompson, 1959; Korzhinskii,
1970). In particular, it is assumed that thermodynamic
equilibrium is maintained across the enstatite-quartz
and across the enstatite-forsterite reaction fronts. This
implies that at given pressure and temperature conditions the chemical potentials of the mobile components, lMgO and lSiO2, are fixed at the enstatite ⁄
forsterite interface through:
lSiO2 ¼ Df GFo þ Df GEn
1
lMgO ¼ Df GFo Df GEn
2
ð4Þ
and at the enstatite ⁄ quartz interface through
lSiO2 ¼ Df GQtz
ð5Þ
1
lMgO ¼ Df GEn Df GQtz
2
DfGi denotes the Gibbs free energy of formation from
the elements of phase i. Note that DfGi = DfGi(P,T),
where P is pressure and T is temperature. The difference in chemical potentials of the respective
components between the enstatite ⁄ quartz and the
enstatite ⁄ forsterite interfaces is as follows:
DlSiO2 ¼ þDf GQtz þ Df GFo Df GEn ¼ Dr G
DlMgO ¼ Df GQtz Df GFo þ Df GEn ¼ þDr G
ð6Þ
where DrG is the Gibbs free energy of the reaction as
written in Eqn (1).
Diffusion of SiO2 and MgO is driven by gradients in
the respective chemical potentials. For the given
spherical geometry only gradients in r must be considered and we can express the flux of component i
through
@li
ð7Þ
@r
where Li is the phenomenological coefficient of diffusion. Assuming that no sources or sinks exist within
the enstatite rim (div(J) = 0) results in a chemical
potential gradient within the enstatite rim, rA £ r £ rC,
that has the following form
Ji ¼ Li
Fig. 2. Radial section illustrating the component fluxes and
hypothetical chemical potentials through the rim. Note that
irrespective of the actual configuration the fluxes coming out of
A are defined positive as indicated by the arrows.
2008 Blackwell Publishing Ltd
86 D. W. SCHMID ET AL.
@li
rA rC D r G
¼
@r
rA rC r2
ð8Þ
Note that both rA and rC change during rim growth.
The sign of the right hand side term follows the convention of Fig. 2. It is negative for cases of i = SiO2
and A = Qtz and for i = MgO and A = Fo; it is
positive for i = SiO2 and A = Fo and for i = MgO
and A = Qtz. The boundary conditions are that the
chemical potentials at the interfaces must correspond
to the values given in Eqs (4) and (5).
The volume of the reactive inclusion A that is consumed per time is through mass balance related to
component fluxes by:
dVA ðtÞ
¼ ðJSiO2 JMgO ÞSA VA
dt
ð9Þ
VA and VA designate the total and the molar volume of
2
is the
the reactive inclusion, respectively. SA = 4p rA
surface area of the inclusion A. Since outward fluxes
are defined positive the sign of the right hand side term
is negative for the olivine in quartz arrangement and
positive otherwise. Combining Eqs (7), (8) and (9) we
obtain:
@lMgO
@lSiO2
dVA ðtÞ
¼ LSiO2
þLMgO
SA VA
dt
@r
@r
ð10Þ
rA rC Dr G V
¼L
S
;
A A
rA rC r2
where L = LSiO2 + LMgO is the combined phenomenological coefficient of diffusion, which accounts for
the combined diffusive fluxes of SiO2 and MgO.
