Journal of Petrology Volume 41 Number 2 Pages 285-304 2000
© Oxford University Press 2000
Symplectic Reaction in Olivine and the Controls of Intergrowth Spacing in Symplectites
J. R. ASHWORTH,* and
A. D. CHAMBERS
SCHOOL OF EARTH SCIENCES, UNIVERSITY OF BIRMINGHAM, EDGBASTON, BIRMINGHAM B15 2TT, UK
Received
January 6, 1999;
Revised typescript accepted
July 23, 1999
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ABSTRACT
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Symplectite growth is analysed in terms of non-equilibrium thermodynamics
and maximum rate of energy dissipation. For a given reaction,
the spacing
of lamellae or rods is predicted to be proportional
to the cube root of L
/v, where v is reaction rate and L is the
Onsager diffusion coefficient of a reference element in the
reaction front of width
. The result is comparable with, but
not identical to, metallurgical theory for discontinuous precipitation
in alloys. It is reasoned that concentration-gradient constraints
place a lower limit on
, which depends on grain-boundary energy
. An upper limit
0·3 J/m
2 is thus estimated using literature
data from experimental oxidation of olivine. Combined with new
observations on exsolution symplectites in olivine from the
Lilloise intrusion, Greenland, this suggests that the exsolution
reaction took place above 800°C. Using previous modelling
of a corona with a symplectic layer,
1 J/m
2 is estimated for
hornblende–spinel symplectite. The energy driving diffusion
plus grain-boundary production in the reaction front was a small
proportion of the overall affinity of the corona reaction. The
theory explains symplectite growth over a wide range of igneous
and metamorphic temperatures and timescales.
KEY WORDS: discontinuous precipitation; energy dissipation; grain-boundary energy; olivine; symplectite
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INTRODUCTION
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It is of general interest in petrology to interpret textures
in terms of the energy terms controlling them, with approximate
quantification where possible. Although processes such as nucleation
are energetically important in some reactions, many other textures
are more influenced by diffusion. Here we develop a general
theory for a common texture of that type, termed symplectic.
Symplectites are mineral intergrowths produced by solid-state
reaction (Sederholm, 1916
), the grains being elongated perpendicular
to the reaction front at which the product minerals grew from
a single reactant mineral. The fine-scale, approximately periodic
spacing of intergrowth reflects difficult diffusion of some
chemical elements. Close analogies have been noted with discontinuous
precipitation (also known as cellular precipitation) in alloys
(Voll, 1982
; Boland & van Roermund, 1983
; Mongkoltip &
Ashworth, 1983
; Boland & Otten, 1985
). An example of a rod
symplectite, with the minor mineral as rods in the major (host)
mineral, is myrmekite (quartz rods in plagioclase). Alternatively,
the minor mineral can be lamellar. Many symplectites occur in
metamorphic coronas (Sederholm, 1916
), which are reaction rims
produced between two reactant minerals. The finer the intergrowth,
the shorter the diffusion range but also the more grain-boundary
area in the symplectite. The scale must be determined by a balance
involving diffusive energy dissipation and grain-boundary energy.
This paper presents a theory of that balance. Semi-quantitative
interpretations follow, for a metamorphic example using literature
data, and for symplectic inclusions in igneous olivine using
new observations.
In olivine, symplectites of clinopyroxene and magnetite are attributed to exsolution (Moseley, 1984). Another process, oxidation, has produced symplectites of Fe oxide or Fe,Mg oxide with silica or Fe,Mg silicate in many igneous olivines (Haggerty & Baker, 1967; Putnis, 1979; Johnston & Stout, 1984). Reaction with atmospheric oxygen at high temperatures can be rapid, for example in lavas (Haggerty & Baker, 1967) and in interplanetary dust particles approaching Earth (Rietmeijer, 1996). Such reactions are therefore amenable to experimental study (Champness, 1970; Kohlstedt & Vander Sande, 1975; Brewster & O’Reilly, 1988). Natural symplectites in olivine are attractive for research because their geometry tends to be simple (lamellar), they form in a system with few major components, and the experimental work aids interpretation. In this context, symplectites of exsolution type were examined in samples from the Lilloise intrusion.
