gibbstut(feb, 2001) pt b - rensselaer polytechnic...

Program GibbsTutorial – Part B

by

Frank S. Spear

© 2001 by Frank S. Spear

F. S. Spear Program Gibbs Tutorial 34

Program Gibbs

Tutorial – Part B

Differential ThermodynamicsThe Gibbs Method

Feb, 2001

Frank S. SpearDepartment of Geology, Rensselaer Polytechnic Institute, Troy, New York 12180

[email protected]

Table of Contents

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Starting the program ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35The Main Menu... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Exercise 1: Analysis of a garnet-grade schist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Geologic setting.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Petrography ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mineral composition and zoning.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Exercise 1.A. Thermobarometry.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Exercise 1.B Contouring divariant regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Exercise 1.C P-T paths from zoned garnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Exercise 1.D Forward modeling of garnet growth.. . . . . . . . . . . . . . . . . . . . . . . 50

Exercise 2: P-T paths calculated from zoned garnet: The Tauern Window... . 53Appendix 1. PostScript graphics files.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


Program Gibbs Tutorial – Part B

Differential ThermodynamicsIntroduction

Program Gibbs has two main calculation modes – integrated (Newton’s method)and differential (Gibbs’ method) thermodynamics. Differential thermodynamics mode isdesigned to calculate changes in dependent thermodynamic variables given user-specifiedchanges in independent variables. The choice of which variables are independent is up tothe user, making this a very powerful algorithm for calculating useful petrologic diagrams.

It is important to recognize that virtually all of the types of calculations that can bedone using differential thermodynamics can also be done using the integrated mode. Theonly “catch” is that consistent enthalpies and solution models are required for integratedmode calculations, and these are not available for all phases. The differential mode isuseful, therefore, in the analysis of “real rocks” where the full complexity of the chemicalsystem is to be considered.

Examples of useful calculations to do with differential thermodynamics include1) Contouring P-T diagrams for mineral composition and modes;2) Forward modeling of garnet zoning3) Inverse modeling of P-T paths from garnet zoning4) Whole-rock reaction balancing

There is a philosophical difference in how to view a rock between the differentialand integrated approaches. The integral approach starts with the end-memberthermodynamic data and activity models and attempts to calculate the mineralogy andmineral composition of the rock under study. In a way, the petrologist attempts to force therock into its proper thermodynamic cubbyhole.

The differential approach starts with the question “what changes caused this rock tobe”? That is, what changes in P and T (i.e. ∆ independent variables) produced the observedreaction textures and mineral zoning? In the differential approach we start at a point near themetamorphic peak and attempt to work backwards.

This section provides step-by-step instructions for these specific types ofcalculations. Where possible, an attempt will be made to integrate the calculations into real-world examples so that the reader can gain an appreciation of the types of problems that canbe solved with differential thermodynamics.

Starting the program

When you start program Gibbs you will see a splash screen with an image of theprogram’s namesake. You will then be asked to select a thermodynamic data file from alist.

Thermodynamic data files in Gibbs.fig file:

1 Gibbs Tutorial Thermo.dat


2 HoPo Thermo (12/20/98) 3 S&C(03/2000) Thermo.dat 4 SPaC(4/2000)_Thermo.dat Pick thermodynamic data file to use in this session

Choose the Gibbs Tutorial Thermo.dat file.

Note that the Gibbs Tutorial Thermo.dat file can be used only in differentialthermodynamics mode. HoPo Thermo, S&C, and SPaC can be used in differential orintegral mode.

The Main menu

The main menu will then appear:

*********************************** Thermo file: Gibbs Tutorial Thermo.dat Gibbs' method *********************************** MAIN MENU OPTIONS: 1 = Begin/save problem ----------------------------- 2 = Single steps 3 = Contour X-Y diagrams 4 = Make my grid 5 = Grow Garnet 6 = DiffGibbs (Garnet growth with diffusion) 7 = Whole rock reaction balancing ----------------------------- 8 = Go to global menu 9 = Plotting menu 10 = Plot digitized reactions 11 = Thermodynamic data menu ----------------------------- CHOOSE OPTION

It is first necessary to open a Gibbs input file. In integrated mode (Tutorial- Part A),it was generally easiest to select minerals from the master input files. Since we arecalculating the equilibrium compositions anyway, it doesn’t matter that the input mineralsaren’t exactly the correct compositions.

In differential mode, we generally open a file that contains the compositions that webelieve represent an equilibrium assemblage at some T and P. These data are obtained fromanalysis of mineral compositions using the electron microprobe. It is very important that themineral compositions are chosen carefully because the program will take these as thereference compositions. In effect, this establishes the ∆H of reaction for every independentreaction in the system (although the ∆H values are never explicitly calculated).

Input files can be constructed using any text editor. Also, there is a routine in Gibbsto help construct an input file from natural data (see Appendix for details).


Exercise 1: Analysis of a garnet-grade schistIntroduction

As a means of introducing the calculation possibilities of Program Gibbs, as well asthe philosophy behind these calculations, this first exercise will focus on analysis of agarnet-grade schist from southwestern New Hampshire.

Geologic setting

The sample to be examined (PUT92-C2) comes from a garnet-grade schist from theDevonian Littleton Formation in eastern Vermont. The region is characterized by west-vergent fold and thrust nappes with later doming. The over all metamorphic gradient acrossthis region is inverted with the higher grade metamorphic rocks located in the structurallyhighest nappes. Sample PUT92 comes from one of the lowest nappes and the P-T historyof the sample has been influenced by higher level nappe emplacement.

