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International Geology Review
2010, iFirst article, 1––40
Porphyroblast crystallization: linking processes, kinetics,
and microstructures
1938-2839 Geology Review,
0020-6814
TIGR
International
Review Vol. 1, No. 1, Jun 2010: pp. 0––0
William D. Carlson*
International
W.D.
CarlsonGeology Review
Department of Geological Sciences, University of Texas at Austin, Austin, TX, USA
Downloaded By: [University of Texas At Austin] At: 21:34 5 August 2010
(Accepted 6 April 2010)
Analysis of the processes, kinetics, and microstructures that characterize porphyroblast
crystallization identifies the primary factors that govern the progress of metamorphic
reactions and highlights the importance of feedbacks among those factors. Commonly,
the kinetics of nucleation and the kinetics of intergranular diffusion are rate-limiting in
porphyroblast crystallization. That finding should inspire petrologic vigilance, as it
implies strong potential for significant thermal overstepping of reactions, crystallization at high levels of chemical affinity, reactions that span protracted intervals of time
and temperature, and limited length scales for chemical equilibration.
Keywords: subsolidus crystallization; kinetics; porphyroblast; nucleation; intergranular
diffusion
Introduction
Porphyroblastic textures are a rich source of information on metamorphic processes,
because they originate via an intricate set of kinetic interactions and feedbacks that can be
identified through careful analysis of a rock’’s microstructural and microchemical features.
This article reviews our current understanding of the key factors that commonly govern
the development of porphyroblastic textures, and explores the linkages among the processes,
kinetics, and microstructures that characterize porphyroblast crystallization.
A brief historical overview reveals that knowledge of the rates and mechanisms of
metamorphic reactions has been sought diligently over several decades. Classic early
treatments were published by Fisher (1978), Walther and Wood (1984), and Rubie and
Thompson (1985), among others. These approaches relied heavily on extrapolations to
nature of reaction rates determined experimentally in the laboratory. But substantiating the
validity of these extrapolations was not easy, inasmuch as there was no convincing way to
verify that the mechanisms and especially the kinetics of reactions studied at laboratory
timescales were relevant to metamorphic recrystallization in nature. Methods of extracting
direct information on reaction processes from observable characters of natural rocks were
pioneered by Kretz (1966, 1969, 1973, 1974); Kretz was among the first to recognize that
crystallization mechanisms could be inferred from the relative spatial disposition of
porphyroblasts and from their crystal size distributions (CSDs), combined with interpretation
of compositional zonation in minerals such as garnet that preserve a record of prograde
chemical changes during growth. But the stochastic nature of porphyroblast crystallization
*Email: [email protected]
ISSN 0020-6814 print/ISSN 1938-2839 online
© 2010 Taylor & Francis
DOI: 10.1080/00206814.2010.496184
http://www.informaworld.com
Downloaded By: [University of Texas At Austin] At: 21:34 5 August 2010
2
W.D. Carlson
requires that large numbers of measurements be amassed to generate a statistically significant result, and the necessary measurements, being inherently 3D, necessitated laborious
grain-by-grain dissection of rocks (Kretz 1973) or tedious grinding for serial sectioning
(e.g. Marschallinger 1998; Spear and Daniel 1998). Consequently, full development of the
techniques of quantitative textural analysis was delayed until the advent of high-resolution
computed X-ray tomography made possible rapid collection of datasets of high accuracy
that encompassed large numbers of crystals (e.g. Carlson and Denison 1992; Ketcham et
al. 2005). As these data have accumulated, they have brought to light the crucial role
played by kinetic feedbacks that determine the rates of nucleation and growth of porphyroblasts, which in turn exert first-order control on primary microstructures and on the style
and extent of chemical equilibration of porphyroblasts with the matrix from which they
grow.
This review attempts to draw together the central insights that have appeared in the
literature on the processes, kinetics, and microstructures of porphyroblast crystallization.
The important cognate topic of controls on chemical equilibration, however, falls largely
outside its scope. Although multiple interconnections preclude examining each topic
wholly in isolation from the others, the treatment below first describes a variety of potential
crystallization processes, then examines the factors that control their kinetics, and finally
explores how porphyroblastic microstructures are related to the interplay among crystallization processes.
Crystallization processes
Crystallization of porphyroblasts results from several distinct grain-scale processes:
generation of an appreciable chemical affinity for reaction, typically by input of thermal
energy into a rock; dissolution of reactant phases into the intergranular medium; nucleation of product phases, in particular the crystals that will grow to become porphyroblasts;
transport of dissolved components through the intergranular medium to the surfaces of the
growing porphyroblasts; and growth by removal of components from the intergranular
medium and their accretion to surfaces of porphyroblasts. If one considers local effects at
the site of crystallization of any single porphyroblast, then these processes occur in
sequence. But in the rock as a whole, they take place concurrently, which permits a variety
of potential interactions among them; the types and extent of interactions depend upon
their relative rates.
Porphyroblastic textures can reflect not only these primary processes of crystallization
that create a population of new crystals from a precursor assemblage, but also subsidiary
or secondary processes that might potentially modify the primary microstructure. These
two categories are treated separately below.
Primary processes
Because local rates of crystallization are governed by the slowest of multiple sequential
processes, several end-member crystallization mechanisms can be defined that are distinguished from one another by different rate-limiting processes, and that are identifiable on
the basis of their macroscopic consequences for porphyroblastic microstructures. Nucleation
of new crystals and growth on pre-existing crystals are competing means of lowering the
reaction affinity during crystallization; they act in parallel, so rates of growth govern the
overall reaction rate when nucleation is comparatively slow, and vice versa. The following
discussion first explores three categories of mechanisms that can control rates of growth,
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International Geology Review
3
then considers the role of nucleation rates by contrasting the feedbacks between nucleation
and growth that arise for growth rates governed by different mechanisms.
In a seminal study, Fisher (1978) estimated the rates of three key processes that might
control crystal growth –– heat flow, intergranular diffusional transport, and accretion on
growth surfaces –– and reasoned that each could become rate limiting over different ranges
of length scales and temperature. He inferred that, although the very first stages of growth
might be limited by the rate of precipitation on the surfaces of micron-scale crystals,
growth of porphyroblasts to macroscopic size at typical metamorphic temperatures should
be limited by the rate of intergranular diffusion of materials from reactants to products.
Generalizing on Fisher’’s (1978) analysis, one can envisage primary mechanisms of
nucleation and growth governed by the rates of change in factors that produce the driving
forces for crystallization, by rates of intergranular transport, or by rates of surficial reaction.
Each of these mechanisms is described below, then a comparison is made between the
macroscopic effects of crystallization controlled by intergranular diffusion and by accretion on growth surfaces.
Driving-force-controlled mechanisms
In common prograde sequences, reaction is initiated when temperature exceeds that of the
equilibrium condition between reactant and product assemblages, and the driving force
behind recrystallization is the chemical affinity for reaction that results from progressively
greater thermal overstepping of the equilibrium. In such cases, heat flow (i.e. the rate of
input of thermal energy into the rock) could conceivably control the overall rate of endothermic reactions, as they can proceed only as rapidly as heat is supplied to the reacting
system. Insofar as many porphyroblasts form via endothermic prograde dehydration
reactions, many circumstances contain at least the potential for heat-flow-controlled
crystallization. Of course, input of thermal energy is not the only means of initiating reaction; any departure from equilibrium produces a chemical affinity for reaction that serves
as a driving force. Thus, retrograde reactions could conceivably be controlled by rates of
heat flow out of the reacting volume; reactions driven by pressure changes might be
limited by rates of compression/decompression; and reactions driven by the influx of a
chemically reactive fluid might be limited by rates of fluid flow or rates of change of fluid
composition by mixing or diffusion. The most general treatment considers the fundamental
driving force to be the chemical affinity for reaction and assesses whether the rate of
change in factors that generate the chemical affinity limits the overall rate of reaction.
Transport-controlled mechanisms
The requirement that nutrients be relocated from sites of reactant dissolution to sites of
porphyroblast crystallization makes it possible for the transport mechanism to become
rate-controlling for the local reaction. In metamorphic systems, this transport is likely to
take place either by advection or by diffusion. Advective transport is commonly pictured
as movement of components dissolved in a mobile fluid phase that migrates through the
rock via a set of multi-scale fractures; microfractures at grain scale may connect to a larger
macroscopic fracture system, and may be transient features that are generated and
destroyed during active deformation or during episodes of fluid generation and escape. In
contrast, diffusive transport of components is envisaged to take place through a static
intergranular medium; the intergranular medium may adopt a range of possible configurations,
from a set of unsaturated grain boundaries with relatively low activities of volatile species
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4
W.D. Carlson
to an interconnected network of fluid-saturated grain edges. If a separate fluid phase is
present at grain junctions, it might be a supercritical mixture of volatile species (H2O,
CO2, CH4, etc.) that bears non-volatile components in solution, or it might, at high
temperatures, be a melt.
Because advection and diffusion are alternative mechanisms of transport that can
proceed in parallel, the dominant transport mechanism in a given situation will be whichever is faster. Advective transport in a mobile fluid phase is expected to dominate in the
case of open-system behaviour, which is typically characterized by large instantaneous
fluid-to-rock ratios, as in hydrothermal systems or where strain localization acts to define
and focus pathways for fluid flow. Diffusional transport through a static fluid phase, on
the other hand, is likely to dominate in closed systems typified by low instantaneous fluidto-rock ratios. The means by which fluids generated by prograde devolatilization reactions
escape the rock, although rarely well-characterized, should therefore play an important
role. Local fluid generation might be episodic, varying as reactant assemblages evolve;
likewise, fluid escape might be sporadic if flow is focused by progressive deformation into
sets of transient microfractures. One can then envisage circumstances in which crystal
growth takes place during periods when the fluid is static and diffusional transport
dominates the kinetics, even though these periods of crystallization are punctuated by
fleeting episodes of fluid escape during which diffusional gradients are flattened by rapid
advective flow.
When evaluating intergranular diffusion as a potential rate-limiting factor, it is important
to recognize that intergranular diffusive fluxes of various elements will differ considerably
from one another under identical conditions, and the key determinant of rates of porphyroblast growth will be the relative availability of the principal components of the crystal.
The rate of porphyroblast growth will be limited by whichever component exhibits the
lowest diffusional flux relative to the required rate of supply, that is, the flux needed to
maintain an appropriate stoichiometry for the crystal. All components with higher fluxes
relative to their proportions in the crystal will be available at the surface of the growing
crystals in amounts adequate to support whatever crystallization is allowed by the diffusional supply of this growth-limiting component. Note that it is not necessarily the slowestdiffusing component that is growth limiting: for instance, if an aluminium-silicate porphyroblast is growing from a quartz-rich matrix, the flux of Al may limit rates of growth even
if the intrinsic diffusivity of Si is lower than that of Al, because abundant local sources
may shorten the required diffusion distances for Si, whereas local depletion of scarcer
sources of Al may require that it be transported from ever-increasing distances as the
porphyroblast grows.
Parenthetically, it is worth considering the possible effects of a build-up in the vicinity
of the porphyroblast of ‘‘waste products’’ –– constituents released from the reactants but not
incorporated in the porphyroblast, as well as reactants present locally in excess of the
amount needed for porphyroblast growth. If they do not diffuse away, these materials
might progressively accumulate in the region undergoing conversion to the new porphyroblast, and if they diffuse slowly, their build-up could conceivably reduce the local chemical
affinity for reaction and impede porphyroblast growth. Poikiloblastic textures may be
interpreted as evidence that the common fate of these materials is local precipitation (or,
for reactants in excess, persistence in the matrix), so that they become inclusions within
the porphyroblast.
Finally, a poorly understood diffusive process must operate to distribute material along
the outer surface of the porphyroblast. As discussed below, common metamorphic fluids
may wet grain edges (three-grain junctions) but not grain surfaces (two-grain junctions).
International Geology Review
5
This implies that transported constituents first arrive at the edges of the porphyroblast, and
must then move from the edges across the surface of the crystal to produce outward
growth of the surface. This ‘‘surface diffusion’’ should exhibit kinetics different from intergranular diffusion, insofar as the diffusive medium –– at least when grain edges are fluid
saturated –– is likely to be distinctly different. Although this process remains largely
unstudied, one might surmise that because of the shorter distances over which movement
is required, surface diffusion is typically capable of keeping pace with the influx of
constituents to grain edges via intergranular diffusion, and indeed this must be true in the
typical case of porphyroblasts that develop facets or outwardly convex shapes, and
concentric zoning patterns.
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Interface-controlled mechanisms
At the nanoscale, porphyroblast-forming reactions require dissolution into the intergranular
medium and precipitation from it; that is, they require removal of constituents from the
solid surfaces of the reactant crystals as they dissolve, and –– after transport –– extraction of
those constituents from solution by precipitation as they are incorporated into the crystal
structure at the surface of the growing porphyroblast. If both of these surficial processes
occur rapidly in comparison to the rate of transport between reactants and products, then
concentration gradients will be established that drive diffusion as described above: concentrations in fluid in contact with reactants will be buffered to the values that represent
local equilibrium between reactant and fluid, whereas concentrations in fluid in contact
with products will be fixed at the values that represent local equilibrium between product
and fluid. Alternatively, if transport rates are rapid enough to flatten any concentration
gradients that would otherwise develop within the intergranular medium, then rates of
crystal growth will be interface controlled, that is, limited by rates of dissolution or precipitation at the interfaces between the intergranular medium and either reactant or product
crystals.
When growth is interface controlled, rapid transport imposes on the fluid a spatially
uniform composition that is undersaturated with respect to reactants but supersaturated
with respect to products. Rates of reactant dissolution should then depend upon the degree
of undersaturation of dissolving constituents at the surfaces of reactants, whereas rates of
precipitation should depend upon the level of supersaturation of precipitating constituents
at the surfaces of the products. If rates of dissolution are rapid in comparison to rates of
precipitation, then the uniform composition of the fluid will approach equilibrium with the
reactants; if rates of precipitation are rapid in comparison to rates of dissolution, then the
uniform composition of the fluid will approach equilibrium with the products.
Feedbacks between nucleation and growth under interfacial versus diffusional controls
Although the processes of dissolution, nucleation, transport, and precipitation may be
considered as sequential steps in the crystallization of a single porphyroblast, a rock-wide
view of crystallization must consider that all of these processes will commonly take place
simultaneously at different sites. This raises the possibility of multiple interactions and
feedbacks that have profound effects on the rock’’s microstructure and the porphyroblasts’’
microchemistry.
To examine these feedbacks, discussion will be restricted to crystallization under
circumstances that are among those most commonly encountered in the study of porphyroblastic textures, namely crystallization driven by input of thermal energy, in rocks saturated
Downloaded By: [University of Texas At Austin] At: 21:34 5 August 2010
6
W.D. Carlson
with a static intergranular fluid that wets an interconnected network of grain edges. An
example would be the prograde crystallization of garnet porphyroblasts from disseminated
fine-grained hydrous precursor phases, of which the simplest case would be the model
reaction of Fe-chlorite with quartz to produce almandine garnet and an aqueous component
of a supercritical fluid.
For kinetic reasons that are discussed below, the mechanisms historically regarded as
most likely to be rate limiting for the growth of aluminous porphyroblasts in common
circumstances are the interface-attachment process that precipitates dissolved constituents
onto the surface of the growing porphyroblast and the transport of Al from reactants to
products by intergranular diffusion. These lead to idealized end-member cases of porphyroblast crystallization that can be identified as ‘‘interface-controlled nucleation and
growth’’ (ICNG) and ‘‘diffusion-controlled nucleation and growth’’ (DCNG). The contrasting macroscopic effects of the feedbacks between nucleation and growth that occur in
these two end-member cases are illustrated in Figure 1.