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SYMPLECTITES IN OLIVINE FROM THE LILLOISE INTRUSION
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The Lilloise is a shallow-level Tertiary mafic intrusion in
East Greenland (Chambers & Brown, 1995
). The symplectites
are found in olivine–chrome spinel cumulates, as thin
platelets of clinopyroxene and magnetite parallel to (100) of
olivine (Fig. 1a). They occur only in olivine less ferroan than
Fo
74, and are best developed in the most magnesian olivines
(Fo
85–87). They are absent from extensively fractured
and serpentinized rocks, are concentrated near the centres of
olivine grains despite a lack of significant zoning of these
grains, and tend to be associated with subgrain boundaries (Fig.
1b). These observations suggest that the symplectites grew after
some deformation, and in those grains or parts of grains where
minor components had been unable to diffuse out of the olivine
to grain boundaries or cracks.
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Fig. 1. Symplectites of clinopyroxene and magnetite in Lilloise olivine. (a)–(c) Optical micrographs, (d) TEM image. (a) and (b) show a thin-section cut approximately parallel to (010) so that the symplectite platelets are viewed edgewise. (a) Plane-polarized light. (b) Oblique polars, showing subgrain structure (bands of different grey shade) in olivine, with platelets on subgrain boundaries. (c) A section approximately parallel to the platelet plane, (100) of olivine (plane-polarized light). The clinopyroxene is not distinguishable from the olivine in this view, but the dark magnetite shows the approximately rectangular boundaries of the symplectic platelets. (d) A platelet viewed obliquely by TEM, approximately along the zone axis [312]; projected directions of x- and y-axes are indicated. Boundaries within the platelet between clinopyroxene and three magnetite lamellae (dark) appear to be non-orthogonal to the (100) platelet boundary, but this is attributable to the obliquity of view.
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Thin-sections cut approximately parallel to (100) show that
the symplectite comprises transparent and dark minerals (Fig.
1c). Parts of specimens L5 and L410 were examined by transmission
electron microscopy (TEM) after ion-beam thinning. The symplectite
minerals were identified by TEM energy-dispersive analysis.
The clinopyroxene is close to pure diopside and the magnetite,
forming lamellae (Fig. 1d), is nearly pure Fe
3O
4.
The average lamellar spacing measured optically in (100) sections (Fig. 1c) is 4·1 µm (SD 0·8 µm, 43 measurements). As the symplectite grew the lamellae branched (Fig. 1c), so as to preserve approximately constant average spacing. TEM images show smaller spacings (2·8 µm in Fig. 1d), but this is a small sample and probably biased towards closely spaced lamellae, because these are easiest to find in a TEM search. TEM gives the best estimate of the proportion of magnetite (20% by volume in Fig. 1d); in optical images, the apparent proportion of the dark mineral is exaggerated by any irregularities in lamellar boundaries or obliquity of these to the direction of view (see Moseley, 1984).
TEM observations on the Lilloise symplectites are entirely consistent with those of Moseley (1984) and Otten (1985) on samples from other intrusions. The reaction is topotactic, i.e. both symplectite minerals have crystallographic orientations related to that of olivine (Moseley, 1984, Table 1), such that the oxygen array is minimally changed between reactant and products. All three minerals have approximately close-packed oxygens. In a close-packed layer, there are three directions of close-packed lines of spheres. In cubic close packing, exemplified by magnetite, the close-packed planes are {111} and the lines of close-packing within them, separated by angles of 60°, are <110> lines. In olivine (Deer et al., 1982), the oxygens are approximately hexagonally close-packed; the close-packing plane is (100) and the lines are in the z direction and ±60° from it. In pyroxene (Cameron & Papike, 1981), the approximately close-packed plane is (100) and the lines are [010] and <013>. The observed topotaxy is such that close-packed planes and lines in olivine are inherited in both product minerals. All the grain boundaries, including the reaction front, are semicoherent rather than crystallographically random (non-coherent).
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PROCESSES OF SYMPLECTITE PRODUCTION IN OLIVINE
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Symplectites of the type described above are attributed to exsolution
of minor components from olivine (Moseley, 1984
). Ca and Fe
3+ form clinopyroxene and magnetite, respectively. The overall
reaction can be summarized as
It
is possible that some oxidation of Fe
2+ to Fe
3+ accompanies
exsolution, especially as the Fe
3+ content of olivine should
usually be small (Nakamura & Schmalzried, 1983
). The retention
of the oxygen sublattice (see previous section), and the low
diffusion coefficient of oxygen in olivine (Morioka & Nagasawa,
1991
), suggest that oxygen is not added; oxidation can be accomplished
by diffusion of electrons and cations:
(see Kohlstedt & Vander Sande, 1975
). Fe should diffuse
to the edge of the olivine grain, as demonstrated by Mackwell
(1992)
in experimental oxidation of fayalite. With or without
oxidation, then, the proportion of pyroxene to magnetite produced
is 3:1 in terms of number of oxygen atoms, and approximately
the same in volume terms. This is consistent with the TEM images
(Fig. 1d).
