1

Towards an Analytical Expression for the Formation of Crystal SizeDistributions (CSDs) in Closed Magmatic Systems

Ronald G. Resmini

The Boeing Company, Chantilly, Virginia 20151 v: (703) 735-3899, f: (703) 735-3305

e: ronald.g.resmini@boeing.com

A model for the formation of crystal size distributions (CSDs) in closed magmatic systems such as sills has been presented [1]. The model generates CSDs and properties of suites of CSDs observed in natural rocks. It is, however, a numerical simulation because of the intractable form of the governing differential equation. Ideally, an analytical expression would facilitate a deeper understanding of the generation of CSDs in closed igneous systems, an important aspect of igneous petrogenesis. The numerical simulation in [1] essentially solves the batch population balance equation (BPBE), a partial differential equation (PDE) describing the formation of CSDs in closed or batch systems. In the model, crystal growth rate, G, is inversely proportional to the second CSD moment (a measure of total surface area in a crystal population and an integral expression) and directly proportional to the amount of mass available for solidification provided by an energy balance expression incorporating latent heat. Nucleation rate is a function of cooling rate. Analytical intractability, and thus the need for numerical methods, is presented by the integro-partial differential equation posed by G being inversely proportional to the second CSD moment in the BPBE. However, the model also facilitates tracking the evolution of the CSD moments (various metrics of the total crystal population) with time. A plot of the second CSD moment vs. the third (a measure of the total amount of precipitated solids) is linear throughout most of the solidification interval. This key relationship suggests substituting the amount of solids formed with time, a quantity more easily calculated, for the time evolution of the total surface area of crystals with time. This substitution converts the intractable integro-partial differential BPBE to an analytically tractable PDE and preserves the inverse dependence of G on the second CSD moment. The linear relationship between the second and third CSD moments is presented. The use of this important relationship to convert the integro-differential BPBE to a normal PDE is then shown. A potential solution is discussed as is its limitations and utility for modeling igneous petrogenesis.

[1] Resmini, R.G., (2001), Eos. Trans. AGU, 82(20), pp. S432.

2

Introduction

• Resmini (1993, 2000, 2001, and 2002) has shown that the batchpopulation balance equation and associated expressions (see slide 3)generate CSDs and properties of CSDs observed in natural rocks.

• The equations on slide 3 must be solved numerically.

• However, it can be shown that the 2nd and 3rd CSD moments varylinearly in time during the formation of a crystal population.

• This is important because the 2nd CSD moment may be cast by aterm that’s easily, analytically calculated: the 3rd CSD moment isrelated to the amount of solids present in the crystallizing system.

• It is then possible to analytically calculate a CSD for a closed system.

• All of the calculations are based on the analytical expression ofJaeger (1957) for the cooling and solidification of an infinite half-sheetof magma with latent heat.

3

00,Ln initial condition

G

It,0n boundary condition, where:

0

L

n

dLt,LnLt

m

t

n

0

2

0

2 dLt,LnLt

mG

0L

nG

t

n

m''

t

TIIgivesthis

t

Tm)Ilog()Ilog(

...from Cashman (1993)

)))t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

t

T 2

22

12

21

23

21

(cooling rate of the liquid...with latent heat)

For an infinite half-sheet of magma; from Jaeger (1957)

Batch PopulationBalance Equation

(BPBE)

Integro-PartialDifferential Equation

BTW...This isAnalyticallyIntractable!

(See slide 15 for definition of all symbols.)

4

The integro-partial differential equation (batch population balance equation)

on slide 2 is intractable.

The BPBE may be cast as a set of nonlinear ordinary differential equations

and solved numerically for moments of the CSD vs. time (Resmini, 2002).

This yields the important result that the 2nd and 3rd CSD moments vary linearly.

This is shown on slide 5 and is from Resmini (2002).

This is important because the 2nd CSD moment (intractable term in the integro-

partial differential equation) may be cast by a term that’s easily, analytically calculated:

the 3rd CSD moment—which is related to amount of solids present

in the crystallizing system.

5

y = 130.71x

R2 = 0.9999

0

5

10

15

20

25

30

35

0.00 0.05 0.10 0.15 0.20 0.25 0.30

3rd CSD Moment, cm3/cm3

2nd C

SD

Mom

ent,

cm

2/c

m3

( to amount of mass present)

( t

o t

otal

am

ount

of

surf

ace

area

)

Increasin

g Time

Key Relationship: The 2nd CSD Moment Varies Linearly with the 3rd Moment

Amount of mass present is easily calculated from the expression of Jaeger (1957).

6

Can Now Recast Integro-Partial Differential Equation as Follows:

m)t

T('IIor)

t

T(m)'Ilog()Ilog(

)))t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

t

T 2

22

12

21

23

21

(of the liquid)

324054.12)t(2

'xerf

)b(erf1TT

)b(erf1)b(erfTT

640.23

640.24240))

)t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

T

m1G

21

21

23

21

2

1221

2

22

12

00,Ln ...but the new boundary condition, n(0,t) = I/G, becomes:

324054.12)t(2

'xerf

)b(erf1

TT

)b(erf1

)b(erfTT640.23

640.24240))

)t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

T

m1

)))t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1'I

t,0n

21

21

23

21

21

23

21

2

1221

2

22

12

m

2

22

12

0L

nG

t

n

where G is now given as

(and it’s not an integral):

The initial condition is still:

7

A Bit More About Recasting G...

