JOURNAL OF PETROLOGY
VOLUME 39
NUMBER 5
PAGES 1063–1076
1998
Abrupt Change in Magma Liquidus
Temperature because of Volatile Loss or
Magma Mixing: Effects on Nucleation,
Crystal Growth and Thermal History of the
Magma
M. HORT∗
ABTEILUNG FÜR VULKANOLOGIE UND PETROLOGIE, GEOMAR FORSCHUNGSZENTRUM, WISCHHOFSTR.1–3,
D-24148 KIEL, GERMANY
RECEIVED APRIL 17, 1997; REVISED TYPESCRIPT ACCEPTED DECEMBER 2, 1997
The liquidus temperature of magma that reaches shallow levels
during its ascent may change abruptly as a result of the release of
volatiles or through magma mixing or a combination of both.
Immediately after this abrupt change of the liquidus temperature
occurs a disequilibrium is observed in the melt, and here it is shown
that the melt rapidly re-equilibrates to its thermodynamic equilibrium.
A quantitative model for cooling and crystallization of a simple
two-component model melt is used to investigate such re-equilibration
processes. The relaxation time, defined as the time required for the
system to re-equilibrate after varying degrees of liquidus temperature
perturbations, is found to be ~3% or significantly less than the
time it takes to crystallize ~40% of the melt, regardless of the
amplitude of the perturbation. Associated with the re-equilibration
process is an increase in crystal fraction that can easily reach
10% depending on the amplitude of the perturbation. During the
compensation of the liquidus temperature perturbation, the temperature of the melt remains nearly constant or may even increase
slightly (depending on the latent heat budget, possibly heat of mixing,
and heat absorbed during volatile exsolution), which suggests
crystallization of the melt without cooling.
INTRODUCTION
magma
On its way from the source region into the Earth’s
crust or even onto the Earth’s surface, magma travelling
through the mantle and crust evolves, and one of the
goals in petrology is to characterize the various processes
that occur during this passage. Of special interest in the
framework of this study are shallow level processes, such
as magma–crust interaction, magma mixing, devolatilization and depressurization, that change the geochemical signature of the melt, its crystal content, its
liquidus temperature, its flow behaviour and its eruptability. Whereas crustal contamination of magma can be
traced through, for example, trace element and isotope
studies (e.g. Wilson, 1989) and is discussed extensively
in the other papers of this special volume, cooling,
depressurization, and possibly magma mixing strongly
affect the thermal history of magma rising in the mantle
and crust. For example, any pressure decrease in volatilesaturated systems (not uncommon in natural systems,
especially for more evolved compositions) will lead to
volatile exsolution and consequent increased liquidus
temperature and change in the melting range. One of
the major consequences of such behaviour is isothermal
crystallization, which was recognized almost 40 years ago
by Tuttle & Bowen (1958) (pp. 67 and 68), and the
importance of the pre-eruptive volatile content in magma
∗Telephone: (49) 431 600 2645. Fax: (49) 431 600 2978. e-mail:
[email protected]
 Oxford University Press 1998
KEY WORDS:
cooling; crystallization; kinetics; liquidus temperature;
JOURNAL OF PETROLOGY
VOLUME 39
for magma evolution has recently been reviewed by
Johnson et al. (1994).
Depending on the specific conditions, changes in the
liquidus temperature caused by depressurization and
degassing may be fairly large: Burnham & Jahns (1962)
and Burnham & Davis (1974) found the liquidus of a
water-saturated albite melt at PH2O = 200 MPa (equivalent to a depth of ~5 km) to be 300°C lower than for
the dry system at the same pressure, and Merrill &
Wyllie (1975) reported a drop by ~180°C of the liquidus
temperature of an olivine nephelinite when saturated
with 5 wt % H2O at 500 MPa. Yoder et al. (1957)
observed a 300°C drop in the liquidus of the plagioclase
system at a PH2O of 500 MPa compared with the system
at 1 bar, and Withney (1975) reported a lowering of the
liquidus temperature in a simplified synthetic rhyolite
system from 1200°C at anhydrous condition and 200
MPa to 835°C at a saturation with 6·6 wt % H2O.
Considering these changes in the liquidus temperature
related to varying amounts of dissolved water, exsolution
of volatiles is an important factor during the final stages
of magma evolution, as it triggers crystallization of the
melt and therefore alters the eruptability and flow behaviour (Eichelberger, 1995).
Most of these abrupt liquidus temperature changes will
occur at fairly shallow levels because, on the one hand,
volatile exsolution only takes place once the saturation
level is reached. This level depends on the magma
composition and volatile species, and a depth of 10 km
appears to be a conservative lower bound for H2O
exsolution, although CO2 exsolution can occur at much
greater depth. Magma mixing, on the other hand, is
expected to occur where magma is stored before eruption,
i.e. also at shallow levels where magma is generally more
evolved and mixes with ascending more primitive melts.
Degassing of the replenishing more primitive melt and
cooling against the more evolved melt drives magma
mixing, and the rising gas pressure may trigger an eruption (e.g. Eichelberger, 1980).
Therefore the main aim of this study is to set some
constraints on the extent of shallow level magmatic
processes caused by abrupt changes of the liquidus temperature. In this context it is not important which process
actually causes the abrupt change in the liquidus temperature, i.e. exsolution of volatiles, change in composition because of magma mixing, or a combination of
both, but only the fact that these changes can commonly
occur. Characteristic to all these processes is a sudden
increase in undercooling of the melt. As a changing
liquidus temperature or change in undercooling has a
direct influence on the nucleation and crystal growth
history of the different phases in the melt, I employ a
kinetic model to investigate the changes in crystallization
history as a result of various degrees of liquidus temperature perturbations. In this context, of special interest
NUMBER 5
MAY 1998
is how long the system takes to re-equilibrate once the
liquidus has risen and how the crystallization path
changes in response to different degrees of liquidus perturbations.