It is evident from the above that Ca, Fe and Mg can diffuse through the olivine during the reaction, whereas Si is conserved in the reaction front, and the topotaxy implies that oxygen hardly moves at all. The lamellar spacing should be related to the limited movement of silicon and oxygen. Their relative immobility is consistent with the smaller diffusion coefficients of Si and O, relative to divalent cations, in bulk olivine (Morioka & Nagasawa, 1991).
Other symplectites in olivine are attributed simply to oxidation. In a pure oxidation reaction producing silica and iron oxide, the proportions of the two minerals should be approximately equal (they contain equal numbers of oxygen atoms):
or
(see Mackwell, 1992
). TEM images (Champness,
1970
), are consistent with this. Some examples lack topotaxy
(Putnis, 1979
). Nevertheless, the oxidation products are so
similar to the exsolution type that Si and O are again likely
to be slow-diffusing elements.
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THEORY OF INTERGROWTH SPACING
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This theory uses the principle of maximum rate of energy dissipation
(or, equivalently, maximum rate of entropy production), which
has been applied to discontinuous precipitation in alloys following
the original suggestion of Cahn (1959)
. It is assumed that reaction
rate is controlled by diffusion together with the grain-boundary
energy term, and is steady state (i.e. short-term fluctuations
are discounted, as is generally acceptable for slow petrological
reactions; e.g. Ashworth & Sheplev, 1997
). Symbols are listed
in Table
1. The total affinity (-
G) is constant for a reaction
at constant pressure, temperature and mineral compositions.
It is divided into the energy that drives processes in the reaction
front and the part driving external processes:
In an open system, the important external process is diffusion
outside the reaction front, supplying components for symplectite
growth. As the chemical-potential gradients driving external
diffusion should be proportional to (-
G)
ext, reaction rate measured
by reaction-front velocity
v (Fig.
2) should also be proportional
to (-
G)
ext, so that
s is a constant in
Within the reaction front, an example of important diffusion
is that of Si (in the exsolution of magnetite + clinopyroxene
from olivine) from the area where magnetite is growing, to be
deposited in the pyroxene. The free energy associated with energy
dissipation by reaction-front diffusion, (-
G)
diff, increases
with increasing spacing of the products, whereas the grain-boundary
area between the symplectite minerals decreases and thus the
energy term (-
G)
gb decreases in the balance of energy at the
reaction front
These energies will
now be examined for lamellar geometry (rod geometry is treated
in the Appendix).
In terms of grain-boundary energy and lamellar spacing ,
where molar volume
V should be an averaged quantity.
With reference to Fig.
2, the diffusive flux
J of an element
in the reaction front varies with distance
x in the region where
the lamellar mineral is growing, thus
Here
kL (mol/m
3) is the effective difference in concentration
of the element between the reactant mineral and the lamellar
product. If there is diffusion through either of these, to or
from the reaction front, this should be taken into account,
but otherwise
kL simply reflects the difference in mineral compositions.
If
kH is the analogous difference between reactant mineral and
the host mineral in the symplectite, then in that region of
the reaction front
If the system is
closed to the element (i.e. there is no external flux),
and, with no diffusion along the host–lamella
grain boundary, there is no discontinuity in
J where the reaction
front crosses that boundary (Fig. 3a). The flux changes direction
at
x = 0 and
x =
/2 (Fig. 3a). Between these points,
If diffusion obeys Fick’s
Law, flux is related to concentration gradient by the diffusion
coefficient
D:
Concentration varies
thus, from
c =
c0 at
x = 0:
This simplifies in a closed system, using equation
(1):
For an
element diffusing from lamella to host,
c0 is the maximum concentration,
and the minimum, at
x =
/2 (Fig. 3b), is
Obviously,
cmin cannot be less than zero. This gives a concentration
limit
Indeed,
cmin for a major element
is unlikely to approach zero. For example, during symplectite
growth in olivine,
cmin for Si occurs in the boundary between
olivine and silica or pyroxene, and should remain at a level
comparable with the Si content of these minerals. Therefore,
the concentration limit should not be closely approached.