In the case of ICNG, rapid diffusion precludes development of any appreciable gradients
in intergranular concentration of the principal constituents needed for crystallization. As a
result, the intergranular medium is a spatially uniform locus of elevated chemical affinity
for reaction, and constitutes a spatially uniform reservoir of the nutrients required for crystal growth. In a matrix that is sufficiently fine grained to be considered homogeneous at a
scale smaller than the spacing between eventual porphyroblasts, potential nucleation sites
will be uniformly disposed throughout the matrix. The spatially uniform chemical affinity
for reaction in the intergranular fluid then results in an equal probability of nucleation at
any point in the rock, except within the volumes already occupied by product crystals; this
results in a near-random spatial disposition of nuclei. Because all crystals derive their
nutrients from the same common rock-wide reservoir, the proximity of crystals to one
another and the relative abundance of reactants in the vicinity of each crystal have no
effect on growth rates.
In the case of DCNG, however, sluggish diffusion leads to appreciable gradients in the
chemical affinity for reaction, manifested as gradients between reactants and products in
the concentration of one or more constituents involved in the reaction. This leads to
numerous feedbacks between nucleation and growth processes. Once a crystal has nucleated and started to grow, a zone that has been diffusionally depleted in nutrients forms
around it and gradually expands outward. Within the depleted zone, the chemical affinity
for reaction is lowered compared with regions not yet affected by diffusion, which
decreases the probability of nucleation in proximity to the growing crystal (Figure 1a).
This localized suppression of nucleation results in a tendency towards spatial ordering of
nuclei, expressed as an increase in the average centre-to-centre spacing of neighbouring
crystals compared with the ICNG case. Also, because each crystal derives its nutrients
from its immediate surroundings, close proximity of crystals to one another leads to lower
growth rates because of competition among them for locally available nutrients (Figure 1b).
This growth suppression is expressed as a tendency towards correlation between the sizes
of crystals and measures of their spatial isolation, an effect not seen in ICNG. Finally,
because nutrients are sourced proximally in DCNG, the local abundance of reactants is a
primary control on the total amount of crystal growth that will occur at any site (Figure 1c).
Reactant-rich regions will yield more total growth for equivalent time than will reactantpoor regions, which results in a direct proportionality between the size of a crystal and its
instantaneous rate of growth; this is again an effect not seen in ICNG.
As an aid to visualization of the feedbacks operating during DCNG, Figure 2 presents
results from a numerical simulation of prograde DCNG in a heterogeneous (layered)
International Geology Review
Interface control
7
Diffusion control
Reaction
affinity
a
Reaction
affinity
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Nucleation
suppression
b
Growth suppression
due to competition
f = 0.01
f = 0.02
c
f = 0.01
f = 0.02
Growth rate and size
scale with local abundance
of nutrients
f = 0.05
f = 0.10
f = 0.05
f = 0.10
Figure 1. Comparison of interface-controlled and diffusion-controlled nucleation and growth.
Porphyroblasts are represented by black polygons. (a) Implications on spatial distribution of nuclei.
During ICNG (left), rapid diffusion precludes development of appreciable gradients in reaction
affinity, so the probability of nucleation is spatially uniform throughout the rock. During DCNG
(right), development of diffusionally depleted zones of low reaction affinity results in suppression of
nucleation in the vicinity of existing crystals, which produces a tendency towards spatial ordering in
the location of crystal centres.
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8
W.D. Carlson
precursor, as described by Carlson and Ketcham (2004) and implemented by Ketcham and
Carlson (2004). With rising temperature, reactant dissolution elevates the reaction affinity
by increasing the concentration of the growth-limiting constituent in the intergranular
medium (Figure 2a and b). Nucleation begins in regions of the highest reaction affinity
and accelerates with increasing temperature (Figure 2b and c). Wherever reactants remain,
the concentration of the rate-limiting constituent is buffered to equilibrium with the reactant, but as reactants are locally exhausted in regions surrounding growing crystals, diffusionally depleted zones develop, suppressing nucleation within those zones of lowered
reaction affinity (Figure 2c and d). Diffusion transports nutrients from reactant-bearing
regions to the surfaces of growing crystals, producing growth (Figure 2c and d); diffusion
also relocates nutrients into originally reactant-free layers, elevating their reaction affinity
and inducing nucleation in them (Figure 2d and e). As the rest of the rock’’s reactants are
consumed by dissolution (Figure 2d), diffusional relaxation of concentration gradients in
the fluid marks the final stage of reaction (Figure 2e), and the system approaches its newly
equilibrated state as reaction ceases (Figure 2f). Crystal separations are greater than would
be the case for a random disposition; sizes of proximal crystals reflect competition for
nutrients during growth; and crystals are larger and more numerous in the regions originally richer in reactants.
Because the feedbacks that operate during DCNG arise from local gradients in the
chemical affinity for reaction, one would expect that such feedbacks would not emerge
when crystallization is limited by rates of change in factors that generate driving forces, by
rates of reactant dissolution, or by rates of advective transport. In those cases, as in the
case of ICNG, the intergranular medium would behave as a spatially uniform region of
elevated chemical affinity for reaction and as a spatially uniform source of nutrients.
Subsidiary or secondary processes
Populations of porphyroblasts whose origin is dominated by the nucleation and growth
processes just described may be subjected to subsidiary or secondary processes that modify the development of microstructural features such as the spatial disposition of porphyroblasts, their CSD, and the sizes of individual crystals in the original population. These
subsidiary or secondary processes may produce effects that complicate or obscure the
microstructural and microchemical signatures of primary crystallization processes, so
their scope and potential impact must be taken into account when attempting to interpret
textures in terms of mechanism.
Figure 1. (b) Implications of crystal locations on growth rates. During ICNG (left), all crystals
grow from the same uniform reservoir of nutrients; growth rates are proportional to each crystal’’s
surface area but are independent of proximity to other crystals. During DCNG (right), growth rates
depend upon the local availability of nutrients, so rates of growth are suppressed for crystals growing in close proximity to one another, leading to a correlation between the size of a crystal and its
degree of isolation in space and time from other crystals. (c) Implications of nutrient heterogeneity
on growth rates. During ICNG (left), diffusion rapidly redistributes constituents from regions of
higher abundance to regions of lower abundance, so all crystals grow from the same uniform reservoir, and rates of growth are independent of local reactant abundance. During DCNG (right), variations from one region to another in volume fraction f of reactants (grey rectangles) produce
differences in growth rate and size of porphyroblasts, because equivalently sized source volumes for
nutrients supplied by diffusion (dashed circles) yield higher nutrient fluxes in more reactant-rich
regions.
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International Geology Review
a
b
c
d
e
f
9
Figure 2. Sequential stages in numerical simulation of diffusion-controlled nucleation and growth.
Results shown are from a 2D version of a 3D model used for analysis of natural porphyroblastic textures (Ketcham and Carlson 2004). Greyscale brightness is proportional to the concentration of the
growth-limiting constituent in the intergranular medium; green tint indicates the presence of reactant;
red regions are porphyroblasts, with deeper red colours corresponding to material accreted earlier in
the crystallization interval. Temperature and time increase from (a) to (f). Boxes in the final panel (f)
call attention to three crystals that nucleated nearly simultaneously [cf. panel (c)]; the larger crystal at
the left grew largely in isolation from other crystals, whereas the pair at the right were limited to
smaller size by competition for nutrients. In panel (f), the locations of the originally reactant-rich
layers are indicated by the faint grey overlay, which allows one to discern that those regions contain
larger and more numerous porphyroblasts, a fact otherwise not evident in the final microstructure.
10
W.D. Carlson
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Bulk deformation
Deformation during crystallization may influence both nucleation and growth. Bell (1985)
and Bell et al. (1986) argued, for example, that nucleation may be localized at crenulation
hinges in rocks rich in phyllosilicates. Transport properties may also be affected: during
active deformation there is a strong potential for focusing of fluid flow, and the anisotropy
of intergranular diffusion pathways in rocks with well-developed planar fabrics may modify the shapes of diffusional domains. Few studies address these questions directly, but the
analysis of syn-kinematic garnet crystallization by Meth and Carlson (2005) suggests that
these processes have only a modest impact on the mechanisms and kinetics of porphyroblast crystallization.
Discounting the extreme case of granulation of porphyroblasts and dispersion of the
resultant particles, the principal impact of bulk rock deformation on a pre-existing population of porphyroblasts would be to redistribute them in space relative to one another in
ways that might alter the signals of nucleation suppression and growth suppression that
are used to identify DCNG. This possibility will be reconsidered below in the section on
porphyroblastic microstructures, after discussion of the methods used to identify suppression
signals.
Asymmetry of growth and impingement during growth
Quantifying the degree of nucleation suppression that takes place during DCNG requires
accurate measurement of the location of each nucleation event that occurs. Asymmetric
growth and impingement of crystals during growth can make it more difficult to locate
properly the site(s) of nucleation.
The nucleation site is commonly regarded as coincident with the morphological centre
of the crystal, and this can be confirmed if the morphological centre also coincides with
the geometrical centre of concentric patterns of compositional zoning (e.g. Chernoff and
Carlson 1997) or the centre of helical symmetry in crystals with spiral internal fabrics
(Robyr et al. 2009). But asymmetric growth, produced for example by inhibition of
growth against nutrient-poor layers or veinlets, can lead to crystals whose morphological
centre does not coincide with the centre of concentric patterns of compositional zoning: an
example would be the ‘‘partial’’ crystals described in Meth and Carlson (2005). It is not
practical, however, to identify nucleation sites on the basis of compositional zoning for all
of the large number of crystals whose locations must be included in a statistically valid
sample set, so the assumption that nucleation occurred at a crystal’’s morphological centre
is routinely made.
Such small offsets in the location of an individual crystal’’s nucleation site relative to
its morphological centre are demonstrably immaterial, but a more significant problem
arises when two or more nucleation events occur close enough to one another in time and
space that the result after growth is an aggregate that may not be easily recognized as
polycrystalline, depending upon the methods used to measure crystals and their locations.
The isometric character of garnet porphyroblasts may preclude definitive recognition of
polycrystalline aggregates by optical means. In garnet, patterns of irregular compositional
zoning, such as multiple erratic Mn-highs inherited from precursor heterogeneities, can be
misleading indicators of polycrystallinity (cf. Hirsch et al. 2003), but electron backscatter
diffraction (EBSD) measurements can provide unambiguous confirmation (or contraindication) of multiple proximal nucleation events, even in rare circumstances in which significant reorientation and/or recrystallization has caused partial amalgamation of multiple
International Geology Review
11
originally independent crystals by reducing misorientations among them (Spiess et al.
2001; Dobbs et al. 2003). For garnet porphyroblasts, polycrystalline aggregates resulting
from impingement during growth are common and well-documented, for example by
Chernoff and Carlson (1997), Hirsch et al. (2003), Meth and Carlson (2005), Whitney et al.
(2008), and others. For QTA, correction techniques have been implemented, based on
observability criteria that account for the likelihood of failing to recognize polycrystalline
aggregates as separate crystals; these improve assessments of nucleation suppression and
growth suppression in such instances (Ketcham et al. 2005).
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Ostwald ripening
The techniques used to identify DCNG are predicated on the notion that the measured
numbers, locations, and sizes of crystals have remained essentially unmodified since primary crystallization occurred. Ostwald ripening is a process that could potentially operate to
modify significantly the primary microstructure of a population of porphyroblasts by
shrinking or eliminating smaller crystals in favour of larger ones, in a process driven by
reduction in total interfacial free energy. Although several authors (e.g. Cashman and
Ferry 1988; Miyazaki 1991, 1996; Eberl et al. 1998) have argued that some porphyroblast
size distributions are generated by Ostwald ripening or substantially altered by it, quantitative evaluation of the kinetics of such a process operating in metamorphic rocks leads to
the conclusion that for times and temperatures appropriate to metamorphic crystallization,
Ostwald ripening can have appreciable effects only at submicron to micron grain sizes
(Carlson 1999); at larger grain sizes, driving forces become negligibly small.
Resorption
Given the progressive nature of metamorphism, a particular mineral may be produced by
reactions at low grade, but may be consumed by reactions at higher grade; more than one
such cycle is possible. This scenario is expected, for example, for garnet in a range of
common pelitic bulk compositions across amphibolite- to granulite-facies conditions; garnet crystals may be partially resorbed if they react to form staurolite but may resume
growth at still higher temperatures (e.g. Loomis 1986; Ridley and Thompson 1986; Pattison and Tinkham 2008). To the extent that these resorption and re-growth reactions modify the ultimate measured crystal sizes, they can impact interpretation of crystallization
mechanisms. In some instances, resorption is easily recognized from thin-section evidence
of coronal reaction rims or from replacement at the margins of originally idiomorphic
crystals (e.g. Denison and Carlson 1997; Carlson 2002), and in such cases it is possible to
correct textural analyses for this effect after proper quantification by HRXCT measurements (Ketcham et al. 2005). If evidence of resorption is instead found only in modification of compositional zoning patterns, valid corrections may not be possible. Because
renewed nucleation is unlikely in an episode of secondary growth following resorption,
crystal sizes will be affected but locations of crystal centres may not, unless resorption is
so extensive that it eliminates altogether some small crystals.
Crystallization kinetics
Quantitative understanding of the kinetics of each key crystallization process is vital
because, as described above, the relative rates of those processes determine the importance
of alternative reaction mechanisms and govern the feedbacks among them. Unfortunately,
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W.D. Carlson
however, few of them are amenable to experimental replication in the laboratory. Quantification of rates of many individual processes has therefore proven difficult, although
progress has been made in determining rates of overall reaction in some circumstances.
Short summaries are given below of current levels of knowledge of the kinetics of key
crystallization processes and their likely role as controlling factors in common metamorphic
environments; this is followed by an assessment of overall rates of porphyroblast crystallization. The discussion will focus on the kinetics of prograde reactions mediated by a
supercritical aqueous fluid.
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Heating rates
The input of thermal energy into rocks, or heat flow, is a fundamental driver of prograde
metamorphic reactions. The rate of energy influx, however, is generally controlled by
external or environmental factors –– tectonics, magma intrusion, effectiveness of conduction
or advection, and so on –– with the result that heating rates can be widely variable. Perhaps
the greatest contrast is the orders-of-magnitude difference between rates of heat flow for
contact metamorphism compared with regional metamorphism.
In contact metamorphic environments, evidence is accumulating that mechanisms of
nucleation and growth are sometimes, perhaps commonly, incapable of keeping pace with
heating rates over a range of timescales and length scales (e.g. Joesten and van Horn 1999;
Waters and Lovegrove 2002; Pattison and Tinkham 2008), strongly suggesting that input
of thermal energy is rarely rate limiting in contact metamorphism.
Likewise, evidence presented below for the dominance of other factors is a good
indication that heat flow is not the rate-controlling process in most common regional
metamorphic environments. But it is nevertheless instructive to consider what one might
seek as macroscopic indicators of heat-flow-controlled crystallization. In heat-flowcontrolled crystallization, endothermic reaction would tend to buffer temperature so that it
could not exceed the equilibrium temperature except by a very small amount. Consequently, the activation energy for nucleation of new crystals of product phases would not
be exceeded to any great degree, with the result that crystallization would take place by
the growth of a very small number of nuclei to very large size, at conditions that never
depart far from equilibrium (cf. Ridley 1985, 1986). An analogous example of crystallization
in such circumstances –– albeit an extreme one –– is the precipitation of gypsum from
aqueous solution at the Naica Mine in Chihuahua, Mexico, where maintenance of nearequilibrium conditions led to exceedingly sluggish nucleation in comparison to growth,
resulting in production of gargantuan crystals of selenite, up to 11 m in length (Garcia-Ruiz
et al. 2007). Although it remains unproven, it is possible that porphyroblasts of extraordinary
dimension in some metamorphic environments represent cases in which heat-flow-controlled crystallization produced exceptionally low rates of nucleation. It may be difficult,
however, to discriminate such cases from an alternative scenario in which rates of growth
are very high relative to the rates of nucleation because of rapid advective transport in
fluid-rich systems, or conceivably, because of an extreme paucity of favourable nucleation
sites.