N. H. and central Mass.

Mesozoic basin rocks

White Mountain magma series

Concord Granite series

New Hampshire magma series

Ord - Sil - Dev cover rocks

Oliverian and dome gneisses

Vt. and western Mass.Connecticut Valley sequence

Late pC - C - Ord rocks

pC of Green Mountains and domes

Explanation

0 10 20Scale (km)

MA

VT

MA

NH

JR

Co

Co

Co

Co

Co

Co

Co

Co

Co

Co

JR

JR

Co

MA

CT

VTNH

MEN

RI

Area of map

Gre

en M

tns

KQM

BG

FM

KD

AD

Ath

D

CD

MD

CY

L

ML

←E

VB

→

←B

HB

→

←M

B→

←O

B→

PUT92


Petrography

The assemblage in sample PUT92-C2 is garnet + biotite + chlorite + muscovite +plagioclase + quartz + ilmenite. Garnet is subhedral and the core contains inclusions of anearlier foliation that is orient roughly the same as the matrix foliation. The dominantfoliation is believed to have formed during formation of early recumbent folds, and thistexture indicates that garnet grew predominantly following foliation formation. Minorflattening of the foliation around garnets suggests that there was some reactivation of thefoliation during thrust emplacement.

Minor retrogression isobserved around the rimof garnet in the form oflate chlorite replacement.The effect of this lateReNTR (retrograde nettransfer reaction) onthermobarometry will bediscussed below.


Mineral compositions and zoning

Garnet from sample PUT92-C2 is zoned with typical garnet-grade zonation.

Fe, Mg and Mn zoning in garnet, samplePUT-92C2, Putney, Vermont. The zoningis typical of prograde growth zonation bythe reactionchlorite + quartz = garnet + H2O.Core-rim zoning are:Alm: 0.59-0.84Prp: 0.028-0.061Sps: 0.22.046Grs: 0.17-0.047

We will use the Gibbs Program todetermine over what P-T paths this type ofzoning is possible.

Click on an image to load TIFF fileinto NIH Image.

FeMg

Mn


Ca zoning decreases from Xgross = 0.17 inthe core to 0.047 on the rim. The change ingrossular content is, in part, a function ofchanging P-T conditions, and, in part, afunction of Rayleigh fractionation. The onlyother Ca-bearing phase in this sample isplagioclase, and growth of garnet requiresconsumption of anorthite component ofplagioclase. As can be seen below,plagioclase is zoned towards more albiticrims.

Ca

Plg

This close view of plagioclase zoningshows the core-rim relations. Themaximum core An content is An(40) andthe minimum rim content is An(04). Notethe irregular (non radial) zoning. This isbecause plagioclase doesn’t homogenize bydiffusion, and consumption of plagioclaseduring garnet growth is not necessarilyradial.


Line traverse across garnet plotted in Excel from data in X-ray map.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Alm

Sps

Grs

Prp

PUT-92 C2


Trace element zoning in garnet

Yttrium in garnet (upper image) can be asensitive monitor of temperature of garnetgrowth and reaction history, especiallywhen xenotime is present in the sample.This garnet has a high Y core (1900 ppm),a zone of lower Y and a higher rim. Thevery upper left margin of the top crystal hasa high Y rim.

Sc (left) increases outward from core andthen decreases again to the rim


Exercise 1.A Thermobarometry

The peak metamorphic temperature of this sample can be inferred from examinationof a petrogenetic grid to be on the order of 475-525 ˚C. Fe-Mg exchange thermometry hasa resolving power of ±25 ˚C, so it is not likely that this estimate can be improved upongreatly. Peak pressure, on the other hand, is not constrained at all by the phase assemblage,and geobarometry is very useful for inferring the crustal thickness at the time ofmetamorphism.

There are two general approaches to thermobarometry currently in practice.• Independently calibrated thermobarometers are equilibria for which a calibration of the

temperature and pressure dependence of the equilibrium constant has been published.Individual equilibrium relations may have several different published calibrations. Forexample there are over ten calibrations of the garnet-biotite Fe-Mg exchangethermometer. Program GTB has been coded to incorporate most of the publishedthermobarometers useful for metamorphic rocks.

• Internally consistent thermobarometers utilize all independent equilibria in ametamorphic assemblage and plot them all on a P-T diagram. If the sample is well-equilibrated, the thermodynamic data set is internally consistent, and activity models areaccurate, then all curves should intersect at a point in P-T space. The degree to whichcurves do not intersect at a point is a measure of the degree of disequilibrium,inconsistency of thermodynamic data, or inaccuracy of activity models. ProgramsThermocalc (Holland and Powell) and TWQ (Berman) are designed to perform thesecalculations. Program Gibbs can also perform these calculations, but currently there isno simply way to translate measured mineral compositions into activity modelsconsistent with the thermodynamic data bases.

Two Fe-Mg exchange thermometers (garnet-biotite and garnet-chlorite) and onegarnet-plagioclase barometer (garnet-plagioclase-muscovite-biotite) apply to samplePUT92-C2. We will demonstrate the use of program GTB with these thermobarometers.

