time scales of magmatic processes from modelling the zoning patterns of crystals

Time scales of magmatic processes from modelling the zoning patterns of crystals

Fidel CostaInst. Earth Sciences ‘Jaume Almera’ CSIC (Barcelona, Spain)

Ralf Dohmen and Sumit ChakrabortyInst. Geol. Mineral. and Geophys., Bochum University (Bochum, Germany)

Table of contents

1. Introduction to zoning in crystals

2. Diffusion equation

4. Modeling natural crystals: isothermal case

* Initial conditions

* Boundary conditions

3. Diffusion coefficient

5. Problems, pitfalls and uncertainities

* Multiple dimensions, sectioning, anisotropy

6. Conclusions and prospects

1. Zoning in crystals

X-ray distribution map of olivine

from lava lake in Hawaii

Moore & Evans (1967)

Development of electron microprobe 1960’, first

traverses and X-ray Maps

Among the first applications to obtain time

scales those related to cooling histories of

meteorites (e.g. Goldstein and Short, 1967)

1. Zoning in crystals

- Major elements, trace elements, and isotopes

- Increasingly easier to measure gradients with good precision and spatial resolution (LA-ICP-MS, SIMS,

NanoSIMS, FIB-ATEM, e-probe, micro-FTIR)

Major and trace element zoning in Plag

Bindeman et al. (2008)

Stable isotope zoning

18O in zircon from Yellowstone magmas

Sr isotope zoning in plagioclase

Tepley et al. (2000)

Diffusion driven by a change in P, T, or composition

distance

t0, T0, P0, X0

distance

con

cen

trat

ion

con

cen

trat

ion

tt, Tt, Pt, Xt

1. Zoning in crystals

- Because D is Exp dependent on T and Ds in geological

materials are slow, minerals record high T events (as

opposed to room T)

- The compositional zoning will reequilibrate at a rate governed by the chemical

diffusion (Fick’s laws)

distance

1. Zoning in crystals

1. Zoning in crystals

Crystals record the changes in variables and environments: gradients are a combined record of

crystal growth and diffusion

2. The diffusion equation

2. Diffusion, flux, and Fick’s law

Diffusion:

(1) motion of one or more particles of a system relative to other particles (Onsager, 1945)

(2) It occurs in all materials at all times at temperatures above the absolute zero

(3) The existence of a driving force or concentration gradient is not necessary for diffusion

SOLIDLIQUIDGAS

2. Diffusion, flux, and Fick’s law

2. Diffusion, flux, and Fick’s law

Random motion leads to a net mass flux when the concentration is not uniform:

equalizing concentration is a consequence, NOT the cause of diffusion

Fick’s first law

Flux has units of ‘mass or moles or volume * distance/time’

Diffusion coefficient has units of ‘distance2/time’

2. Diffusion, flux, and Fick’s law

More general formulation by Onsager (1945) using the chemical potential

There might be other contribution to fluxes; e.g., crystal growth or dissolution

2. Diffusion, flux, and Fick’s law

Fick’s second law: mass balance of fluxes

Analogy: gain or loss in your bank account per month =

Your salary ($$ per month) - what you spend ($$ per month)

2. Diffusion, flux, and Fick’s law

Fick’s second law: mass balance of fluxes

1. We need to solve the partial differential diffusion equation. (a) analytical solution (e.g., Crank, 1975) or (b) numerical methods

(e.g., Appendix I of chapter)

2. We need to know initial and boundary conditions. This is straightforward for ‘exercise cases’, less so in nature.

3. We need to know the diffusion coefficient

3. Diffusion coefficient

3. Diffusion coefficient: Tracer

e.g., diffusion of 56Fe in homogenous olivine

Di*= tracer diffusion coefficientw = frequency of a jump to an adjacent site

l = distance of the jumpf = related to symmetry, coordination number

3. Diffusion coefficient: multicomponent

Multicomponent formulation (Lasaga, 1979) for ideal system, elements with the same charge and exchanging in the

same site

e.g., diffusion of FeMg olivine

3. Diffusion coefficient

Perform experiments at controlled conditions to determine D* or DFeMg

Q = activation energy (at 105 Pa), ΔV =activation volume, P = pressure in Pascals, R is the gas constant,

and Do = pre-exponential factor.

3. Diffusion coefficient

New experimental and analytical techniques allow to determine D at the conditions (P, T, fO2, ai) relevant for the magmatic

processes without need to extrapolation

e.g., Fe-Mg in olivine along [001]

Dohmen and Chakraborty (2007)

3. Diffusion coefficient

New theoretical developments allow a deeper understanding of the diffusion mechanism and thus to establish the extend to which experimentally determined D apply to nature (e.g.,

impurities, dislocations, etc).