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Fig. 3. Variation of flux (a) and concentration (b) with distance x in the reaction front of Fig. 2, for an element that diffuses from the lamella region (to be deposited in the host mineral). It is assumed that this element diffuses in the reaction front only, i.e. equation (1) applies.
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The following treatment of diffusive energy dissipation uses
the Onsager diffusion coefficient
L, which relates flux
J to
chemical-potential gradient:
A useful
approximation, given that
c should not approach zero, is
If
L is measured at
c =
c0, equation
(2) can be
written
The condition for the concentration
limit (3) not to be approached becomes
Using the principles of irreversible thermodynamics (Fisher & Lasaga, 1981), the rate of energy dissipation per unit volume is T
= -J(dµ/dx):
Multiplying by
.d
x and integrating from
x = 0 to
x =
/2 gives the energy dissipation rate in an area of reaction
front measuring
/2 by 1 m. Then dividing by
v/2 gives the energy
dissipated per cubic metre of symplectite produced, and multiplying
by
V gives (-
G)
diff per mole of symplectite:
The above applies
to a single diffusing element. For the general case of several
diffusing elements
i,
where
(The use of ratios
LSi/
Li is convenient for the
interpretation of coronas below.)
K effectively measures the
strength of diffusive segregation between the product minerals.
Maximum rate of energy dissipation
Summarizing the working above and in the Appendix,
where
and
For a given reaction under fixed conditions [i.e. for given
values of
a,
b,
s and (-
G)], values of
v and
can be found that
maximize either reaction rate or rate of energy dissipation.
This is illustrated in Figs
4 and
5, by treating
v and
as variables,
although in physical reality they cannot vary arbitrarily (it
will be argued that they should, in fact, be constrained to
the values for maximum energy dissipation rate, demonstrated
by Figs 4c and 5b). In Fig.
4, with reaction rate
v treated
as variable, the solution shown for
(Fig. 4a) is the largest
root of the cubic equation
(8), which gives the largest (-
G)
diff and hence the fastest energy dissipation at given
v. Figure
5 shows this solution for
(as the large filled circle) at two
selected values of
v. In Fig. 4a, as
v increases, the faster
transport of material requires shorter diffusion distances,
so
decreases. This implies more grain boundaries, so (-
G)
gb increases (Fig. 4b); (-
G)
ext increases proportionally to
v,
whereas (-
G)
diff decreases, maintaining constant sum (-
G). Figure
4 shows a velocity limit, beyond which there is no solution
for
; the sum of (-
G)
diff and (-
G)
gb would exceed the available
energy (-
G) -
sv for all
. At this velocity limit,
vmax (Fig.
5a), the above sum just falls to the required amount at one
value of
, where
This situation
maximizes reaction rate, but does not quite maximize the rate
of energy dissipation. The latter involves the diffusive term
(-
G)
ext as well as (-
G)
diff. We require the maximum of
Differentiating with respect to
v and equating
to zero,
Also, differentiating equation
(8) with (-
G) constant,
Combining these
two equations with equations
(7),
and rearrangement gives
Half of the reaction-front energy (-
G)
rf goes into each term.
This situation is illustrated graphically in Figs 4c and 5b.
The result remains valid in a completely closed system, i.e.
if (-
G)
ext is zero. Equating (-
G)
diff with (-
G)
gb in equations
(7) gives the solution for
:
or
at maximum rate of energy dissipation. Equation
(11) is the key result for the Applications sections below.
Although inequality (4) indicates that the concentration limit would be approached with increasing if other quantities could be held constant, relation (11) among these quantities shows that the concentration limit actually sets a minimum value of for a given reaction. Combining equation (11) with inequality (4) for the lamellar closed-system case shows that, for the concentration limit not to be approached for element i,
The rod-geometry equivalent, not necessarily in
a closed system, from inequality (A1) of the Appendix, is
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COMPARISON WITH DISCONTINUOUS PRECIPITATION IN ALLOYS
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In discontinuous precipitation, an alloy decomposes to two purer
metals in a rod or lamellar intergrowth, when held at a temperature
below the relevant solvus. Zoning is detectable in the host
product, and reflects the concentration profile in the reaction
front. In the petrological analogue, such zoning is not generally
possible. Let us consider the growth of myrmekite. In the reaction
front, Na and Ca must both diffuse away from growing quartz.