COOLING AND CRYSTALLIZATION
MODEL
To keep things simple, a sheet-like body intruded somewhere into the crust is assumed. The sheet-like body is
of infinite horizontal extent and is sandwiched between
two infinite half-spaces of wall-rock. This is certainly
not an uncommon situation, as many intrusive bodies
observed at the Earth’s surface are broadly sheet like,
with a height to length ratio of 0·5 or significantly less
(e.g. Marsh, 1988; Jaupart & Tait, 1995). Upon intrusion
the magma is at its liquidus temperature and, once
emplaced, it is assumed to be always well mixed (i.e. it
cools convectively), such that the temperature decreases
uniformly with time throughout the magma body. Overall
cooling of the intrusion is restricted only by the rate of
conductive heat loss into the country rock above and
below the intrusive sheet. This kind of model gives the
fastest possible cooling—initially twice as fast as for
the same body cooling by conduction only (see Marsh,
1989)—and therefore yields the largest possible disequilibrium in the melt because of cooling.
Upon cooling below its liquidus, nucleation and crystal
growth take place everywhere in the melt, and after an
initial period of nucleation and crystal growth the melt
evolves closely along its liquidus. If nothing extraordinary
happens from now on, the melt will slowly solidify while
being cooled until it is completely crystallized [this case
has been investigated in earlier studies (e.g. Kirkpatrick,
1976; Brandeis et al., 1984; Spohn et al., 1988; Hort &
Spohn, 1991a) and is therefore not repeated here]. In
the case of this study, however, it is assumed that instead
of slowly crystallizing the undercooling increases abruptly
as a result of exsolution of volatiles, or magma mixing,
or a combination of both. This type of abrupt perturbation yields the largest possible disequilibrium state
a degassing or mixing process can create, and both the
cooling model and rise in liquidus temperature are geared
towards creating a disequilibrium.
Energy balance
The energy balance for a well-stirred, isothermal, infinite
intrusive sheet sandwiched between two half-spaces, both
being initially at the same temperature, is (e.g. Hort et
al., 1993)
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HORT
CRYSTALLIZATION IN RESPONSE TO LIQUIDUS CHANGES
j d
d
T=−2
T∗
dt
H dz
K
+
z=0,H
Ld
φ
cp dt
and
(1)
where T is the temperature everywhere in the intrusive
sheet and T∗ is the temperature in the country rock
above and below the intrusive sheet, where heat transport
is assumed to be the same and by conduction only. H is
the thickness of the sheet, z is the vertical coordinate
(zero being at the upper contact), t is time, and j and cp
are the thermal diffusivity and specific heat capacity of
the melt and the country rock, which are for simplicity
not distinguished here. For the reader’s convenience,
Table 1 also lists all the symbols used in this study. The
left-hand side (LHS) of (1) represents the rate of change
in temperature in the sheet-like magma body. This is
equal to the actual heat loss into both half-spaces [first
term on right-hand side (RHS) of (1)], reduced by the
production of latent heat of crystallization (second term
on RHS). Here L is the latent heat and dφ/dt is the rate
at which melt solidifies (see next section).
Phase diagram and crystallization kinetics
Although magma is a multicomponent system, the simplest meaningful phase diagram including a changing
liquidus is a two-component system (Fig. 1a). In this
model the liquidi of the phases A, Tma, and B, Tmb,
Tma,b(c), are expressed by two quadratics that intersect at
the eutectic point (see Fig. 1a)
Tma(c)=DTr+a1a(c2−2c)+a2ac+a3a,
Tmb(c)=DTr+a1bc2+a2bc+a3b
(2)
with DTr≠0 for cΖcr
where ana,b are coefficients (see Table 2) and DTr is the
prescribed change in liquidus temperature occurring at
a certain composition cr. c is the composition of the melt
in terms of volume per cent of component A, and without
loss of much generality it is assumed that no volumetric
effects caused by mixing occur during the crystallization
process. It should be noted that, in the following, all
compositions, solid fractions, or degrees of crystallization
are related to the liquid and solid volumes of the system
unless specifically stated otherwise.
Crystallization in the melt proceeds through nucleation
and crystal growth, their associated rates I (nucleation)
and U (growth) both being described as thermally activated processes (e.g. Kingrey et al., 1976; Rao & Rao,
1978)
A B A B
DGt
DG
exp − c
RT
RT
I(T)=Io ·exp −
(3)
A BC
DGt
RT
U(T)=Uo ·exp −
A
Dhm(Tm−T)
RTmT
1−exp −
BD
.
(4)
Here DGt and DGc are the activation energies for atomistic
self diffusion and for the development of a critical-sized
nucleus; Dhv is the enthalpy difference between the melt
and the crystals, R is the gas constant, and Io and Uo are
rate constants, respectively. At the liquidus temperature
both rate functions are equal to zero and with increasing
undercooling they both increase, pass through a maximum [I(Ti) = Im, U(Tu) = Um; see below and Fig. 1b],
and then decrease (see Fig. 1b). Whereas the decrease of
both rate functions after passing through the maximum
is governed by the increasing ‘viscosity’ of the melt (first
term on the RHS in both rate functions), the increase of
the rate functions in the case of the nucleation rate
function is given by the decreasing activation energy of
a critical-sized nucleus, DGc, and in the case of the crystal
growth rate function by the gain in energy through the
phase transformation.
