JOURNAL OF PETROLOGY
VOLUME 38
NUMBER 8
PAGES 1003–1020
1997
The Skaergaard Layered Series.
Part III. Non-dynamic Layering
ALAN E. BOUDREAU1∗ AND ALEXANDER R. McBIRNEY2
1
DEPT. OF GEOLOGY, DUKE UNIVERSITY, DURHAM, NC 27708-0227, USA
2
DEPT. OF GEOLOGICAL SCIENCES, UNIVERSITY OF OREGON, EUGENE, OR 97403, USA
RECEIVED MAY 6, 1996 REVISED TYPESCRIPT ACCEPTED MARCH 18, 1997
Layering in the Skaergaard Intrusion has been divided into two
general types, one produced by magmatic flow and another by
processes resulting from variations of rates of nucleation and crystallization, and, in the case of the Layered Series, by compactionrelated processes. Modal variations caused by shifts of cotectic
proportions produce thick layers which, in the Layered and Upper
Border Series, are diffuse and normally lack strong foliation and
lineation. In the Marginal Border Series, the layers are thinner and
sharper; possibly because the rate of accumulation was slower.
Oscillatory nucleation may have played a role in producing finescale cyclic layers, but it was less important than solution and
reprecipitation during slow cooling and Ostwald ripening. Evidence
for compaction is found in deformed plagioclase laths and a relative
deficiency of incompatible elements in rocks formed on the floor.
Layering related to compaction becomes sharper with increasing
height in the Layered Series until it suddenly disappears above the
trough horizon near the base of Upper Zone b. Mechanical sorting
during compaction may have produced crude layering, but if it did
the evidence has long since been destroyed by the superimposed effects
of solution and reprecipitation when interstitial liquid rose through
the overlying crystals and re-equilibrated with them. Numerical
simulations illustrate how small differences of surface energy caused
by variations of grain size, textural dependence of solubility, and
pressure solution can cause segregation of minerals into layers during
solution and reprecipitation.
KEY WORDS:
compaction; layering; metasomatism; pressure solution
∗Corresponding author. Telephone: (919) 684-5646. Fax: (919) 6845833. e-mail: [email protected]
INTRODUCTION
Recent studies of the structural and textural features of
the Skaergaard Intrusion (McBirney & Nicolas, 1997)
have distinguished two broad types of layering, one
produced by the dynamic effects of magmatic flow and
another by processes that operate in situ such as varied
nucleation and growth of crystals, recrystallization, or by
compaction-related mechanisms. We refer to layering
formed by these latter processes as ‘non-dynamic’ to
emphasize that it is not the result of fluid dynamic
processes. Although most layering combines elements of
more than one process, the contribution of each mechanism can usually be recognized from its distinctive form
and setting.
The distinguishing features of dynamic magmatic layering have been described in Part II of this series (McBirney & Nicolas, 1997). They are best seen near the steep
margins of the Layered Series in what Wager & Brown
(1968) referred to as the ‘cross-bedded zone’ where the
layering is disrupted by slumping and channeling and
the rocks have a marked foliation and lineation.
Layering not associated with magmatic flow takes a
variety of forms, but we can distinguish two general
end-members, each of which has a distinct origin and
characteristics. The first results from varied rates of
nucleation and crystal growth; it can be seen throughout
the intrusion. The second is associated with compaction
and is confined to rocks formed on the floor. Although
these mechanisms are closely associated and tend to
reinforce one another, each has its own distinctive characteristics.
 Oxford University Press 1997
JOURNAL OF PETROLOGY
VOLUME 38
GENERAL FEATURES OF NONDYNAMIC LAYERING
Layering produced by variations of
intensive parameters
The basic mechanism Wager (1961) credited for welllayered rocks that formed on the steep walls was also
responsible for the indistinct layering in the interior of
the Layered Series and nearly all the layering in rocks
that crystallized under the roof. It is thought to be
the result of transitory excursions about the cotectic
proportions of precipitating minerals (Harker, 1909;
Wager, 1959; Maaløe, 1978). These variations may have
been brought on by any of a variety of events including
convective overturn, invasions of new magmas, contamination with country rocks, gain or loss of volatiles,
and any other factor affecting intensive parameters, such
as the composition, temperature, or oxygen fugacity of
the magma (Hort et al., 1993; Naslund & McBirney, 1996).
[A mechanism based on double-diffusive convection was
thought to be an important effect of this kind (McBirney
& Noyes, 1979), but closer examination of its theoretical
basis (McBirney, 1985) raised doubts as to its importance
in natural magmas.]
In the case of rocks formed on the floor or under
the roof, these layers tend to be diffuse and, though
conspicuous when viewed from a distance, may be far
from apparent at close range. The most conspicuous
example is the Triple Group (Fig. 1), a set of three felsic
layers near the top of Middle Zone. From almost any
part of the intrusion they are seen extending for hundreds
of meters across almost the entire width of the intrusion,
but on an outcrop scale they are so indistinct that they
easily pass unnoticed.
We find no systematic spacing of these layers; their
distribution seems totally random. Modal variations are
normally gradational on a scale of tens or hundreds of
centimeters, but in the Marginal Border Series individual
layers are thinner and much sharper, possibly because
the sequence accumulated more slowly and is relatively
compressed. Similarly, grading from mafic to felsic minerals is much more pronounced in the Marginal Border
Series where the outer, wallward side of an individual
layer is normally more mafic. Except in the special
case of crescumulate textures produced by constitutional
supercooling close to the contact, fabrics are essentially
isotropic.
An unusual variety of this diffuse layering, found in a
few local parts of the Layered Series, is cyclical on a
scale of one or two centimeters. It is confined to small
areas, mainly in Upper Zone a (Fig. 2a), and is also
observed in a nearby rhyolitic dike (Fig. 2b). The rhythmic
spacing was first ascribed to oscillatory nucleation (McBirney & Noyes, 1979), but we now assign more importance
to competitive growth of crystals during slow cooling
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AUGUST 1997
(Boudreau, 1987, 1994, 1995). Small initial differences
of grain size or modal proportions are accentuated and
repeated by cyclical solution and crystal growth under
conditions similar to those of Ostwald ripening. None of
the examples we have found in the Skaergaard Intrusion
measures more than a few meters in vertical or horizontal
extent, but in larger bodies, such as the Stillwater Complex, they are much more extensive (Boudreau, 1987;
Naslund & McBirney, 1996).