Rate law for reaction rim growth in spherical geometry
It is convenient to define reaction progress, n(t), as the
fraction of the reactive inclusion that has been
consumed at time t:
nðtÞ ¼ 1 VA ðtÞ
V0
ð11Þ
where V0 = VA(t = 0) is the initial volume of the
reactive inclusion. Correspondingly r0 = rA(t = 0) is
the initial radius of A. From this the following volume
and radius relations are obtained:
VA ðtÞ ¼ V0 ð1 nðtÞÞ
VB
VB ðtÞ ¼ V0 nðtÞ VA
1
rA ðtÞ ¼ r0 ð1 nðtÞÞ3
ð12Þ
1
VB 3
rB ðtÞ ¼ r0 1 þ nðtÞ ð13Þ
VA
3
VC
VC
VC ðtÞ ¼ V0 nðtÞ rc ðtÞ ¼ r0 1þnðtÞ 1
VA
VA
1
ð14Þ
Note that VB(t) designates the volume of matrix
phase B consumed. rB is a hypothetical radius that
corresponds to the inner interface of the matrix if there
would be no mechanical adjustment of the matrix to
the volume change. Note that in general there will be a
difference between the volumes of the reactant phases
consumed and the resulting product, i.e. rC „ rB, as
symbolized in Fig. 1.
Expression (12) and (14) can be combined with (10)
and evaluated at r=rA to obtain a rate law for diffusion controlled reaction rim growth in spherical
geometry:
dnðtÞ 3LVA
¼
XFðnÞ
ð15Þ
dt
r20
where
FðnÞ ¼
1
1
3
ð1 nðtÞÞ3 1 þ nðtÞ VVAC nðtÞ
1
3
ð1 nðtÞÞ 1 þ
nðtÞ VVC
A
nðtÞ
1
ð16Þ
3
W is a generalized reaction affinity. For a stress free
transformation, it is equal to the reference reaction
affinity
X ¼ X0 ¼ Dr G
ð17Þ
However, if mechanical effects are considered W will
have to be augmented with the relevant terms and will
become a function of reaction progress.
Viscous adjustment of the matrix
The problem of a growing or shrinking sphere in an
infinite matrix is a frequent one. In its mechanical form
it is related to the Lamé problem (Lamé, 1852) the
analysis of which was extended to include elastoplasticity, phase transitions, and mineral reactions (e.g.
Hill, 1950; Lee et al., 1980; Morris, 1992, 2002; Fischer
et al., 1994; Mosenfelder et al., 2000; Fletcher &
Merino, 2001).
The mechanical equilibrium expression in the considered spherical geometry is greatly simplified, since
for symmetry reasons the tangential displacements and
shear stresses are all zero and the radial velocity v only
depends on the radial distance r. In the absence of
body forces we can write:
drr 2ðrr rh Þ
¼ 0;
þ
r
dr
ð18Þ
where rr and rh are the radial and the tangential stress
components, respectively. Because of the symmetry,
only one tangential stress component is required, i.e.
r/ ¼ rh
ð19Þ
These symmetry arguments also hold for strains and
strain rates.
2008 Blackwell Publishing Ltd
COUPLED DIFFUSION AND CREEP MODEL 87
Note that in the following mechanical analysis the
rim is treated as part of the inclusion, and therefore
does not explicitly account for the mechanical properties of the rim. The volume mismatch introduced by
the rim growth is assumed to occur at the outer
enstatite rim interface and must be compensated by the
mechanical response of the matrix. This essentially
reduces the mechanical problem to a single domain
since only the response of the matrix material to
boundary conditions must be investigated. Note that
this represents a simplification with respect to the
experiments of (Milke et al., 2009) since they observed
that the actual volume change is introduced at the
inner enstatite interface in the case of quartz inclusions
immersed in a forsterite matrix. However, the
mechanical effect of the rim is likely to be negligible as
the bulk volume compensation must occur by creep of
the surrounding matrix.