PLEASE CHECK APPENDIX II OF THE CHAPTER

and

AGU Oral presentation, Tuesday 8h30’

4. Solving the diffusion equation

Initial and boundary distribution (conditions)

4. Initial distribution (conditions)

Shape of profile may retain info about initial distribution

4 strategies for initial distribution

Initial conditions

1. Use slower diffusing elements to constrain shape

of faster elements

Examples

-An for Mg in plagioclase (Costa et al., 2003)

-P for Fe-Mg in olivine (Kahl et al., 2008)

-Ba for Sr in sanidine (Morgan and Blake, 2006)

Initial conditions

1. Use slower diffusing elements to constrain shape

of faster elements

Examples

-An for Mg in plagioclase (Costa et al., 2003)

-P for Fe-Mg in olivine (Kahl et al., 2008)

-Ba for Sr in sanidine (Morgan and Blake, 2006)

OLIVINE Examples

X-Ray Map of P

X-Ray Map of Fe

Initial conditions

2. Using arbitrary maximum initial concentration range in natural samples

(this provides maximum time estimates)

Examples

* Sr in plagioclase (Zellmer et al., 1999)

* Fe-Mg in Cpx (Costa and Streck, 2003; Morgan et al., 2006)

* Ti in Qtz (Wark et al., 2007)

Initial conditions

3. Using a homogeneous concentration profile

Examples

* Fe-Mg, Ca, Ni, Mn in olivine (Costa and Chakraborty, 2004; Costa and Dungan, 2005)

* O in zircon (Bindeman and Valley, 2001)

Blue Creek flow, Post-HRT

0

2

4

6

20 400

2 ky

0.1 ky

10 ky

25 ky

5 ky

Radius, μm

core

rim

18 ‰ O VSMOW

, Radius μm20 40

5 ky

core

rim0

0.5 ky

10 ky

25 ky

,Middle Biscuit Basin flow -Post LCT

18 ‰ O VSMOW

-2

0

2

4

Ion Probespots

core

-airabraded

Initial conditions

4. Use a thermodynamic (e.g., MELTS) and kinetic model to

generate a growth zoning profile

Examples

* Plagioclase (Loomis, 1982)

* olivine- Chapter and AGU Poster, Tuesday afternoon

time distance

Eq

uil

%

Con

c.C

onc.

100

0

Initial conditions

Initial conditions: effects on time scales

1. Despite the difference in shapes of the initial profiles the maximum difference on calculated time scales is a factor of ~1.5

2. Although the initial profile that we assume controls the time that we obtain, the error can be evaluated and is typically not

very large

3. When in doubt perform models with different initial conditions to asses the range of time scales

4. Boundary conditions

Characterizes the nature of exchange of the elements at the boundaries of the crystals (e.g., other crystals or melt).

Two end-member possibilities

Boundary conditions

1. Open: the crystal exchanges with the

surrounding (e.g., Fe-Mg in melt-olivine interface)

X-Ray Map of Fe

Boundary conditions

2. Closed: no exchange.

(a) D of the element of interest is much slower in the surrounding than in the mineral

X-Ray Map of Ca

Olivine

Plag

e.g., Ca in olivine-plag contact

Boundary conditions

2. Closed: no exchange.

(b) the mineral is surrounded by a phase where the element does not partition (e.g. Fe-Mg: olivine/plag)

Kahl et al. (2008)

X-Ray Map of Mg

Olivine

Plag

time distance

Eq

uil

%

Con

c.C

onc.

Closed boundary

Open boundary

100

0

Boundary conditions

Equilibration in the closed system occurred much faster

Incorrectly applying a no flux condition to an open system can lead to underestimation of time by factors as large as an order of magnitude. But in general not difficult to recognise which type of boundary applies

to the natural situation

Boundary conditions: effects on time scales

Boundary conditions: effects of crystal growth or dissolution

(a) Neglecting crystal growth tends to overestimate time scales

t2

distance

Con

c.

(b) Neglecting crystal dissolution tends to underestimate time scales (e.g., shortening of diffusion profiles)

distance

t1

Con

c.

Non-isothermal process

* If there is no overall cooling of heating trend, results from a single intermediate T are correct, likely for some

volcanic rocks (e.g., Lasaga and Jiang, 1995)

CC 1960 plag1_t1

30

35

40

45

50

0 100 200 300 400 500 600

distance from rim in micrometers

An mol %

Non-isothermal process

* If there are protracted cooling and reheating (e.g., plutonic rocks) we need to have a T-t path

*This affects: (a) the diffusion coefficient, (b) the diffusion equation, and (c) the boundary conditions.