Through parameterizing equations (3) and (4), Spohn
et al. (1988) have shown that four parameters, Im, Um, Ti,
and Tu, are sufficient to completely specify the rate
functions given above, which makes direct knowledge of
difficult to assess parameters such as DGc obsolete. Here
Ti and Tu are the temperatures at which the nucleation
and growth rate functions pass through their global
maxima (see Fig. 1b), and Im and Um are the corresponding
maximum amplitudes for nucleation and growth (see
Fig. 1b and above). Numerical values for Ti, Tu, Im, and
Um for various melts have been compiled by Dowty
(1980). In addition to the parameterization, Spohn et al.
(1988) have shown that the ratios Ti/Tm and Tu/Tm (Tm
being the liquidus temperature) can be treated as constant
for each phase as long as the composition of the melt
does not change considerably during the crystallization
process. This type of kinetic model is based on interfacecontrolled growth [equation (4)] and homogeneous nucleation [equation (3)]. Heterogeneous nucleation processes can to some extent be accounted for semiempirically by increasing the temperature Ti and the
amplitude Im, which is equivalent to a reduction of DGc
(Hort & Spohn, 1991b).
Because the intrusion is assumed to remain isothermal
[see equation (1)] during cooling, nucleation and crystal
growth take place everywhere in the melt once it has
cooled below its liquidus temperature. While the crystals
grow the solid fraction increases and the change in
fraction solid of component A, dφa/dt, is determined
from the Johnson–Mehl–Avrami equation (see e.g. Kirkpatrick, 1976)
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JOURNAL OF PETROLOGY
VOLUME 39
NUMBER 5
MAY 1998
Table 1: Symbols used
Parameter
Meaning
Units
First app.
2
ana,b
Constants (see Table 2)
vary
Av
Avrami number
—
10
c
Composition of the melt
vol %
2
c ao
Initial concentration of component A in the melt
vol %
5
cp
Specific heat of melt and country rock
J/kg per K
1
cr
Composition at which the liquidus temperature rises
vol %
2
DG c
Activation energy for the development of a critical-sized nucleus
J/mol
3
DG t
Activation energy for atomistic self diffusion
J/mol
3
1
H
Thickness of the intrusion
m
Dh v
Enthalpy difference between melt and crystals
J/mol
4
I, I∗
Nucleation rate
m–3s–1, —
3, 8
Io
Nucleation rate constant
m–3s–1
3
I a, I a∗
Nucleation rate of component A
m–3s–1, —
5, 11
I m = I(T i)
Maximum nucleation rate at temperature T i
m–3s–1
text
L
Latent heat
J/kg
1
R
Gas constant
J/mol per K
3
r, q
Crystal radius
mm, —
Fig. 5
S
Stefan number
—
9
t, s
Time
s, —
1, 7
t kin
Kinetic time scale
s
10
t r, sr
Relaxation time
s, —
14, text
t th
Thermal time scale
s
7
t∗, s∗
Time at which a certain crystal nucleates
s, —
5, 11
T, H
Temperature of the melt in the intrusion
K, —
1, 6
T∗, H∗
Temperature in the country rock
K, —
1, 12
T 0, H0
Initial temperature of melt in intrusion
K, —
6, Fig. 1a
T i, Hi
Temperature at which the nucleation rate is at its maximum
K, —
text, Fig. 1a,b
T m, Hm
Liquidus temperature
K, —
4, text
T ma(c), Hma(c)
Liquidus temperature of component A
K, —
2, Fig. 1a
T mb(c), Hmb(c)
Liquidus temperature of component B
K, —
2, Fig. 1a
T u, Hu
Temperature at which the crystal growth rate is at its maximum
K, —
text, Fig. 1a, b
Tw
Initial temperature of the country rock
K
6
DT r, DHr
Rise of liquidus temperature at composition c r
K, —
2, text
U, U∗
Crystal growth rate
ms–1, —
4, 8
U a, U a∗
Crystal growth rate of component A
ms–1, —
5, 11
Uo
Crystal growth rate constant
ms–1
4
U m = U(T u)
Maximum crystal growth rate occurring at temperature T u
ms–1
text
z, f
Vertical coordinate, zero being at the top of the sill
m, —
1, 7
φ
Fraction crystallized of the melt
—
1
5
φa
Fraction crystallized of component A in the melt
—
φb
Fraction crystallized of component B in the melt
—
text
j
Thermal diffusivity of melt and country rock
m2/s
1
k, k∗
Time
s, —
5, 11
r
Shape factor of crystals precipitating from the melt
—
5
Subscript m denotes the liquidus temperature; subscripts a and b refer to the different components in the melt. If there are
two parameters given in one line, the second one is always the corresponding dimensionless parameter. The numbers
given in the column ‘First app.’ refer to the equation numbers.