Layering associated with compaction
At the other extreme of non-conventional layering is a
more conspicuous variety seen throughout most of the
interior of the Layered Series. It consists of sharply
defined mafic and felsic layers, a few centimeters or
decimeters thick and separated by thicker intervals of
homogeneous gabbro of widely differing thickness. We
will not describe this layering in detail, for it has been
the subject of several detailed studies (Wager & Deer,
1939; Wager & Brown, 1968; McBirney & Noyes, 1979;
Irvine, 1987; Conrad & Naslund, 1989). We attribute
this layering to the mineral segregation that accompanies
or is significantly enhanced by compaction. Although we
discuss several mechanisms that can give rise to this type
of layering, its development is broadly analogous to that
of metamorphic banding in the sense that it is the result
of solution and reprecipitation of minerals in response
to the stresses of compaction.
On a broad scale, these layers become sharper and
more numerous with increasing height in the Layered
Series until they suddenly disappear above the trough
horizon near the base of Upper Zone b. On a local scale,
however, they are particularly well developed in the
vicinity of blocks that fell from the roof and disturbed
the mush of crystals on the floor. Because this sharp,
intermittent layering is normally more mafic at the base
and felsic at the top and has a superficial resemblance
to sedimentary deposits laid down by turbidity currents,
it was once thought to be the result of crystal-laden
magma descending from the walls and sweeping across
the floor. The objections to this explanation are now
well known. No gaps, unconformities, or other evidence
have been found to support the notion that sections of
the walls supplied the large masses of crystals forming
extensive layers in interior parts of the floor. Similar layers
are equally well developed in many larger intrusions, such
as the Bushveld and Stillwater Complexes, where the
walls were less steep and turbidity currents could scarcely
have had sufficient energy to flow for tens or hundreds
of kilometers across the floor. Although they may have
a well-developed foliation (Brothers, 1964), the rocks
rarely have a strong lineated fabric in which elongated
minerals have a preferred orientation within the plane
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NON-DYNAMIC LAYERING OF THE SKAERGAARD INTRUSION
Fig. 1. Triple Group at the top of Middle Zone is clearly visible from a distance (a) but is scarcely noticeable at close range (b). In places the
plagioclase-rich units have fine-scale rhythmic layers.
of foliation or layering. This is not to say that currents
moving across the floor did not influence the orientation
of mineral grains, but that they did not deposit them by
conventional sedimentary processes.
Perhaps the most curious type of layering is seen in a
set of trough-like structures near the boundary between
Upper Zones a and b. Wager & Deer (1939) originally
proposed that these features resulted from turbidity
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Fig. 2. (a) Fine-scale cyclic layering near the top of Upper Zone a. Layering of this kind is restricted to areas of limited extent, mainly in the
upper half of the Layered Series. (b) and (c) Fine concentric layers in a rhyolitic dike ~500 m beyond the eastern margin of the Skaergaard
Intrusion (McBirney et al., 1990).
currents sweeping across the floor, an explanation that
Irvine (1987) has elaborated in detail. We see several
objections to this explanation. The most obvious is the
total absence of any of the features normally associated
with high-energy, sedimentary deposits. The troughs are
concordant synforms with sides that become steeper
upward in narrowing stacks of layers. The cross-bedding,
scour and fill, and lateral migration of channels so typical
of sedimentary deposits laid down by agrading distributory streams are conspicuously absent. Taylor &
Forester (1979) and Sonnenthal (1992) have commented
on isotopic and trace-element relations they find difficult
to reconcile with the turbidity current hypothesis.
STRUCTURES AND TEXTURES
PRODUCED BY COMPACTION
Although ‘filter-pressing’ has been considered a potential
mechanism of differentiation since it was first proposed
by Bowen (1928), it is only in recent years that compaction
has been given serious consideration. This neglect is due
mainly to early impressions that its effectiveness would
be severely limited by the small density difference between
crystals and iron-rich interstitial liquids and by the inferred low permeability resisting the force of this weak
density contrast (Sparks et al., 1985; Morse, 1986). However, these objections have not considered reactions
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between liquid and the crystals through which it rises,
nor the growth of large crystals at the expense of small
ones during crystal coarsening, both of which can augment permeability manyfold. Numerical simulations
(Sonnenthal, work in progress) show that, when these
factors are taken into account, the process is effective
even in thick sills and ponded lavas. This conclusion
finds support in the facility with which liquids ooze into
fractures in partly crystallized lavas and shallow sills
where the pressure differences driving flow are very small.
Although rarely obvious in the field, the effects of
compaction have been recognized in a growing number
of studies of ponded lava flows (Helz, 1987; Helz et al.,
1989; Philpotts et al., 1996) and thick sills (Shirley, 1987).
In the case of the Skaergaard Intrusion, it appears to
have been pervasive throughout most of the Layered
Series, where it was a primary factor in the compositional
evolution of the magma (McBirney, 1995). The evidence
for this is both chemical and textural.
Fig. 3. Concentrations of Ba, a representative incompatible element
in the principal series of the Skaergaard Intrusion.
Geochemical effects
The geometric configuration of the intrusion provides a
convenient way of identifying the relative importance of
any petrologic mechanism driven by gravity. By comparing the compositions of rocks that crystallized simultaneously on the floor, walls, and under its roof, one
can relate their differences to the geometrical orientation
in which they formed. On doing this, one sees that
the roof series has consistently larger concentrations of
incompatible elements than equivalent units on the floor
or walls (Fig. 3). Long thought to be the result of contamination, isotopic evidence shows that the greater
abundance of lithophile elements, such as Ba, Rb, Zr,
and Nd, in the Upper Border Series could not have
come from assimilated Archean gneiss, for the strontium
isotopic ratios of these same rocks are, on average,
even less radiogenic than those of the Layered Series
(McBirney, 1995).
The relatively depleted character of rocks formed on
the floor is more logically attributed to a gravitational
process, either compaction or convective fractionation,
or possibly both. Density relations make convective fractionation less likely, at least during the early stages of
crystallization when the differentiating liquid increased
in density and would have been heavier than the overlying, less-differentiated magma. These same relations
probably account for the relative depletion of the Marginal Border Series where heavy, differentiated liquids
are thought to have flowed down the wall to pond on
the floor. At a later stage after iron enrichment passed
its peak and concentrations of silica and volatiles greatly
reduced the density of the residual liquid, extraction of
these late liquids by convective fractionation would have
augmented the effects of compaction.