Garnet-biotite thermometry1) Start program GTB2) Open mineral file PUT 92C2.asm (option 1)…you will see a listing of key words in the

file. Hit return3) Choose option 3 (plot Keq lines)4) Once in the thermometer menu, choose option 1 (Grt-Bt), then choose the Hodges and

Spear calibration (#2). Do not correct for Fe3+ (option 0)5) You now proceed to choose Grt-Bt pairs, one at a time. To choose a mineral, highlight

the specific analysis, and either double click, or click OK. Garnet will be chosen first,then biotite, and after the selection of biotite, a KEq line will be plotted on the P-Tdiagram. Choose these pairsa) Grt 5 + Bt 16b) Grt 5 + Bt 17c) Grt 5 + Bt 28d) Grt 5 + Bt 29e) Grt 5 + Bt 30f) Grt 5 + Bt 31

The keys next to the mineral analyses indicate the textural position of the analysis.For example, "chl_rim_grt" is a chlorite in contact with a garnet rim, "mtrx_chl", isa matrix chlorite, etc. For this exercise, plot garnet analysis 5 (rim of garnet againstall biotites


6) Once done selecting Grt-Bt pairs, select cancel, and then select option 0 (return to RxnMenu)

7) Choose option -1 (Next Menu). This will bring up the geobarometer menu8) Choose barometer 2 (Grt-Pl-Ms-Bt)9) Choose the Hodges and Crowley calibration (option 3)10) In the same manner as before, select Grt-Pl-Bt-Ms pairs. For this exercise, select rim

garnet (5), a single matrix biotite (28) and muscovite (27), and pair these with allplagioclase analyses (33, 34, 35,36,37,38), as we are interested in pressure variation,which is indicated by variations in grossular content of garnet and anorthite content ofplagioclase. The results should look like the diagram on the left.

DiscussionThe small range of garnet-biotite temperatures and garnet-plagioclase-muscovite-

biotite pressures suggests that the matrix of the sample is well-equilibrated. Although thechlorite on the rim of the garnet suggests there is some of retrograde hydration, it does notseem to have affected the biotite compositions, inasmuch as the biotite touching the garnetgives the same temperature as the matrix biotite far from the garnet rim.

Garnet-chlorite thermometryRepeat the above exercise using garnet (5) + every chlorite. Chlorites numbered 14,

15, 18, and 20 are the retrograde chlorites touching the garnet rim and chlorites 21, 22 and26 are in the matrix away from the rim. The chlorites that are touching garnet give a highertemperature than the chlorites in the matrix because they were produced by a retrograde nettransfer reaction (ReNTR). The reaction probably only operated very locally (right on therim of the garnet) because there are no late, crosscutting chlorites in the matrix. In somesamples, pervasive operation of ReNTRs can result in production not only of Fe-richchlorite, but also Fe-rich biotite (biotite maintains exchange equilibrium with chlorite duringretrogression). If this had happened, then garnet-biotite Fe-Mg exchange thermometrywould have yielded a temperature above the metamorphic peak.

400 500 600 700 800 9000

2

4

6

8

10

T C

P k

bar

PUT-92C2 Rim thermobarometry

1

Grt-Bt

Grt-Plg-Bt-M

s

400 500 600 700 800 9000

2

4

6

8

10

T C

P k

bar

PUT92-C2 Grt-Chl thermometry

Grt rim+

retro Chl

Grt rim+

matrix Chl


YAG-xenotime and YAG-monazite thermometry

Recently calibrated YAG-xenotime (Pyle and Spear, 2000) and YAG-monazite(Pyle et al., in press) thermometers have been applied to the sample. Application of theYAG-Xno thermometer to the core (1902 ppm Y) yields a temperature estimate of 497°C.This estimate is a maximum, as xenotime was not observed in the sample. Application ofthe YAG-monazite thermometer, using rim compositions of garnet, plagioclase, monazite,and assuming an X OH-Ap of 0.1 yields a temperature estimate of 496°C. The garnet-monazite temperature is consistent with the temperatures obtained from garnet-biotite andgarnet-chlorite Fe-Mg exchange. The garnet-xenotime temperature from the garnet coremight suggest little temperature change during garnet growth, but this temperature is amaximum because xenotime was not observed in the garnet core. P-T path calculationsdiscussed below suggest that the core of the garnet nucleated at approximately 450 ˚C.

350 450 550 6500

2

4

6

8

10

T (°C)

P K

bar

0

1000

2000

3000

4000

5000

450 500 550 600 650 700T ( °C)

pp

m Y

in

ga

rne

t

: σY = 100 ppm

PUT 92C2Grt Core: 1902 ppm Y(maximum T)

KE

q=

0.4

YAG-MnzT estimate,Grt rim +matrix Mnz

TmaxYAG-Xno

YAG+OH Ap+Qtz=Grs+AN+Mnz+W [Y]Grt vs. TXenotime-bearing samples Monazite-bearing samples

PUT 92C2: σT based on σY


Exercise 1.B Contouring divariant regions using differentialthermodynamics (Gibbs method) with mass balance constraints

A useful application of the Gibbs Program to understanding this sample is toconstruct a contour diagram that shows how mineral composition and modal mineralogychanges with P and T. For this purpose we will use differential mode and the compositionson the rim of the garnet as reference conditions. These compositions have been stored in aGibbs input file named PUT92C2_Gibbs.in. We will draw contours for garnet andplagioclase compositions that reflect both the rim and core compositions, as listed in thetable below:

