Transcript Poster_Graphics

The Crystal Size Distribution Intercept vs. Slope
Relationship: A Numerical Simulation
Ronald G. Resmini
The Boeing Company
Chantilly, Virginia 20151
[email protected]
A crystal size distribution (CSD) is a quantitative representation of the population of crystals
in a rock. Presented as a spectrum of the natural log of crystal population density, ln(n), vs.
crystal size, L, natural CSDs for minerals in igneous rocks are generally linear with negative
slope in ln(n) vs. L space. A plot of the intercepts of a suite of CSDs vs. the corresponding
slopes of the same CSDs also yields a linear trend with negative slope. Suites of CSDs of
minerals in igneous rocks from different settings show this trend though with varying ranges
and magnitudes of CSD slope and intercept. A possible mechanism for this intercept vs.
slope relationship is presented. Numerically simulated CSDs are generated for successively
increasing depths within a solidifying infinite half-sheet of magma. Cooling rate is calculated
analytically with an expression that incorporates latent heat of crystallization. The CSDs are
generated using the log(nucleation rate) vs. log(cooling rate) kinetic relationship of Cashman
(1993) combined with a mass balance-based growth rate that is inversely proportional to the
amount of surface area on previously nucleated and growing crystals for deposition of solids.
Thus, the amount of solids crystallized as a function of time is derived from the cooling rate
expression; the number of crystals is determined by the nucleation rate. The intercept vs.
slope relationship results when the intercepts of the individual numerical CSDs are plotted
against the corresponding slopes. The numerical CSDs show progressively lower intercepts
and lower magnitudes of the CSD slope with increasing distance from the half-sheet/wallrock
contact. The numerically derived trend is similar to those obtained from CSDs of natural
rocks. Implications of the trend for constraining cooling history and crystal nucleation kinetics
are presented.
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A Typical CSD
AT-67
ln(n) (no./cm4)
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
0.0
0.5
1.0
1.5
2.0
L (mm)
A CSD of plagioclase in a high-alumina basalt from Atka Island, Alaska,
from Resmini (1993). CSDs may be characterized by their slope and
intercept. Numerous CSDs from a suite of samples may be represented
as points on a plot of CSD intercept vs. slope as shown next (pl. 3).
2
The CSD Intercept vs. Slope Relationship
Intercept (no./cm4)
20.0
17.5
15.0
Dome Mountain, NV
(plagioclase)
Atka, AK
(plagioclase)
12.5
10.0
Crater Flat, NV
(olivine)
7.5
5.0
-20.0
-15.0
-10.0
-5.0
0.0
Slope (mm-1)
Intercept and slope values for numerous CSDs (from Resmini, 1993).
Note the linear trends. The modal abundance of plagioclase in the
Dome Mtn. rocks is ~6.5 vol.%.
3
Method
• Build CSDs for increasing distances from the contact of the
Jaeger (1957) infinite-half sheet of magma (a sill proxy) (eq. 1);
CSDs generated for 1, 5, 50, 100, and 500 meters from contact
• Use the Cashman (1993) nucleation rate, I, expression (eq. 2)
• Crystal growth rate, G, is by the “Distribution of Mass” method
(see next slide)
Cooling rate expression from Jaeger (1957) for an infinite half-sheet of magma:
T 1  T2  T1
x '  3 2
x'
2
 
exp( (
1 t
1 ) )
t 2 1  erf (b) (2 ) 2 
2(  2 t ) 2
(of the liquid) eq. 1
Crystal nucleation rate (I) relation of Cashman (1993):
T
T m
log( I)  log( I' )  m( ) or I  I' ( )
t
t
Symbols and values are given in the Symbol Table, below.
eq. 2
4
Method
Growth Rate By “Distribution of Mass”
As indicated previously, nuclei are activated according to eq. 2 on pl. 4. Crystal growth,
however, proceeds by applying the solids formed cooling (minus the amount due to nucleation) to
each pre-existing nucleus and crystal. The amount of solids formed is easily calculated with the
linear fraction of solids (f) vs. temperature relationship within the solidification interval used in the
modified heat capacity method employed by Jaeger (1957). Thus, a single growth rate is
calculated such that if every pre-existing crystal, crystallite, or nucleus in the system grows at that
rate, all newly formed solid (again, minus that amount of solid due to new nucleation) is consumed
in crystal growth (referred to as the "distribution of mass" growth rate mechanism). All particles
grow at the same rate at each time-step; growth rate dispersion or size-dependent growth rate
mechanisms are not employed. The model is thus a numerical simulation that tracks batches of
spherical crystals as they nucleate and grow during the solidification interval. The spherical crystal
assumption, also made by Marsh (1998), is adequate for the current analysis.
Note that the temperature vs. time information and the linear fraction of solids (f) vs.
temperature relationship within the solidification interval allows a calculation of the amount of
solids precipitated as a function of time which further facilitates the calculation of a growth rate
based on, and constrained by, the consumption of newly precipitated solids (minus the amount of
solids consumed in nucleation) and the extant crystal population.
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Method
CSDs are thus generated for various positions within an infinite
half-sheet of magma. All CSDs are calculated assuming
complete solidification (i.e., 100% solids). From each CSD, the
slope and intercept parameters are extracted and subsequently
plotted.