For large reaction progress the volume change will
exceed what can be accommodated elastically. Under
the given conditions this will cause viscous creep of the
matrix. In this case the strain rates are related to the
radial velocity by
e_r ¼
dv
;
dr
e_h ¼
v
r
v
The coupling between rim growth by diffusion and
viscous creep of the matrix is accounted for by the fact
that the Gibbs free energy at the inclusion-matrix
interface is a function of the normal stress acting on
the interface (Kamb, 1961). This can be expressed in
terms of reaction affinity as
ð26Þ
X ¼ X0 ðrr ðr ¼ rC Þ þ p1 ÞDr V
where Dr V ¼ VC VA VB is the molar volume
change of reaction. Note that it is assumed that the
normal stress across the product rim is constant and
therefore the change in flux is essentially due to the
different molar volumes. The reaction affinity (26)
combined with Eq. (15) yields the following rate law
for rim growth in a viscous matrix:
dn
¼
dt
X0
r20
4gr0 MðnÞDr V
rC
3LVA FðnÞ
X0
¼
r20
4gMðnÞDr V
NðnÞ
3LVA FðnÞ
p¼
rr þ 2rh
3
ð22Þ
The boundary conditions are that at the inclusion ⁄ matrix interface the matrix must creep with a
velocity that compensates the volume change induced
by the reaction, v(rC)=vreact, and that far away the
stress state is equal the ambient pressure (p¥),
rr(¥) = )p¥. Solving the system of ordinary differential equations, Eqs (18), (21) and (22), under the
given boundary conditions we obtain
v¼
vreact r2C
r2
4r2C
ð23Þ
r3
2r2
rh ¼ p1 þ vreact g 3C
r
The expression for the velocity that results from the
volume change of the reaction is
rr ¼ p1 vreact g
2008 Blackwell Publishing Ltd
ð27Þ
with
1
3
VC
NðnÞ ¼ 1 þ nðtÞ 1
VA
p is the pressure defined by
ð24Þ
2 3
VC
VC
1 VC
MðnÞ ¼
1
1
þ
nðtÞ
1
1
3 VA
VA
VB
ð25Þ
ð20Þ
ð21Þ
drc dn VB VC
df
¼
¼ r0 MðnÞ
dt
dn dt
VB
with
Assuming a viscous only response we can write the
constitutive equations for a linear, incompressible
viscous material with viscosity g as:
dv rr þ p
¼
e_r ¼
dr
2g
v rh þ p
e_h ¼ ¼
r
2g
react
ð28Þ
Note that W0, F(n), M(n), and DrV are all negative
quantities for the studied system and therefore Eq. (27)
is positive.
DISCUSSION
The rate law for coupled reaction and creep, Eq. (27),
can be represented in a simplified form as
dn
dt
Dr G
r20
diffusivity
ð29Þ
þ viscosityDr V
The rate of progress is directly proportional to DrG,
which only takes into account the far-field conditions
and therefore expresses the degree of reaction overstepping in terms of ambient conditions (pressure and
temperature). The rate of progress is modified by the
relative efficiency of the diffusion (effective component
diffusivity) and the viscous response of the matrix,
88 D. W. SCHMID ET AL.
Table 1. Thermodynamic data and molar volumes from
Holland & Powell (1998) calculated for the experimental conditions of 1000 C and 1 GPa.
V [cm3 mol)1)
Quartz
SiO2
23.38
Forsterite
Mg2SiO4
44.87
Duration [h]
1
9
)7873
3.85e-16
7e-5
where the former is accounted for by the first term in
the denominator and the latter by the second term. The
ÔslowerÕ term will be the rate limiting one. If there is no
volume change involved, the reaction is entirely diffusion controlled. Interestingly only the diffusion term
in the denominator shows size (radius) dependence.
This size dependency implies that small reactive
inclusions will tend to be creep controlled while larger
ones will rather be diffusion controlled.
Using the values given in Table 1, a time integration of
the ordinary differential Eq. (27) was performed to
obtain the evolution of the rim thickness with respect to
time, Figure 3. From the analysis of the quartz in forsterite case the phenomenological coefficient of diffusion is estimated to be L = 3.85e-16 [mol2 s)1 m)1 J)1],
which corresponds to an effective combined component
diffusivity of D = 1.6e-16 [m2 s)1].1
Note that this effective component diffusivity was
determined from the rim growth rate that was
observed for the quartz in olivine arrangement under
the assumption that matrix creep effects are negligible,
i.e. Dr V ¼ 0. The obtained value is in agreement with
the findings of Abart et al. (in press), who derived
effective component diffusivities for the same rim
growth experiments using a different approach that is
based on the thermodynamic extremum principle.