Put92-C2. Garnet and plagioclase rim and core compositionsPlagioclase GarnetXan Xprp Xalm Xsps Xgrs Fe/(Fe+Mg)

Rim 0.15 0.061 0.84 0.046 0.054 0.932Core 0.35 0.028 0.59 0.22 0.17 0.955

1) Start Program Gibbs2) Select the Gibbs Tutorial thermo.dat file3) From the Gibbs main menu, select Begin/Save new problem

a) Select 1 (Open Gibbs input file)b) Open the file PUT92C2_Gibbs.inc) Type 0 to return to the main menu

4) From the main menu, select 3 (Contour x-y plots)We will use the “Absolute contour routine” to make the contour diagram. Theprocedure will be as follows:

5) From the Contour menu, select 3 (Absolute contour routine)a) Select the variable to contour – start with Xalmandineb) Click “Reset”c) Click “Select”d) Type in the value to contour (the rim composition first)e) Click “Compute contour”

If the program beeps, just hit returnf) Type in the core value to contourg) Click “Compute contour”h) Click “Cancel” to exit the routine

6) Now reset the data to the starting conditions (rim compositions). To do this:a) Select 6 (Reference Points)b) Select 3 (Return to start)

7) Now repeat 5-16 for each of the composition variables in the table. Do spessartine last(this requires some special handling)

8) To contour spessartine:You will discover when you try to contour spessartine at the core composition that theprogram can’t find the contour. This is because the contour curves back on itself anddoesn’t have a solution at P = 6 kbar. If we know this is what is happening, we canhelp the program along in the following way.a) After resetting to the starting conditions, select 5 (Move to a new contour). This

will bring you to the Steps menub) Select 2 (Choose monitors)

i) Click “Reset” and select T and P as monitorsii) Type in –3000 for ∆P. This will decrease P from 6000 to 3000 bars at constant

temperature.


iii) Type in 30 for “Number of finite difference iterations” (100 bar steps).iv) Click “Done”

c) From the steps menu, select 4 (Compute one increment). This will lower thepressure at constant temperature.

d) Select 0 (return). This will bring you back to the contour menue) Now continue contouring Spessartine. Select 2, specify the spessartine contour and

compute. The program should be able to find the core spessartine contour at P =3000 bars.

9) If you wish, you can also contour moles of garnet. Repeat 5-16 and on 6, selectMgarnet. Type in the value of moles of garnet at the rim and compute the contour. Nowtype in 0.0 moles (the garnet isograd) and click compute. It probably won’t work usingthis routine because

10) When you are done, you can save the file if you wish. The final result, after somecleaning up in Illustrator, looks like this:

P-T contour diagram for sample PUT92-C2. Each color is a different contour. Solid linesare rim contours, dashed lines are core contours.

350 400 450 500 550 600 6500

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T (C)

P k

bar

PUT92-C2(+ mass balance)

An 0.15

An

0.40

Sps 0.046

MG

rt 0.043

Sps 0

.22

Grs 0.054

Grs

0.1

7

Alm

0.84

Fe/(F

e+M

g) 0.955

Fe/(F

e+M

g) 0.932

Prp 0.061

Prp 0.028

Alm

0.59

MGrt 0.000


DiscussionThe contour diagram is useful to help understand how this garnet-zone schist will

react along different P-T paths. The rim contours all intersect at a point as they must,because this is the reference conditions. The cyan contours (Mgrt) show that a wide rangeof P-T paths will produce garnet, including isothermal loading, isobaric heating, andheating with decompression.

Ideally, this type of diagram could be used to infer the P-T path during garnetgrowth, but this doesn’t work quantitatively because the diagram is drawn at constant bulkcomposition, but the rock changes effective bulk composition as it evolves due tofractionation into garnet (e.g. Mn and Ca). Examination of the core contours reveals thatthere is no P-T region in which they all intersect, as there should be if the diagram was aperfect model for the evolution of this rock. In fact, the region subtended by the garnet corecompositions, shown as a purple region, is rather large. Because Mn and Ca are twoelements that fractionate extensively into garnet, it is probable that these contours are mostaffected by the changing bulk composition. If we weight more heavily the core contour forFe/(Fe+Mg), then the P-T path appears to involve a bit of heating (10-15 ˚C) and possiblysome loading.

Exercise 1.C P-T paths from zoned garnet (Spear and Selverstone, 1983)

A method was developed in the early 1980’s to calculate the P-T path that wouldproduce a measured zoning profile in a garnet. This is the inverse of the contour methodshown above because now we will use the changes in composition along the zoning profileas input and compute changes in P and T. It is easy to do this in Program Gibbs becausethe user has complete freedom over which variables are chosen to be independent. For thisexercise, we will choose the garnet and plagioclase compositions to be independent and Pand T to be dependent.

For this exercise we will not use mass balance constraints. This will result in anincrease in the variance of the system of equations from two (Duhem’s theorem) to thephase rule variance (four in this example). Therefore, we will need four independentvariables to calculate the P-T path. Garnet contains three independent composition variables(we use Alm, Sps, and Grs) and plagioclase contains one (we use anorthite). These arealso a good choice because, as we saw in the contour diagram, there is a high angle ofintersection between these contours, so changes in both P and T will not be subjects of lowangle intersections.