Their chemical-potential gradients are therefore in the same
direction, whereas the only possible zoning in the growing plagioclase
would have Na and Ca varying antithetically. In some symplectites,
zoning parallel to the reaction front is observed, but can possibly
be attributed to modification of the minerals after their growth
(Ashworth
et al., 1992
).
Discontinuous precipitation in metals generally does not require diffusion beyond the reaction front, so (-G)rf equals the total free energy change (-G). This may also be true for some symplectite-forming reactions, such as the decomposition of augite into two pyroxenes studied by Boland & Otten (1985). However, in the metals, the zoning means that (-G) is only a fraction of the free energy change (-G)eq that would be realized if equilibrium, unzoned products grew: this fraction is denoted P(-G)eq. The overall energy balance is
and
P is a function
of
v and
(Cahn, 1959
), decreasing as
and
v increase.
The reactions in metals can be studied experimentally. Commonly, estimates are available for (-G)eq and . Zoning is measured by analytical TEM (Zieba & Gust, 1998). From the zoning profiles, P can be calculated. This depends on the thermodynamic model used for solid solution; nevertheless, P(-G)eq and (-G)gb can be estimated more readily than (-G)diff, and the latter is not usually calculated explicitly. The conditions for maximum v(-G)diff are found by maximizing v times the left-hand side of equation (14). Solorzano & Purdy (1984) demonstrated a clearly defined maximum as a function of v at constant . Bögel & Gust (1988) reasoned that v and can be optimized simultaneously. They obtained the result that (-G)gb is twice (-G)diff at maximum energy dissipation rate, independently of the model used for P. This is comparable with, but clearly not identical to the present result: it corresponds to equation (9) for maximum reaction rate in this work, rather than equation (10) for maximum rate of energy dissipation. The different result of Bögel & Gust (1988) is not due to zero (-G)ext, which did not invalidate equation (10): the difference lies in the zoning.
Bögel & Gust (1988) claimed good agreement between their theory and experiments, particularly at small departures from equilibrium. However, data of Duly et al. (1994) seem to require revision of the theory of Bögel & Gust (1988). The metallurgical problem may be complicated by additional factors, such as non-steady-state variations in v between zero and about five times average velocity (Abdou et al., 1996). These may be related to variations in the zoning pattern, in an oscillatory manner perpendicular to the reaction front (Zieba & Gust, 1998, p. 80).
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APPLICATION TO SYMPLECTITES IN OLIVINE
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For a natural symplectite, the growth rate
v could be estimated
from equation
(11) and measurements of
, if
,
LSi and
K were
known. Although there is an additional grain-boundary energy
associated with the large boundary area between platelets and
olivine [their shape being presumably controlled by low-energy
semicoherent (100) boundaries], this does not depend on
and
need not be considered here.
A value for K can be calculated by equation (6) if the relative magnitudes of diffusion coefficients for the major diffusing elements can be estimated. The lowest diffusion coefficients should be for Si and O; as O is inferred not to diffuse at all, the diffusion term is considered to be dominated by Si. Using the quantities for Si alone, equation (6) becomes
It is reasonable to assume a closed system for
Si, so equation
(1) applies, giving
From stoichiometry,
pL 
0·25 (see section on processes
above). As no Si enters magnetite, (
kL)
Si equals the concentration
of Si in olivine (neglecting volume change in reaction). The
molar volume of olivine is approximately 4·4
x 10
-5 m
3/mol
(Holland & Powell, 1998
) and contains one mole of Si, so
(
kL)
Si 2·3
x 10
4 mol/m
3, and
K 2·8
x 10
6 mol
2/m
6.
The concentration limit (3) places some constraints on the interpretation of the Lilloise symplectites. It is assumed that the maximum concentration, c0, approximates that of Si in olivine, which also equals (kL)Si so that cmin > 0 implies
It can immediately be shown that
DSi in the reaction
front must exceed that in bulk olivine. The largest values reported
for the latter (Houlier
et al., 1990
) are around 8
x 10
-22 m
2/s
at 1250°C (as high an initial temperature as could be considered
for the Lilloise intrusion). Grain-boundary widths
are
1–3
nm for non-coherent boundaries in olivine (Farver
et al., 1994
),
and should be smaller for semicoherent ones. These values would
lead to a maximum of 2·4
x 10
-30 m
3/s for
DSi which,
with
pL 0·25 and
4·1 µm, would mean
v < 5
x 10
-18 m/s from inequality (16). The platelets would
take more than 10
5 years to grow 20 µm. This is longer
than would be available at temperatures near 1250°C (maximum
possible intrusion temperature).