Petrographic and structural effects
Our co-worker, Robert Hunter, has drawn our attention
to several significant petrographic features of igneous
‘cumulates’ that have strongly influenced our view of
how these rocks form (Hunter, 1987; McBirney & Hunter,
1995). Planar fabrics that have long been taken as a
natural consequence of sedimentation and compaction
must now be viewed with caution. As Higgins (1991)
pointed out, foliation, in itself, is not a reliable criterion
for mechanical rotation of plagioclase by compaction
alone. This conclusion has been reinforced by geochemical evidence that the strong foliation to which
Wager & Deer (1939) gave the name ‘igneous lamination’
has little if any correlation with the amount of interstitial
liquid the rocks retained (McBirney & Hunter, 1995).
Quantitative petrofabric and compositional analyses of
rocks from the Stillwater Complex have shown that
development of these fabrics is a function not only of
compaction but also of a number of other factors including interaction with exsolved fluids (Meurer & Boudreau, 1997).
The origin of this strong foliation in the Skaergaard is
still unclear, but we suspect that rotation of grains can
be aided by differential solution and reprecipitation of
crystals with different crystallographic orientations with
respect to the principal stresses. Two distinct mechanisms
contribute to the development of foliation during compaction, one mechanical rotation and the other selective
pressure solution and recrystallization (Meurer & Boudreau, 1997). In the first, crystals with an initially random
orientation (Fig. 4a) are rotated toward a weak planar
orientation of their long axis (Fig. 4b). This effect is then
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plagioclase laths, that are unequivocal signs of anisotropic
strain. Unfortunately, evidence of this kind is far less
common in large intrusions that cooled slowly enough
to permit extensive recrystallization and annealing. Individual crystals are weakly deformed, if at all, when
they grow in the presence of interstitial liquids that take
up much of the strain caused by differential stresses, and
even when grain-to-grain contacts make crystals self
supporting, dissolution and reprecipitation can accommodate a large measure of deformation.
Lineation, although it is an important consequence
of magmatic flow, does not normally develop during
compaction unless there is a component of simple shear.
In the interior regions of the Skaergaard Layered Series
lineation is weak or undetectable, except where foundered
blocks have caused local deformation or where irregular
compaction has led to strong warping of the layers
(McBirney & Nicolas, 1997). With this exception, foliation
without lineation is more likely to reflect pure shear
and compaction than the simple shear expected from
magmatic flow.
LAYERING RELATED TO
COMPACTION
Fig. 4. Schematic diagram illustrating the formation of igneous lamination during compaction. In this diagram the number, sizes and
shapes of crystals remain unchanged during mechanical rotation (a to
b), whereas only the total cross-sectional area is constant in the transition
from b to c.
Demonstrating that the Skaergaard gabbros have undergone compaction is one thing; proving that this compaction resulted in modal layering is quite another. The
evidence is largely circumstantial and stems not only
from the inadequacies of the alternative explanations
but also from an improved understanding of magmatic
crystallization. We know of only two mechanisms that
have been seriously proposed to account for compactionrelated layering, one through mechanical sorting and the
other through solution and reprecipitation.
accentuated by solution of stressed corners, edges and
small crystal faces, and by equivalent growth that favors
large, sub-horizontal faces (Fig. 4c).
We find (Park & McBirney, work in progress) that the
large (010) faces of plagioclase are normally more stable
than the smaller (001) faces and that the boundaries
between the two tend to lie close to the plane of foliation.
From this we infer that plagioclase crystals with long
axes at steep angles to the horizontal are less stable and
tend to dissolve and contribute to growth of other grains
with stable faces normal to the direction of maximum
stress (Fig. 5). This is supported by observations from the
Stillwater Complex, where the aspect ratio of plagioclase
shows a strong positive correlation with a quantitative
measure of foliation (Meurer & Boudreau, 1997).
This interpretation is also in accord with other signs
of mechanical deformation, such as bent or broken
More than half a century ago, Coats (1936) observed
that crystals of differing sizes and densities tend to sort
themselves in crude layers as they settle from a suspension
and compact under the force of gravity. His simple
experiments were largely ignored, possibly because they
had no satisfactory theoretical basis. Apart from a few
studies of industrial materials that show that particles
can be segregated by upward-infiltrating liquids (e.g.
Font, 1990), no adequate explanation has been given
for the layering produced during compaction. We are
convinced, however, that the phenomenon is real, for our
experiments have fully verified all of Coats’ observations.
To do this, we used natural minerals in a size range
of 0·1–0·5 mm and bromoform diluted with acetone to
a density slightly less than that of the lightest mineral. A
Mechanical segregation during compaction
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Fig. 5. Distorted and broken laths of plagioclase are a conspicuous result of compaction of the layering. Twinning tends to be much more
intense where the crystals have been deformed. Vertical thin sections were cut from an oriented sample of Middle Zone. The top of the
photograph is toward the top of the specimen. Width of field is 5 mm.
50:50 mixture of plagioclase and pyroxene was placed
in a 250 ml mixing cylinder with a somewhat smaller
volume of bromoform, brought into suspension by vigorous shaking, and allowed to settle. Little if any segregation was observed during the initial stage of settling,
but as the bed of crystals continued to compact, irregular
aggregates of plagioclase began to form. Even though
the crystals formed a self-supporting framework, they
continued to compact, reducing the pore space and
driving out interstitial liquids. The rising liquid seems to
entrain the lighter crystals of plagioclase, carrying some
to the top of the bed but segregating others into crude
layers (Fig. 6).
As Coats noted, the effectiveness of separation is a
function of several factors, including: (1) density contrast
of the crystal species; (2) density contrast of the solids
and liquid; (3) proportions of the crystal species; (4) grain
size and shape; (5) proportions of liquid and solids; (6)
viscosity of the liquid; (7) flow velocity of the liquid. The
last parameter is not independent of the others. To assess
these various effects properly, one should conduct a series
of experiments in which each factor could be varied
independently. To date, however, our efforts to do this
have had only limited success, owing mainly to the
problem of finding materials with appropriate physical
properties. At this stage, we can only offer a few broad
generalizations.
Alternating layers develop best when the density contrast is small and the proportion of liquid is less than
that of the crystals. In mixtures in which the density
contrast between liquid and crystals is very large or the
proportion of liquid is greater than that of the crystals,
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the extensive recrystallization that subsequently affected
more slowly cooled bodies. Lacking any firmer evidence,
one can only speculate on the importance of this unusual
type of mechanical sorting.