The compositions we will use are in the table below. Note that the rim is not exactlythe same as specified above, but this is well within analytical uncertainty. Only the core andrim compositions are listed. The numbers we will need for this calculation are the changesin composition from rim to core, which are listed as the increments numbered 1-5.

Put92-C2. Garnet and plagioclase zoning for P-T path calculationPlagioclase Garnet

An Alm Sps GrsRim 0.015 0.840 0.046 0.054

1 0.050 -0 .037 0.027 0.0152 0.050 -0 .052 0.030 0.0313 0.050 -0 .119 0.086 0.0484 0.050 -0 .030 0.021 0.0115 0.050 -0 .015 0.009 0.009

Core 0.040 0.586 0.218 0.168


1) Start program Gibbs2) Choose the “Gibbs Tutorial” thermodynamic data set3) Select Begin/Save from the main menu4) Select 1 (Open input file) and open the file named “PUT92C2_Gibbs(noMB).in”5) Select 0 to return to main menu6) Select 2 (Single Steps)7) From the Steps menu, select 2 (Choose monitors/set deltas)

a) Click “Reset”b) Select as monitors XAn, XAlm, XSps, and XGrs.c) Type in the ∆ values for the first step (0.05, -0.037, 0.027. 0.015)d) Set Number of Finite difference steps to 10e) Click “Done”

8) Select 4 from the Steps menu (Compute). You should see a segment of the P-T pathdrawn.

9) Repeat 6-8 for each of the 5 steps in the table.

DiscussionThe calculated P-T path shows approximately 50 degrees of heating and 1.5 kbar of

loading. Comparison with the contour diagram reveals that the contour diagram gives agood first approximation to the path, although the path calculated directly from the garnetzoning is much more precise.

The calculated ∆T is most sensitive to zoning of Fe, Mg and Mn and the calculatedchange in pressure is most sensitive to the grossular and plagioclase zoning. In thecalculation above, it was assumed that the An content of plagioclase changed from 0.15 onthe rim to 0.40 in the core. It is always a good idea to examine the sensitivity of thecalculated P-T path to uncertainty in the plagioclase composition because it is generally notsimple to correlate the core of the garnet with a specific plagioclase composition.Calculating the path using a range of assumptions permits ready evaluation of thissensitivity.

350 400 450 500 550 600 6500

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T (C)

P k

bar

PUT92-C2Garnet zoning P-T path

Rim

Core


It should be noted that the placement of the path in P-T space is more uncertain thanthe ∆P and ∆T along the path. Garnet-biotite thermometry has a resolution of approximately±25 ˚C, but the ∆T of the path is precise to approximately 3-8 ˚C (cf. Kohn, 1993, CMP).

Exercise 1.D Forward modeling of garnet growth

As a final exercise on this sample, it is useful to attempt to model the observedzoning by forward modeling. In this example, we start at the conditions of the garnet core,and grow garnet along the inferred P-T path, calculating the zoning profile as we go. Thefinal zoning profile can then be compared with the observed profile to see if we have agood match. In more sophisticated modeling, simultaneous diffusion and growth can besimulated to account for homogenization of garnet by diffusion. Interested readers shouldrefer to the DiffGibbs manual for instructions.

The initial conditions in the forward model of sample PUT92-C2 are those of thegarnet core. We can measure the core composition, but the compositions of other phases inequilibrium with the core are not known (above we assumed that the plagioclase core wasalso in equilibrium with the garnet core). Our inverse modeling of the garnet zoning profilein Exercise 1.C results in calculating the values of all dependent variables at the garnet core,and we can use these compositions as our initial conditions for the forward modeling. Theeasiest way to save the results at the garnet core is to use option 5 (Save current problem)from the Begin/Save menu. This has been done and the input file is namedPUT92C2_Gibbs_Core.in.

We will use the “Grow Grt” module for this exercise, although we could also usethe “DiffGibbs” module. There are some idiosyncrasies in how the input files must beorganized for these two routines. For the Grow Garnet routine, garnet must be the lastmineral, and “New Variables” must bet turned on for garnet only. I’ll try to clean them upin the future. Also, it is in principle possible to do these calculations using Newton’smethod, but I think the code will balk if you try at this point. So the Grow Garnet routineforces you to use the Gibbs Tutorial data set and differential thermodynamics.

1) Start Program Gibbs2) Select “Gibbs Tutorial” thermodynamic data file3) From the main menu, select 5 (Grow Garnet)4) From the Grow Garnet menu, Select 1 (Open problem/add grt/draw plot)

a) Select 1 (Open new input file) and select the file named PUT92C2_Gibbs_Core.inb) Select 3 (draw axes) to redraw the plotc) Select 4 (Pick variables to plot). Type in 1 for each element it requests (this will plot

each one)d) You can set the way new garnet is distributed to multiple garnets that nucleate

sequentially using option 5. Since we are only modeling a single garnet, this isirrelevant here.

e) Select 0 (return)5) From the Grow Garnet menu, select 2 (Choose monitors).

a) Select T and P (if not already selected)b) Type in for ∆T = 50 and ∆P = 500. This will get us to the rim conditions along a

linear P-T path. You can, of course, select any P-T path you desire.c) Set “Number of finite difference steps” to 100 (1/2 degree/step)d) Click Done

6) From the Grow Garnet menu, select 4 (Compute one increment). The garnet zoningprofile should show on the screen.