To proceed further requires some constraint on or LSi/v. Literature values for grain-boundary energies in minerals are scarce, but Spry (1969) tabulated surface energies that are likely to be about three times the values of interface (grain-boundary) energies, and indicate in the range 0·03–2 J/m2. Hay & Evans (1988) experimentally estimated = 130 mJ/m2 in calcite at 800°C. By treating grain boundaries as dislocation arrays, Penn et al. (1999) calculated = 0·3–1·0 J/m2 in sillimanite (varying with misorientation angle between the grains). Combining this type of theory with experimental data on solid–liquid dihedral angles, Cooper & Kohlstedt (1982) estimated = 0·9 ± 0·4 J/m2 for high-angle (misorientation >15°) grain boundaries in olivine. Also in olivine, they found = 0·08 ± 0·02 J/m2 for low-angle (0·2°) subgrain boundaries, in which the individual dislocations are easily resolved in TEM. The semicoherent boundaries considered here, in which individual dislocations can also be resolved (Moseley, 1984, Fig. 2e), are probably intermediate in kind between low-angle subgrain boundaries and incoherent grain boundaries. This is consistent with metallurgical analogy, which indicates that a semicoherent boundary might have about one-fifth the energy of a non-coherent one (Kretz, 1994, p. 302), i.e. possibly in the range 0·006–0·4 J/m2.
Some further information can be gleaned from the oxidation experiments in olivine. These have mostly produced symplectites comprising iron oxide and silica (references in Table 2). They provide a constraint on as follows. Although some diffusion of oxygen seems possible in these reactions, it is assumed that Si diffusion still dominates the energetics. For a symplectite of iron oxide + silica, pL 0·5 by stoichiometry (see section on processes above), (kL)Si 2·3 x 104 mol/m3 as in the exsolution case, and K 1·1 x 107 mol2/m6 from equation (15). From some photomicrographs, can be estimated (Table 2). Rearrangement of inequality (12) then gives an upper limit for , if the concentration limit for Si is not to be approached:
The low-temperature experiments give the most stringent limits
(Table
2). Those of Khisina
et al. (1995
, 1998)
are complicated
by the production of two oxide minerals, and also ‘ferriolivine’
between the oxide–silica regions, and so may not represent
the simple reaction analysed here. ‘Ferriolivine’
(the Fe end-member of which is laihunite, Fe
2+2–3xFe
3+2xSiO
4)
tends to be produced in preference to symplectites in low-temperature
oxidation of olivine (<600°C approximately, in air: Kondoh
et al., 1985
; Khisina
et al., 1998
). Iishi
et al. (1997)
produced
probable symplectites at unusually low temperature (300°C),
but in demounted thin-sections immersed in an aqueous medium,
where species from that medium enter the grain boundaries and
may enhance diffusion so that the concentration limit (16) is
satisfied. Reaction to ‘ferriolivine’ instead of
symplectite may reflect approach to the concentration limit,
or possibly that
becomes so small (<10 nm) that symplectites
are ‘mostly grain boundary’ and have higher than
normal free energy. The experiment of Champness (1970)
at 650°C,
which produced the two-phase symplectite without ‘ferriolivine’,
is taken as setting the upper limit to be used here:
0·3
J/m
2. This is used in contouring the plot of log
vs log
v (Fig.
6). Admittedly, the energy may be smaller for semicoherent boundaries
than for those between haematite and amorphous silica produced
by Champness (1970)
.
Where the size of the symplectite is reported, this and the duration of the experiment give an estimate for v (Table 2). [This obviously varies greatly with experimental conditions: for example, Mackwell (1992) achieved high rates in producing a surface layer of iron oxide + silica, rather than inclusions inside olivine.] Then equation (11) gives an estimate for LSi, which combined with the upper limit for gives a lower limit for LSi (Table 2). This indicates that LSi increases strongly with temperature, as would be expected from the Arrhenius law. Because of differences of experimental approach, detailed comparisons are not possible. However, the rough estimates of LSi in Fig. 6 are comparable with separate experimental results on grain-boundary diffusion in olivine. Fisler & Mackwell (1994) and Yund (1997) studied diffusion-controlled growth of an olivine layer. The rate-controlling element is not directly identified, but Si is probably indicated. These studies indicate DSi 10-23 m3/s in non-coherent, dry olivine boundaries at 1000°C, corresponding to LSi 2 x 10-23 mol2/J per s. At 1400°C in enstatite grain boundaries, results of Fisler et al. (1997) give DSi 10-22 m3/s.