Solution and recrystallization
Interstitial liquids escaping from a compacting pile of
crystals rise to levels where temperature and cotectic
relations differ from those deeper in the section. The
liquids are therefore reactive and must re-equilibrate,
both thermally and chemically, with the new environment
through which they rise. Under conditions of slow crystallization this equilibration can proceed, even when the
differences of chemical potential are small. In addition
to temperature differences, several factors affect the relative free energy of crystals and may alter the fabric and
modal proportions of the original liquidus assemblage.
Just as the textures of igneous rocks are governed by
the physical and chemical properties of the liquid from
which they crystallize, liquids in equilibrium with crystals
have local differences that stem from three factors: (1)
differential pressure solution; (2) differences of grain size;
(3) the affinities of like and unlike crystals.
Fig. 6. Crude layers produced by segregation of plagioclase and
pyroxene during compaction of a slurry suspended in dilute bromoform.
Pressure solution
the light and heavy crystals are able to separate completely
and form two layers with the light mineral overlying the
heavy.
Although the forces responsible for this sorting are
poorly understood, they seem to be related to some form
of self-organization of particles according to their drag
coefficients in a viscous fluid. The fact that the mechanism
seems to operate only within a restricted range of conditions may explain why it is not more common. Until
more has been learned about the phenomenon, we can
only speculate on its importance, but we can point to a few
possible cases of what could be interpreted as ‘Coatsian
layering’. All known examples are in sills that have a
basal zone of coarse mafic minerals that were carried in
suspension at the time of intrusion and settled to the
floor as a single mass. Bruce Marsh has shown us plagioclase-rich lenses in the Yorkhaven Sill that closely
resemble the crude layers produced in our experiments,
and has supplied photographs of similar layers in the
Ferrar Dolerites of Antarctica (Fig. 7; B. D. Marsh,
personal communication, 1995). In both places, the plagioclase-rich lenses developed in a coarse bronzite-rich
mass that was brought in as a dense suspension and
underwent gravitational compaction on the floor.
The scarcity of this type of layering in larger intrusions
may be due to the slower accumulation of crystals or to
Because the surface energy of a crystal increases with
stress, points where stress is concentrated tend to dissolve,
whereas those under smaller stress grow. The effect of
pressure differs from one mineral species to another. In
a mixture of two minerals, the more pressure-sensitive
phase has a greater free energy when the proportions of
that mineral are large than when they are small and
stress is taken up by grains of a more resistant mineral.
Pressure solution has long been considered an important
factor in metamorphic rocks (Fyfe, 1976), but Dick &
Sinton (1979) seem to have been the first to suggest that
it could produce layering in igneous rocks. They proposed
that some of the layering in the ultramafic rocks of
ophiolites developed when olivine and pyroxene were
segregated into separate layers of dunite and pyroxenite
in a zone of strong tectonic deformation close to the base
of the crust. Because olivine dissolves and reprecipitates
more readily than pyroxene, individual crystals of pyroxene in olivine-rich rocks bear a disproportionately
greater share of the total stress than do crystals of
pyroxene in an olivine-poor rock. Thus, they tend to
dissolve in regions where they are less abundant and
reprecipitate where they are modally more important.
As the relative sizes of grains are reduced by pressure
solution, the chemical potential difference is further increased by the size-dependent difference of surface energy. Thus modal and grain-size differences are
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Fig. 7. Example of ‘Coatsian’ layering in a dolerite sill in Antarctica (photo courtesy of B. D. Marsh).
enhanced, and an initially weak inhomogeneity can develop into layers that are increasingly monomineralic.
This mechanism could be equally effective under either
simple shear, as Dick and Sinton proposed, or under
pure shear associated with compaction.
The stability of crystals is also a function of the crystallographic face on which stress is concentrated. As we
observed in an earlier section, the large 010 faces of
tabular plagioclase crystals appear to be more resistant
than other smaller faces. When contacts bear stress, the
latter tend to dissolve whereas the former will grow and
increase the proportion of grains oriented with 010
normal to the direction of maximum stress. We credit
much of the foliation in plagioclase-rich rocks to this
effect.
Grain size
Owing to their greater volume-to-surface ratio, the free
energy of small crystals (and hence their solubility) is
greater than that of larger ones. This difference is the
driving force by which Ostwald ripening leads to a
general coarsening of grain size as large crystals grow at
the expense of smaller ones. Crystal size distributions of
slowly crystallized rocks, which typically show a pronounced paucity of smaller grain sizes, are one line of
evidence that their minerals have undergone modification
by the aging process (Waters & Boudreau, 1996). The
positive correlation between grain size and modal abundance of olivine and chromite seen in olivine chromitites
of the Stillwater Complex ( Jackson, 1961) was attributed
by Boudreau (1995) to more rapid aging of minerals in
those rocks in which the mineral is modally dominant
and probably explains a similar correlation observed in
graded layers of the Skaergaard Middle Zone (Conrad
& Naslund, 1989). Crystal aging can occur in a noncompacting assemblage, but would aid any other compaction-driven effects.
The tendency for large crystals to grow at the expense
of smaller ones is a function of the size-dependent concentration variations defined by the Gibbs–Thomson
equation (see Table 1 for explanation of symbols):
C=C xexp
A B
2r
.
RTr
(1)
A small difference of surface energy gives larger grains
a competitive advantage, so that small initial variations
owing to some random effect are magnified and can lead
to cyclic layering. Examples of such layering have been
described in detail and successfully modeled by computer
simulations (Boudreau, 1987, 1994, 1995; McBirney et
al., 1990).