DiscussionThe model (shown with black lines) is a good match for the data (colored symbols),

lending support to the proposed P-T path. Of course, this result is to be expected, becausewe used the observed zoning profile to calculate the path in the first place. The oneexception is grossular towards the rim, where the model calculation is slightly higher thanthe observed. This is most likely due to the amount of plagioclase that is reacting with thegarnet as the reaction proceeds. The input model has 10 modal percent plagioclase, and it isassumed that this plagioclase remains homogenous throughout (despite the observation thatthe plagioclase is zoned). If the amount of plagioclase involved in the reaction decreaseswith time, then the grossular in garnet will decrease more than with the homogeneousplagioclase model, consistent with the observed zoning trend.

The literature contains discussions about whether the “bell shaped” Mn profile isdue to Rayleigh fractionation, or P-T-X phase equilibrium constraints. A way to evaluatethis question is to construct two forward models, one assuming fractional crystallizationand one not. An input file similar to the one used above, but with fractional crystallizationturned off, is included (PUT92C2_Gibbs_Core(noFx).in). The interested student can rerunthe previous exercise with this input file (don’t replot the axes because this will erase thescreen). The resulting plot looks like this:

Alm

Sps

Grs

Prp

0 2000 4000 6000 8000 10000 120000.00

0.20

0.40

0.60

0.80

1.00

Radius (µm)

Mol

e fr

actio

nPUT92-C2


Fractional crystallization results in a decrease the radius of the garnet, and a decrease in therim composition of Mn and Ca and an increase in the rim composition of Fe, relative to theno fractionation model. Of course, without fractional crystallization, garnet would behomogeneous throughout its growth, so a zoned garnet would not actually be produced.The zoning profile shown in red above is actually the trace of the homogeneous garnetcomposition as it grows.

0 2000 4000 6000 8000 10000 120000.00

0.20

0.40

0.60

0.80

1.00

Radius (µm)

Mol

e Fr

actio

nPUT92-C2

Fe/(Fe+Mg)

Alm

Sps

Grs

Prp

No fractional crystallization

withfractional

crystallization (black)


Exercise 2: P-T paths calculated from zoned garnet:The Tauern Window

One of the first published P-T path calculated using the Gibbs method was for agarnet from the Tauern Window by Selverstone et al. (1984, J. Petrol., v. 25, pp. 501-534). At the time this path was calculated, the idea that Barrovian metamorphism typicallyfollowed clockwise P-T paths was relatively new. Furthermore, no P-T path had ever been

calculated from a natural sample thatdisplayed a clockwise path.

The sample studied by Selverstone etal. (1984) came from a unit called the LowerSchieferhulle series in the southwest TauernWindow, Austria. This unit is part of thePennine basement and sits with tectoniccontacts between the Zentralgneis core andthe Upper Schieferhulle.

Photo to right: J. Selverstone sampling inthe Tauern Window.

The sample studied in detail for garnet zoningis from a rock unit called “garben schist” because ofthe large hornblende crystals that roughly resemblebundles of wheat (“garben” in German) (see photo atleft). Many samples contain spectacularly largegarnets

Zürich

München Wien

Helvetic, Flysch,and Penninic Units

Australoalpine Units

Southern Alps

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Basel

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FH-1M


Garben schists contain a range of assemblages that include subsets of the phaseshornblende, kyanite, staurolite, garnet, biotite, epidote, plagioclase, ankerite, quartz,ilmenite, and rutile with either paragonite or chlorite. The sample studied in detail containedthe sub-assemblage hornblende + kyanite + staurolite + garnet + biotite + epidote +plagioclase + quartz + chlorite which, with fluid, is divariant in the systemCNKFMnMASH.

Photomicrograph of sample FH-1M. Field of view is approximately 1 cm.

Garnet in sample FH-1M is strongly zoned with bell-shaped spessartine andgrossular, increasing pyrope, and almandine that shows a maximum. Fe/(Fe+Mg)

decreases monotonically fromcore to rim suggesting Tincreased throughout garnetgrowth.

The input data for therim P-T conditions arecontained in the files namedIN-TAUERN WIND rim. Wewill use these data to reproducethe calculations published bySelverstone et al. (1984).

0.15

0.10

0.15

0.10

0.10

0.05

0.60

0.65

0.70

Xsp

sX

alm

Xgrs

Xprp

0.80

0.85

0.90

Fe/(F

e+M

g)

Xsps

Xalm

Xgrs

Xprp

Fe/(Fe+Mg)

1234

57

6 12

45 3

7

6

Distance mm

rim rimcore

0 1 2 3 4 5

FH-1M


1) From the MAIN menu, select option 1 (BEGIN/SAVE) and open the disk file named“IN-TAUERN WIND rim”. The assemblage present in the matrix of the rock is garnet+ biotite + chlorite + kyanite + staurolite + hornblende + plagioclase + quartz.Evidence was given by Selverstone et al. (1984) that this assemblage was presentduring entire growth of garnet.