For the Lilloise reaction, values 4 µm, K 2·8 x 106 mol2/m6 and 0·3 J/m2 imply LSi > 3 x 10-10 v. A minimum for v is set by noting that 1 ky is the maximum time that would be available for reaction in the cooling, shallow-level intrusion. Growth of 20 µm (to give platelet dimensions 40 µm as in Fig. 1c) then implies v > 6 x 10-16 m3/s, approximately. This indicates minimum LSi 10-25 mol2/J per s, only two orders of magnitude below the experimentally based estimates at 1000°C. Although the temperature dependence of LSi is not yet well constrained experimentally, indications are that LSi would be < 10-25 mol2/J per s at T < 800°C (Table 2 and following sections). Thus, the exsolution reaction in the Lilloise olivine probably took place at T > 800°C.
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APPLICATION TO METAMORPHIC CORONAS
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Studies of the distribution of minerals among layers in coronas
have led to an understanding of their diffusion-controlled growth
(Ashworth & Sheplev, 1997
). In one case with a symplectic
layer, the ratio
LSi/
v is fairly well constrained for diffusion
through the grain boundaries across layers. Assuming that
LSi in the reaction front is approximately the same as in those
boundaries, an order-of-magnitude estimate can be obtained for
. This case is the hornblende–spinel symplectite originally
described by Mongkoltip & Ashworth (1983)
. It grew from
plagioclase in a reaction with olivine (Fig.
7). The rocks are
troctolites from
500 Ma gabbroic intrusions in NE Scotland.
Unlike the Lilloise, these were emplaced during regional metamorphism,
and cooled slowly, undergoing retrograde reaction in the amphibolite
facies (Mongkoltip & Ashworth, 1983
). The symplectite has
rod geometry (Fig.
8).
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Fig. 8. Rod symplectites in troctolite from Belhelvie, NE Scotland. (a) Hornblende–spinel symplectite, forming the hornblende–spinel layer of Fig. 7, in which spinel (lighter) is the rod mineral. Many rods are sectioned approximately perpendicular to length, and show nearly circular cross-sections. (Reflected light, oil immersion.) (b) A more complex area in backscattered electron image (from Mongkoltip & Ashworth, 1983, Fig. 4f). In hornblende–spinel symplectite (left), the spinel rods (bright) are sectioned oblique to length and therefore show elongation. In anorthite–hornblende symplectite (right), polygonal hornblende is the minor or rod mineral. This area of anorthite–hornblende symplectite is unusual in also containing blebs of spinel.
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The estimate for LSi/v comes from previous work. Using the slight zoning of hornblende across the monomineralic layer as a measure of minimum concentration gradient for Al in the grain boundaries, Ashworth (1993) estimated a maximum value for (DAl)eff.t, where (DAl)eff is the effective or bulk diffusion coefficient for Al and t is the duration of reaction. Following the reasoning of Ashworth & Sheplev (1997, p. 3681), this gives a maximum estimate for (LSi)eff.t, 3·8 x 10-10 mol2/J per m, corresponding to a minimum overstep of equilibrium temperature (by 100°C), and minimum estimate for (-G)ext (10 kJ per mole of symplectite produced). It is not possible that (LSi)eff.t was an order of magnitude smaller than this, because the temperature overstep would become geologically impossible (many hundreds of degrees). It is conceivable that concentration gradients were even smaller than considered above, in which case (LSi)eff.t is underestimated, and (-G)ext is overestimated by the same factor. However, Ashworth et al. (1998, p. 245), comparing the present reaction at T 600°C with a granulite-facies one at T 720°C, noted that the suggested difference in (LSi)eff is consistent with experimental activation energies for grain-boundary diffusion in this temperature range (Farver & Yund, 1995, 1996).