CHECK PAGES 13-21 of the CHAPTER

AGU POSTER, 13h40’, Tuesday,

5. Potential pitfalls and errors

Errors and uncertainties associated with time determinations

Two types:

(1) those associated with how well we understand and reproduce the natural physical conditions (e.g., multiple

dimension etc), and

(2) those associated with the parameters used in the model (e.g., T, D)

Effects of geometry and multiple dimensions

These are important depending on the type, shape, and size of the crystal that we are studying and on the diffusion time

Off core sections; 2D effects on sectioning

Kahl et al. (2008)

X-Ray Map of MgX-Ray Map of Mg

Effects of geometry and multiple dimensions

Neglecting multidimensional effects tends to overestimate time scales

Careful with the orientation of the diffusion front with respect to the traverse- one can create unnecessary and artificial 2D effects

Not OK

OK

Data acquisition

Effects of geometry and multiple dimensions

0

100

200

300

400

500

600

700

800

0 50 100 150 200

microns

Mg ppm

ini

calc

equi

1D, t = 225 y

Example of plagioclase

Costa et al (2003), GCA 67

0100200300400500600

700

800

Mg ppm

ini

0100200300400500600

700

800

Mg ppm

calc

1D, t = 225 y ------> 2D, t = 60 y

almost a factor of 4!

Effects of geometry and multiple dimensions

Fe-Mg zoning in olivine: diffusion anisotropy and 2 dimensional effects

Mg zoning

b

c

150 µm

Fo zoning

ModelInitial

Anisotropy of diffusion

EBSD

T6: , , ?? ~0, ~0, ~0 Dt6 ?? Dt6 ~ Da

2 Da

c

Anisotropy of diffusion

Fe-Mg diffusion in olivine : Dc ~ 6 Da ~ 6 Db

This can be used as a test for diffusion-controlled zoning

1. T uncertainties from geothermometers are ~ 30 oC

E (kJmol-1) 500 200-300 100

Uncer.Factor 4 2 1.3 At magmatic T

Lower uncertainties if T is determined experimentally

Errors and uncertainties on the parameters

2. Experimental D determination at a given T: within a factor of 2

3. Uncertainties in other variables that D depends on, e.g., oxygen fugacity, pressure. Typically much smaller

4. One can expect overall uncertainties on calculated times s of a factor of 2 to 4, e.g. 10 years might mean between 5 - 20 years

6. Summary of times and contrast with other types of data

Turner and Costa (2007), Elements

The Bishop Tuff: comparing crystal diffusion studies with radioactive data and residence

times

Residence time in ky

BT

dOC

dOL

dOD

dYGdY

dYA

WMC

dDMdMK

100

200

300

400

500

600

700

800 Long Valley system

Rb-Sr (felds)U-Pb (Zrc)U-Th dis. (Zrc)U-Th dis. (other)

00

500

Eruption age ( Ar/ Ar, K-Ar, C) in ka40 39 14

1000 1500 2000

Bishop Tuff geochronological data

Residence time using zircon ages ca. 50 to 100 ky;

Time from diffusive equilibration of Ti in quartz ca. 100 y

Is there something wrong?

t1 = 100000 y

Radioactive decay vs. chemical diffusion time scales

Cooling and crystallization;Small intrusions or eruptionsTime recorded by isotopes is100 000 y

Major intrusion; entrainment of old cumulates. Crystal overgrowth drive diffusion

t2 = 100 y

Radioactive time = t1 + t2

Diffusion time = t2

7. Conclusions and Perspectives

Conclusions and perspectives

(1) Modeling zoning patterns of crystals can be used to obtain time scales of magmatic and volcanic processes. The uncertainties can be limited by careful petrological analysis,

multiple time determinations in a single thin section, and improved D data.

(2) The ranges of time scales and types of processes which can be determined is almost unlimited thanks to the large

variety of elements, minerals, and crystals that can be exploited. In the future smaller gradients will be exploited,

NanoSIMS, FIB-ATEM

New developments: FIB + ATEM

Vielzeuf et al., 2007

Conclusions and perspectives

(3) Time scales determinations from modeling the zoning patterns are numerous but still not very much exploited

method.

Results so far indicate that many volcanic processes are short, e.g., < 100 years. This is typically shorter and within the error of radiogenic isotope determinations and more

studies using both methods should be performed.