Rates of dissolution and precipitation
Experimental approaches have been successful in quantifying the kinetics of surficial
processes, that is, dissolution from the surfaces of reactants and precipitation on the
surfaces of products. In a classic synthesis of published data, Wood and Walther (1983)
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13
evaluated rates of dissolution of a variety of minerals in a large number of hydrothermal
experiments in which several weight percent of H2O (±CO2) was present to provide a
freely circulating fluid capable of transporting dissolved constituents efficiently from
reactants to products. Given such rapid transport, the kinetics of dissolution and precipitation –– operating simultaneously –– become rate limiting and can therefore be estimated
from measured rates of overall reaction. The experimental data demonstrate that these
surficial processes occur at rates that are proportional to the degree of undersaturation/
supersaturation and to the surface area of the phases involved; overall, the reaction rate is
limited by the reactant or product with the smallest surface area. Both processes proceed
rapidly on laboratory timescales when driven by reaction affinities comparable to those in
natural circumstances, so in nature they ‘‘are not rate-determining except under exceptional circumstances’’ and ‘‘transport rather than surface reaction is generally the slowest
step’’ (Walther and Wood 1984, p. 249). Because these experiments were conducted at
fluid-to-rock ratios higher than those encountered in nature –– except perhaps in hydrothermal systems –– the conclusion that dissolution and precipitation mechanisms are not rate
limiting is even more compelling for low fluid-to-rock ratios, as smaller quantities of fluid
would require even less reaction progress to equilibrate with either reactants or products.
In the light of this unambiguous and longstanding demonstration that surficial processes are not commonly rate limiting in nature, it is puzzling that so many later analyses of
metamorphic reaction kinetics have continued to employ formulations with explicit or
implicit interfacial controls, relying upon rate equations that depend directly on surface
area (Lasaga and Rye 1993; Balashov and Yardley 1998; Ague and Rye 1999; Lasaga et al.
2000; Baxter 2003).
At least one clear demonstration does exist, however, of precipitation rates governing
the reaction kinetics for porphyroblast formation in nature, albeit in unusual circumstances. Wilbur and Ague (2006) provide a compelling analysis of the origin of extraordinary garnet crystals with chemical features and inclusion patterns that reveal initial
dendritic to branched morphologies. They conclude that these features reflect the initiation
of growth at unusually high levels of supersaturation, and they apply an elegant model
relating morphology to supersaturation and to the energetics of bonding at the garnet
surface. The observed textures and their model indicate that, after a period of initial
growth at high supersaturation, the consequent reduction in the chemical affinity for reaction caused the reacting system to evolve towards one in which surficial kinetics were no
longer rate limiting. As a noteworthy aside, this study also calls attention to the importance to crystal morphology of the energetics of bonding at the crystal surface; the observation that many porphyroblasts commonly adopt idioblastic habits, and their consequent
position in the ‘‘crystalloblastic series’’, derives principally from the intensity of surface
bond strengths, and need not reflect specific mechanisms of growth.
Nucleation rates
Nucleation of a stable new crystal requires that the resulting reduction in free energy
exceeds the energetic cost of creating interfaces at the surface of the new crystal. Thus,
rates of nucleation depend principally upon the magnitude of the reaction affinity and
upon the values of the free energies of interfaces between the nucleus and its surroundings.
Kinetic theories of nucleation, originating largely in studies of metallurgical and ceramic
systems, quantify these dependencies.
The fundamentals of these theories are outlined below; a more detailed explanation
and development can be found, for example, in Christian (1975, Chapter 10), or in Kelton
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14
W.D. Carlson
et al. (1983). These theories proceed from the postulate that small clusters of atoms (or
molecules) of the product phase grow or shrink by the incremental addition or removal of
single atoms (or molecules). The kinetics and energetics of these incremental changes,
considered in aggregate, govern the evolution of the overall population of variably sized
clusters.
If a cluster of atoms of a product phase, representing a potential new nucleus, is very
small, then for a slight increment of its further growth, the energetic cost of creating additional interfacial area will exceed the reduction of energy due to the incremental decrease
in reaction affinity, so the cluster will be unstable. But if a cluster of sufficient size should
form, then because of its smaller surface-to-volume ratio, its further growth can result in a
net decrease in free energy. The smallest cluster whose further growth will produce a net
decrease in free energy is therefore said to be a nucleus of critical size, and the work
required to create a critically sized nucleus is the activation energy that must be overcome
for nucleation to occur. Rates of porphyroblast nucleation per unit volume (dNV/dt) can be
expected to obey an equation of the general form
 − QN 
dN V
γ3
,
= ν CN exp 
, with QN = s

2
dt
 kB T 
(∆GV )
(1)
in which the first factor (n) can be regarded as the frequency with which single atoms (or
molecules) are accreted to critically sized clusters to convert them to stable nuclei, and the
product of the two remaining factors (CN and the exponential) gives the concentration
(number per unit volume) of critically sized clusters that represent potential sites of nucleation; CN is the concentration of subcritical clusters of all sizes, and the exponential factor
describes the fraction of them that are of critical size. In the exponential factor, kB is
Boltzmann’’s constant, and the activation energy for nucleation (QN) contains a geometrical factor (s) for the shape of the critically sized nucleus, and is proportional to the cube of
the term for the interfacial energy per unit surface area of nucleus (g3) but inversely
proportional to ( ∆GV)2, with ∆GV being the reduction in free energy per unit volume of
the nucleus, which is the negative of the reaction affinity per unit volume.
Homogeneous nucleation, in which nuclei form directly from the host intergranular
medium without the involvement of a third phase, generates greater interfacial energy
costs than does heterogeneous nucleation, in which nuclei form on favourable substrates
that have low interfacial energy against the nucleus. In the case of heterogeneous nucleation,
the energetic contributions due to the various interfaces and to the shape of the nucleus
will be functions of the absolute and relative magnitudes of the interfacial energies between
each pairing among the nucleus, its host, and the substrate. Consequently, rate equations
for heterogeneous nucleation, although similar in overall form to Equation (1), must take
on greater complexity to account properly for the effects of interfacial energies; no completely general form can be specified.
In a prograde crystallization event, the pre-exponential factor in Equation (1) will
remain sensibly constant, whereas the exponential term will increase from nearly zero
towards a maximum of unity as the reaction affinity increases during progressive overstepping of the reaction. As a result, the behaviour of the nucleation rate in such an event will
resemble that shown in Figure 3. Rates of nucleation asymptotically approach a maximum
value fixed by the number density of potential nucleation sites; larger reaction affinities
(resulting from greater reaction entropies, or greater amounts of thermal overstepping, or
International Geology Review
Nucleation rate (cm–3·sec–1)
Nucleation rate (cm–3·sec–1)
4.0E-13
4.0E-13
3.5E-13
Larger density of
potential nucleation sites
3.0E-13
2.5E-13
3.0E-13
Larger reaction affinity
Smaller interfacial energy
2.0E-13
1.5E-13
Smaller density of
potential nucleation sites
1.0E-13
5.0E-14
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3.5E-13
2.5E-13
2.0E-13
0.0E+0
435
15
1.5E-13
Smaller reaction affinity
Larger interfacial energy
1.0E-13
5.0E-14
455
475
495
Temperature (°C)
515
535
0.0E+0
435
455
475
495
Temperature (°C)
515
535
Figure 3. Dependence of nucleation rates on temperature during prograde metamorphism, calculated from Equation (1). Specific values are for illustration only, but fall in the range determined by
Kelly and Carlson (2008b) in numerical simulations that match natural textures. Calculated rates
give the number of nuclei per unit volume per unit time in regions of the rock with maximum reaction affinity, that is, regions unaffected by diffusion. In each case, the rate rises exponentially at first,
then asymptotically approaches a limiting value determined by the number density of nucleation
sites; the rapidity with which the limiting value is approached increases with greater reaction affinity
and decreases with greater interfacial energies.
both) and smaller interfacial energies will both cause the maximum value to be
approached more rapidly. If the precursor contains a variety of potential nucleation sites
with a range of interfacial energies, then rates will be different for each type of site –– faster
for sites with lower interfacial energies, slower for sites with higher interfacial energies ––
and the overall nucleation rate would be the sum of the rates at each type of site. However,
as Figure 3 illustrates, rates will be exponentially higher on sites with lower interfacial
energies, so the overall rate will be dominated by nucleation on those sites.
Nucleation rates given by Equation (1) will not obey an Arrhenius-type relationship, in
which the logarithm of the nucleation rate is a linear function of inverse absolute temperature, because the activation energy is not temperature-independent. Instead, because the
value of ∆GV increases progressively above the equilibrium temperature, the activation
energy for nucleation decreases with increasing temperature, and the nucleation rate (in
regions of maximum reaction affinity) approaches a limiting value that is invariant over
time and temperature.
Application to natural systems of the classical theory of nucleation outlined above is
problematic, particularly when dealing with polyphase metamorphic reactions in which
the nucleation of more than one product mineral is required. To take this into account,
Equation (1) would have to be modified to include several terms that account for the new
surfaces produced by creation of new nuclei of each of the newly formed products, but it is
unclear how the overall reduction in free energy should be partitioned across each of these
phases. (Fortunately, although the overall energetics of the reaction should also take into
account the decrease in interfacial energy resulting from dissolution of reactant crystals,
this should be negligible, as the amount of dissolution needed to supply nutrients for the
critical nuclei is so small that the reduction in interfacial energy is immaterial when the
precursor minerals are already macroscopic in size.) Nonetheless, the classical theory of
nucleation outlined above likely captures the essential relationships among the factors that
affect nucleation rates, and thus provides a useful kinetic framework.
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W.D. Carlson
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Effects of reaction affinity
The role of reaction affinity in nucleation is best considered by focusing again on a prograde crystallization event in which fluid-saturated grain junctions are the principal means
of chemical communication between reactants and products. In this case, the reaction
affinity will be manifested in a fluid in which key components are supersaturated with
respect to the product phase, and the rate of nucleation will depend upon the degree of
supersaturation.
In the case of ICNG, the supersaturation will be spatially uniform; it will rise as the
reaction is overstepped, will fall as crystallization proceeds, and will eventually return to
zero as the reaction reaches completion. The degree of supersaturation over time –– and
thus the rate of nucleation –– will depend upon the relative rates of heating and reaction. If
interface-controlled growth is rapid enough to nearly keep pace with heating, the supersaturation will remain low and nucleation rates will be modest. Conversely, if interface-controlled growth is slow, thermal overstepping will drive the supersaturation to high values,
and the nucleation rate will quickly approach its maximum value; the resulting increase of
crystallization because of growth on the new nuclei will provide a negative feedback by
reducing the reaction affinity and thus decreasing the nucleation rate. At all times, nucleation rates are spatially uniform throughout the rock.
In the case of DCNG, however, the degree of supersaturation will vary in both space
and time because concentration gradients will develop between reactants and products. In
the vicinity of growing crystals, where supersaturation is low, nucleation rates will be
negligible; in portions of the rock not yet affected by reaction, the presence of reactant
phases will buffer fluid compositions, supersaturation will rise continually with thermal
overstepping, and local rates of nucleation in these regions will increase accordingly
towards the limiting value.
Effects of interfacial energies
Equation (1) illustrates that nucleation rates are extremely sensitive to interfacial energies,
because interfacial energies affect the activation energy so dramatically. Consequently,
because activation-energy barriers are lower, heterogeneous nucleation should be strongly
favoured over homogeneous nucleation if suitable substrates exist. This is especially true
when the reaction affinity is small: with a small driving force, the only sites that can be
activated are those that present the lowest energy barriers. At large reaction affinities,
however, nucleation can occur simultaneously at all sites for which the activation-energy
barrier is exceeded; nonetheless, it will occur at exponentially higher rates on lower-barrier
sites than on higher-barrier sites, so heterogeneous nucleation on low-energy sites should
again predominate.
Nucleation rates may therefore be expected to show great variation in nature, insofar
as precursor materials may present greatly different types and densities of potential nucleation sites that provide diverse degrees of reduction of interfacial energies. This effect was
invoked, for example, to account for variations in rates of nucleation of biotite porphyroblasts in adjacent lithologic layers in the study of Hirsch and Carlson (2006). The potential
for epitaxial nucleation is high, and strong evidence for it exists in the case of crystals of
garnet that bear orientational relationships to precursor micas or other phases (Frondel
1940; Powell 1966; Spiess et al. 2007; Robyr et al. 2009).
Quantification of the interfacial energies pertinent to porphyroblast nucleation remains
elusive; only order-of-magnitude estimates are available. For heterogeneous nucleation on
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a substrate by precipitation from a fluid, the activation energy depends not only on the
interfacial energy between the nucleus and the substrate, but also on the interfacial
energies between each of the solids and the host fluid. Ashworth and Chambers (2000)
summarized a variety of earlier estimates (p. 294––295) and their own work with the
conclusion (p. 301) that ‘‘grain-boundary energies of minerals are ∼0.1 to 1 J/m2’’, and
values in that range were also estimated for quartz––quartz and garnet––quartz interfaces by
Miyazaki (1996, p. 280). The interfacial energy between quartz and water was estimated
by Parks (1984) as 0.36 J⋅m−2, which is slightly above the maximum in the range of values
tabulated by Walton (1967) for a variety of soluble sulphates, carbonates, and halides
precipitated from aqueous solution. These estimates have limited predictive value, however,
because nucleation rates vary exponentially with the cube of the interfacial energies. As a
consequence, very precise knowledge of interfacial energies would be required to make
quantitative predictions of nucleation kinetics, even if the mechanism –– including the
identity of the substrate and any epitaxial relationships –– were known.
Experimental determination
The sensitivity of nucleation rates to interfacial energies also makes it extremely difficult
to conceive and execute laboratory experiments that can provide reproducible rates of
nucleation while replicating the essential character of the process as it occurs during
porphyroblast crystallization. In fact, no experimental determinations of nucleation rates
have been published that have direct relevance to the nucleation of porphyroblasts.
The few data on nucleation rates that do exist (e.g. Liu and Yund 1993; Kerschhofer
et al. 1998) are restricted to solid-state polymorphic transformations, and even in these
relatively simple cases, reproducibility across a variety of starting materials has not been
demonstrated. Such reproducibility is essential, because without it, the likelihood is high
that measured rates of nucleation are governed principally by the abundance of heterogeneities of various types within the starting materials: even in the simple case of the solidstate topotaxial transformation of aragonite to calcite, for example, the presence of free
surfaces, cleavages, fractures, twins, and inclusions was found to exert dominant control
over nucleation rates (Carlson and Rosenfeld 1981; Carlson 1983).
Consequently, the applicability of experimentally determined nucleation rates to natural
environments hinges critically upon demonstration that the experiments properly replicate
the mechanisms operating in nature. This is a daunting requirement, especially in the polyphase systems relevant to porphyroblast crystallization, so it is not surprising that successful experimental approaches to the nucleation of porphyroblasts have not yet been reported.