2) Go to SINGLE STEPS menu (option 2 from the MAIN MENU).Note that the variance of the assemblage is 2 and we have 3 independent compositionalparameters from the zoning in the garnet (XAlmandine, XSpessartine, XGrossular). Also,we know the composition of the plagioclase in the matrix of the rock (An32) and in thecore of the garnet (An18).There are several ways we could tackle this problem.(A) We could select 2 of the 4 possible monitor parameters and check to see how theother two calculate (compare the computed core composition with the actual corecomposition). We could then pick a different pair of monitors and see how the resultscompare with the first set. This is the procedure we will follow first using firstalmandine and grossular as monitors and then using spessartine and plagioclase. Notethat it would not be a good idea to choose almandine + spessartine or grossular +anorthite as monitors because almandine and spessartine are both T sensitive and Pinsensitive and grossular and anorthite are both P sensitive but relatively T insensitive.This could generate large errors in ∆T or ∆P from low-angle intersections.(B) Alternatively, we could try and find a "sensible" way to increase the variance(wouldn't it be nice if rocks were always be this well constrained). One possibility isto allow Pfluid to be an independent variable. In this way we could guarantee that 3 ofthe 4 monitors fit perfectly, and use the fourth as a check. We'll try this too.

Below is a table containing the garnet and estimated plagioclase composition from thegarnet rim to the core, expressed as changes in composition for 6 increments.

Increment ∆ XAlm ∆ XGrs ∆ XSps ∆ XAnor (estimate)

1 0.024 0.018 -0.018 -0.02332 0.004 0.011 -0.003 -0.02333 0.010 0.000 0.002 -0.02334 -0.005 0.024 0.002 -0.02335 -0.009 0.010 0.015 -0.02336(core) -0.038 0.007 0.051 -0.0233

3) Select SINGLE STEPS option 2 (CHOOSE MONITORS). Choose ∆XAlm and∆XGrs as monitors and set ∆XAlm = 0.024 and ∆XGrs = 0.018. Choose NSTEP =10.

4) Select SINGLE STEPS menu option 4 (COMPUTE 1 INCREMENT). The P-T pathfor the first increment should be drawn on screen.

5) Repeat (7) and (8) using zoning increments 2-6 to draw the entire P-T path. Your plotshould look like this (red curve):


The P-T path can be compared with that published by Selverstone et al. (1984) (notedifferent scales). You will notice that the published P-T path has a shape very similar to theone we just calculated, but the absolute values of ∆T and ∆P are different. The publishedpath shows ∆T ≈ -20 ˚C, ∆P ≈ +3.5 kbar (at maximum pressure) whereas the one wecomputed shows ∆T ≈ -75 ˚C, ∆P ≈ +3.5 kbar. The discrepancy in the calculated ∆T mostlikely results from the different thermodynamic data set used. In particular, Selverstone etal. (1984) used a constant value for the entropy of water whereas the current program usesa value that is a function of P and T.

It is useful to compare the compositions that are computed for the core of the garnet withthe measured compositions. Because the variance of the sample is 2, we have this checkon the calculations.

6) Select SINGLE STEPS option 7 (PRINT DATA TO SCREEN AND FILE).The current status of the assemblage will appear:

************************************************************************T P Pfluid START 550.0 7000.0 7000.0 CURRENT 473.3 10584.0 0.1

--MONITOR PARAMETERS-- NSTEP= 10 IMASS= 0 4 X_Alm 6 X_Grs -0.038 0.007

System components Si.. Al.. Mg.. Fe.. Mn.. Ca.. Na.. K... H2O. Mass balance = OFF

# OF PHASES, NPH= 9 Mineral compositions

400 500 600 700 8000

2

4

6

8

10

12

14

16

18

20

T (C)

P k

bar

Sample FH-1MTauern Window

rim

coreAlm+Grs monitors

Sps+Plgmonitors


PhCo X(i) dX(i) sumdX(i) 1 Quartz-(a-b) abQz 1.0000 0.0000 0.0000 2 Water H2O 1.0000 0.0000 0.0000 5 Kyanite Ky 1.0000 0.0000 0.0000 93 Plagioclase-ideal_solution Ab 0.8983 0.0000 0.2183 An 0.1017 0.0000 -0.2183 30 Garnet-ideal_1-site_mixing Prp 0.0947 -0.0017 -0.0913 Alm 0.6330 -0.0038 -0.0140 Sps 0.1053 0.0048 0.0353 Grs 0.1670 0.0007 0.0700 18 Biotite-(FE-Mg-Mn)_ideal_1_site_ Phl 0.6004 -0.0018 -0.0746 Ann 0.3919 0.0014 0.0719 MnBt 0.0078 0.0004 0.0028 40 Staurolite-ideal MgSt 0.1245 -0.0014 -0.0855 FeSt 0.8475 0.0000 0.0775 MnSt 0.0279 0.0013 0.0079 21 Chlorite-(4.5-3.-2.5)-ideal_1-si Chlin 0.6458 -0.0020 -0.0692 Daph 0.3373 0.0011 0.0623 MnChl 0.0170 0.0009 0.0070 61 Hornblende(Mg-Fe-Mn)-ideal_1-sit Mg-Hb 0.5250 -0.0019 -0.0910 Fe-Hb 0.4722 0.0018 0.0902 Mn-Hb 0.0028 0.0001 0.0008

The calculated concentrations for the core of the garnet for spessartine, anorthite andFe/(Fe+Mg) in the amphibole are 0.1036, 0.121, and 0.47, respectively, compared withthe measured values of 0.122, 0.18, and 0.51. This agreement is fair, and all of thesechanges are in the correct direction.