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HORT
CRYSTALLIZATION IN RESPONSE TO LIQUIDUS CHANGES
Table 2: Numerical values for the coefficients of equation (2) used to
calculate the liquidi shown in Fig. 1 [note that the coefficients have
already been normalized, to be consistent with equation (6)]
Component
a1
a2
a3
A
−0·466
0·079
0·545
B
−0·137
−0·106
0·953
Fig. 1. (a) Phase diagram. Here the liquidi of components A (Hma) and (B) (Hmb) and the solidus are drawn with bold continuous lines. The bold
dashed and dotted lines give the compositional variation of temperatures of the maximum of the crystal growth rate, Hu, and of the maximum of
the nucleation rate, Hi, respectively. It should be noted that all temperatures have been normalized according to equation (6), where T0 = Tma
(ca0 = 0·65) was used. (b) Dimensionless nucleation and crystal growth rates [equation (8)]. The nucleation rate (dotted line) and the growth rate
(dashed line) are plotted for a melting temperature of hm = H = 1 as a function of the normalized temperature H [equation (6)]. The two
diagrams are connected by a couple of fine lines, the horizontal fine continuous line indicating the liquidus temperature of the melt, the vertical
fine continuous line indicating the initial composition of the melt, the horizontal fine dashed line connecting the maximum of the crystal growth
rate in (b) with the compositional variation of the maximum of the crystal growth rate in (a), and the horizontal fine dotted line connecting the
maximum of the nucleation rate in (b) with the compositional variation of the maximum of the nucleation rate in (a), respectively.
Dimensional analysis of the equations
t
t
d
1
φa=3rUa(t) /Ia [T (t∗) ] w /Ua [T (k) ] dkx2dt∗
(cao−φa) dt
0
t∗
I
II
It is convenient to reduce the number of unknowns in
the equations introduced by employing nondimensional
parameters. For temperature, T, we define
T−Tw
H=
T0−Tw
(5)
where the first part of the integral in (5) (i.e. part I) gives
the number of crystals nucleated between time 0 and t,
and part II, together with the factor 3rUa(t) (r being a
shape factor of the crystals), gives the time rate of change
of solid fraction caused by the growth of the existing
crystals in the melt. The solid fraction is related to the
composition through the lever rule.
(6)
where T0 is the temperature of the magma at time t =
0 and Tw is the initial temperature of the wall-rock; time,
t, and distance, z, are normalized by the thermal diffusivity
j and H
1067
s=
t
z
j
t= ’ f=
H2 tth
H
(7)
JOURNAL OF PETROLOGY
VOLUME 39
and tth is a typical cooling time based on conduction.
Finally, for nucleation rate, I, and growth rate, U, we
define
I
U
I∗= , U∗= .
Im
Um
(8)
Now two nondimensional groups of parameters are introduced:
S=
L
cp(T0−Tw)
(9)
AB
(10)
and
Av=(rImU 3m)1/4
t
H2
= th .
j
tkin
The first parameter (9) is the well-known Stefan number,
here expressed as the ratio between the latent heat and
the specific heat stored in the melt. The second parameter,
the Avrami number, in a slightly different form introduced earlier by Spohn et al. (1988), can be interpreted
as the ratio of the thermal time scale, tth, to the kinetic time
scale, tkin = (rImUm3)–0·25, and measures the importance of
heat transfer relative to kinetics of crystallization. For
large enough values of Av (e.g. Av → x), cooling is slow
relative to the time of nucleation and crystal growth, and
the kinetics is relatively unimportant. In contrast, for
small enough values of Av, kinetics becomes increasingly
important relative to cooling such that crystallization
may be incomplete after cooling to low temperatures
and glass may develop. Replacing the time scale for
conductive cooling, tth, with that for convective cooling
converts (10) to the equation used by Brandeis & Jaupart
(1986) in their parameterization of crystallization kinetics.
Now all previous equations can be nondimensionalized
using (6)–(10). The Avrami number appears in the rate
of change of fraction crystallized (here written for component A)
s
1 dφa
=−3Av4U∗a (s) /I∗a [T(s∗) ] ·
ca0−φa) ds
0
(11)
s
w/ U∗a [T(k∗) ]dk∗x2ds∗
s∗
which itself enters into the energy equation (1), which
now becomes
K
d
dH∗
dφ
H=−2
+S
ds
df f=0
ds
(12)
where dφ/dt is the sum of the contributions to changes
in fraction crystallized from both phases (A and B), dφa/
dt and dφb/dt, respectively.
NUMBER 5
MAY 1998
The equations are solved with a finite difference technique. Before (12) is integrated it is rewritten into an
integro-differential equation (see, e.g. Marsh & Maxey,
1985; Ghiorso, 1991), because in the case of latent
heat no analytical solution for (12) can be found. The
integration excluding latent heat was tested successfully
against an analytical solution (Carslaw & Jaeger, 1959).
Equation (11) is rewritten into a system of four ordinary
differential equations (ODEs) following a suggestion of
Daessler & Yuen (1993) and then solved with an algorithm
for the integration of a set of stiff ODEs (Press et al., 1992).
The resulting numerical approach was satisfactorily tested
for stability and convergence using constant time steps
between 10–8 and 10–7.
RESULTS
The model presented in the last section involves seven
parameters. Except for the Stefan number, S, and the
initial composition of the melt, cao, which are kept constant
in all model calculations at 0·12 and 0·65, respectively,
all other parameters (DTr, cr, Av, Ti and Tu) have been
varied over reasonable ranges. DTr is the amplitude of
the instantaneous increase of the liquidus temperature
that occurs at the prescribed composition cr. DHr, the
dimensionless equivalent to DTr [see equation (6)], has
been varied between 0·01 and 0·1, which corresponds
to a DTr of ~10–100 K using typical values for a magmatic
system. With the initial composition of the melt cao being
0·65, cr has been varied between 0·61 and 0·55, with the
value in the standard model being 0·58. Using the lever
rule, this corresponds to a solid fraction in the melt of
about 0·1–0·22, with the solid fraction in the standard
model being about 0·17.
