1 Towards an Analytical Expression for the Formation of Crystal Size Distributions (CSDs) in Closed Magmatic Systems Ronald G. Resmini The Boeing Company, Chantilly, Virginia v: (703) , f: (703) e:

2 Introduction Resmini (1993, 2000, 2001, and 2002) has shown that the batch population 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 2 nd and 3 rd CSD moments vary linearly in time during the formation of a crystal population. This is important because the 2 nd CSD moment may be cast by a term that’s easily, analytically calculated: the 3 rd CSD moment is related 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 of Jaeger (1957) for the cooling and solidification of an infinite half-sheet of magma with latent heat.

3 initial condition boundary condition, where:...from Cashman (1993) (cooling rate of the liquid...with latent heat) For an infinite half-sheet of magma; from Jaeger (1957) Batch Population Balance Equation (BPBE) Integro-Partial Differential Equation BTW...This is Analytically Intractable! (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 2 nd and 3 rd CSD moments vary linearly. This is shown on slide 5 and is from Resmini (2002). This is important because the 2 nd CSD moment (intractable term in the integro- partial differential equation) may be cast by a term that’s easily, analytically calculated: the 3 rd CSD moment—which is related to amount of solids present in the crystallizing system.

5 3 rd CSD Moment, cm 3 /cm 3 2 nd CSD Moment, cm 2 /cm 3 (  to amount of mass present) (  to total amount of surface area) Increasing Time Key Relationship: The 2 nd CSD Moment Varies Linearly with the 3 rd Moment Amount of mass present is easily calculated from the expression of Jaeger (1957).

6 Can Now Recast Integro-Partial Differential Equation as Follows: (of the liquid)...but the new boundary condition, n(0,t) = I/G, becomes: 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. time in a cooling and solidifying half-sheet of magma—with latent heat: This expression is based on a linear relationship between the amount of solids precipitated vs. temperature (within the solidification interval). Thus, temperature may be easily used to estimate amount of solids present within the solidification interval—which in turn is linearly related to the 2 nd CSD moment. Note the expression for T Liquid, above, in the equation for G on slide 6— with some terms for the linear relationship between the 2 nd and 3 rd CSD moments.

8 The Solution: { { { { { { { { { { { a q c h d/t f g c sqrt(d)/sqrt(t) j e...and the values of the other variables are:

9 L (mm) ln(n), no./cm 4 The Analytical CSD All values used to generate this CSD are the same as those used to generate the CSD from Resmini (2002) and shown next... CSD, after complete solidification, for a position 1 meter from the Jaeger (1957) half-sheet/wallrock contact

10 Numerical CSD L (mm) ln(n), no./cm 4 CSD, after complete solidification, for a position 1 meter from 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 2 nd and 3 rd 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; note the initial condition that should be employed; f(L) = n(L) early in the solidification interval.

12 3 rd CSD Moment 2 nd CSD Moment...it’s not linear at the very beginning of the solidification interval... 3 rd CSD Moment 2 nd CSD Moment...quickly becomes linear...

13 Should really solve: (of the liquid) 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) in closed magmatic systems such as sills has been presented. The model is based on noting (from previous modeling efforts) that the 2 nd and 3 rd CSD moments vary linearly in time during the formation of a crystal 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 closer coupling of quantitative petrography with geochemical thermodynamic models of igneous petrogenesis.

15 Symbol Table

16 References Cited