During time t, the reaction front advanced into plagioclase by a distance of typically 100 µm. Although growth rate should decrease parabolically with time (Ashworth & Sheplev, 1997), three-quarters of the symplectite should have grown at a rate within a factor of two of (100/t) µm/s. Thus dividing (LSi)eff.t by 100 µm gives, as a viable approximation, (LSi)eff/v 3·8 x 10-6 mol2/J per m. The approximate relation between (LSi)eff and LSi is
where
N is the number of grain boundaries per metre of line
transect (Ashworth
et al., 1998
). Measurements give
N 3
x 10
4 m
-1 in the hornblende layer (Ashworth, 1993
), leading to
LSi/
v 6·3
x 10
-11 mol
2/J per m.
The value of K is to be estimated next. Following the previous diffusion modelling by Ashworth & Birdi (1990) and Ashworth & Sheplev (1997), 24-oxygen moles are used for all minerals. For each mole of plagioclase consumed, more than one mole of symplectite is produced (Fig. 7), because material diffuses along grain boundaries through the corona to the symplectite-forming reaction front. (Implicit in this is that oxygen diffuses along with the cations. Oxygen diffusion is not treated explicitly in the modelling, where the components are oxides, so that Si stands for SiO2. Relatively easy oxygen diffusion, in comparison with the reaction in olivine, is consistent with the generally non-coherent nature of grain boundaries in the coronas.) This addition of material is incorporated into the calculation of K, by using the stoichiometric coefficients 
j of the minerals and their molar contents nij of components i. Diffusion to the reaction front being predominantly grain-boundary diffusion through the symplectite layer, diffusion through bulk minerals is neglected: addition or subtraction of component i at the reaction front is due solely to the local reaction. Advance of the reaction front by 1 m corresponds to production of 1/V moles of symplectite per m2 and consumption of (-Pl/
j)/V moles of plagioclase, where Pl is the stoichiometric coefficient of plagioclase and the summation is over the product minerals. This releases (-Pl/j) niPl/V moles of component i. The proportion of the reaction front at which mineral j is growing is pj. The deposition of component i in mineral j corresponding to the above release from plagioclase is (j/pjj)nij/V. Thus the net addition of component i to the reaction front where j is growing is given by
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Table
3 shows (
kj)
i values,
leading to
K from equation (A2) of the Appendix.
The above working follows corona models using LSi/LAl = 0·1 (Table 3). It is possible to model the coronas with other LSi/LAl ratios, as high as 0·3 and with no lower limit; LSi/Li (i = Mg, Fe, Ca, Na) then varies proportionally to LSi/LAl (Ashworth & Sheplev, 1997, Fig. 3). So, approximately, does the value estimated for LSi/v; thus the ratio of the two that appears in LSi/Kv for equation (11) varies relatively little, and trial calculations indicate that the estimates presented here are reliable to within a factor of two, provided that LSi/LAl > 0·03. It seems unlikely that the two diffusion coefficients differ by more than an order of magnitude, given that both Si and Al diffuse within the reaction front but hardly at all beyond it (Mongkoltip & Ashworth, 1983).
The remaining quantity required is , which is measured in micrographs such as Fig. 8. If n rods are counted in an area A m2, each rod being associated with an area of approximately 
(/2)2, the estimate for is
The average result
is 5·0 ± 0·5 µm from 322 rods. Micrographs
giving anomalously large
(up to 11 µm) were discounted
as probably from sections cut misleadingly oblique to the rods,
or material coarsened after growth. This is supported by the
following observation on the anorthite–hornblende symplectite,
which occurs as patches adjacent to the hornblende–spinel
(Fig. 8b). The geometry is more regular, and consistently gives
= 10·0 ± 0·4 µm. It has a smaller
K because Al and Si are less strongly separated than in hornblende–spinel
(Table
3). The factor of
18 in
K suggests a factor of (18)
1/3 = 2·6 in
if
v was the same, which compares well with
the factor of 2·0 in the above data.
Using all the above information, the result for from equation (11) is 1·3 J/m2. This is an order-of-magnitude estimate only, but is satisfactorily within the range of values indicated by the literature reviewed in the last section. The measurement of is one obvious possible source of error. Errors could also arise in LSi/v; if the estimate of (-G)ext is too large, or if LSi in the reaction front is larger than in grain boundaries within layers, then the value used for LSi/v will be too small and the estimate of too large. Using = 1·3 J/m2, Fig. 9 shows the suggested ranges of LSi for the reaction, and also that the concentration limit was not approached.
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