Affinity of like crystals
The magnitude of surface free energy of a given mineral
is not a unique value but depends on the nature of the
crystal’s surroundings. In general, one would expect the
surface energy of a crystal to increase as the surrounding
material, whether it be a silicate liquid or other crystalline
phases, becomes less compatible with the surface of the
crystal structure. For example, McLean (1957; in Spry,
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Table 1: List of symbols and units
A, B
component A or B, respectively
C
liquid concentration of component A (mol/cm3)
Cx
equilibrium liquid concentration for a crystal of infinite (large) radius
C0
equilibrium liquid concentration of component A for the case in which
mineral a has 0% contact with other a mineral grains (mol/cm3)
C 100
equilibrium liquid concentration of component A for the case in which
mineral a has 100% contact with other a mineral grains (mol/cm3)
C(W)
concentration of component A in a liquid that is in equilibrium with a rock of
H
ratio of mass of mineral a to that of mineral b (non-dimensional)
Q
source term for component A (mol/cm3/s)
R
gas constant (J/mol/K)
T
temperature (K)
V
liquid velocity (cm/s)
a, b
denotes mineral a or b, respectively
f
liquid fraction (nondimensional)
fa
fraction of solids composed of mineral a
n
crystal number density (per cm3)
q
crystal growth constant (cm/s)
r
radius (cm)
s
supersaturation (non-dimensional)
t
time (s)
z
distance (cm)
mode/texture characteristics W
K
maximum mode/texture-dependent concentration variation (mol/cm3)
U
fraction crystallized (non-dimensional)
W
variable describing the rock mode/texture
q
crystal density (mol/cm3)
r
surface tension (J/cm2)
′
non-dimensional value of a variable
¯ (overbar)
characteristic quantity of a variable
1969) noted that the surface energy of copper may vary
by as much as two orders of magnitude, and is generally
lower when in contact with other copper phases (crystals
or Cu liquid). This is also supported by the tendency for
monomineralic strings of minerals to be among the last
material to melt during fusion (Philpotts & Carroll, 1996).
It is thus apparent that the solubility of a mineral depends
not only on grain size and stress but also on the nature
of its contact with surrounding phases. Specifically, it is
probable that certain precipitating mineral grains have
lower solubility against ‘like’ mineral grains than against
‘unlike’ ones. As with crystal aging, this phenomenon
can occur regardless of whether the system is undergoing
compaction, as it is driven by interfacial energy effects
alone.
Mineral segregation can occur in the following manner.
Cotectic crystallization of two minerals a and b will
nominally result in random variations of the distribution
of a and b with no preferred arrangement of a–a, a–b,
and b–b contacts. However, minor irregularities may
result in zones or ‘protolayers’ in which, for example,
a–a contacts are slightly more abundant than in the
immediately surrounding rock (e.g. ‘Coatsian’ layers).
Because of this small difference, crystals of a in this zone
will have a lower solubility and will grow at the expense
of crystals in the surrounding rock. Similarly, for a twocomponent liquid, the local abundance of a–a contacts
means that mineral b is relatively less stable than in the
surroundings, where it is slightly more abundant and
hence will dissolve as components of b migrate out into
the surrounding rock. The result is that the initial textural
irregularity will not only continue to grow with time but
will accelerate as the number of a–a contacts increases
and the number of b–b contacts decreases. Furthermore,
this mechanism is self-propagating as regions dominated
by a–a contacts cause the surrounding regions to be
dominated by b–b contacts and vice versa. This growth
of textural irregularities is similar to that proposed by
Ortoleva and others for formation of metamorphic banding (e.g. Ortoleva et al., 1987).
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In layered intrusions, some of the most striking layered
mineral assemblages involve spinels (chromite or
magnetite). The most extreme examples consist of alternating layers of almost pure spinel and silicate minerals
(Fig. 8). Textural evidence indicates that initial modal
variations may have been enhanced by surface energy
differences at spinel–spinel contacts (i.e. contacts involving an isotropic mineral) and spinel–silicate contacts.
The relative ease by which chromite grains nucleate
together and even 1ink together to form extensive chains
during growth (e.g. Hunter, 1987) suggests that spinel–
spinel contacts have relatively lower surface energies
than other types. Because of the lower surface energy,
spinel–spinel contacts would have lower equilibrium
liquid concentrations than would sites of spinel–silicate
mineral contacts. This would result in spinel grains at
mixed contacts dissolving in favor of grains in which
spinel–spinel contacts predominate. Over time, this might
lead to a nearly complete segregation of spinel from
silicate minerals.
Combined effects
It is important to note that the effects of these three factors
(differential pressure solution, grain size and affinity of
like crystals) tend to reinforce one another, so that any
small initial inhomogeneity is enhanced as crystals coarsen, the relative abundance of one of the minerals increases, and more grains of the dominant mineral are in
direct contact with one another. In this process, the
intergranular liquid plays a crucial role. If the liquid does
not move and mass transport is solely by diffusion within
the liquid, the scale of the effects is on the order of
centimeters only (Boudreau, 1994, 1995). In the presence
of a moving fluid, however, the efficiency and scale of
mass transfer is potentially much greater. Thus, liquid
expelled by compaction and rising through the crystal
mush surmounts the limitations of diffusion and increases
the vertical dimensions and intensity of the layering.
QUANTITATIVE MODEL OF LAYER
GROWTH
Presented here is a very general model for separation of
two crystals in a two-component system into modally
segregated layers. It is much simplified from the more
detailed treatment of the general phenomenon of selforganization in geologic systems discussed by Ortoleva
(1994), but illustrates the pertinent mechanisms of interest
as they might occur in a large layered intrusion.
Following a similar derivation by Boudreau (1994), we
consider a two-component system (components A and
B) which crystallizes two minerals a and b. For a twocomponent system, any change in the liquid concentrations of component A will result in an equal but
opposite change in the liquid concentration of component
B. Hence, one can relate all changes of the liquid solution
to that of component A alone, as is done in the following
formulations (i.e. the quantity C refers to the concentration of component A). It is further assumed that
both minerals are in contact with an interstitial liquid,
that the liquid is moving through the system under
the influence of a mechanism such as compaction or
convection, and that mass transport by advection is
dominant over that by diffusion and hence diffusional
transport is ignored (but see the scaling discussion, below).
At any location, the change in liquid concentration of
component A is equal to its net rate of flux of liquid into
or out of the location, plus the rate at which it is produced
or destroyed within the system by reaction with the solid
assemblage. For a one-dimensional transport, one has
∂C
∂C
=−V −Q.
∂t
∂z
(2)
The term Q is a source term defined by the gain or loss
of solution component A by growth or dissolution of
minerals a and b (assumed to be spherical with average
radius r):
∂C ∂U ∂r
Q=
∂U ∂r ∂t
=−naqa 4p(ra)2
(3)
∂ra
∂r
+nbqb4p(rb)2 b .