Observational constraints
Despite great difficulty in quantifying nucleation rates, some important observational
constraints on nucleation kinetics do exist. The illuminating study by Joesten and van
Horn (1999) of crystallization kinetics at the margins of basalt dikes intruded into
siliceous-carbonate hosts illustrates that even when heating and cooling rates are very
rapid, the exponential rise in nucleation rate with thermal overstepping can yield appreciable
numbers of nuclei. Crystals of product phases (grossular and wollastonite) appeared in
host rocks at the margins of all dikes except the very smallest, those for which the time
spent above the equilibrium temperature for reaction was measured in tens of days.
In regional metamorphic environments, garnet crystals in pelitic and mafic rocks can
preserve a record of their relative times of nucleation in the chemical composition at their
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W.D. Carlson
cores; for example, in common circumstances, Mn is sequestered in garnet during
progressive crystallization, so that later-nucleating crystals have lower Mn concentrations
in their cores than do early-nucleating crystals. If it can be demonstrated that Mn has diffused rapidly enough through the intergranular medium to achieve rock-wide equilibration, then the central Mn content of garnet crystals can serve as an index to each crystal’’s
time of nucleation. If care is taken to section crystals precisely through their centres to
measure compositions at their cores, a record can be obtained of the relative time of nucleation of each crystal. Studies using this approach have documented both nearly ‘‘instantaneous’’ nucleation, in which all crystals form very early in the crystallization event and
then grow concurrently (e.g. Meth and Carlson 2005), as well as the alternative case in
which nucleation is a protracted process that spans nearly the entire crystallization interval, with new crystals appearing even in the final stages of reaction (e.g. Chernoff and
Carlson 1997; Denison and Carlson 1997).
An emerging observational approach with considerable promise is the extraction of
nucleation rates from numerical models of natural porphyroblastic textures, constrained
by HRXCT measurements of key microstructural features. Early models of DCNG (Carlson
et al. 1995) retrieved reasonable values for rates of intergranular diffusion, but in some
cases required unrealistically large values for activation energies for nucleation. More
sophisticated models (Ketcham and Carlson 2004; Kelly and Carlson 2008b) that actually
compute the spatial and temporal variation of the reaction affinity during a prograde crystallization event, employing a nucleation rate law equivalent to that in Equation (1) and a
diffusion rate law equivalent to that in Equation (2) below, have met with much greater
success. These models quantify within a narrow range both nucleation rates and their variation with time and temperature, and place constraints on the values of key physical factors that determine nucleation kinetics in nature. Results from this approach are model
dependent, so accuracy requires reliable constraints on reactions, metamorphic conditions,
and heating rates in the natural occurrences, but results to date indicate that estimated
uncertainties in inputs translate to uncertainties in retrieved nucleation rates (and the physical factors that determine them) that are quite small compared with the wide ranges
spanned by other estimates.
Diffusion rates
The principles governing rates of intergranular diffusion are relatively well understood,
although quantification of diffusion rates remains a challenge. Diffusive fluxes during a
crystallization event are dynamic functions of temperature and the intrinsic mobilities of
species; of concentration gradients, solubility, and speciation in the intergranular medium;
and of the nature of the intergranular network that constitutes the set of diffusional pathways during reaction.
Intergranular diffusional fluxes are given by a variant of Fick’’s first law, namely
J = − Deff ⋅!c ⋅ φ ⋅ τ .
(2)
For bulk diffusion through a uniform medium, the flux (J) is proportional to the concentration gradient (!c), and the constant of proportionality is the diffusion coefficient (D); but
intergranular diffusive fluxes are also proportional to the volume fraction of the intergranular medium (f, which is the interconnected porosity in a fluid-saturated system), and the
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constant of proportionality is an effective diffusion coefficient (Deff), which can be
thought of as the bulk diffusion coefficient in a hypothetical material consisting entirely of
the intergranular medium. Because intergranular diffusion proceeds along irregular grain
junctions, the paths between sources and sinks are not straight lines, so the concentration
gradient (!c) between source and sink must be reduced by a tortuosity factor (τ).
Temperature and intrinsic mobilities
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The temperature dependence of the effective diffusion coefficient is formulated in terms
of the Arrhenius expression
 −Q 
Deff = D∞ exp  D 
 RT 
(3)
in which D∞ is the fictive effective diffusion coefficient at infinite temperature, and QD
the activation energy for diffusion through the intergranular medium. Diffusivities therefore increase exponentially with temperature; the magnitude of this temperature dependence is given by the activation energy.
The values of D∞ and QD define what may be called the intrinsic mobility of a diffusing species; they are fundamental in the sense that they reflect features characteristic of
the diffusing substance such as the size of the diffusing molecular group (e.g.
ions + ligands ± spheres of hydration) and the strength of the bonds within the intergranular medium that must be broken and reformed during diffusional transport.
Concentration gradients, solubilities, and speciation
Equation (2) illustrates that fluxes are directly proportional to gradients in concentration
(!c). The ratio between concentrations in equilibrium with reactants and products scales with
the reaction affinity, with the result that diffusional fluxes scale with solubilities (e.g. Frantz
and Mao 1976). Thus, highly soluble species can be more efficiently transported by intergranular diffusion than relatively insoluble species. Consequently, the nature and chemistry of the
intergranular medium are key factors in the kinetics of intergranular diffusion.
The activity of H2O in the intergranular medium exerts first-order control on the diffusivity of components needed for growth of typical porphyroblasts. Carlson (2002) compared
the diffusivity of Al during H2O-saturated crystallization versus hydrous-but-H2O-undersaturated crystallization under amphibolite-facies conditions, and inferred a difference between
effective diffusion coefficients of four orders of magnitude. A later study of almost completely anhydrous reaction suggested a further decrease in diffusivity by two to three orders
of magnitude compared with the hydrous-but-H2O-undersaturated case (Carlson, in press),
so the full range of the effects of H2O activity may span 6––7 orders of magnitude.
The chemistry of intergranular aqueous fluids must also have a direct effect on diffusivity. Cursory comparison of the relative solubilities in typical metamorphic fluids of the
dominant cations involved in metamorphic reactions (e.g. Yardley 2005) would suggest
rapid transport of monovalent cations, somewhat more restricted rates of diffusion for
most divalent cations, and limited diffusivity of trivalent, quadrivalent, and pentavalent
cations. This general pattern for relative rates of diffusion has been confirmed in numerous
studies of natural coronal textures and metasomatic zones, for which analytical models
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W.D. Carlson
based in non-equilibrium thermodynamics (Joesten 1977; Ashworth and Birdi 1990; Johnson
and Carlson 1990) place quantitative constraints on the ratios of elemental diffusivities
(e.g. Johnson and Carlson 1990; Carlson and Johnson 1991; Ashworth et al. 1992; Ashworth
and Sheplev 1997; Markl et al. 1998). The general trends in relative diffusivity probably
reflect the relative solubilities determined by the chemistry of the most commonly
encountered fluids, but a wide range of variation should be possible. If ligands are present
(e.g. Cl−, OH −, F −, CO3−2, SO4−2, PO4−3, etc.) that enhance the ability of a fluid to draw
certain species into solution, diffusional transport of those species will be increased, and
conversely the absence of such ligands will reduce diffusional transport of those species.
Identifying situations in which unusual fluid chemistry controls diffusivity is complicated
by the fact that fluids may not leave behind in the rock an unambiguous record of their
content of key ligands. Further effects may be expected from variations in solubility and
speciation as a result of temperature, pH, and oxidation state, although Yardley (2005)
argues persuasively that in deep crustal fluids, the latter two factors are likely to be buffered within narrow ranges by equilibration of fluids with common mineral assemblages.
Intergranular network
The type and abundance of diffusional pathways are determined by the geometry of the
intergranular network and its volume fraction (f). During prograde dehydration, when
reactions are actively evolving an aqueous component of a fluid phase, the distribution of
the fluid in the rock may or may not represent an energetically stable configuration. But if
thermodynamic equilibrium of interfacial tensions is presumed, then the existence of an
interconnected intergranular network depends upon the relative magnitudes of the interfacial
energies of fluid and minerals, which determine the ability of the fluid to wet intergranular
junctions. The available experimental data, however, paint a very convoluted picture in
which the wetting characteristics of deep crustal fluids depend in highly complex ways
on fluid composition, mineralogy, mineral chemistry, mineral orientation, temperature,
and pressure (Watson and Brenan 1987; Hay and Evans 1988; Brenan 1991; Laporte and
Watson 1991; Lee et al. 1991; Holness 1993; Watson and Lupulescu 1993; Wark and
Watson 1998; Hiraga et al. 2001; Yoshino et al. 2002; Mibe et al. 2003). Furthermore,
most data come from work on monomineralic aggregates of quartz, plagioclase, pyroxene,
olivine, or carbonate, so their ability to predict the distribution of fluids in polymineralic
natural assemblages may therefore be limited. One highly relevant result stands out, however: for saline aqueous fluids in quartzose assemblages under deep crustal conditions,
interfacial energies dictate that fluids in the rock will aggregate along three-grain junctions, making up an interconnected network along grain edges rather than segregating into
unconnected pores (Watson and Brenan 1987).
The sources just cited all agree that the wetting characteristics of metamorphic fluids preclude the stable existence of fluid films covering two-grain junctions (grain boundaries). But
grain boundaries in quartzose rocks can nonetheless host appreciable concentrations of
hydrous components; in the presence of H2O, quartz surfaces and quartz––quartz grain boundaries terminate in silanol (SiOH) groups, which are themselves covered by adsorbed molecules of H2O (Holness 1993). Similar features may be expected also to characterize two-grain
or three-grain junctions involving silicates other than quartz, over a range of H2O activities.
This hydroxylation of grain junctions will enhance intergranular diffusion in hydrous-butfluid-undersaturated rocks, and could assist surface diffusion (transport across grain surfaces).
Diffusional fluxes scale in proportion to the volume fraction of the interconnected grainedge network, which is the mathematical product of the cross-sectional area of the channels
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International Geology Review
21
along three-grain junctions and the total length of such channels. Watson and Brenan (1987,
p. 498) observe in their experiments that the length of the channel is determined by the grain
size of the material, whereas the channel diameter (or cross-sectional area) is controlled primarily by the fraction of fluid present. Finer-grained rocks will have a higher spatial density
of grain edges than coarser-grained ones, so if the volume fraction of fluid in the rock is considered to be fixed by some external constraint, the cross-sectional areas of the channels
must be smaller in finer-grained rocks than in coarser-grained rocks.
But the open-system behaviour of fluids during metamorphism may differ importantly from the closed-system behaviour in laboratory experiments. In particular, the
amount of fluid in the intergranular network might well be determined directly by the
scale of the network. Because the wetting of grain-edge channels via capillarity is in
itself evidence that wet grain edges have a lower free energy than dry ones, wetting
should cause retention of small quantities of fluid along grain edges, even though substantial quantities of fluids produced by dehydration reactions will escape the rock
through microscale and macroscale fracture systems. In that case, the equilibrium volume
fraction of fluid in the rock will be a function of the abundance of capillaries, that is, a
function of the size and abundance of the grain-edge channels. Finer-grained precursors,
because of their greater spatial density of grain edges, would contain a larger volume
fraction of fluid within their intergranular network than would coarser-grained precursors
under the same conditions. The evidence developed by Carlson and Gordon (2004) suggests that this is true, and that diffusional transport in fine-grained rocks is enhanced
accordingly. They showed that, if the cross-sectional areas of grain-edge channels are
assumed to be sensibly independent of grain size, then the volumes of porphyroblasts
whose growth is controlled by diffusion along grain edges in the precursor should vary in
proportion to the inverse square of the precursor grain diameter. Their data for mean
sizes of garnets in six biotite-quartzites of variable matrix grain size are consistent with
that predicted relationship.
If diffusive fluxes do depend upon matrix grain size, then it is important to consider
the effects of grain coarsening during metamorphism. Precursor assemblages may coarsen
appreciably during the prograde crystallization of porphyroblasts, reducing the volume of
the intergranular network, which in turn will diminish diffusional fluxes. This effect runs
counter to the increase in diffusivity produced by rising temperature. An attempt to assess
the relative magnitudes of these effects suggests that, in certain circumstances, grain
coarsening of the matrix may profoundly reduce diffusional fluxes, leading to an overall
reduction in diffusivity despite increases in temperature (Figure 4 in Carlson and Gordon
2004). This reduction is most likely to occur if crystallization involves relatively finegrained precursors (with sub-millimetre grain diameters) and relatively high temperatures (>500––600°C), but even then it may not be significant if coarsening is inhibited by
the presence of secondary phases or inclusions that impede the migration of grain boundaries (cf. Herwegh and Kunze 2002; Berger and Herwegh 2004).
Tortuosity
Tortuosity (t) in polygranular aggregates is difficult to measure, but it can be calculated
for idealized geometries. Calculations of this type dealing with diffusion through grain
boundaries (two-grain junctions) rather than grain edges (three-grain junctions) suggest
that values of t range from approximately 0.3 to 0.9 (cf. Brady 1983); this multiplicative
factor is small compared with the uncertainty in other quantities contributing to estimates
of diffusional fluxes, so effects of tortuosity are commonly ignored.
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W.D. Carlson
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Determination of intergranular diffusivities
It has proven very challenging to measure intergranular diffusivity experimentally in ways
that allow reliable extrapolation to natural systems. A major obstacle is to produce an
experimental situation in which reaction is clearly limited by intergranular diffusion yet
occurs to a measurable extent at laboratory timescales. As an example, Brady (1983)
demonstrated that single crystals of periclase immersed in quartz powder at 700°C at an
H2O pressure of 0.1 GPa are armoured by an approximately 20 µm-thick polycrystalline
rim of forsterite in a matter of minutes, but that further reaction, which requires intergranular diffusion across the forsterite rim, is negligible even after 8 weeks. An approach
sometimes taken in such circumstances is to conduct experiments at higher temperatures,
then extrapolate back down to natural temperatures of interest (e.g. Tanner et al. 1985;
Fisler et al. 1997; Liu et al. 1997; Yund 1997; Milke et al. 2001; Milke and Heinrich
2002; Keller et al. 2008), but this is commonly not feasible for studies relevant to the
crystallization of most porphyroblasts because the appropriate assemblages are not stable
at the high temperatures required for appreciable reaction. There is much promise in the
experimental approach taken by Farver and Yund (1995a,b, 1996, 2000a,b), who
successfully quantified rates of intergranular diffusion of K, Ca, Si, and O by measuring
the bulk penetration of isotopically labelled tracers into very fine-grained polycrystalline
aggregates of feldspar, quartz, calcite, and forsterite. However, no data have yet appeared
on some vital elements (most notably Al), sparse information is available on the effect of
variable H2O activity, and the role of fluid chemistry remains largely unexplored.
The paucity of experimental data on intergranular diffusivity has engendered attempts to
extract such data from natural occurrences in which intergranular diffusion can be identified
as the rate-limiting process and for which sufficient constraints exist on temperature and
heating/cooling rates to allow quantification of diffusivities. Three studies of coronal
textures (Ashworth 1993; Carlson 2002, in press) produce reasonably well-constrained
estimates for the diffusivity of Al, but the results are pertinent only to the anhydrous or H2Oundersaturated conditions under which the coronas formed, and the determinations for H2Oundersaturated conditions are likely to underestimate transport during H2O-saturated
prograde porphyroblast growth by roughly 4 orders of magnitude (Carlson, in press). Extraction
of estimates for intergranular diffusivity of Al from numerical simulation of the evolution of
natural porphyroblastic textures (Carlson et al. 1995; Kelly and Carlson 2008b) has led to a
reasonably consistent set of estimates at temperatures characteristic of amphibolite-facies
crystallization, but as with similar estimates of nucleation rates, these determinations are
model dependent and rely upon accurate characterization of metamorphic conditions.