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ln(n)
Wallrock
Intercept
Slope
ln(n)
Contact
L (mm)
L
3
Magma
L
Intercept
ln(n)
1
ln(n)
Infinite Half-Sheet of Magma
ln(n°)
Slope
L
7
20.0
20.0
18.0
18.0
ln(n), no./cm4
ln(n), no./cm4
Results: CSDs Generated From The Model
5 Meters
16.0
14.0
12.0
10.0
8.0
6.0
4.0
0.00
0.25
0.50
0.75
1.00
1.25
14.0
12.0
10.0
8.0
6.0
4.0
0.00
1.50
50 Meters
16.0
0.25
L (mm)
0.50
0.75
1.00
1.25
1.50
L (mm)
Depth
Slope
Intercept
(meters)
(mm-1)
(no./cm4)
1
5
50
100
500
-17.02
-12.82
-7.87
-6.99
-4.53
19.43
18.29
16.22
15.59
13.75
Typical model CSDs and a table of all CSD parameters.
8
Typical Model CSD Evolution
The CSD Located 5 Meters From the Contact
20
16
19
Values refer to
percent solids.
ln(n), no/cm4
14
12
10
8
100%
6
10%
5%
50%
4
2
0.0
0.2
0.4
0.6
L (mm)
0.8
1.0
CSD Intercept, no/cm4
18
18
17
16
y = 1.1737Ln(x) + 13.37
15
2
R = 0.9954
14
13
12
1.2
0
20
40
60
80
100
Percent Solids
The evolution of the CSD located 5 m from the sill contact. Note that
CSD slope is constant throughout the solidification interval and that CSD
intercept evolves vs. percent solids as shown. This behavior is important to
note because in subsequent plates, model results for 100% solids will be
compared to natural CSDs calculated for minerals with significantly lower
modal abundances.
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The CSD Intercept vs. Slope Relationship
20.0
Model CSDs
Intercept (no./cm4)
1m
5m
17.5
50 m
15.0
100 m
500 m
12.5
10.0
Atka, AK
Dome Mountain, NV
7.5
5.0
-20.0
Crater Flat, NV
-15.0
-10.0
-5.0
0.0
Slope (mm-1)
The plot of plate 3 now with the model CSDs included. The modal
abundance of plagioclase in the Dome Mtn. rocks is ~6.5 vol.%
whereas the model CSDs are for 100% solids. The model CSDs
define a trend similar to that of the natural CSDs.
10
Offset of Model CSD Trend Due to Higher Modal Abundance
12.5
10.0
7.5
5.0
-20.0
-15.0
Increasing % Solids
15.0
Model CSDs
Increasing Time
17.5
Constant CSD Slope
Intercept (no./cm4)
20.0
-10.0
-5.0
0.0
Slope (mm-1)
As indicated in plate 9, a CSD evolves throughout the solidification interval
with constant slope. Thus, a point on a plot of CSD intercept vs. slope
evolves in time (i.e., as a function of increasing % solids) by “moving”
vertically along the intercept axis, as shown. Intercept value maps modal
abundance.
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Discussion
• The model intercept vs. slope trend shows concavity; the natural sample
trends are apparently linear. The scatter inherent in the natural data may
be masking a curved trend.
• Though not shown here, different values of I’ and m in eq. 2 of plate 4 will
yield suites of CSDs with trends different from that shown in plate 10.
Thus, the definition of intercept vs. slope trends for suites of samples
may constrain nucleation rate kinetic parameters.
• The intercept vs. slope trend of the model CSD data indicates that lower
overall CSD intercepts and low absolute values of the slope are due to
longer, slower cooling.
• Thus, in addition to providing information on nucleation kinetics, the CSD
intercept vs. slope relationship for a suite of samples may bound cooling
times. Such bounds may then be related to magmatic system size.
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Summary and Conclusions
Numerically simulated CSDs are generated for successively increasing depths (1, 5, 50,
100, and 500 meters) within a solidifying infinite half-sheet of magma. Cooling rate is
calculated analytically with an expression that incorporates latent heat of crystallization
(Jaeger, 1957). The CSDs are generated using the log(nucleation rate, I) vs. log(cooling
rate, T/t) kinetic relationship of Cashman (1993) combined with a mass balance and
extant crystal population balance-based growth rate mechanism ("distribution of mass").
The CSD intercept vs. slope relationship results when the intercepts of the individual
numerical CSDs are plotted against the corresponding slopes. The numerical CSDs show
progressively lower intercepts and lower magnitudes of the CSD slope with increasing
distance from the half-sheet/wallrock contact. The numerically derived trend is similar to
those obtained from CSDs of natural rocks. The intercept vs. slope trend of the model
CSD data indicates that lower overall CSD intercepts and low absolute values of the slope
are due to longer, slower cooling. Thus, the interplay between I vs. T/t, the distribution
of mass growth rate mechanism, and the mass balance imposed by crystallizing solids
produces the trends evident in a plot of CSD intercept vs. slope.
13
Symbol Table
14
Acknowledgements
Partial funding for this work provided by The Boeing Company.
References
Cashman, K.V., (1993). Relationship between plagioclase crystallization and
cooling rate in basaltic melts. Contrib. Mineral. Petrol., v. 113, pp. 126-142.
Jaeger, J.C., (1957). The temperature in the neighborhood of a cooling intrusive
sheet. Am. J. Sci., v. 255, pp. 306-318.
Marsh, B.D., (1998). On the interpretation of crystal size distributions in magmatic
systems. J. Petrol., v. 39, no. 4, pp. 553-599.
Resmini, R.G., (1993). Dynamics of magma within the crust: A study using
crystal size distributions. Ph.D. Dissertation, Johns Hopkins University,
329 pp.
Additional Information
Pre-prints of a manuscript currently in review at the Journal of
Volcanology and Geothermal Research are available below.
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