Based on the obtained phenomenological coefficient
of diffusion we estimate the viscosity of the olivine
matrix to be approximately 1e14 [Pas] or less. For such
a case the rim thickness
pffiffi shows the expected linear
growth with respect to t. However, if the matrix viscosity is higher, then the rim growth is substantially
reduced. The 1e17 [Pas] curve shows an example which
is primarily creep controlled throughout the experiment. Yet, the 1e15 [Pas] curve shows both diffusion
and viscous rate controlling effects. Initially the matrix
viscosity is rate determining, but as time and rim
growth progress diffusion becomes the limiting factor
1
The phenomenological coefficient of diffusion is related
to the effective component diffusivity, Di, through Li ¼
ðXi =VÞðDi =RTÞ, where Xi is the mole fraction, V the molar
volume of the material within which the diffusion takes place, R
the gas constant, and T the temperature. Individual component
diffusivities cannot be determined. However, an effective
be obtained from
combined component
diffusivity, D, can
en
L ¼ ð1=Ven RTÞ Xen
SiO2 DSiO2 þ XMgO DMgO , which yields D ¼
2LVen RT:
Rim thickness [μm]
8
DrG [J mol)1]
L [mol2 s)1 m)1 J)1]
r0[m]
6
12
18
24
36
48
80
Qtz in Ol
log10(η [Pas])
14 (O/Q)
14
15
16
17
10
Enstatite
Mg2Si2O6
83.85
3
7
6
Ol in Qtz
5
4
3
2
1
0
0
50
100
150
200
250
300
350
400
450
500
sqrt (Duration [s])
Fig. 3. Time required for the growth of an enstatite rim of
certain thickness for different effective matrix viscosities. Rim
growth becomes substantially decelerated for matrix viscosities
in excess of 1e15 [Pas]. Open circles indicate the rim thickness
obtained for the two different matrix-inclusion arrangements in
a single experiment. Note that with the exception of the 14
(O ⁄ Q) line this plot is produced for the quartz in olivine setting.
14 (O ⁄ Q) corresponds to an olivine in quartz experiment with a
matrix viscosity of 1e14 [Pas]. The difference between the two
configurations is marginal. The maximum duration given here is
equal to the 80 h of the experiments of Milke et al., 2009.
as can be seen from the
pffiffilinear relation that develops
between growth and t for longer run durations.
Hence, creep control is especially important for the
initial stages of rim growth when the diffusion
distances are short and diffusion is therefore more
efficient than during the later stages of rim growth
when the reaction rim becomes thicker. The data from
the olivine in quartz experiment plots between the
1e15 [Pas] and the 1e16 [Pas] lines, therefore putting
the difference between quartz and olivine rheologies at
one to two orders of magnitude under the given conditions. Hence, the relative viscosities of the quartz and
the olivine matrix materials can be determined from
this type of experiment. If the effective component
diffusivities are known one can use them as rheometers. Note that the present formulation assumes a
Newtonian (linear) viscous creep behaviour of the
matrix. A power-law rheology would be more appropriate and hence the usage of these experiments as
rheometers more intricate as pointed out by Morris
(2002). In any case, it is important to realize that the
rim thickness may contain information regarding the
efficiency of diffusion as well as viscous creep.
