1

The Crystal Size Distribution Intercept vs. SlopeRelationship: A Numerical Simulation

Ronald G. Resmini

The Boeing CompanyChantilly, Virginia 20151

ronald.g.resmini@boeing.com

A crystal size distribution (CSD) is a quantitative representation of the population of crystalsin 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 negativeslope in ln(n) vs. L space. A plot of the intercepts of a suite of CSDs vs. the correspondingslopes of the same CSDs also yields a linear trend with negative slope. Suites of CSDs ofminerals in igneous rocks from different settings show this trend though with varying rangesand magnitudes of CSD slope and intercept. A possible mechanism for this intercept vs.slope relationship is presented. Numerically simulated CSDs are generated for successivelyincreasing depths within a solidifying infinite half-sheet of magma. Cooling rate is calculatedanalytically with an expression that incorporates latent heat of crystallization. The CSDs aregenerated 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 theamount 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 rateexpression; 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 plottedagainst the corresponding slopes. The numerical CSDs show progressively lower interceptsand lower magnitudes of the CSD slope with increasing distance from the half-sheet/wallrockcontact. The numerically derived trend is similar to those obtained from CSDs of naturalrocks. Implications of the trend for constraining cooling history and crystal nucleation kineticsare presented.

2

2.01.51.00.50.00.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

L (mm)

ln(n

) (n

o./c

m4)

AT-67

A Typical CSD

A CSD of plagioclase in a high-alumina basalt from Atka Island, Alaska,from Resmini (1993). CSDs may be characterized by their slope andintercept. Numerous CSDs from a suite of samples may be representedas points on a plot of CSD intercept vs. slope as shown next (pl. 3).

3

5.0

7.5

10.0

12.5

15.0

17.5

20.0

-20.0 -15.0 -10.0 -5.0 0.0

The CSD Intercept vs. Slope Relationship

Dome Mountain, NV(plagioclase)

Crater Flat, NV(olivine)

Atka, AK(plagioclase)

Slope (mm-1)

Inte

rcep

t (n

o./c

m4)

Intercept and slope values for numerous CSDs (from Resmini, 1993).Note the linear trends. The modal abundance of plagioclase in theDome Mtn. rocks is ~6.5 vol.%.

4

Crystal nucleation rate (I) relation of Cashman (1993):

Method

m)t

T('IIor)

t

T(m)'Ilog()Ilog(

Cooling rate expression from Jaeger (1957) for an infinite half-sheet of magma:

Symbols and values are given in the Symbol Table, below.

)))t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

t

T 2

22

12

21

23

21

(of the liquid)

• 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)

eq. 2

eq. 1

5

MethodGrowth 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.

6

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.

7

Wallrock

Magma

Contact

Infi

nit

e H

alf-

Sh

eet

of

Mag

ma

L

ln(n

)

L

ln(n

)

L

ln(n

)

L (mm)

ln(n

)

ln(n°)

Slope

1

Slope

Inte

rcep

t

3

Intercept

8

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Results: CSDs Generated From The Model

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

20.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50

5 Meters 50 Meters

L (mm)L (mm)

ln(n

), n

o./c

m4

Typical model CSDs and a table of all CSD parameters.

Depth Slope Intercept

(meters) (mm-1) (no./cm4)

1 -17.02 19.435 -12.82 18.2950 -7.87 16.22

100 -6.99 15.59500 -4.53 13.75

ln(n

), n

o./c

m4

9

Typical Model CSD EvolutionThe CSD Located 5 Meters From the Contact

y = 1.1737Ln(x) + 13.37

R2 = 0.9954

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Percent SolidsC

SD

Inte

rcep

t, no

/cm

4

2

4

6

8

10

12

14

16

18

0.0 0.2 0.4 0.6 0.8 1.0 1.2

5%10%

50%

100%

L (mm)

ln(n

), n

o/cm

4

The evolution of the CSD located 5 m from the sill contact. Note thatCSD slope is constant throughout the solidification interval and that CSDintercept evolves vs. percent solids as shown. This behavior is important tonote because in subsequent plates, model results for 100% solids will becompared to natural CSDs calculated for minerals with significantly lowermodal abundances.

Values refer topercent solids.

10

5.0

7.5

10.0

12.5

15.0

17.5

20.0

-20.0 -15.0 -10.0 -5.0 0.0

Dome Mountain, NV

Crater Flat, NV

Atka, AK

Model CSDs

The CSD Intercept vs. Slope Relationship

Slope (mm-1)

Inte

rcep

t (n

o./c

m4)

The plot of plate 3 now with the model CSDs included. The modalabundance of plagioclase in the Dome Mtn. rocks is ~6.5 vol.%whereas the model CSDs are for 100% solids. The model CSDsdefine a trend similar to that of the natural CSDs.

1 m5 m

50 m 100 m

500 m

11

5.0

7.5

10.0

12.5

15.0

17.5

20.0

-20.0 -15.0 -10.0 -5.0 0.0

Model CSDs

Incr

ea

sing

Tim

e

Co

nst

ant

CS

D S

lop

e

Incr

ea

sing

% S

olid

s

As indicated in plate 9, a CSD evolves throughout the solidification intervalwith constant slope. Thus, a point on a plot of CSD intercept vs. slopeevolves in time (i.e., as a function of increasing % solids) by “moving”vertically along the intercept axis, as shown. Intercept value maps modalabundance.

Offset of Model CSD Trend Due to Higher Modal Abundance

Slope (mm-1)

Inte

rcep

t (n

o./c

m4)

12

Discussion

• The model intercept vs. slope trend shows concavity; the natural sampletrends are apparently linear. The scatter inherent in the natural data maybe masking a curved trend.

• Though not shown here, different values of I’ and m in eq. 2 of plate 4 willyield suites of CSDs with trends different from that shown in plate 10.Thus, the definition of intercept vs. slope trends for suites of samplesmay constrain nucleation rate kinetic parameters.

• The intercept vs. slope trend of the model CSD data indicates that loweroverall CSD intercepts and low absolute values of the slope are due tolonger, slower cooling.

• Thus, in addition to providing information on nucleation kinetics, the CSDintercept vs. slope relationship for a suite of samples may bound coolingtimes. Such bounds may then be related to magmatic system size.

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Summary and ConclusionsNumerically 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.

14

Symbol Table

15

Acknowledgements

References

Partial funding for this work provided by The Boeing Company.

Cashman, K.V., (1993). Relationship between plagioclase crystallization andcooling rate in basaltic melts. Contrib. Mineral. Petrol., v. 113, pp. 126-142.

Jaeger, J.C., (1957). The temperature in the neighborhood of a cooling intrusivesheet. Am. J. Sci., v. 255, pp. 306-318.

Marsh, B.D., (1998). On the interpretation of crystal size distributions in magmaticsystems. J. Petrol., v. 39, no. 4, pp. 553-599.

Resmini, R.G., (1993). Dynamics of magma within the crust: A study usingcrystal size distributions. Ph.D. Dissertation, Johns Hopkins University,329 pp.

Additional InformationPre-prints of a manuscript currently in review at the Journal ofVolcanology and Geothermal Research are available below.