The Avrami number has been varied between 103 and
107, with its value in the standard model being 105.
The variation of this parameter over several orders of
magnitude is mainly due to the large variability of Im,
Um, and H. The ratio of Ti/Tm (the location of the
maximum of the nucleation rate function) was varied
between 0·94 and 0·88, with the standard model value
being 0·94. This ratio has been chosen close to unity to
semi-empirically account for heterogeneous nucleation
processes in the melt (see above). Values for purely
homogeneous nucleation are represented by the lower
end of the range investigated. The ratio of Tu/Tm was
chosen to be always 0·02 larger than the ratio Ti/Tm to
accommodate the fact that the crystal growth rate function passes through its maximum almost always closer to
the liquidus than the nucleation rate (e.g. Kirkpatrick,
1981). Of course, other differences between these two
values are possible, but they would not change the general
conclusions drawn from the calculations presented here.
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CRYSTALLIZATION IN RESPONSE TO LIQUIDUS CHANGES
In the following, I will first discuss the general thermal
history of the melt before turning to the parameter
variation. In all model calculations it is assumed that the
melt is emplaced instantaneously at its melting temperature. Then it starts to cool by losing heat through
the top and the bottom of the system, and upon cooling
nuclei can develop and grow in the melt. In the model
I let this crystallizing system evolve for a while, mainly
to reach an equilibrium state so as to minimize the
influence of the initial conditions imposed. Once this
equilibrium is reached I impose the abrupt rise in the
liquidus temperature DHr (see discussion on cr above), in
order to investigate the system’s response to the increased
undercooling. After this perturbation I let the system reequilibrate, and all model calculations are stopped once
the melt composition reaches 0·45. This was found to
be sufficient to study the effect of the changing liquidus
temperature and in addition reduces the amount of
computer time needed. Furthermore, at this degree of
crystallization (37%) the melt can still be transported and
erupted (Marsh, 1981), as it is only partially crystallized.
General thermal and crystallization history
Figure 2a–c shows the temporal evolution of the temperature in the intrusion, the undercooling below the
liquidus temperature, the nucleation and crystal growth
rate, and the solid fraction in the melt. In Fig. 3 the
crystallization path has been plotted directly into the
phase diagram. Upon initial cooling the melt starts to
undercool below its liquidus (Fig. 2a, dotted line) and
with increasing undercooling nucleation in the melt sets
in (Fig. 2b, s ≈ 2 × 10–5, continuous line). Initially, the
number of nuclei is small and no significant change in
solid fraction can be observed (Fig. 2c) as a result of
growth of these nuclei. Therefore, the undercooling increases further for a small amount of time (Fig. 1a, dotted
line) until enough crystals are formed. Because of the
growth of these small crystals (Fig. 2b, dotted line),
crystallization of component A becomes significant
(Fig. 2c, s ≈ 10–4). This crystallization is accompanied
by the production of latent heat (see the rise in temperature early on; Fig. 2a, continuous line), which, together with the growth of crystals, drives the residual
melt composition back to the liquidus (Fig. 3, c >0·6).
Upon asymptotically approaching the liquidus (Fig. 3,
dash–dotted line) the undercooling decreases (Fig. 2a,
dotted line), and nucleation ceases and crystal growth
slows down (Fig. 2b). From now until the rise of the
liquidus temperature occurs at cr = 0·58, the melt slowly
crystallizes by growth of the existing crystals in the melt
(Fig. 2b, dotted line, Fig. 2c). During this stage the
undercooling below the liquidus is <10–3 (Fig. 2a, dotted
line), which converts back to an undercooling of <1 K
using typical values for a magma. This means that
the system is evolving very close to its thermodynamic
equilibrium.
After about 5·5 × 10–4 time units, when the melt
composition is equal to 0·58, the liquidus temperature
increases abruptly (here by 0·05, which is equivalent to
~50 K using typical values for a magmatic system; see
also Fig. 3, dash–dotted line). As explained above, this
rise in liquidus temperature is imposed on the system
from the outside and is assumed to be related to release
of volatiles, depressurization or magma mixing, or a
combination of these processes. In passing, I would like
to mention that in this model any compositional change
as a result of these abruptly changing conditions has
been neglected, as well as any change in the amplitudes
of the nucleation and crystal growth rate (for a discussion
of this, see below). It is important to recognize that the
amount by which the liquidus temperature has risen is
equivalent to the amount by which the melt is now
undercooled (Fig. 2a, dotted line). Therefore nucleation
instantaneously restarts in the melt and crystal growth
accelerates significantly (Fig. 2b). Taken together, crystallization in the melt becomes very efficient and the
solid fraction increases almost instantaneously from 17
to 27% (Fig. 2c). The whole disturbance of the system
lasts only for 2 × 10–5 dimensionless time units and we
call this time interval the relaxation time. It can be
defined as the time the undercooling needs to evolve back
to the value it had immediately before the perturbation of
the liquidus temperature occurred.
In this case, the relaxation time is about 2 × 10–5,
which is equivalent to ~1% of the time it takes to
crystallize the first 37% of the melt. However, in that
comparatively short amount of time ~10% of the melt
crystallizes (see above). This means that the system can
rapidly compensate for liquidus temperature perturbations through crystallization, i.e. high nucleation
and crystal growth rates and release of latent heat.
Immediately after the perturbation, the system returns
to its equilibrium situation as rapidly as possible. Once
it regains the raised liquidus temperature (dash–dotted
line in Fig. 3), the system crystallizes further (Fig. 2c)
only by growth of the existing crystals (Fig. 2b, dotted
line), a situation equivalent to that described above for the
time after the initial period of nucleation (0·60 < c < 0·58).