Rates of advective transport
At high fluid-to-rock ratios in open systems, advection can be a highly effective means of
transport of dissolved constituents in fluids. Fluid fluxes may be channelized either by
fracture systems or by localization in layers undergoing reduction in solid-phase volume
because of devolatilization. But to the extent that a pervasive component of fluid flow acts
at grain scale, advective fluxes will be contributing factors in porphyroblast growth. When
advective flows are rapid, the attendant fluxes are large, and factors other than transport
become rate limiting for the reaction. In that case, accepting that rates of surficial processes
are rapid, as experimental determinations seem to dictate (Walther and Wood 1984), then
rates of nucleation may remain as the only factor capable of throttling reaction rates. Each
nucleus that appears should be expected to grow rapidly, and the reaction can run to
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completion before large numbers of nuclei form; the expected result is then a texture
consisting of a small number of large porphyroblasts.
An intriguing and well-known set of occurrences that is likely to illustrate such crystallization is the suite of localities in the Adirondack Mountains, New York, in which
mafic amphibolites host unusually large garnet porphyroblasts, some ranging up to a
metre or more in diameter. McLelland and Selleck (2008) noted that the unusually large
porphyroblasts are restricted to portions of the amphibolite unit that are in proximity to
granitic pegmatites; they attribute the growth of large garnet crystals to the influx of aqueous fluids sourced by the intrusives. However, it is also noteworthy that these large
crystals are commonly surrounded by a monomineralic zone of either amphibole or plagioclase that represents a portion of the matrix amphibolite that has been depleted of one
mineral or the other during growth of the garnet. Such zonation indicates local sourcing of
key nutrients, and is characteristic of diffusion-controlled growth. This zonation opens the
door to speculation that these occurrences are examples of the hybrid control mechanism
alluded to above, in which episodic fluid influx and escape allows for repeated development of local gradients in chemical affinity during low-flow or no-flow periods, punctuated by advective flows that flatten these gradients. If periodic flushing of the system
imposes a low level of supersaturation on the intergranular fluid, then once the original
number and spacing of nuclei is established, further nucleation would be negligible, all
growth would take place on the existing nuclei, and the diffusional gradients would
continually be re-established in the same spatial pattern after each flushing event. In this
way, depleted zones surrounding the porphyroblasts could develop during episodic intervals of diffusion-limited growth, interspersed with intervals of rapid advective transport.
Composite rates of overall reaction
The descriptions above call attention to several feedbacks that act to determine overall rates
of reaction in prograde sequences. The most elemental of them is the interplay between rising temperature (which acts to increase the reaction affinity) and crystallization via nucleation and growth (which acts to reduce the reaction affinity). Nucleation rates increase
exponentially to a limiting steady-state value in response to thermal overstepping of a reaction; likewise, rates of intergranular diffusion increase exponentially with absolute temperature. Thus, a linear increase in temperature provokes an exponential rise in crystallization
rate, which effectively places an upper limit on the magnitude of reaction affinities and a
lower limit on the rate of crystallization in most circumstances. Because of the exponential
response of crystallization rate to thermal overstepping, the most fundamental control on the
rate of prograde metamorphism is simply the rate of temperature increase. Consequently, the
crystallization of metamorphic minerals can span millions of years (e.g. Christensen et al.
1989) when driving forces change slowly, or can be over in a matter of days (e.g. Joesten
and van Horn 1999) when driving forces change rapidly.
From this perspective, the difference in overall reaction rates between contact and
regional metamorphic environments (e.g. Baxter 2003) is easily understood as a reflection
of the contrast in rates of heat flow that characterize intrusive systems on one hand and
tectonism in the deep crust on the other. In both cases, overall rates of reaction are mediated
by rates of change of temperature. This offers an alternative to the position taken by Baxter
(2003, p. 193), who states that the ‘‘discrepancy between regional and contact metamorphic
reaction rates probably relates to differences in fluid content’’.
In passing, it is worth noting that venerable formalisms appear in the metallurgical and
ceramics literature that account for overall reaction kinetics by integrating over time the
24
W.D. Carlson
effects of nucleation rates and growth rates (e.g. Avrami 1939, 1940, 1941; Christian
1975, Section 4). Unfortunately, application of these formalisms generally requires a suite
of assumptions (e.g. isothermal crystallization, constant rates of nucleation, constant rates
of radial growth, random spatial dispositions of nuclei, etc.) that make them poorly suited
to analysis of the inherently complex problem of porphyroblast crystallization.
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Porphyroblastic microstructures
Porphyroblastic microstructures are direct manifestations of the mechanisms that operate
at sub-grain scales during crystallization, and of the kinetic feedbacks among them. Consequently, microstructures can reveal much about the processes that formed them, when
they are analysed quantitatively and interpreted in the context of reaction mechanisms and
kinetics. Information on processes is contained in the porphyroblasts’’ CSDs; in their
relative positions, and in correlations between their sizes and their relative positions; and
in the relative rates of growth of crystals as inferred from features of their compositional
zoning. Furthermore, in DCNG, differences in diffusivity of various elements will impose
kinetic controls on the evolving composition of porphyroblasts, which has led to novel
appreciation of the rates of chemical equilibration and the scales of chemical disequilibrium attending metamorphic crystallization.
Assessment of these microstructural features has become known as ‘‘quantitative textural analysis’’ or QTA. Nearly all QTA of porphyroblastic microstructures has been concerned with garnet crystallization, because garnet is widespread in rocks of variable bulk
composition over a broad range of metamorphic conditions, has an equant habit that facilitates assessments based on spherical symmetry, and is capable of preserving in its chemical
zoning a detailed record of conditions during growth. Thus, garnet crystallization will be the
focus of the discussion that follows.
Crystal size distributions
The relative frequency of occurrence of porphyroblasts of different size –– the
porphyroblasts’’ CSD –– encodes a record of rates of nucleation and growth integrated over
the crystallization interval, simply because the size of each crystal is determined by how
long and how fast it grew. To decode this record, forward models of crystallization
governed by one or more chosen mechanisms are used to generate model CSDs that can
be compared to CSDs measured in natural rocks. The results, however, cannot identify
mechanisms unambiguously, because nearly identical size distributions can arise from
more than one governing mechanism, and because a single such mechanism operating
under different metamorphic conditions can produce a wide array of CSDs.
As an example, consider that, although there is a general tendency towards larger
porphyroblasts in rocks of higher grade, porphyroblast CSDs tend to be self-similar over a
range of grades (e.g. Cashman and Ferry 1988); in other words, when the CSDs from
rocks metamorphosed under a range of conditions are normalized to mean radius, they are
nearly invariant, despite increases in mean radius and decreases in crystal number density
at higher grade. Cashman and Ferry (1988) and Miyazaki (1996) interpreted this selfsimilarity to be the result of Ostwald ripening, inferring that larger crystals grew at the
expense of smaller ones to produce a net reduction in interfacial energy, and that this
occurred to a greater degree in rocks of higher grade. That interpretation is in agreement
with annealing theory (the LSW formulation: Lifshitz and Slyozov 1961; Wagner 1961),
which predicts eventual attainment of a steady-state normalized CSD, invariant over time.
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Carlson (1999), however, interpreted the same characteristic of porphyroblast CSDs to be
the result of DCNG, because nucleation rates increase with the degree of thermal overstepping of a reaction, whereas diffusion rates, which control crystal growth, increase with
absolute temperature. The result is that the reaction kinetics at high temperature favours
rapid growth on existing nuclei because of faster diffusion, whereas the reaction kinetics
at low temperature favours nucleation of new crystals, because rates of growth are limited
by slower diffusion. Hirsch (2008) demonstrated that models of DCNG could generate
CSDs matching those in the same rocks for which Cashman and Ferry (1988) invoked
Ostwald ripening as the dominant mechanism, illustrating the ambiguity that is inherent in
attempts to deduce crystallization mechanisms from analysis of CSDs alone. Although
there are subtle but important features of the CSDs themselves that can often distinguish
between the two interpretations (in particular, the sense of skewness and the expected
frequency of crystals that depart greatly from the mean radius), both interpretations are
broadly consistent with the general shape and character of measured natural CSDs, so the
conclusion that Ostwald ripening is not a significant modifier of CSDs at macroscopic
grain sizes comes primarily from a quantitative analysis of its kinetics (Carlson 1999).
As the preceding example illustrates, sophisticated analyses that seek to extract crystallization mechanisms from CSDs (e.g. Eberl et al. 1990, 1998; Kile et al. 2000; Kile and
Eberl 2003) are unlikely to meet with success when applied to porphyroblasts, because of
the great complexity of factors that influence porphyroblast crystallization kinetics. This is
particularly true for the case of DCNG. CSDs spanning the full range of published behaviours can arise from this single process, owing to the intricate feedbacks among heating
rates, nucleation rates, and growth rates, when combined with the limitless variety of heterogeneous reactant distributions in precursors (Carlson and Ketcham 2008; Kelly and
Carlson 2008a). For instance, DCNG from a heterogeneous precursor with rapid saturation
of nucleation sites (Meth and Carlson 2005) gives rise to size-proportional growth rates
that would be interpreted as the result of an entirely different mechanism using the
approach of Kile and Eberl (2003).
The non-uniqueness of porphyroblastic CSDs means that reaction mechanisms must
be inferred from a variety of microstructural and microchemical evidence, and not from
CSDs alone. Nonetheless, CSDs do provide important constraints on inferred mechanisms
and kinetics, because any proposed mechanism must be consistent with observed CSDs.
Agreement between forward models and measured CSDs is therefore a necessary, but not
sufficient, condition for correct identification of crystallization mechanisms.
Spatial disposition of crystals and size-isolation correlations
Statistical analysis of the spatial disposition of porphyroblasts, and of the degree of correlation between crystal sizes and measures of their isolation can be a robust means of identifying the operation of DCNG mechanisms. These techniques seek observational
evidence for the nucleation-suppression and growth-suppression effects illustrated in
Figure 1a and b, and seen in Figure 2.
Methods
This type of analysis was pioneered by Kretz (1969), who examined a variety of statistical
assessments of the spatial disposition of crystals, and in particular presented methods for
comparing measured mean centre-to-centre nearest-neighbour distances to those expected
for a random distribution of points, which constitutes a test for nucleation suppression.
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W.D. Carlson
Additional tests for non-randomness in the positions of crystal centres were described in
Carlson (1989, Appendix 1). Rudimentary tests for growth suppression appeared in Carlson
(1989, 1991), based on comparison of the size of crystals with the distances to their nearest
neighbours; improved techniques were developed by Denison et al. (1997).
These measures have been largely superseded by more sophisticated approaches based
on correlation functions, which are capable of testing for relationships over a range of
length scales, rather than depending upon global averages. The pair correlation function
(PCF) and mark correlation function (MCF) of Stoyan and Stoyan (1994) were introduced
into the crystallization literature by Raeburn (1996), developed more fully by Daniel and
Spear (1999), and further refined by Hirsch et al. (2000), to which the reader is referred
for a full explanation. These functions quantify spatial relationships among crystals over a
range of length scales. The PCF is sensitive to the short-range ordering of crystal centres
that can arise from suppression of nucleation in the vicinity of growing porphyroblasts,
but also to the clustering at any scale that can arise from heterogeneity in the locations of
favourable sites for nucleation, which may be influenced by reactant abundance (Figure 4a).
The MCF is sensitive to the size-isolation correlations that can arise from retardation of
growth among crystals competing with one another for nutrients (Figure 4b). In homogeneous rocks, the PCF and the MCF should be equally capable of revealing crystallization
mechanisms, but spatial heterogeneity in the abundance of nucleation sites or reactants
leads to clustering of crystals, which in the PCF is superimposed upon and thus may
obscure the short-range ordering imposed by diffusion-controlled nucleation suppression.
Thus, in practice, the MCF has proven to be the more useful metric for identifying DCNG;
it is actually preferentially sensitive to growth competition in clustered arrays of crystals,
as competition increases with proximity.
Interpretation of these correlation functions, however, requires careful attention to
proper calculation of Monte Carlo simulations, which are used to identify values of the
functions that constitute a null-hypothesis region for comparison to samples in which
spatial dispositions of crystals are to be evaluated. As applied to porphyroblast crystallization, these functions test the null hypotheses that nucleation is spatially random (except
within the interiors of pre-existing crystals) and that nutrients for growth are supplied by a
spatially uniform reservoir. Consequently, these functions can invalidate the hypothesis
that a microstructure formed by ICNG, if their measured values depart to a statistically
significant degree from the values representing random nucleation and growth from a
spatially uniform reservoir; and if the measured values reveal spatial ordering of crystal centres and competition for nutrients, then DCNG is implied. The converse, however, is not
true: failure to reject the null hypothesis does not demonstrate the operation of ICNG; it
could equally well reflect a DCNG signal that is too weak to be discriminated from the statistical noise. Weakening of the DCNG signal occurs if too few crystals are measured, or if
the crystals are sparse and thus small compared with their mean nearest-neighbour spacing.
So far, it has not proven practical to invert this procedure so as to test the null hypothesis of
DCNG, which if rejected would demonstrate the operation of alternative crystallization
mechanisms –– to do this would require large numbers of Monte Carlo simulations of DCNG,
which are computationally too intensive –– so rigorous demonstration of ICNG or other
mechanisms that yield near-random spatial dispositions has remained elusive.
Summary of accumulated results
Application of correlation-function analysis, with high-quality data and proper attention
to the need for properly constructed null-hypothesis envelopes and observability criteria
International Geology Review
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Test distance (cm)
0.10
0.20
0.30
0.40
0.50
0.60
PCF
1.00
0.75
0.50
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0.25
1.00
MCF
0.90
0.80
0.70
0.60
0.50
0.40
2.5
5.0
7.5
Test distance (mean radii)
10.0
Figure 4. Examples of the use of spatial correlation functions to identify reaction mechanisms.
Data are from garnet crystals in blueschist from Franciscan Complex of California. (Top) Pair correlation function (PCF). Values of the PCF on the vertical axis are calculated from the spatial disposition of crystal centres, over a range of length scales shown on the horizontal axis. Measured
PCF values (shown as dots) are compared to the values obtained from Monte Carlo simulations of
dispositions of a set of crystals of equivalent sizes that would result from an episode of ICNG;
those simulations fall within an envelope indicated by the lightly shaded region. The centre of the
darker shaded vertical bar marks the value of the mean nearest-neighbour distance between crystals, and the bar’’s width is twice the standard deviation of that distance; thus the bar locates the
approximate maximum limit out to which interactions among crystals will occur. At distances out
to this limit, values falling below the envelope reveal a statistically significant departure from randomness towards ordering, consistent with the operation of DCNG. (Bottom) Mark correlation
function (MCF). Values of the MCF on vertical axis are calculated from sizes of crystals and their
distances from their neighbours, over a range of length scales shown on the horizontal axis. As with
the PCF, measured MCF values are compared to the values obtained from Monte Carlo simulations
of ICNG. At distances out to the mean nearest-neighbour limit and just beyond it, values falling
below the envelope reveal a statistically significant departure from randomness that indicates a correlation between the size of crystals and their distances from each other, consistent with the operation of DCNG.
(e.g. Ketcham et al. 2005; Hirsch and Carlson 2006; Hirsch 2008), has provided rigorous confirmation of the common operation of DCNG during porphyroblast crystallization that emerged in earlier studies (Carlson 1989; Carlson and Denison 1992; Chernoff
and Carlson 1997; Denison and Carlson 1997; Carlson and Gordon 2004; Meth and
Carlson 2005).
