A spreadsheet to model magma crystallization
Although the concept of crystal fractionation is mentioned frequently in books and papers, the process is less well-understood (and less well-defined) than one might imagine, given its importance in petrology over the past century. In particular, students have a difficult time understanding how crystallization of a given set of phases affects the composition of the liquid from which they are being crystallized (keeping in mind that just crystallizing stuff isn’t by itself crystal fractionation).
To help get this across I developed a simple spreadsheet that allows one to “crystallize” by typing the initial of a given phase (e.g., p for plagioclase) into a 10x10 grid of liquid (l) cells. The spreadsheet keeps track of the composition of the liquid, the bulk composition of the crystals, and how much material remains to be crystallized. The initial setup looks like this:
The pie charts below the “magma chamber” show the modal compositions of the magma, and the remaining phases available to crystallize. The oxide-oxide plots show the bulk composition of the starting material (blue; never moves), the liquid (here hidden behind the bulk composition because they are the same) and the bulk composition of the crystals (undefined here, since there aren’t any). Here is what things look like after a handful of cells have been filled with plagioclase and mafic minerals (m stands for hornblende, biotite, magnetite, augite,…):
Now the liquid (red) is separated from the bulk composition, and the crystals (yellow) are on the graphs for 3 of the 4 plots. We can see that the crystals are much lower in SiO2 than the bulk composition, and as a consequence the liquid is heading for higher SiO2.
The small pale blue dots on the plots are data from the Lassen and Shasta volcanic centers in California; these give the trends for typical calc-alkaline, subduction-related assemblages. We can see that our crystals are lower in CaO and higher in FeO+MgO than this trend (and way too high in K2O, but we’ll worry about that later). We can improve on the fit by crystallizing more plagioclase and (relatively) less mafic material as we go along.
Here is what the system might look like if we crystallize (by typing) minerals in proportions that mimic real dacites (see inset photomicrograph):
The match between liquid and crystals and the Lassen-Shasta trend is much better. At this point one might ask the students if the crystals (the “cumulate”, although that term should be avoided…) should match the calc-alkaline trend. Lots of good material for discussion there.
At this point, near the rheologic lock-up point, only a little quartz, and no sanidine, has been crystallized. Taking the magma to full crystallization is a revealing exercise.
Pedagogical Value
This exercise helps students to understand the compositional consequences of crystallizing minerals from magmas. I’ve found it difficult to get this across, e.g., if you crystallize a lot of ferromagnesian minerals, then the liquid rapidly gets depleted in FeO and MgO, and SiO2 goes up rapidly. This exercise makes the point that by the time you’ve reached 50% or so crystals, the liquid is high-silica rhyolite.
The exercise also is an example of the lever rule in action; the bulk composition is always on the line between the bulk crystals and the liquid, and the line segment lengths can be used to calculate the proportions of each.
I originally developed this exercise in part to show that early crystallization of K-feldspar (to make megacrysts free-swimming in abundant magmatic liquid) doesn’t work, and that is a good student exercise, too.
If nothing else, this is a graphical exercise in differentiation and the lever rule, and less messy than the very successful M&M experiment that Karl Wirth devised so many years ago.
Implementing the Exercise
Here are some notes on using this exercise.
The goal is to virtually crystallize the virtual magma chamber in the spreadsheet by typing the initial of a mineral in a space currently occupied by liquid (l). You must obey mass balance (i.e., you can’t use more of a mineral than the bulk composition contains), and you should follow the general crystallization sequence that petrologists have worked out over the last century or so.
Your job is to further crystallize the magma by typing a letter corresponding to one of the remaining phases (e.g., “s”) in place of a liquid cell (“l”). Some things to keep in mind:
You might want to keep both the liquid and bulk crystals near the fields defined by the Shasta-Lassen rocks, as these would be the liquid and the cumulate left behind, and hence observable and represented in the rock record. Or ignore those sequences and see what you make.
See if you can make any sanidine megacrysts, as this magma is the same composition as megacryst-bearing granodiorites.
More detail:
There are 4 minerals, plagioclase (p), quartz (q), sanidine (s), and mafics (m), plus liquid (l). Mafics are a mash-up of biotite, hornblende, pyroxenes, and magnetite. The particular mix I used is based on modal proportions in Sierran granites, and is thus rather rich in biotite (and consequently K2O).
Spreadsheet is locked so that the user can only change cells in the box.
Crystal compositions used in the calculations are in the gray "other stuff" below the good stuff on the spreadsheet. Feel free to modify.
There are 100 cells, each of which were occupied with liquid at the start. I’ve provided two versions of the spreadsheet: one with all liquid, and one in which about half the cells are pre-filled, mostly with plagioclase and mafic minerals in accord with the mineralogy of crystal-rich dacites such as the one in the photo (see also Uturuncu, Cerro Galan, etc.).
Table in the center keeps track of what has crystallized from the magma and what is remaining to be crystallized.
Plots on the right show the bulk composition in the center and the liquid and crystals at either end of the tie line (i.e., the lever rule).
Blue dots in background of these plots are points from the Lassen Peak and Mount Shasta volcanic fields in northern California—good arc-related volcanic rocks for reference.
Table at bottom center gives the current compositions of crystals and liquid, plus the bulk composition of the system.
Pie charts are a graphical look at the magma and the remaining mineral phases to be crystallized.
There is no size scale in this exercise, but the crystal aggregates in the box are meant to mimic crystals in magma that crystallizes to make a typical granodiorite, so they would be mm-scale and the box might be a cm or two on a side. Feel free to put your own scale interpretation on it.
Feel free to unlock the sheet and monkey around with the bulk composition and the compositions of the minerals. The compositions I used were chosen to resemble those of continental arc granodiorites and dacites.
Comments, corrections, suggestions greatly appreciated!