∂t
∂t
Assuming a first-order reaction mechanism, the rate of
precipitation or dissolution of crystals can be expressed
as
∂ra qa
= [C−C(w)]
∂t
qa
(4)
∂rb
q
= − b [C−C(w)]
∂t
qb
(5)
and
where q is the crystal growth constant and q is the density
of the crystalline phase. C(W) is the solution concentration
in equilibrium with a rock with modal/textural characteristics W. That is, it is assumed that all the mechanistic
mineralogical/textural phenomena that affect local equilibrium liquid concentrations as described previously can
be summarized in a general phenomenological expression
for C(W). In detail, W is a function of mineral modes as
well as grain size, shape and orientation. As a first
approximation, C(W) is taken to be a simple function of
local mineral mode, which is itself a function of grain
size, r, and the crystal number density, n, such that
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JOURNAL OF PETROLOGY
VOLUME 38
NUMBER 8
AUGUST 1997
Fig. 8. (a) Layers of chromitite alternating with anorthosite at the Dwars River in the Bushveld Complex of South Africa. (b) Detail of coarse,
plagioclase-rich segregation or residual clot within a magnetite layer.
C(W)=C100+(1−fa)K
(6)
where
fa=
C
D
na(ra)3
na(ra)3+nb(rb)3
(7)
and K is a constant. The quantity C 100 is the equilibrium
liquid concentration for the situation where mineral a is
completely surrounded by a. (It is noted that the liquid
can be crystallizing at the eutectic but that in any small
region there may be no a–b contacts.) K is a constant
equivalent to the maximum concentration difference
between an assemblage where mineral a is surrounded
entirely by a and one where a is surrounded entirely by
b. The value of K can be small if one is considering
solubility variations arising from surface energy effects
alone, but could be substantial for the case where crystals
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BOUDREAU AND McBIRNEY
NON-DYNAMIC LAYERING OF THE SKAERGAARD INTRUSION
of one phase are experiencing most of the strain during
compaction.
A qualitative view of the situation is shown schematically in a two-crystal, two-component system, where
the location of the eutectic is assumed to be a function
of the local mode/texture as shown in Fig. 9. In this
figure, the eutectic for the case where a is surrounded
completely by a is at a lower concentration (i.e. at C 100)
than is the case for which a is surrounded by b (i.e. at
C 0). It should be noted that we are assuming that the
eutectic does not change its location with respect to
temperature, although a temperature dependence could
be readily added (Boudreau, 1994). Also, the effect of
the rock mode/texture term W as defined in equations
(6) and (7) includes grain size and number but not
orientation of the mineral grains as would be required if
one were not considering spherical grains.
Finally, liquid velocity will not be constant but will
change as the porosity changes. It is assumed that for
small changes in total solid fraction, f, then
V f
.
V=
f
(8)
Nondimensional equations
Additional dimensionless equations equivalent to the
above, derived in the Appendix, are presented below. It
is assumed that mineral a is the more reactive mineral
phase, and parameters of mineral b are scaled to the
properties of mineral a. Thus, the reaction/transport
equations (2) and (3) become
1 ∂s′
∂r ′
∂r ′
∂s′
=−V ′ −na′(ra′)2 a +Hnb′(rb′)2 b
b ∂t′
∂z′
∂t ′
∂t ′
(9)
n r 3q
H= b b3 b
n ar a q a
(10)
The quantity H is the ratio of the characteristic amounts
of the two minerals a and b initially present and, for most
cotectic crystallization systems, the ratio is close to one.
The grain growth equations (4) and (5) become
∂ra′
=[s′−(1−fa)]
∂t ′
(11)
∂rb′
=−qb′[s′−(1−fa)]
∂t ′
(12)
Fig. 9. Schematic representation of the expected shift of the eutectic
position in a two-component system as a function of mineral mode. It
is assumed that minerals a and b both have lower free energies when
in contact with like grains than unlike grains. Where mineral a is
surrounded by b (i.e. b is modally dominant), b–b contacts are predominant and hence a is relatively soluble and b less soluble. The
location of the eutectic is then as shown by the continuous lines. Where
a is surrounded mainly by a (i.e. a is modally dominant), then a–a
contacts predominate and hence b is relatively soluble and a is relatively
insoluble. The location of the eutectic is then shifted to the location
shown by the dashed lines. The maximum concentration difference
between rocks rich in a (at C 100, where 100% of the contacts are a–a
contacts) and those rich in b (at C 0, where 0% of the contacts are a–a
contacts) defines the maximum mode/texture-dependent concentration
difference term, K.
V′=
The liquid velocity equation (8) becomes
(14)
The mode/texture dependence of the equilibrium
liquid concentration defined by equations (6) and (7) is
expressed through the term fa in equations (11) and (12).
The b term in equation (9) is a scaling constant and is
given by
where
r 2q q
qb′= a2 a b .
r b qbqa
1
f′
4pnaqa(ra)3
.
K
b=
(13)
(15)
This scaling constant, b, is simply the ratio of the moles
of component A initially present as crystals a (per unit
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JOURNAL OF PETROLOGY
VOLUME 38
volume) to the maximum molar liquid concentration
difference in component A that can develop by a change
in rock mode or texture between regions (as illustrated
in Fig. 9).
Finally, the characteristic time and length scales are
given by
t=
ra qa
qa K
AB
(16)
t V
.
b
(17)
and
z=
Estimates of characteristic times and
scaling constant, b
Rearranging equation (17), the characteristic time can
be expressed as a function of the characteristic length,
scaling constant and interstitial liquid velocity:
bz
.
V
t=
(18)
Thus, the time it takes for a layer to develop increases
as either the thickness of the layer increases or as the
scaling constant increases, both of which imply there is
a relative increase in the amount of material that must be
transferred between regions to effect mineral segregation
relative to the maximum amount of material that can be
transported in a unit volume of liquid. In contrast, the
time it takes a layer to develop is inversely proportional
to the velocity of the interstitial liquid, as a higher liquid
velocity speeds up transfer of material between regions.
From estimates of the cooling time and interstitial
liquid velocities one can estimate permissible values of
the scaling constant, b, and from this the required
supersaturation differences, K, required to effect layer
formation. A plot of characteristic times as a function of
velocity of interstitial liquid and values b ranging from
103 to 106 is shown in Fig. 10 for a characteristic length
of 1 cm as a typical length scale for non-dynamic igneous
layering. An interstitial liquid velocity of 10–6−10–7 cm/s
is equivalent to 3–30 cm/yr, or in the range of estimated
values for compaction-driven fluid velocities in a typical
large intrusion (Shirley, 1987; Sonnenthal & McBirney,
1997). Shirley (1987) estimated a minimum compaction
time of 200 years for the Muskox intrusion, and this can
be taken as a minimum time scale for non-dynamic
layering to develop.