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W.D. Carlson
Looking in detail at the work done in the author’’s laboratory, the sources above
combined with unpublished data and theses (Denison 1995; Barnett 1999; Meth 1999;
Schneider 1999) document examination of a total of 26 individual samples; of these, 23
generated spatial statistics that depart significantly from randomness in the direction consistent with DCNG. Porphyroblasts studied were garnet (21) and biotite (2). Both pelitic
(18) and mafic (5) lithologies are included, and although these studies were focused
predominantly upon middle-amphibolite-facies assemblages (18), DCNG was also
deduced for one sample each with an assemblage representing the lower greenschist,
upper greenschist, upper amphibolite, blueschist, and eclogite facies. Thus, it appears that
DCNG governs the crystallization of aluminous porphyroblasts across a wide range of
metamorphic conditions in both pelitic and mafic bulk compositions.
The three samples that did not exhibit evidence for DCNG were chosen specifically as
likely counterexamples to the common trend. In all three cases, values of the PCF and
MCF fall within the null-hypothesis envelope, and thus the textures do not depart significantly from random dispositions of nuclei and spatially uniform sources of nutrients. The
first case examined centimetre-scale grossular-andradite porphyroblasts that grew in a siliceous marble in response to the influx of large amounts of aqueous fluid; the second
examined sparsely distributed coarse-grained magnetite crystals in a fine-grained chlorite
schist; the third examined coarse-grained diopside crystals in a marble. In the first two
cases, the small numbers of porphyroblasts examined and their sparseness leave open the
possibility that DCNG operated to form these microstructures but that its signal did not
emerge from statistical noise. It seems more likely, however, that in the first case rapid
advective fluxes eliminated transport as a rate-controlling step, and that in the two other
cases, which do not involve aluminous porphyroblasts, relatively high solubilities for Ca,
Mg, and Fe facilitated rapid transport of these species.
The evidence accumulated to date from statistical analysis of the spatial distribution of
crystals points strongly towards the dominance of DCNG mechanisms during crystallization of aluminous porphyroblasts. In some instances, the evidence from spatial statistics
may be –– or may appear to be –– at variance with evidence from other methods of analysis,
and it is vital to include evidence from as many sources as possible when seeking to identify
crystallization mechanisms. Nonetheless, it appears that spatial statistics, when properly
applied, are among the least ambiguous indicators of mechanism.
Potentially spurious signals
The use of spatial statistics to extract crystallization mechanisms from textures relies upon
two key assumptions, so it is important to consider if and when those assumptions are
likely to be viable.
First, these statistical analyses proceed from the premise that the measured locations
and sizes of crystals are not appreciably different than they were when the crystals nucleated and grew. Yet bulk deformation will modify the relative positions of crystals. If
deformation yields greater randomness in the relative positions of crystals, it will have the
effect of diminishing the strength of the nucleation- and growth-suppression signals used
to identify DCNG. The robustness of these signals to bulk deformation has not been examined
rigorously, but the characteristic signature of DCNG has been observed to persist as a
statistically significant feature in rocks in which garnet crystallization is demonstrably
syn-kinematic and records significant strain (e.g. Meth and Carlson 2005). Considering
the prevalence of DCNG in analyses based on spatial statistics, perhaps the more pertinent
question is this: can syn- or post-crystallization deformation create spurious evidence for
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ordering or competition? Creation of a spurious ordering signal would require dispersion
of crystals that happened by chance to have nucleated in closer-than-random proximity.
Creation of a spurious competition signal would require preferential congregation of crystals with sizes smaller than the local average. The improbability of generating such effects
as a consequence of the strain regimes that typify metamorphic deformation suggests that
these signals are relatively robust with respect to syn- or post-crystallization deformation,
but quantitative tests, based perhaps on forward models, are an obvious need.
Second, these statistical analyses proceed from the premise that the suppression of
nucleation because of the progressive development of depleted domains surrounding
growing crystals dominates over other factors as a means of localizing nuclei. But if crystals
that act as substrates for epitaxial nucleation are coarse-grained or are inhomogeneously
distributed in the precursor, then the spatial disposition of these substrates may potentially
exert control over the location of porphyroblasts, as claimed by Spiess et al. (2007). The
importance of this effect depends entirely on the grain size and degree of lithologic homogeneity in the antecedent rock from which the porphyroblasts originate. For relatively
fine-grained and reasonably homogeneous precursors, it would be of little consequence,
and it would be very difficult to discern the effects of such localization of nuclei in the
eventual porphyroblastic texture.
Microchemical features
Garnet porphyroblasts commonly preserve in their compositional zoning a record of the
changing chemistry of the surrounding matrix, and this record can sometimes be exploited
to glean information on reaction mechanisms. A key notion in this approach is that some
constituents of the growing crystal (e.g. Mn, Fe, Mg, Ca) may diffuse through the intergranular medium rapidly enough to flatten any gradients in their chemical potential, even
though diffusion of another single species (e.g. Al) may be rate limiting for crystal growth.
So, for example, if diffusion of Mn is rapid enough compared with Al for the intergranular
medium to be regarded as a spatially uniform reservoir for Mn, then in all crystals, zones
of equivalent Mn content must have crystallized at the same time. Mn content then
becomes a time-marker that can be used to assess relative times and rates of nucleation
and growth.
Normalized radius––rate relations
Kretz (1974) introduced the ingenious technique of normalized radius––rate relations as a
means of testing the predictions of a specified growth rate law against observed zoning
patterns. If all garnets in a specimen are precipitating the same composition at any specified
point in time, then in all crystals the radial distance between any two close compositional
contours is proportional to the average rate of radial growth during the crystallization
interval bounded by those contours. The average radius of the garnet during that growth
interval is the radial distance to the midpoint between the two contours. When the growth
rate and the average radius for that interval in a garnet crystal are both normalized to the
equivalent quantities measured in the largest crystal in the specimen, the normalized values
are related to one another in a specific way for any chosen growth rate law (Figure 5). As
discussed in Carlson (1989, p. 7––10), measurements by Kretz (1974) and by Finlay and
Kerr (1987) failed to conform to predictions of rate laws for the growth of isolated spherical
crystals from a homogeneous precursor under conditions of interface control, isothermal
diffusion control, or heat-flow control. But when the effects of an exponential increase in
30
W.D. Carlson
Normalized rate
7
Normalized rate
7
6
6
5
5
4
4
3
3
2
2
1
1
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1
0
Normalized radius
MnO
FeO
MgO
1
0
Normalized radius
Figure 5. Normalized radius––rate relations, after Carlson (1989). Patterns of chemical zoning in
crystals of different size, for elements that diffuse rapidly enough to maintain rock-wide equilibrium, define curves on a normalized radius––rate diagram that can be diagnostic of crystallization
mechanisms. Application of this technique is limited by the fact that it relies upon the assumption
that all crystals are growing in isolation from one another (i.e. without diffusional competition) and
that all are growing from the same homogeneous matrix. (Left) Horizontal dashed line shows relations resulting from ICNG; dashed curve shows relations resulting from isothermal DCNG. Family
of solid curves shows relations for thermally accelerated DCNG, in which increasing temperature
during prograde metamorphism produces more rapid diffusion; larger ranges of temperature produce
greater departure from dashed isothermal curve. (Right) Example of data from garnets of Picuris
Range, New Mexico, that confirm thermally accelerated DCNG.
intergranular diffusivity with temperature in a prograde event are considered, comparable
measurements yielded excellent agreement with prediction (Figure 6 in Carlson 1989).
Subsequent attempts to apply this approach have brought to light important limitations. It is straightforward to derive the normalized radius––rate functions for isothermal
growth of isolated crystals from a homogeneous precursor, and simple models of polythermal
crystallization can also be implemented easily. But the assumptions that crystals are growing
in isolation from one another and from a homogeneous matrix are problematic for many,
perhaps most, rocks. Invalidity of those assumptions would have no effect on normalized
radius––rate relations in the case of ICNG, as there is no competition for nutrients, but that
is emphatically not the case for DCNG. Competition for nutrients among neighbouring
crystals will reduce growth rates compared with the isolated case, and heterogeneities in
the distribution of reactants will cause crystals in nutrient-rich regions to grow more
rapidly than those in nutrient-poor regions. Relaxing these assumptions is not possible,
because one cannot write a radial growth rate law that will describe crystallization under
circumstances in which each crystal grows within its own unique set of surroundings.
An illuminating finding that has come from forward models of DCNG (Carlson and
Ketcham 2004; Ketcham and Carlson 2004; Kelly and Carlson 2008b) is that the net effect
of diffusional competition and prograde temperature increases is an averaging of nutrient
supply over time that yields nearly constant rates of radial growth for large portions of a
crystal’’s history (Figure 6). This has the pernicious effect of mimicking the constant rate
of radial growth that might otherwise be considered a characteristic feature of interfacecontrolled growth kinetics. In consequence, inferences of ICNG that are rooted in constant
rates of radial growth are open to question, and the possibility that they instead reflect
International Geology Review
31
Radius (cm)
0.08
0.07
0.06
0.05
0.04
0.03
0.02
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0.01
0
0
2
4
6
8
10
Time (million years)
12
14
Figure 6. Rates of competitive, thermally accelerated, diffusion-controlled growth for four crystals, from numerical simulation. Modelled radii are shown as grey curves, solid over the interval
from 20% to 90% of radial growth history. Dashed black lines show that modelled radii increase
nearly linearly with time over the interval from 20% to 90% of the crystal radius. Thus, the effect of
increasing temperature and mutual competition is that each crystal displays, for a significant fraction
of its growth history, a nearly linear increase in radius with time. Because constant radial rates of
growth also characterize interface-controlled growth, inferences of mechanism based on growth
rates alone are ambiguous.
competition during prograde DCNG should be entertained. This might be the explanation
for occasional reports of data from spatial statistics and normalized radius––rate relations
that are in conflict, with the former suggesting DCNG and the latter suggesting ICNG (as
perhaps in Daniel and Spear 1999).
Size-proportional growth
A related method of utilizing compositional zoning patterns to identify crystallization
mechanisms relies on the concept that only in DCNG are nutrients sourced locally. Thus,
if it can be shown that growth was more rapid for crystals in reactant-rich zones, DCNG is
implied. Commonly, the effects of reactant distribution are convolved with the effects of
diffusional competition and of nucleation at different times, so unambiguous demonstration
of such a control on growth rates is precluded. But Meth and Carlson (2005) encountered
an instance in which all garnet porphyroblasts nucleated very near the beginning of the
crystallization interval and then grew syn-kinematically from a highly heterogeneous
precursor. Using widths of narrow compositional zones as proxies for growth rates over
short intervals of time, they were able to document a direct proportionality between the
radii of porphyroblasts and their rates of radial growth. This is consistent with intergranular
diffusion that preserves local inhomogeneities in the distribution of reactants, causing
crystals to grow at rates dependent on the local supply of nutrients: thus faster-growing
crystals were larger at all stages of the process. Simplified calculations (ignoring diffusional competition) confirmed that DCNG in heterogeneous precursors should lead to this
form of ‘‘size-proportional’’ growth, in which crystal size is a passive consequence of
32
W.D. Carlson
growth rates that are determined by local differences in nutrient availability (Meth and
Carlson 2005, p. 172).
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Rates and scales of chemical equilibration
The growing recognition that intergranular diffusion is commonly a rate-limiting factor in
the growth of porphyroblasts has focused attention on the rates at which metamorphic
assemblages and the minerals within them attain chemical equilibrium, as this is fundamental to the rocks’’ ability to register accurately their pressure––temperature––time evolution. Slow intracrystalline diffusion allows some porphyroblasts, most notably garnet, to
preserve an extraordinary variety of patterns of internal compositional zoning, and these
patterns can be connected to relative rates of intergranular diffusion of the major, minor,
and trace components that make up the crystal. Those relative rates in turn determine the
degree to which chemical equilibrium is achieved and recorded in the mineral chemistry.
A detailed review of this important topic cannot be included here, but because no discussion of porphyroblast crystallization would be complete without at least brief mention
of it, some highlights of current interest are described below. A more complete exposition
appears in Carlson (2002).
As noted above, studies of coronal structures have documented appreciable differences in the diffusivities of different species, but in terms of porphyroblast microchemistry, the vital distinction is between species that diffuse more rapidly and species that
diffuse more slowly than the growth-limiting component, that is, the one that exhibits the
lowest diffusional flux relative to the rate of supply needed for porphyroblast growth. The
evidence reviewed above points to the intergranular diffusional flux of Al as the most
common determinant of rates of crystal growth and thus of overall rates of metamorphic
reaction, so Al occupies a pivotal role in terms of chemical equilibration: elements whose
intergranular diffusivity is higher than that of Al will appear to equilibrate rock-wide
because they will always be present at the surface of the growing crystal in the amount
needed to complement the limiting Al flux, whereas those elements whose intergranular
diffusivity is lower than that of Al may exhibit disequilibrium behaviour, as their incorporation into the porphyroblast will be subject to a variety of kinetic controls. The net result
is a state of ‘‘partial chemical equilibrium’’, meaning equilibrium for some elements, but
not for others, as illustrated in Figure 7.
Some instances of partial chemical equilibrium are revealed by unusual or irregular
zoning in garnet crystals that is readily interpreted as overgrowth by the porphyroblast of
Chemical potential gradients
Very slow Slow
Medium
µi in regions not affected by diffusion
Fast
µi
Figure 7. Schematic chemical potential gradients around a growing porphyroblast for elements
with very slow, slow, medium, and fast rates of intergranular diffusion. Only if rates of intergranular
diffusion are rapid enough to flatten chemical potential gradients will equilibrium concentrations of
an element be incorporated into the growing crystal. For comparative purposes only, chemical
potentials for different elements are shown as spanning the same range of values; actual values
would have different magnitudes and ranges. After Carlson (2002).
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International Geology Review
33
chemical heterogeneities in the precursor (e.g. Yang and Rivers 2001; Hirsch et al. 2003).
Species that are capable of only very limited intergranular diffusion, yet are compatible
with the garnet’’s chemistry, are incorporated without having moved appreciably down
their chemical potential gradients. Similarly immobile but incompatible or saturated
species (e.g. Ti), or components present in excess without appreciable gradients to drive
diffusion (e.g. Si), may remain in the form of mineral inclusions within the porphyroblast
(e.g. ilmenite, quartz).
More cryptic evidence of partial chemical equilibrium appears in zoning patterns such
as those described by Chernoff and Carlson (1997, 1999), in which compositional anomalies
within garnet crystals are shown to have originated not simultaneously, but rather at equivalent stages in the growth of each crystal, that is, at equivalent extents of local reaction
progress. The implication is that chemical communication is limited to the immediate
vicinity of each growing porphyroblast for some elements (in this case Ca and many trace
elements), but not others (in this case, Mn, Fe, and Mg).
A similar explanation has been advanced by Skora et al. (2006) to explain patterns of
REE incorporation in eclogitic garnet, in which HREEs display peak abundance in the
core of crystals, but progressively lighter REEs have progressively lower core concentrations, matched by annular rings of progressively higher abundance at progressively greater
distances from the core. Models based upon diffusion-limited uptake of REE with increasing temperature during garnet crystallization quantitatively replicate the observed zoning
patterns, with the implication that at no point in the growth of the garnet did the system
achieve rock-wide equilibration for REEs. Konrad-Schmolke et al. (2008), however,
attributed very similar features in different rocks to equilibration with a changing
sequence of different reactant assemblages.