The point that the matrix viscosity effect is more
pronounced in the early stages of rim growth is also
2008 Blackwell Publishing Ltd
COUPLED DIFFUSION AND CREEP MODEL 89
10–6
log10 (Growth Velocity [mm/yr] (log10(Strain Rate [s–1]))
10–8
10
log10 (L [mol2 s–1 m–1 J–1])
Reaction rate dξ/dt [s–1]
10–7
–9
10–10
10–11
10–13 10
10
10
12
10
14
10
16
18
10
10
20
22
10
Viscosity (η [Pas])
4 (–2)
–13
3 (–3)
–14
2 (–4)
–15
1 (–5)
–16
0 (–6)
–17
–1 (–7)
–18
–2 (–8)
–19
–3 (–9)
–20
–4 (–10)
–21
10
11
Creep Controlled
12
n
tio
si
an
Tr
10–12
–12
Quartz
–11
0.1
0.5
0.9
Olivine
ξ
Diffusion Controlled
13
14 15 16
log10(η [Pas])
17
18
19
20
Fig. 4. Reaction rate as a function of effective matrix viscosity
for various stages of reaction progress n; horizontal segments of
the curves represent diffusion controlled regime, segments with
negative slope indicate creep controlled regime. Note that the
transition from diffusion control to creep control shifts to higher
viscosity with reaction progress.
Fig. 5. Contour plot of log10(growth velocity [mm yr)1]) in g
(viscosity) and L (phenomenological coefficient of diffusion)
space; isocontours are also labelled for log strain rate (in brackets).
Olivine and quartz flow laws are plotted for the calculated
strain rates and the experimental temperature of 1000 C.
illustrated in Fig. 4 where for a given phenomenological coefficient of diffusion the progress rate is plotted
v. viscosity. The three curves represent different stages
of progress for the quartz in olivine configuration. For
small viscosities the rate is independent of viscosity and
entirely determined by the efficiency of diffusion. Once
the viscosity reaches a certain transition value the
progress rate drastically drops with further increase
in matrix viscosity. The transition from diffusion
controlled to creep controlled growth occurs at higher
viscosities the more progressed the rim growth is.
While Fig. 4 illustrates how the viscosity of the
matrix restricts the growth of the rim it must be kept in
mind that a total reaction stop will not be achieved in
this idealized model since linear viscous materials
always creep when a differential stress is applied.
Therefore the question arises how the growth behaves
with respect to time in the creep controlled regime.
This end member case can be analysed by setting
L fi ¥ in Eq. (27). The result is that the reaction rim
grows exponentially in time. This was already derived
by Morris (1992) for a similar case where he developed
a model that combines the solid-state transformation
of a spherical inclusion with the mechanical response
of the surrounding matrix. Morris (1992) results show
that in the limit of infinitely fast phase change kinetics,
the inclusion growth is mechanically restricted and
exponential growth of the sphere radius with time is
observed.
The dependence of the reaction rate on diffusion
efficiency and viscosity and the competition between
the two mechanisms is best illustrated with a map plot.
Figure 5 contours the growth velocity of the outer
interface (@rC =@t) for n = 0.01 and the values in Table
1 for the quartz in olivine configuration. The domains
of the two rate controlling factors are separated by a
relatively sharp transition zone as indicated by the
dashed ÔTransitionÕ line. In the diffusion controlled
domain the reaction velocity is independent of viscosity changes; in the creep controlled domain the reaction velocity does not depend on the efficiency of
diffusion. The isocontour values are given for velocity
as well as rate of deformation that is caused by the
reaction progress (in brackets). These strain rates
together with the temperature of the experiment can be
used to evaluate flow laws for the two matrix materials.
We use the laws from Hirth & Kohlstedt (2003) for wet
olivine and Paterson & Luan (1990, synthetic quartz
from silicic acid powder origin) for quartz. These flow
laws restrict the region of applicability of the map in
Fig. 5. For the estimated phenomenological coefficient
of diffusion of L = 3.85e-16 [mol2 s)1 m)1 J)1] and
n = 0.01 a strain rate of approximately 1e-6 [s)1]