Dependence of the relaxation time on
various parameters
In general, the solid fraction was found to increase
significantly upon perturbing the liquidus temperature,
with the relaxation time being extremely small. Figure 4a
shows that this relaxation time changes only by a factor
of two while varying the amplitude of the perturbation
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Fig. 2. (a) Temporal evolution of the temperature in the melt (continuous line) and the undercooling below the liquidus temperature (dotted
line), (b) the nucleation rate (continuous line), the crystal growth rate (dotted line), and (c) the fraction crystallized.
(DHr is represented by the circles in Fig. 4a) and the
composition at which the perturbation occurs (cr, triangles
in Fig. 4a), although the increase in solid fraction as a
result of the perturbation changes (from 2% for DHr =
0·01 to 14% for DHr = 0·1). The main reason for the
relaxation time being independent of the amplitude of
the perturbation is the exponential increase of the nucleation and crystal growth rate with undercooling. That
is, the larger the perturbation, DHr, the larger are the
instantaneous nucleation and crystal growth rates because
of the increased undercooling below the liquidus (see
Fig. 1b), securing a rapid return of the system to its
equilibrium state. It is therefore hidden in the exponential
nature of the nucleation and the crystal growth rate
function that the relaxation time is nearly independent
of the perturbation. Perturbations significantly larger than
1 – Ti/Tm will yield increased relaxation times, but such
perturbations are geologically unreasonably large and
therefore not considered.
Apart from the relaxation time being independent of
cr and DHr, it is found to be slightly dependent on the
location of the maxima of the nucleation and crystal
growth rate function, Ti/Tm and Tu/Tm (see Fig. 4b,
circles) and extremely dependent on the Avrami number
(see Fig. 4b, triangles). The small increase in relaxation
time with decreasing Ti/Tm can be related to the fact
that for a given liquidus temperature perturbation, DHr,
the nucleation and crystal growth rate in direct response
to this perturbation decrease when Ti/Tm and Tu/Tm
decrease. Therefore it takes slightly longer for the system
to compensate for the perturbation with decreasing values
of Ti/Tm and Tu/Tm.
The dependence of the relaxation time on the Avrami
number can be divided into two different regimes. For
Av < 5 × 104 the relaxation time increases nonlinearly,
whereas for Av > 5 × 104 the relaxation time, sr, decreases linearly with Av and is fitted nicely (R2 = 0·9997)
by
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CRYSTALLIZATION IN RESPONSE TO LIQUIDUS CHANGES
Fig. 3. Crystallization path (continuous line) plotted into the phase diagram. All other lines are as in Fig. 1a, except now the dash–dotted line
gives the liquidus temperature including the perturbation that occurs at cr = 0·58.
log(Av)=−1·0335 log(sr)+0·08
(13)
which is shown by the continuous line in Fig. 4b. Clearly,
for increasing values of Av the relaxation time tends
towards zero, as the kinetic time scale becomes shorter
and shorter with increasing Av. As Av → 1, cooling of
the system begins to dominate the thermal history, keeping the system in disequilibrium during most of the
crystallization process. As cooling and kinetics now both
influence the relaxation time, the dependence on Av
becomes nonlinear, as shown in Fig. 4b by the triangles
for <5 × 104.
In response to the perturbation of the liquidus, a
significant increase in the solid fraction was observed.
The increase in solid fraction is associated with the
growth of already existing crystals as well as the nucleation
and growth of new crystals (see above). It is therefore of
interest to determine by how much the crystals have
increased their radii during this period of rapid solidification. Figure 5 shows the radius increase with
respect to the radii the crystals had before the perturbation of the liquidus temperature occurred. The field
marked in Fig. 5 summarizes the results in response to
the variation of the parameters discussed above. Most of
the crystals increase their radii by 2–20% and only the
very small ones nearly double their radii. Therefore
even fairly large perturbations of the liquidus should not
express themselves too much in older, larger crystals, but
in the fine-grained groundmass that develops at the burst
of crystallization in response to the liquidus perturbation,
as described by Swanson et al. (1989) for microlite crystallization (see below).
DISCUSSION AND CONCLUSIONS
This model combines the cooling of a well-stirred infinite
sheet of melt sandwiched between two half-spaces with
crystallization in the melt. Related studies on pure cooling
and crystallization have been carried out before (e.g.
Brandeis et al., 1984; Hort et al., 1993), and here we focus
on the response of the system to an increase in the
liquidus temperature. The main goal of the study is to
investigate the general relaxation behaviour of the system
in response to a perturbation of the liquidus temperature
rather than model the crystallization history of a specific
melt. In the following, I will first discuss the assumptions
inherent to this specific model and then I will turn to a
comparison with observations and final conclusions.
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Fig. 4. Dependence of the relaxation time of the system on (a) the amplitude of the perturbation, DHr (Χ), and the composition at which the
perturbation occurs, cr (Μ), and (b) the Avrami–number, Av (Μ), and the location of the nucleation and crystal growth rate function, Ti/Tm and
Tu/Tm (Χ). (Note the logarithmic scale of the x-axis in (b).] The continuous line shows the relation given by (13).
Assumptions
Most of the assumptions included in the cooling and
crystallization part of the model have been discussed
earlier (e.g. Spohn et al., 1988; Hort & Spohn, 1991b)
and will not be repeated here. Inherent to this model
are the following three main assumptions, which will be
discussed subsequently below:
(1) the change in the liquidus temperature occurs
abruptly;
(2) the kinetics does not change while the liquidus
temperature changes (i.e. Ti/Tm, Tu/Tm, Im, and Um
remain constant);
(3) the heat of exsolution of volatiles has been neglected.