Discussion: connections to igneous processes
The thrust of the other articles in this thematic issue encourages a look at the intriguing
possibilities for furthering understanding that may lie at the interface between igneous and
metamorphic processes. Given the extensive array of possible mechanisms, kinetics, and
feedbacks that can operate at low subsolidus temperatures in metamorphic systems,
should we not expect to observe phenomena that are in some ways comparable when we
examine igneous systems in the subsolidus portions of their cooling histories? Despite
obvious differences, recrystallization in subsolidus igneous systems may have important
features in common with metamorphic crystallization. Organized around a brief inventory
of mechanisms to highlight both differences and similarities, this short discussion simply
offers a number of questions in hope of stimulating fruitful comparative investigation of
the mechanisms and kinetics of crystallization processes in metamorphic and subsolidus
igneous systems.
Heat input to drive prograde metamorphic reactions is imposed on the system by
external processes, whereas heat loss proceeds spontaneously as a subsolidus igneous
system evolves; does this make it more likely that subsolidus recrystallization in igneous
systems will be rate-limited by heat loss to the surroundings? To what extent does (potentially quite rapid) volatile loss act as a driving force for subsolidus recrystallization in
igneous systems? Given that products of igneous crystallization (especially in granitic systems) may remain thermodynamically stable and thus not far from chemical equilibrium
while cooling through a considerable range of subsolidus temperatures, eliminating the
need for nucleation of new phases, but also minimizing the magnitude of chemical affinity
for reaction, what role is played by nucleation kinetics during subsolidus recrystallization?
34
W.D. Carlson
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Because of the abundance of late-evolving juvenile fluids and the size and scale of
hydrothermal flow systems in and around intrusions, are advective flows dominant over
intergranular diffusion as a transport mechanism in subsolidus igneous recrystallization?
Do the wetting characteristics of residual melt films or of ligand-rich late-stage aqueous
fluids produce the interconnected networks needed for efficient intergranular diffusion
in subsolidus igneous rocks? What volume fraction do such networks commonly
occupy? In comparison to intergranular diffusion in metamorphic rocks, how rapid is
intergranular diffusion through such melt films or through fluids capable of such high
solubility for key elements? Is there reason to believe that dissolution/precipitation
involving such melt films or fluids would ever become rate limiting for subsolidus
recrystallization?
Conclusion
The foregoing review leads to the conclusion –– recognized, admittedly, as an overgeneralization –– that the dominant controls on porphyroblast crystallization are typically
rates of nucleation and of intergranular transport, noting that transport is commonly
diffusive through a static fluid-saturated grain-edge network, but may be advective in
highly open systems characterized by actively flowing fluids. In contrast, rates of heat
flow and rates of dissolution and precipitation are likely to be subsidiary factors in
nearly all cases.
The dominance of nucleation and transport as kinetic controls has several vital implications: (1) strong potential for appreciable overstepping of reactions before the onset of
nucleation, and a significant likelihood of subsequent crystallization at high levels of
chemical affinity; (2) reaction extending across protracted intervals of time and temperature; and (3) a propensity for development of partial chemical equilibrium, with concomitant impacts on the reliability of equilibrium-based methods of thermochemical analysis.
Particularly important areas for further investigation include quantification of rates of
nucleation, and explicit identification of epitaxial relationships and their corresponding
values of interfacial energies; better characterization of the controls on solubility of
species –– major, minor, and trace –– in fluids attending porphyroblast crystallization, and
improved characterization of the chemistry of those fluids; and quantitative determination
of the effective diffusivities of important species, especially Al, inasmuch as it occupies a
pivotal position in determining the ability of porphyroblasts to record accurately their
metamorphic histories.
Acknowledgements
The author is greatly indebted to Richard Ketcham for his vital contributions during their longstanding collaborative research efforts on several of the topics covered here, and as the list of cited
references shows, the content of this review is also built upon the ideas and findings of many other
colleagues and students, too numerous to list individually. To all of them the author is grateful for
discussions, concepts, and interactions that have sustained his prolonged interest in the mechanisms
and kinetics of metamorphic reactions. The National Science Foundation has supported this work
over the years in a succession of much-valued grants: EAR-9118338, EAR-9417764, EAR9902682, EAR-0635375. The author is particularly grateful to Allen Glazner for providing essential motivation by inviting this contribution and for exhibiting remarkable editorial patience and
forbearance. Finally, the author thanks Richard Ketcham, Mark Cloos, Eric Kelly, and Stephanie
Moore for careful informal reviews of the manuscript, and David Hirsch and an anonymous reader
for formal reviews, all of which substantially improved many elements of both substance and
presentation.
International Geology Review
35
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References
Ague, J.J., and Rye, D.M., 1999, Simple models of CO2 release from metacarbonates with implications for interpretation of directions and magnitudes of fluid flow in the deep crust: Journal of
Petrology, v. 40, p. 1443––1462.
Ashworth, J.R., 1993, Fluid-absent diffusion kinetics of Al inferred from retrograde metamorphic
coronas: American Mineralogist, v. 78, p. 331––337.
Ashworth, J.R., and Birdi, J.J., 1990, Diffusion modelling of coronas around olivine in an open
system: Geochimica et Cosmochimica Acta, v. 54, p. 2389––2401.
Ashworth, J.R., Birdi, J.J., and Emmett, T.F., 1992, Diffusion in coronas around clinopyroxene:
Modelling with local equilibrium and steady state, and a non-steady-state modification to account
for zoned actinolite-hornblende: Contributions to Mineralogy and Petrology, v. 109, p. 307––325.
Ashworth, J.R., and Chambers, A.D., 2000, Symplectic reaction in olivine and the controls of intergrowth spacing in symplectites: Journal of Petrology, v. 41, p. 285––304.
Ashworth, J.R., and Sheplev, V.S., 1997, Diffusion modelling of metamorphic layered coronas
with stability criterion and consideration of affinity: Geochimica et Cosmochimica Acta, v. 61,
p. 3671––3689.
Avrami, M., 1939, Kinetics of phase change. I: General theory: Journal of Chemical Physics, v. 7,
p. 1103––1112.
Avrami, M., 1940, Kinetics of phase change. II: Transformation-time relations for random distributions of nuclei: Journal of Chemical Physics, v. 8, p. 212––224.
Avrami, M., 1941, Kinetics of phase change. III: Granulation, phase change, and microstructure:
Journal of Chemical Physics, v. 9, p. 177––184.
Balashov, V.N., and Yardley, B.W.D., 1998, Modeling metamorphic fluid flow with reaction-compaction-permeability feedbacks: American Journal of Science, v. 298, p. 441––470.
Barnett, A., 1999, Quantitative textural analysis used to determine garnet porphyroblast nucleation
and growth mechanisms in a blueschist from the Franciscan Complex, Jenner, California [B.S.
thesis]: Austin, University of Texas at Austin, 26 p.
Baxter, E.F., 2003, Natural constraints on metamorphic reaction rates: Geological Society Special
Publication, v. 220, p. 183––202.
Bell, T.H., 1985, Deformation partitioning and porphyroblast rotation in metamorphic rocks: A radical
re-interpretation: Journal of Metamorphic Geology, v. 3, p. 109––118.
Bell, T.H., Rubenach, M.J., and Fleming, P.D., 1986, Porphyroblast nucleation, growth, and dissolution in regional metamorphic rocks as a function of deformation partitioning during foliation
development: Journal of Metamorphic Geology, v. 4, p. 37––67.
Berger, A., and Herwegh, M., 2004, Grain coarsening in contact metamorphic carbonates: Effects of
second-phase particles, fluid flow and thermal perturbations: Journal of Metamorphic Geology,
v. 22, p. 459––474.
Brady, J.B., 1983, Intergranular diffusion in metamorphic rocks: American Journal of Science,
v. 283A, p. 181––200.
Brenan, J.M., 1991, Development of metamorphic permeability: Implications for fluid transport
processes: Reviews in Mineralogy, v. 26, p. 291––319.
Carlson, W.D., 1983, Aragonite-calcite nucleation kinetics: An application and extension of Avrami
transformation theory: Journal of Geology, v. 91, p. 57––71.
Carlson, W.D., 1989, The significance of intergranular diffusion to the mechanisms and kinetics of
porphyroblast crystallization: Contributions to Mineralogy and Petrology, v. 103, p. 1––24.
Carlson, W.D., 1991, Competitive diffusion-controlled growth of porphyroblasts: Mineralogical
Magazine, v. 55, p. 317––330.
Carlson, W.D., 1999, The case against Ostwald ripening of porphyroblasts: Canadian Mineralogist,
v. 37, p. 403––413.
Carlson, W.D., 2002, Scales of disequilibrium and rates of equilibration during metamorphism:
American Mineralogist, v. 87, no. 2––3, p. 185––204.
Carlson, W.D., in press, Dependence of reaction kinetics on H2O activity as inferred from rates of
intergranular diffusion of aluminum: Journal of Metamorphic Geology, DOI: 10.1111/j.15251314.2010.00886.x.
Carlson, W.D., and Denison, C., 1992, Mechanisms of porphyroblast crystallization: Results from
high-resolution computed X-ray tomography: Science, v. 257, p. 1236––1239.
Carlson, W.D., Denison, C., and Ketcham, R.A., 1995, Controls on the nucleation and growth of porphyroblasts: Kinetics from natural textures and numerical models: Geological Journal, v. 30, p. 207––225.
Downloaded By: [University of Texas At Austin] At: 21:34 5 August 2010
36
W.D. Carlson
Carlson, W.D., and Gordon, C.L., 2004, Effects of matrix grain size on the kinetics of intergranular
diffusion: Journal of Metamorphic Geology, v. 22, p. 733––742.
Carlson, W.D., and Johnson, C.D., 1991, Coronal reaction textures in garnet amphibolites of the
Llano Uplift: American Mineralogist, v. 76, p. 756––772.
Carlson, W.D., and Ketcham, R.A., 2004, More meaningful models of diffusion-controlled nucleation and growth: Moving beyond the diffusional continuum: Geological Society of America
Abstracts with Programs, v. 36, p. 203.
Carlson, W.D., and Ketcham, R.A., 2008, Non-uniqueness of porphyroblast CSDs, Abstract V33B2215: Eos, v. 89.
Carlson, W.D., and Rosenfeld, J.L., 1981, Optical determination of topotactic aragonite-calcite
growth kinetics: Metamorphic implications: Journal of Geology, v. 89, p. 615––638.
Cashman, K.V., and Ferry, J.M., 1988, Crystal size distribution (CSD) in rocks and the kinetics and
dynamics of crystallization. III. Metamorphic crystallization: Contributions to Mineralogy and
Petrology, v. 99, p. 401––415.
Chernoff, C.B., and Carlson, W.D., 1997, Disequilibrium for Ca during growth of pelitic garnet:
Journal of Metamorphic Geology, v. 15, p. 421––438.
Chernoff, C.B., and Carlson, W.D., 1999, Trace-element zoning as a record of chemical disequilibrium during garnet growth: Geology, v. 27, p. 555––558.
Christensen, J.N., Rosenfeld, J.L., and DePaolo, D.J., 1989, Rates of tectonometamorphic processes
from rubidium and strontium isotopes in garnet: Science, v. 244, p. 1465––1469.
Christian, J.W., 1975, The theory of transformations in metals and alloys: Part I –– Equilibrium and
general kinetic theory: Oxford, Pergamon Press, 586 p.
Daniel, C.G., and Spear, F.S., 1999, The clustered nucleation and growth processes of garnet in
regional metamorphic rocks from north-west Connecticut, USA: Journal of Metamorphic Geology, v. 17, p. 503––520.
Denison, C., 1995, Three-dimensional quantitative textural analysis of metamorphic rocks using
high-resolution computed X-ray tomography: Methods, techniques, and application to natural
samples [Ph.D. dissertation]: Austin, University of Texas at Austin, 309 p.
Denison, C., and Carlson, W.D., 1997, Three-dimensional quantitative textural analysis of metamorphic rocks using high-resolution computed X-ray tomography. Part II: Application to natural
samples: Journal of Metamorphic Geology, v. 15, p. 45––57.
Denison, C., Carlson, W.D., and Ketcham, R., 1997, Three-dimensional quantitative textural
analysis of metamorphic rocks using high-resolution computed X-ray tomography. Part I:
Methods and techniques: Journal of Metamorphic Geology, v. 15, p. 29––44.
Dobbs, H.T., Peruzzo, L., Seno, F., Spiess, R., and Prior, D., 2003, Unraveling the Schneeberg
garnet puzzle: A numerical model of multiple nucleation and coalescence: Contributions to
Mineralogy and Petrology, v. 146, p. 1––9.
Eberl, D.D., Drits, V.A., and Srodon, J., 1998, Deducing growth mechanisms for minerals from the
shapes of crystal size distributions: American Journal of Science, v. 298, p. 499––533.
Eberl, D.D., Srodon, J., Kralik, M., Taylor, B.E., and Peterman, Z.E., 1990, Ostwald ripening of
clays and metamorphic minerals: Science, v. 248, p. 474––477.
Farver, J.R., and Yund, R.A., 1995a, Grain-boundary diffusion of oxygen, potassium and calcium in
natural and hot-pressed feldspar aggregates: Contributions to Mineralogy and Petrology, v. 118,
p. 340––355.
Farver, J.R., and Yund, R.A., 1995b, Interphase boundary diffusion of oxygen and potassium in
K-feldspar/quartz aggregates: Geochimica et Cosmochimica Acta, v. 59, p. 3697––3705.
Farver, J.R., and Yund, R.A., 1996, Volume and grain boundary diffusion of calcium in natural and
hot-pressed calcite aggregates: Contributions to Mineralogy and Petrology, v. 123, p. 77––91.
Farver, J.R., and Yund, R.A., 2000a, Silicon diffusion in a natural quartz aggregate: Constraints on
solution-transfer diffusion creep: Tectonophysics, v. 325, p. 193––205.
Farver, J.R., and Yund, R.A., 2000b, Silicon diffusion in forsterite aggregates: Implications for
diffusion accommodated creep: Geophysical Research Letters, v. 27, p. 2337––2340.
Finlay, C.A., and Kerr, A., 1987, Evidence for differences in growth rate among garnets in pelitic
schists from northern Sutherland, Scotland: Mineralogical Magazine, v. 51, p. 569––576.
Fisher, G.W., 1978, Rate laws in metamorphism: Geochimica et Cosmochimica Acta, v. 42,
p. 1035––1050.
Fisler, D.K., Mackwell, S.J., and Petsch, S., 1997, Grain boundary diffusion in enstatite: Physics and
Chemistry of Minerals, v. 24, p. 264––273.
Downloaded By: [University of Texas At Austin] At: 21:34 5 August 2010
International Geology Review
37
Frantz, J.D., and Mao, H.K., 1976, Bimetasomatism resulting from intergranular diffusion: I. A
theoretical model for monomineralic reaction zone sequences: American Journal of Science,
v. 276, p. 817––840.
Frondel, C., 1940, Oriented inclusions of staurolite, zircon, and garnet in muscovite: Skating crystals
and their significance: American Mineralogist, v. 25, p. 69––87.
Garcia-Ruiz, J.M., Villasuso, R., Ayora, C., Canals, A., and Otálora, F., 2007, Formation of natural
gypsum megacrystals in Naica, Mexico: Geology, v. 35, p. 327––330.
Hay, R.S., and Evans, B., 1988, Intergranular distribution of pore fluid and the nature of high-angle
grain boundaries in limestone and marble: Journal of Geophysical Research, v. 93, p. 8959––8974.
Herwegh, M., and Kunze, K., 2002, The influence of nano-scale second phase particles on deformation of fine-grained calcite mylonites: Journal of Structural Geology, v. 24, p. 1463––1478.
Hiraga, T., Nishikawa, O., Nagase, T., and Akizuki, M., 2001, Morphology of intergranular pores
and wetting angles in pelitic schists studied by transmission electron microscopy: Contributions
to Mineralogy and Petrology, v. 141, p. 613––622.