A long characteristic time is consistent with the observation that well-developed modal segregation layering
is not common or well developed in small or relatively
thin intrusions such as the Palisades sill. It is only in
intrusions the size of the Skaergaard or larger that a
NUMBER 8
AUGUST 1997
liquid+crystal mush zone is both thick enough and
persists long enough for the processes that drive selfsegregation to operate. Even within the Skaergaard, the
observation that layering becomes more defined with
height is consistent with the interpretation that a thicker
crystal mush enhances non-dynamic layer formation.
Assuming characteristic time scales from 1000 to 10 000
years, values of the scaling constant, b, would need to
be in the range of 103 – 104. However, the value of b
will not remain constant over the course of crystallization.
The scaling constant is initially small during the early
nucleation growth period, when the volume of crystalline
material is low and solubility differences driven by surface
energy are relatively large. At this liquid-dominant stage,
the value of b can be less than one (Boudreau, 1995). It
increases rapidly, however, as grains become larger and
size-driven liquid concentration differences decrease. It
then decreases as loading by overlying crystals increases
and crystals begin to deform during compaction. Thus,
it is during the early growth phase and then during
compaction that non-dynamic layering is most likely to
develop, as these are the times when b is smallest.
It should be noted that the phenomenon occurs
whether the transport is by advection (as is modeled
here) or by diffusion [as modeled by Boudreau (1994)].
Including a diffusion transport term in equation (2) would
not change qualitatively the observed segregation, as the
diffusive gradients are such that they also will favor
mineral segregation. Indeed, for slow infiltration rates,
the addition of a diffusional mode of transport along with
advection would lead to more rapid material exchange
between developing layers and hence actually accelerate
the time scale for layer development. For example, for
the 1 cm length scale under consideration, the low liquid
velocities at longer characteristic times are such that the
product V L approaches values for diffusion mass transport appropriate for silicate liquids. For layering that
develops on a finer length scale, such as is illustrated in
Fig. 2, the system can evolve on diffusion length and time
scales and hence could occur without liquid migration.
However, for layers with longer characteristic lengths
(i.e. thickness), such as the decimeter-scale mafic–felsic
layers of the Skaergaard Layered series, diffusion alone
would be insufficient. Compaction-aided advection or
interstitial liquid convection is required in addition to
diffusion to effect the necessary mass transport between
regions.
Numerical model of layer development
A one-dimensional numerical model of grain size and
textural evolution with time using a finite-difference
analog of the nondimensional equations (9)–(13) is shown
in Fig. 11. [Because growth of one mineral phase is
1016
BOUDREAU AND McBIRNEY
NON-DYNAMIC LAYERING OF THE SKAERGAARD INTRUSION
Fig. 10. Plot of characteristic times (in seconds) as a function of velocity of intercumulus liquid, for a range of values of the scaling constant, b,
all calculated for a characteristic length of 1 cm.
matched by dissolution of the other mineral phase, porosity changes little and hence velocity variations expressed
through equation (14) are not considered in the following
simulations.] In this calculation, the system starts as a
uniform distribution of a and b mineral grains, all of the
same average size but with a local ‘bump’ in the grain
size of mineral a. For the calculation, the scaling constant,
b, is taken to be 104. The calculation follows the evolution
of this initial bump over ten nondimensional space steps
and over nine nondimensional time units. The initial
bump of larger grains of mineral a is located at a
(nondimensional) distance of 3·3 units: grains of mineral
a in the peak of this bump are 5% larger than the average
elsewhere. Plotted in the four graphs of Fig. 11 are, from
top to bottom, the radius of mineral a, the radius of
mineral b, the volume of mineral a as a percent of total
solids volume (i.e. the modal abundance of mineral a),
and the scaled liquid concentration of component A—all
plotted as dimensionless quantities. The interstitial liquid
is taken as flowing from left to right. The plot shows
the evolution of the profiles for these quantities at the
dimensionless times of 0, 3, 6 and 9 time units.
Because the number of grains is constant, the region
of larger grains centered at a distance of 3·3 units causes
a small increase in the modal abundance of a in this
bump, which in turn affects the local equilibrium liquid
concentration. Let us consider first what occurs as the
interstitial liquid, moving from left to right, begins to
encounter the region where the modal abundance of a
is beginning to increase. Because of the textural dependence of the eutectic position, the equilibrium concentration for component A is lower where mineral a is
modally more abundant. Hence the liquid finds itself
oversaturated in mineral a as it encounters the upstream
side of the bump and begins to precipitate more mineral
Fig. 11. Numerical model of layer development. Plotted against the
dimensionless distance are, from top to bottom, the nondimensional
radius of mineral a (ra′), the nondimensional radius of mineral b (rb′),
the volume percent of mineral a as a percentage of total solids volume
(fa), and the scaled nondimensional concentration (s′). Shown is the
evolution of the profile at dimensionless time (t ′) equal to 0, 3, 6 and
9 time units. (See text for additional discussion.)
a. For mineral b it is just the opposite; the liquid finds
itself undersaturated in mineral b as it encounters the
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JOURNAL OF PETROLOGY
VOLUME 38
upstream side of the bump and hence b begins to dissolve.
Once the liquid passes the peak of the bump, however,
the situation is the reverse. On the downstream side of
the bump, the liquid is moving into assemblages that
have progressively more b than a. In this case, the liquid
is always oversaturated in b but undersaturated in a as
it moves to the right of the initial bump, and thus a
dissolves whereas b precipitates. The net effect is that a
becomes more abundant on the upstream side whereas
b becomes more abundant on the downstream side. The
original peak in grain size of a grows but it also migrates
upstream. In addition, the initiation and growth of the
b peak itself induces a new a peak to form downstream
from the first peak, which in turn induces yet other peaks
to form. After nine nondimensional time steps, three a
peaks and three b peaks have developed.
We note that regions in which a or b are modally
dominant tend to become sharply defined from their
neighboring regions, as seen in the modal plot of volume
percent of mineral a. Also, grains in the individual layers
may be size-graded. That is, the larger grains are at the
downstream side of each layer (this is better developed
in the induced layers than in the layer formed from the
initial bump). If the layering were developed horizontally
in response to vertical movement of liquid (as in a
compacting pile of cumulus crystals), then the size grading
would be similar to the size distribution produced by
Stokes’ law gravitational separation of larger from smaller
grains during a crystal settling event. In this case, however,
the grain sizes need not be hydraulically equivalent as
would be expected for layering formed by crystal settling.