results, which corresponds, for the given temperature,
to an effective viscosity of 1e14-1e15 [Pas] for the two
matrix materials. This range of effective viscosities
corresponds very well with the results that were
obtained (Fig. 3) without the knowledge of any actual
flow law. For the experimental conditions, the two flow
laws lie in the transition region from diffusion to creep
control and hence small changes in the viscosity can
have significant effects on the rate of rim growth. As
already pointed out the actual flow laws are non-linear
and vary with strain rate, i.e. the effective viscosity
2008 Blackwell Publishing Ltd
90 D. W. SCHMID ET AL.
decreases with increasing effective strain rate. Since the
strain rate induced by the reaction decreases with
reaction progress we expect that the effective viscosities
are initially lower and towards the end of the experiment higher than predicted by the plot in Fig. 3 that
assumes a constant viscosity value. While the two flow
laws largely support our findings they contradict our
experiments in terms of which of the two matrix
materials is the stronger, i.e. more viscous one. The
experiments show slower rim growth for olivine grains
in quartz matrix but the flow laws predict the opposite.
This may be due to the inaccurate knowledge of how
strong the synthetic quartz is (see discussion in Luan &
Paterson, 1992).
Reactions that involve relatively large volume
changes require matrix adjustments that cannot be
accounted for by elastic material response and therefore require creep flow. The experimental conditions of
1 GPa and 1000 C correspond to relatively deep levels
of the lithosphere. Regions that are colder and closer
to the EarthÕs surface will be characterized by higher
viscosities and therefore the closure due to viscous
creep will be even more important.
CONCLUSIONS
Motivated by the experiments of (Milke et al., 2009)
we have developed a model for the growth of an
enstatite reaction rim at olivine-quartz interfaces that
accounts for the systematic differences in observed rim
thickness depending on whether olivine grains are
embedded in a quartz matrix or vice versa. The reaction is characterized by a bulk volume change of )6%,
too much to be compensated by an elastic matrix
response. Rock deformation experiments that are run
under the experimental conditions of 1000 C and
resulting strain rates of 1e-6 [s)1] show that both
quartz and olivine creep viscously (Paterson & Luan,
1990; Hirth & Kohlstedt, 2003). Therefore, our model
couples diffusion (transport of species through the
growing enstatite rim) with the creep response of the
matrix. The resulting growth law shows the following
characteristics.
1. The progress rate is proportional to the reaction
overstepping.
2. The progress rate is controlled by the slower one of
the two processes, either diffusion or creep. The transition zone between the two regimes is quite narrow.
3. Only the diffusion term is sensitive to the initial size
of the reactive inclusion. For small sizes diffusion is
efficient and reaction rate is likely to be creep controlled, depending on the actual values of the kinetic
parameters (viscosity and efficiency of diffusion). For
large sizes the inverse is true.
4. With progressive rim growth the diffusion distance
grows and diffusion tends to become more important
as a rate controlling mechanism.
5. If rim growth occurs without any volume change
then the progress rate is not affected by viscous creep
of the matrix and the usual linear proportionality
between rim thickness and the square root of time
results. This progress rate represents the upper limit;
reactions with volume change and non-negligible
matrix creep response will yield progress rates that fall
below this maximum.
6. The importance of the viscous creep is directly
proportional to the volume change of the reaction.
7. In the creep controlled regime the rim growth is
exponential in time (cf. Morris, 1992).
In terms of the experiments of Milke et al., 2009 our
model explains the difference in rim thickness with a
difference between the two effective matrix viscosities
of about one order of magnitude. The actual values,
1e14-1e15 [Pas], correspond well with the range of
published flow laws for quartz and olivine under the
given conditions. Our experiments show that the synthetic quartz was mechanically stronger (more viscous)
than the San Carlos olivine.
The results of our analysis imply that natural coronas and reaction rims may contain information on
both component diffusion as well as viscous creep of
the matrix. Analogous to thermal closure (Dodson,
1973) reactions can be closed mechanically. As compared to our experiments, mechanical closure will be
even more prevalent throughout crustal conditions
where the effective viscosities are larger.
ACKNOWLEDGEMENTS
We thank three anonymous reviewers and K. Schulmann for their constructive reviews. This work was
supported by a Center of Excellence grant from the
Norwegian Research Council to PGP and through
DFG project AB 314/2-1 (FOR 741).
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Received 8 May 2008; revision accepted 24 November 2008.