There are many natural systems where the changes in
the liquidus temperature will certainly be abrupt, e.g.
eruption processes. However, even when changes in the
liquidus temperature are gradual, the results presented
here should still hold, as the calculations presented in
this paper explored the worst possible situation by imposing an abrupt perturbation of the liquidus temperature. It was shown above that the relaxation time is
fairly independent of the amplitude of the perturbation
(Fig. 4a) and therefore raising the liquidus temperature
gradually over an extended period of time can be treated
as a series of small perturbations. One of the differences
between a gradual and an abrupt change of the liquidus
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CRYSTALLIZATION IN RESPONSE TO LIQUIDUS CHANGES
Fig. 5. Increase in crystal radius as a result of the increased solidification during the relaxation process as a function of the crystal radius before
the perturbation of the liquidus temperature occurs. The field summarizes the results of various model calculations where the parameters have
been varied over a wide range (see text). The dimensionless crystal radius q can be converted to the crystal radius r by multiplying q by Um × tth.
With typical values q = 10–4 is equivalent to a radius of ~1 mm.
temperature is that in the case of a gradually changing
liquidus temperature only crystal growth compensates
for the disequilibrium situation, whereas in the case of
an abrupt change nucleation and crystal growth contribute to the re-equilibration process (see below).
In the case of magma mixing, the assumption of the
kinetics not changing much when the liquidus changes
poses no problem. However, in the case of a degassing
process, the assumption that the kinetic rate functions
are independent of the actual amount of water being
dissolved in the melt needs further discussion. At a first
glance, this assumption appears to be critical, as it is well
known that the viscosity of a magma is strongly dependent
on the actual amount of water dissolved in the melt (e.g.
Dingwell et al., 1993), and one would expect the amplitude
of the nucleation and crystal growth rate function to
be dependent on the water dissolved in the melt too.
According to the compilation of Dowty (1980) (see his
table 1a and b), however, the amplitudes Im and Um of
the nucleation and crystal growth rate functions for a
given melt appear more or less constant regardless of the
amount of water dissolved, whereas Ti/Tm and Tu/Tm
are found to decrease with decreasing water content.
Newer data on the crystal growth rates of plagioclase by
Muncill (1988) support this observation and show that
the maximum crystal growth rates for An10 and An30 are
the same in 2 and 5 kbar H2O-saturated melts, with Tu/
Tm also decreasing with decreasing water content.
The main reason for this observed invariability of the
maximum nucleation and crystal growth rates may be
hidden in two counteracting effects. First, the viscosity
decreases with increasing water content, which should
increase nucleation and crystal growth rates. Second, the
absolute temperatures during crystal growth in watersaturated systems are lower. This will lower the nucleation
and crystal growth rates, as both are thermally activated
processes and the rates depend on absolute temperature
[see equations (3) and (4)]. These two effects (lower
viscosity and lower absolute temperature) may well nearly
cancel each other and lead to the invariability of the
amplitudes discussed above.
The ratios Ti/Tm and Tu/Tm, which clearly depend on
water content (see discussion above) have also been
assumed to be constant. The calculations presented have
shown for decreasing values of Ti/Tm and Tu/Tm a slight
increase of the relaxation time (see Fig. 4a, circles) and
therefore incorporating a variable Ti/Tm and Tu/Tm will
not change the general constancy of the relaxation time.
The detailed effect of H2O on the system may, however,
vary from mineral to mineral depending on the complexity of the crystallizing structure (Sekine & Wyllie,
1983), but this is beyond the scope of this study as we
consider only one phase. However, I expect the results
of generally short relaxation times not to change much,
because of the near invariability of the amplitudes of the
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nucleation and crystal growth rate function discussed
above.
Finally, it was assumed that the devolatilization does
not affect the temperature of the melt. Sahagian &
Proussevitch (1996) have shown theoretically that oversaturation degassing at 10–20 bar can cause cooling of
an albite melt by 8 K/wt % of exsolved water. In
terms of this model such behaviour would increase the
undercooling below the liquidus after degassing and
therefore accelerate nucleation and growth, which would
result in a slight reduction of the relaxation time. The
cooling because of exsolution of volatiles may also partially compensate for the increase in temperature because
of the release of latent heat upon the burst of crystallization following the perturbation of the liquidus. In
the standard model (Figs 2 and 3) an increase in the
melt’s temperature of 0·006 dimensional temperature
units (equivalent to ~6°C using typical values) occurs in
response to the rapid solidification following the liquidus
perturbation of ~50 K.
Comparison with observations and
applications
The model calculations were intended to investigate the
general behaviour of a crystallizing system in response
to a liquidus temperature perturbation and thereby serve
as a guide towards the possible crystallization features
during shallow level processes rather than allow a direct
quantitative application of the results to rocks. The latter
task is beyond the scope of this paper, as it involves a
model for the crystallization of a multicomponent melt
including several degrees of freedom, which makes it
rather difficult to extract some general features. These
can, however, easily be extracted from the calculations
presented in this study, and in the following I will first
discuss some examples of enhanced crystallization (i.e.
sudden increase in crystal size and a burst of nucleation)
as a result of liquidus temperature changes and then I
will focus on some consequences of the calculation for
real rocks.