The scaled concentration profile tends to mirror the
modal abundance profile for mineral a. This is because
the mass of solid material is large as compared with the
maximum texture-induced concentration differences and
causes the scaled concentration profile to be strongly
controlled at the equilibrium values defined by the local
rock mode/texture.
Finally, on observing the texture and modal variations
evolve, one tends to be taken by the propagation of the
pattern. What is perhaps more important, however, is
the fact that minor textural irregularities tend to become
more sharply defined over time. In a rock composed of
initially weakly defined layers formed by a variety of
nucleation or mechanical segregation mechanisms, the
processes outlined above will continue to enhance the
modal and textural contrast between layers. This modal
enhancement of preexisting modal variations may be the
principal cause of non-dynamic layering.
CONCLUSIONS
Skaergaard layering produced by non-dynamic processes
differs from that caused by magmatic flow. If it results
NUMBER 8
AUGUST 1997
from variations of intensive parameters that alter rates
of nucleation and crystal growth, the layers have diffuse
boundaries and the minerals have little if any lineation.
A notable exception is the layering in the Marginal Border
Series, which advanced more slowly and is relatively
compressed. Most of the sharp layering in the interior
of the Layered Series is thought to be related in some
way to compaction and other processes involving porous
flow of interstitial liquids. Although mechanical segregation may be effective during the initial stages of
accumulation, differential pressure-solution seems to have
been the principal mechanism. Initial modal variations
are strongly enhanced and sharpened as the ascending
liquid transfers components from one level to another.
The driving force of segregation is the free-energy difference resulting from combined effects of grain size, pressure solution, and the relative affinities of like and unlike
minerals. Where compaction produces a regionally uniform upward percolation of liquid, this segregation leads
to formation of planar layers. However, focused flow
or non-uniform compaction may cause more irregular
structures. The coincidence of the disappearance of extensive strataform layering with the beginning of the
trough layers in the Upper Zone is consistent with a
change from uniform to focused flow of interstitial liquid
at this level.
ACKNOWLEDGEMENTS
This work has been supported by grants from the National
Science Foundation to A. E. Boudreau (NSF EAR 9217664 and 95-17144). McBirney’s 25 years of work on
the Skaergaard Intrusion would not have been possible
without the financial support provided by a series of
grants from the National Science Foundation. Review
by W. P. Meurer, Bruce Marsh and an anonymous
reviewer is acknowledged and much appreciated.
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Wager, L. R. & Deer, W., 1939. Geological investigations in East
Greenland, Part III. The petrology of the Skaergaard Intrusion,
Kangerdlugssuaq, East Greenland. Meddelelser om Grønland 105, 1–
352.
Waters, C. & Boudreau, A. E., 1996. A reevaluation of crystalsize distributions in chromite cumulates. American Mineralogist 81,
1452–1459.
APPENDIX
The characteristic and dimensionless forms of the various
values are defined as follows:
ra=rara′
rb=rbrb′
t=tt ′
na=nana′
nb=nbnb′
z=zz′
V=V V ′
f=f f ′
(A1)
For the radius, crystal number density, and liquid velocity,
one can take the typical, or system-averaged, initial values
as the characteristic quantities. One can also define a
dimensionless supersaturation:
C=C 100(1+s).
(A2)
The characteristic time and length are to be derived
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VOLUME 38
(A3)
(A4)
VC
n (r )3
∂r ′
∂s
C100 ∂s
=−V ′ 100
−4pqa a a na′(ra′)2 a
x ∂t ′
z ∂z′
t
∂t ′
(A12)
nb(rb)3
2∂rb′
nb′(rb)
.
+4pqb
t
∂t ′
qb′=
or
C
D
∂ra′ t qa C100s
=
−(1−fa) .
K
∂t ′ r aqa
K
With further substitution of equation (A6), one has
K ∂s′
V K ∂s′
n (r )3
∂r ′
=−V ′
−4pqa a a na′(ra′)2 a
x ∂t ′
z ∂z′
t
∂t ′
One can then define a characteristic time as follows:
t=
q a ra
qa K
(A5)
In addition, one can define a ‘scaled supersaturation’ in
which the supersaturation is scaled to the maximum
mode/texture-induced concentration difference:
sC
s′= 100 .
K
n (r )3
∂r ′
+4pqb b b nb′(rb)2 b .
t
∂t ′
∂ra′
=[s′−(1−fa)] .
∂t ′
∂s′
V t ∂s′
n (r )3
∂r ′
=−V ′
−4pqa a a na′(ra′)2 a
∂t ′
z ∂z′
K
∂t ′
n (r )3
∂r ′
+4pqb b b nb′(rb)2 b .
K
∂t ′
∂rb′
t q
=− b {C100(1+s)−[C100+(1−fa)K]}
∂t ′
rb qb
(A8)
4pqana(ra)3
K
b=
D
t V
=b .
z
(A16)
Then, on substitution of (A15) and (A16) into equation
(A14), one arrives at
∂r ′
∂r ′
1 ∂s′
∂s′
=−V ′ −na′(ra′)2 a +Hnb′(rb′)2 b
b ∂t ′
∂z′
∂t ′
∂t ′
where
(A9)
On substitution of the expression for the characteristic
time and the scaled supersaturation [equations (A5) and
(A6)], one has
∂rb′
=qb′[s′−(1−fa)]
∂t ′
(A15)
and furthermore let
or
tq
C s
∂rb′
=− b K 100 −(1−fa)
∂t ′
rb qb
K
(A14)
One can then define the scaling constant, b:
(A7)
A similar treatment for mineral b—substitution of equations (A1), (A2) and (6) into equation (5)—gives
(A13)
or, on rearranging, one has
(A6)
The non-dimensional crystal growth rate for mineral a
is then given by
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AUGUST 1997
ra qa qb
.
(A11)
r b qb qa
For the transport–reaction equation (2), substitution of
equations (3), (A1) and (A2) gives
below. Substitution of the equations (A1), (A2) and (6)
into equation (4) and rearrangement gives
∂ra′ t qa
=
{C100(1+s)−[C100+(1−fa)K]}
∂t ′ r aqa
NUMBER 8
(A10)
H=
qbnb(r b)3
.
qana(r a)3
(A18)
Finally, for equation (8), substitution of equations (A1)
gives
V V ′=
or
V ′=
where
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Vf
ff′
1
f′
(A19)
(A20)