Hirsch, D.M., 2008, Controls on porphyroblast size along a regional metamorphic field gradient:
Contributions to Mineralogy and Petrology, v. 155, p. 401––415.
Hirsch, D.M., and Carlson, W.D., 2006, Variations in rates of nucleation and growth of biotite porphyroblasts: Journal of Metamorphic Geology, v. 24, p. 763––777.
Hirsch, D.M., Ketcham, R.A., and Carlson, W.D., 2000, An evaluation of spatial correlation functions
in textural analysis of metamorphic rocks: Geological Materials Research, v. 2, p. 1––21.
Hirsch, D.M., Prior, D.J., and Carlson, W.D., 2003, An overgrowth model to explain multiple,
dispersed high-Mn regions in the cores of garnet porphyroblasts: American Mineralogist, v. 88,
p. 131––141.
Holness, M.B., 1993, Temperature and pressure dependence of quartz-aqueous fluid dihedral angles:
The control of adsorbed H2O on the permeability of quartzites: Earth and Planetary Science
Letters, v. 117, p. 363––377.
Joesten, R., 1977, Evolution of mineral assemblage zoning in diffusion metasomatism: Geochimica
et Cosmochimica Acta, v. 41, p. 649––670.
Joesten, R., and van Horn, S.R., 1999, Numerical modeling of calcite coarsening in the aureoles of
en echelon dikes: Analysis of the kinetic control of isograd geometry in contact metamorphism,
in Jamtveit, B., and Meakin, P., eds., Growth, dissolution and pattern formation in geosystems:
Dordrecht, Kluwer Academic, p. 109––141.
Johnson, C.D., and Carlson, W.D., 1990, The origin of olivine-plagioclase coronas in metagabbros
from the Adirondack Mountains, New York: Journal of Metamorphic Geology, v. 8, p. 697––717.
Keller, L.M., Wunder, B., Rhede, D., and Wirth, R., 2008, Component mobility at 900°C and 18
kbar from experimentally grown coronas in a natural gabbro: Geochimica et Cosmochimica
Acta, v. 72, p. 4307––4322.
Kelly, E.D., and Carlson, W.D., 2008a, Crystal-size distributions from numerical simulations of porphyroblast crystallization: Constraints on reactant distribution: Geological Society of America
Abstracts with Programs, v. 40, p. 254––255.
Kelly, E.D., and Carlson, W.D., 2008b, Kinetics of nucleation and intergranular diffusion determined from numerical simulations of crystallization in regionally metamorphosed rocks,
Abstract V24C-07: Eos, v. 89.
Kelton, K.F., Greer, A.L., and Thompson, C.V., 1983, Transient nucleation in condensed systems:
Journal of Chemical Physics, v. 79, p. 6261––6276.
Kerschhofer, L., Dupas, C., Liu, M., Sharp, T.G., Durham, W.B., and Rubie, D.C., 1998, Polymorphic transformations between olivine, wadsleyite, and ringwoodite: Mechanisms of intracrystalline nucleation and the role of elsatic strain: Mineralogical Magazine, v. 62, p. 617––638.
Ketcham, R.A., and Carlson, W.D., 2004, Two- and three-dimensional modeling of metamorphic
crystallization controlled by diffusion: Geological Society of America Abstracts with Programs,
v. 36, p. 203.
Ketcham, R.A., Meth, C.E., Hirsch, D., and Carlson, W.D., 2005, Improved methods for quantitative analysis of three-dimensional porphyroblastic textures: Geosphere, v. 1, p. 42––59.
Kile, D.E., and Eberl, D.D., 2003, On the origin of size-dependent and size-independent crystal
growth: Influence of advection and diffusion: American Mineralogist, v. 88, p. 1514––1521.
Kile, D.E., Eberl, D.D., Hoch, A.R., and Reddy, M.M., 2000, An assessment of calcite crystal
growth mechanisms based on crystal size distributions: Geochimica et Cosmochimica Acta,
v. 64, p. 2937––2950.
Downloaded By: [University of Texas At Austin] At: 21:34 5 August 2010
38
W.D. Carlson
Konrad-Schmolke, M., Zack, T., O’’Brien, P.J., and Jacob, D.E., 2008, Combined thermodynamic
and rare earth element modelling of garnet growth during subduction: Examples from ultrahighpressure eclogite of the Western Gneiss Region, Norway: Earth and Planetary Science Letters,
v. 272, p. 488––498.
Kretz, R., 1966, Grain size distribution for certain metamorphic minerals in relation to nucleation
and growth: Journal of Geology, v. 74, p. 147––173.
Kretz, R., 1969, On the spatial distribution of crystals in rocks: Lithos, v. 2, p. 39––66.
Kretz, R., 1973, Kinetics of the crystallization of garnet at two localities near Yellowknife: Canadian
Mineralogist, v. 12, p. 1––20.
Kretz, R., 1974, Some models for the rate of crystallization of garnet in metamorphic rocks: Lithos,
v. 7, p. 123––131.
Laporte, D., and Watson, E.B., 1991, Direct observation of near-equilibrium pore geometry in
synthetic quartzites at 600––800°C and 2––10.5 kbar: Journal of Geology, v. 99, p. 873––878.
Lasaga, A.C., Luttge, A., Rye, D.M., and Bolton, E.W., 2000, Dynamic treatment of invariant and
univariant reactions in metamorphic systems: American Journal of Science, v. 300, p. 173.
Lasaga, A.C., and Rye, D.M., 1993, Fluid flow and chemical reaction kinetics in metamorphic
systems: American Journal of Science, v. 293, p. 361––404.
Lee, V.W., Mackwell, S.J., and Brantley, S.L., 1991, The effect of fluid chemistry on wetting
textures in novaculite: Journal of Geophysical Research, v. 96, p. 10023––10037.
Lifshitz, I.M., and Slyozov, V.V., 1961, The kinetics of precipitation from supersaturated solid solutions: Journal of the Physics and Chemistry of Solids, v. 19, p. 35––50.
Liu, M., Peterson, J.C., and Yund, R.A., 1997, Diffusion-controlled growth of albite and pyroxene
reaction rims: Contributions to Mineralogy and Petrology, v. 126, p. 217––223.
Liu, M., and Yund, R.A., 1993, Transformation kinetics of polycrystalline aragonite to calcite: New
experimental data, modelling, and implications: Contributions to Mineralogy and Petrology,
v. 114, p. 465––478.
Loomis, T.P., 1986, Metamorphism of metapelites: Calculations of equilibrium assemblages and
numerical simulations of the crystallization of garnet: Journal of Metamorphic Geology, v. 4,
p. 201––229.
Markl, G., Foster, C.T., and Bucher, K., 1998, Diffusion-controlled olivine corona textures in
granitic rocks from Lofoten, Norway: Calculation of Onsager diffusion coefficients, thermodynamic modelling and petrological implications: Journal of Metamorphic Geology,
v. 16, p. 607––623.
Marschallinger, R., 1998, A method for three-dimensional reconstruction of macroscopic features in
geological materials: Computers and Geosciences, v. 24, p. 875––883.
McLelland, J.M., and Selleck, B.W., 2008, The role of fluids in the genesis of megacrystic Gore
Mountain type garnet amphibolite, Adirondack Mountains, New York: Geological Society of
America Abstracts with Programs, v. 40, p. 235.
Meth, C., 1999, Quantitative textural analysis in determining crystallization control mechanisms of a
diopside marble from Cascade Slide, Adirondack Mountains, New York [B.S. thesis]: Austin,
University of Texas at Austin, 30 p.
Meth, C.E., and Carlson, W.D., 2005, Diffusion-controlled synkinematic growth of garnet from
a heterogeneous precursor at Passo del Sole, Switzerland: Canadian Mineralogist, v. 43, p.
157––182.
Mibe, K., Yoshino, T., Ono, S., Yasuda, A., and Fujii, T., 2003, Connectivity of aqueous fluid in
eclogite and its implications for fluid migration in the Earth’’s interior: Journal of Geophysical
Research, v. 108, p. ECV 6-1––6-10.
Milke, R., and Heinrich, W., 2002, Diffusion-controlled growth of wollastonite rims between
quartz and calcite: Comparison between nature and experiment: Journal of Metamorphic
Geology, v. 20, p. 467––480.
Milke, R., Wiedenbeck, M., and Heinrich, W., 2001, Grain boundary diffusion of Si, Mg, and O in
enstatite reaction rims: A SIMS study using isotopically doped reactants: Contributions to Mineralogy and Petrology, v. 142, p. 15––26.
Miyazaki, K., 1991, Ostwald ripening of garnet in high P/T metamorphic rocks: Contributions to
Mineralogy and Petrology, v. 108, p. 118––128.
Miyazaki, K., 1996, A numerical simulation of textural evolution due to Ostwald ripening in metamorphic rocks: A case for small amount of volume of dispersed crystals: Geochimica et Cosmochimica Acta, v. 60, p. 277––290.
Downloaded By: [University of Texas At Austin] At: 21:34 5 August 2010
International Geology Review
39
Parks, G.A., 1984, Surface and interfacial free energies of quartz: Journal of Geophysical Research,
v. 89, p. 3997––4008.
Pattison, D.R.M., and Tinkham, D., 2008, Interplay between equilibrium and kinetics in metamorphism of pelites in the Nelson Aureole, British Columbia: Geological Society of America
Abstracts with Programs, v. 40, p. 453.
Powell, D., 1966, On the preferred crystallographic orientation of garnet in some metamorphic
rocks: Mineralogical Magazine, v. 35, p. 1094––1109.
Raeburn, S.P., 1996, New methods in quantitative metamorphic petrology: 1, In situ determination
of iron valence in minerals; 2, The application of 3-D textural analysis to the study of crystallization kinetics [Ph.D. dissertation]: University Park, Pennsylvania State University, 281 p.
Ridley, J., 1985, The effect of reaction enthalpy on the progress of a metamorphic reaction, in
Thompson, A.B., and Rubie, D.C., eds., Metamorphic reactions: Kinetics, textures, and deformation: New York, Springer-Verlag, p. 80––97.
Ridley, J., 1986, Modelling of the relations between reaction enthalpy and the buffering of reaction
progress in metamorphism: Mineralogical Magazine, v. 50, p. 375––384.
Ridley, J., and Thompson, A.B., 1986, The role of mineral kinetics in the development of metamorphic microtextures, in Walther, J.V., and Wood, B.J., eds., Fluid-rock interactions during metamorphism: New York, Berlin, Heidelberg, Tokyo, Springer, p. 154––193.
Robyr, M., Carlson, W.D., Passchier, C., and Vonlanthen, P., 2009, Textural, chemical, and
microstructural records during growth of snowball garnet: Journal of Metamorphic Geology,
v. 27, p. 423––437.
Rubie, D.C., and Thompson, A.B., 1985, Kinetics of metamorphic reactions at elevated temperatures and pressures: An appraisal of available experimental data, in Thompson, A.B., and Rubie,
D.C., eds., Metamorphic reactions –– kinetics, textures and deformation: Advances in Physical
Geochemistry: New York, Springer-Verlag, p. 27––79.
Schneider, R., 1999, 3-D textural analysis of a garnet-amphibole rock from the Franciscan Complex
of California: Nucleation and growth mechanism of porphyroblastic crystals using high-resolution X-ray computed tomography [B.S. thesis]: Austin, University of Texas at Austin, 26 p.
Skora, S., Baumgartner, L.P., Mahlen, N.J., Johnson, C.M., Pilet, S., and Hellebrand, E., 2006, Diffusion-limited REE uptake by eclogite garnets and its consequences for Lu––Hf and Sm––Nd geochronology: Contributions to Mineralogy and Petrology, v. 152, p. 703––720.
Spear, F.S., and Daniel, C.G., 1998, Three-dimensional imaging of garnet porphyroblast sizes and
chemical zoning: Nucleation and growth history in the garnet zone: Geological Materials
Research, v. 1, p. 1––44.
Spiess, R., Groppo, C., and Compagnoni, R., 2007, When epitaxy controls garnet growth: Journal of
Metamorphic Geology, v. 25, p. 439––450.
Spiess, R., Peruzzo, L., Prior, D.J., and Wheeler, J., 2001, Development of garnet porphyroblasts by
multiple nucleation, coalescence and boundary misorientation-driven rotations: Journal of Metamorphic Geology, v. 19, no. 3, p. 269––290.
Stoyan, D., and Stoyan, H., 1994, Fractals, random shapes, and point fields: Methods of geometrical
statistics: New York, Wiley.
Tanner, S.B., Kerrick, D.M., and Lasaga, A.C., 1985, Experimental kinetic study of the reaction:
Calcite + quartz = wollastonite + carbon dioxide, from 1 to 3 kilobars and 500° to 850°C: American Journal of Science, v. 285, p. 477––620.
Wagner, C., 1961, Theorie der Alterung von Niederschlagen durch Umlosen: Zeitschrift Elektrochemie,
v. 65, p. 581––591.
Walther, J.V., and Wood, B.J., 1984, Rate and mechanism in prograde metamorphism: Contributions to Mineralogy and Petrology, v. 88, p. 246––259.
Walton, A.G., 1967, The formation and properties of precipitates: New York, Interscience, 232 p.
Wark, D.A., and Watson, E.B., 1998, Grain-scale permeabilities of texturally equilibrated, monomineralic rocks: Earth and Planetary Science Letters, v. 164, p. 591––605.
Waters, D.J., and Lovegrove, D.P., 2002, Assessing the extent of disequilibrium and overstepping of
prograde metamorphic reactions in metapelites from the Bushveld Complex aureole, South
Africa: Journal of Metamorphic Geology, v. 20, p. 135––149.
Watson, E.B., and Brenan, J.M., 1987, Fluids in the lithosphere, 1. Experimentally-determined
wetting characteristics of CO2––H2O fluids and their implications for fluid transport, host-rock
physical properties, and fluid inclusion formation: Earth and Planetary Science Letters, v. 85,
p. 497––515.
Downloaded By: [University of Texas At Austin] At: 21:34 5 August 2010
40
W.D. Carlson
Watson, E.B., and Lupulescu, A., 1993, Aqueous fluid connectivity and chemical transport in
clinopyroxene-rich rocks: Earth and Planetary Science Letters, v. 117, p. 279––294.
Whitney, D.L., Goergen, E.T., Ketcham, R.A., and Kunze, K., 2008, Formation of garnet polycrystals during metamorphic crystallization: Journal of Metamorphic Geology, v. 26, p. 365––383.
Wilbur, D.E., and Ague, J.J., 2006, Chemical disequilbrium during garnet growth: Monte Carlo simulations of natural crystal morphologies: Geology, v. 34, p. 689––692.
Wood, B.J., and Walther, J.V., 1983, Rates of hydrothermal reactions: Science, v. 222, p. 413––415.
Yang, P., and Rivers, T., 2001, Chromium and manganese zoning in pelitic garnet and kyanite:
Spiral, overprint, and oscillatory (?) zoning patterns and the role of growth rate: Journal of
Metamorphic Geology, v. 19, p. 455––474.
Yardley, B.W.D., 2005, Metal concentrations in crustal fluids and their relationship to ore formation: Economic Geology, v. 100, p. 613––632.
Yoshino, T., Mibe, K., Yasuda, A., and Fujii, T., 2002, Wetting properties of anorthite aggregates:
Implications for fluid connectivity in continental lower crust: Journal of Geophysical Research,
v. 107, p. ECV 10-1––10-8.
Yund, R.A., 1997, Rates of grain boundary diffusion through enstatite and forsterite reaction rims:
Contributions to Mineralogy and Petrology, v. 126, p. 224––236.