Observations
Crystallization caused by exsolution of volatiles is a
common observation in, for example, crystallization textures of obsidian dome systems, where the loss of H2O
results in microlite crystallization (Swanson et al., 1989),
or in acid shallow level intrusives, where the ubiquitous
granophyric texture may actually be a quench texture
brought about by abrupt volatile loss (M. O’Hara, personal communication, 1997). In both cases an abrupt
change of the liquidus temperature leads to an increased
nucleation and a rapid growth period, which is exactly
what has been observed in the model calculations. The
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disequilibrium period in both cases is, according to the
model calculation, extremely short, with the possibility
of developing a dendritic texture (high crystal growth
rates). The breakdown of amphibole phenocrysts of the
post 18 May eruption of Mount St Helens, where the
amphibole phenocrysts are surrounded by reaction rims
of small plagioclase, pyroxene and Fe–Ti oxides crystals,
is, according to Rutherford & Hill (1993), generally
agreed to result from the decrease in H2O content of the
coexisting melt produced during magma ascent, but other
explanations are also possible (Rutherford & Hill, 1993).
This feature may be explained also through increased
nucleation as well as crystal growth because of the increased undercooling. The harrisitic textures observed in
Rhum, Scotland, are also assumed to be a result of a
sudden supersaturation of the melt during the crystallization process (Donaldson, 1982). However, there
are many possible reasons for this supersaturation, and
two of the most plausible reasons are magma mixing (hot
primitive magma mixing with more evolved basaltic
magma, e.g. Huppert & Sparks, 1980) or exsolution of
volatiles (Donaldson, 1982). Both of these processes involve again an abrupt increase in undercooling associated
with a perturbation of the liquidus temperature. The
period of rapid crystal growth may explain the skeletal–
dendritic olivine growth observed in the harrisitic textures
found in the Rhum rocks.
Applications
In the model calculations it was found that small perturbations of the liquidus temperature are compensated
only through crystal growth, whereas large perturbations
cause both nucleation and crystal growth. As explained
above, a gradual change of the liquidus temperature can
be thought of as being a series of small perturbations,
i.e. a gradual change of the liquidus temperature will
only be compensated through crystal growth. In terms
of rock texture, I expect small or gradual changes of the
liquidus temperature to lead to textures with large, in
some circumstances possibly even dendritic crystals (see
the example above for the Rhum intrusion), whereas
large perturbations will result in a fine-grained groundmass with some large crystals (e.g. phenocrysts) swimming
in it (see example for microlite crystallization above).
While investigating the dependence of the relaxation
time on the Avrami number, a linear relationship between
these two quantities [see equation (13) and Fig. 4b] was
found. As indicated by the fit (13), the slope of the line
is nearly –1 and therefore (13) can be simplified to
1·2×tkin=
1·2
≈tr
(rImU 3m)0·25
using (7) and (10). tr is the relaxation time in seconds.
This relationship clearly shows that the relaxation time
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CRYSTALLIZATION IN RESPONSE TO LIQUIDUS CHANGES
is controlled only by kinetics and is independent of the
actual size of the intrusion, provided Av > 5 × 104. Using
typical values (r = 1, Im = 100–106 m–3s–1 and Um =
10–10–10–6 m/s) gives relaxation times between about 1
month and minutes. These times are significantly smaller
than typical cooling times of systems characterized by
larger values of Av. Large igneous systems with lower
SiO2 content (e.g. lava lakes) are typically characterized
by large Avrami numbers, whereas smaller, SiO2-rich
systems (e.g. obsidian domes) are characterized by smaller
Avrami numbers, where the simple relationship (14) may
not hold.
Another important observation in the model calculations is that the temperature of the melt does not
necessarily have to drop to keep crystallization going.
This is an interesting aspect of the calculation, as it
suggests that, while crystallizing and exsolving volatiles,
the melt may actually not cool, as release of latent heat
increases the temperature of the melt and exsolution of
volatiles would decrease the temperature of the melt.
Spinning this concept further, in a mature magmatic
feeder system the temperature of a rising magma may
therefore remain nearly constant as the melt gradually
solidifies [see also Tuttle & Bowen (1958)].
CONCLUSIONS
In summary, the modelling has shown that one of the key
observables, the relaxation time, is nearly independent of
the parameters varied in the model, and the following
conclusions for the behaviour of magma during shallow
level processes can be drawn from the results:
(1) A magma that is slightly crystallized can accommodate an increase of the liquidus temperature
caused by devolatilization extremely rapidly, probably in
significantly less than 3% of the time it takes to crystallize
about half of the melt. This means that melts on their
way towards the Earth’s surface can accommodate perturbations of the thermodynamic equilibrium very rapidly
and in response change flow behaviour, eruptability, and
hazardous potential at the same rate.
(2) The relaxation time of the system is nearly independent of the amplitude of the liquidus temperature
perturbation, because of the exponential nature of the
nucleation and crystal growth rate function.
(3) A system exsolving volatiles is crystallizing, and in
doing so the actual temperature of the system is not
necessarily decreasing but may remain constant or even
increase. Therefore a melt moving up a feeder system
may crystallize because of an increase in the liquid’s
temperature, but at the same time it may also maintain
its temperature. This study allows a quantitative assessment of this process.
ACKNOWLEDGEMENTS
I thank J. Gardner for valuable discussions, R. Werner
for reading an earlier draft of the manuscript, and T. Hansteen and P. Sachs for pointing out some references.
Many thanks go to M. O’Hara, M. Carrol and A. Klügel,
as well as an anonymous referee, for very helpful and
insightful comments. Part of this research was funded
through the European Union under ENV4-CT96-0259.
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