It is of course possible to "force" any two calculated "core" compositional variablesto match the measured values by choosing those variables as monitor parameters. Forexample, XAnorthite and XSpessartine could be forced to fit perfectly if these values werechosen as monitors. This exercise is worth doing because it gives a feel for the sensitivityof the P-T path to choice of monitor parameters.

7) Reset the mineral assemblage to the starting conditions (SINGLE STEPS option 6, thentype 3).

8) Recalculate the P-T path, this time using XSps and XAnor as monitor parameters. Selecta square for the symbol. Choose option 2 to set the appropriate monitor parameters anddeltas, then calculate the path using option 4. Repeat options 2 and 4 with theappropriate values of ∆XSps and ∆XAnor. The final P-T path is shown above in green.

************************************************************************T P Pfluid START 550.0 7000.0 7000.0


CURRENT 504.3 8886.9 0.2

--MONITOR PARAMETERS-- NSTEP= 10 IMASS= 0 5 X_Sps 3 X_An 0.051 -0.023

System components Si.. Al.. Mg.. Fe.. Mn.. Ca.. Na.. K... H2O. Mass balance = OFF

# OF PHASES, NPH= 9 Mineral compositions PhCo X(i) dX(i) sumdX(i) 1 Quartz-(a-b) abQz 1.0000 0.0000 0.0000 2 Water H2O 1.0000 0.0000 0.0000 5 Kyanite Ky 1.0000 0.0000 0.0000 93 Plagioclase-ideal_solution Ab 0.8198 0.0023 0.1398 An 0.1802 -0.0023 -0.1398 30 Garnet-ideal_1-site_mixing Prp 0.1224 -0.0029 -0.0636 Alm 0.6269 -0.0035 -0.0201 Sps 0.1190 0.0051 0.0490 Grs 0.1317 0.0013 0.0347 18 Biotite-(FE-Mg-Mn)_ideal_1_site_ Phl 0.6273 -0.0025 -0.0477 Ann 0.3638 0.0020 0.0438 MnBt 0.0089 0.0004 0.0039 40 Staurolite-ideal MgSt 0.1537 -0.0024 -0.0563 FeSt 0.8130 0.0010 0.0430 MnSt 0.0333 0.0015 0.0133 21 Chlorite-(4.5-3.-2.5)-ideal_1-si Chlin 0.6689 -0.0026 -0.0461 Daph 0.3124 0.0017 0.0374 MnChl 0.0187 0.0009 0.0087 61 Hornblende(Mg-Fe-Mn)-ideal_1-sit Mg-Hb 0.5597 -0.0027 -0.0563 Fe-Hb 0.4369 0.0026 0.0549 Mn-Hb 0.0034 0.0002 0.0014

Note that the final P-T conditions differ somewhat from those computed earlier. Also notethat now the spessartine and anorthite compositions match the measured core compositions,but the computed XAlm = 0.627 compared with the measured value of 0.633, the computedand measured grossular core values are 0.131 and 0.167, respectively, and the computed andmeasured hornblende core values are 0.44 and 0.518, respectively. This match is not perfect,as it should be if the rock were being modeled perfectly and hints that there may have beenother factors involved in the metamorphism of this sample.


Appendix: Making a Gibbs input file

This first exercise will demonstrate the making of making an input file for the Gibbsprogram. The file we will make will be for the assemblagemuscovite + biotite + Al2SiO5 + quartz + K-feldspar + H2Oin the KFASH system. This assemblage is univariant and we will use the input file inexercise 2 to draw a univariant curve (the muscovite breakdown reaction).

1) Start program Gibbs2) Select the “Gibbs Tutorial” thermodynamic data file3) Choose menu item 1 (Begin/save a problem)4) Choose submenu item 3 (Create a new input file)

a) In the dialog box, give the assemblage a title (e.g. Ms+Bt+AlSi+Qtz+Kfs+H2O -KFASH), Click the appropriate boxes for the KFASH system and specify thestarting T and P to be 770. ˚C, 8733 bars. The dialog box should look like this:

b) Select from the mineral listi) For quartz, H2O, AlSi and K-feldspar, there are no composition data to enter,

so when the mineral dialog box appears, just click "done"ii) a-b quartz (mineral number 1). This mineral will switch between alpha and beta

quartz according to their relative stability fields.iii) H2O (mineral number 2)


iv) Al-Silicate (mineral number 8). This mineral will switch between kyanite,sillimanite and andalusite, depending on the P-T conditions.

v) K-feldspar (mineral 92). This mineral contains disorder terms, so it will changestructural state with temperature.

vi) Muscovite (mineral number 117). Muscovite 117 is a muscovite-celadonitesolid solution. Input values of XFecelad and Xms of 0.1 and 0.9, respectively:

vii) Biotite (mineral 119). Input values of Si = 2.5, Al(iv) = 1.5, Al(vi) = .5, Mg =0, Fe = 2.5.

viii) Click cancel on the mineral selection list.5) Click on the "New Input File" window and save using "Saveas" command from the file

menu. Name the file "KFASH MsBtAlSiKfsQtzH2O".6) Click on the "Command" window and hit return to close "New Input File" window.

Now you can open the data file for use in Gibbs.

gibbstut(feb, 2001) pt b - rensselaer polytechnic...

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