Recasting G is based on the Jaeger (1957) expression for temperature vs. timein a cooling and solidifying half-sheet of magma—with latent heat:

2

1

)t(2

'xerf

)b(erf1

TT

)b(erf1

)b(erfTTT

2

1221Liquid

This expression is based on a linear relationship between the amount of solidsprecipitated vs. temperature (within the solidification interval).

Thus, temperature may be easily used to estimate amount of solids present withinthe solidification interval—which in turn is linearly related to the 2nd CSD moment.

Note the expression for TLiquid, above, in the equation for G on slide 6—

with some terms for the linear relationship between the 2nd and 3rd CSD moments.

8

The Solution:

324054.12)t(2

'xerf

)b(erf1TT

)b(erf1)b(erfTT

640.23

640.24240))

)t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

T

m1G

21

21

23

21

2

1221

2

22

12

{ { { { {{

{ { { { {a q c h d/tf

g c sqrt(d)/sqrt(t) j

e

j't

derfcgf

e)'t

dexp('thcqa

)'t

dexp('thcq'I

t,Ln

21

21

m

23

23

2

heqa

dfC

cg

cfj10

inverf

d't

1

df

gtd

erfcfjlogheqa

LC1

...and the values of the other variables are:

9

y = -47.761x + 21.194

R2 = 0.9983

0

5

10

15

20

25

0.00 0.10 0.20 0.30 0.40 0.50 0.60

L (mm)

ln(n

), n

o./c

m4

The Analytical CSD

All values used to generate this CSD are the same as those used togenerate the CSD from Resmini (2002) and shown next...

CSD, after complete solidification, for a position 1 meterfrom the Jaeger (1957) half-sheet/wallrock contact

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Numerical CSD

y = -17.055x + 19.445

R2 = 0.9855

0

5

10

15

20

25

0.00 0.20 0.40 0.60 0.80 1.00 1.20

L (mm)

ln(n

), n

o./c

m4

CSD, after complete solidification, for a position 1 meterfrom the Jaeger (1957) half-sheet/wallrock contact.

11

Discussion

• The analytical CSD resembles those observed in natural rocks.

• It is not, however, equivalent to that generated by the numerical model.

• This is because of the initial condition of n(0,L) = 0.

• Note on slide 12 that the linear relationship between the 2nd and 3rd

moments does not hold at the very beginning of solidification.

• The linear relationship is established very early in the solidification interval.

• The problem that should be solved is that given on slide 13; notethe initial condition that should be employed; f(L) = n(L) early inthe solidification interval.

12

3rd CSD Moment

2n

d C

SD

Mo

me

nt

y = 130.71x

R2 = 0.9999

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.000 0.005 0.010 0.015 0.020 0.025

...it’s not linear at the very beginningof the solidification interval...

y = 130.71x

R2 = 0.9999

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

0.015 0.030 0.045 0.060 0.075 0.090 0.105 0.120

3rd CSD Moment2

nd

CS

D M

om

en

t

...quickly becomes linear...

13

Should really solve:

m)

t

T(' I I or )

t

T( m )' I log( )I log(

0L

nG

t

n

) ))t ( 2

' x( exp( t

) (

' x

) b( erf 1

T T

2

1

t

T2

2 2

1 2

21

23

21

(of the liquid)

324054.12)t(2

'xerf

)b(erf1

TT

)b(erf1

)b(erfTT640.23

640.24240))

)t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

T

m1G

21

21

23

21

2

1221

2

22

12

324054.12)t(2

'xerf

)b(erf1

TT

)b(erf1

)b(erfTT640.23

640.24240))

)t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1

T

m1

)))t(2

'x(exp(t

)(

'x

)b(erf1

TT

2

1'I

t,0n

21

21

23

21

21

23

21

2

1221

2

22

12

m

2

22

12

L f 0, L n

Note Different Initial Condition ...but same B.C., n(0,t) = I/G :

14

Summary and Conclusions

• An analytical model for the formation of crystal size distributions (CSDs) inclosed magmatic systems such as sills has been presented.

• The model is based on noting (from previous modeling efforts) that the 2nd

and 3rd CSD moments vary linearly in time during the formation of acrystal population.

• This important relationship facilitates the conversion of an intractable integro-partial differential equation (the batch population balance equation)to a tractable form.

• The model produces CSDs similar to those observed in natural rocks.

• However, the model must incorporate non-zero initial conditions (i.e.,n(0,L) = f(L)); this is a direction of future work.

• An analytical expression for the generation of CSDs can facilitate closercoupling of quantitative petrography with geochemical thermodynamicmodels of igneous petrogenesis.

15

Symbol Table

16

References Cited

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. Resmini, R.G., (2002). Modeling crystal size distributions with a moment-transformed

batch population balance equation. EOS, Trans., A.G.U.; v. 83, no. 19, p. S380. Resmini, R.G., (2001). The crystal size distribution (CSD) intercept vs. slope relationship:

a numerical simulation. EOS, Trans., A.G.U., v. 82, no. 20, p. S432. Resmini, R.G., (2000). Numerical simulation of crystal size distributions (CSDs) in sills.

EOS, Trans., A.G.U., v. 81, no. 19, p. S435. Resmini, R.G., (1993). Dynamics of magma within the crust: A study using crystal size

distributions. Ph.D. Dissertation, Johns Hopkins University, 329 p.