Time scales of magmatic processes from modelling the

Time scales of magmatic processes from modelling the zoning patterns of crystals Fidel Costa Inst. Earth Sciences ‘Jaume Almera’ CSIC (Barcelona, Spain) Ralf Dohmen and Sumit Chakraborty Inst. Geol. Mineral. and Geophys. , Bochum University (Bochum, Germany)

Table of contents 1. Introduction to zoning in crystals 2. Diffusion equation 3. Diffusion coefficient 4. Modeling natural crystals: isothermal case * Initial conditions * Boundary conditions 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, Nano. SIMS, FIB-ATEM, e-probe, micro-FTIR)

Major and trace element zoning in Plag

Stable isotope zoning 18 O in zircon from Yellowstone magmas Bindeman et al. (2008)

Sr isotope zoning in plagioclase Tepley et al. (2000)

1. Zoning in crystals Diffusion driven by a change in P, T, or composition t t, T t, P t, X t concentration t 0, T 0, P 0, X 0 distance

1. Zoning in crystals - The compositional zoning will reequilibrate at a rate governed by the chemical diffusion (Fick’s laws) - Because D is Exp dependent on T and Ds in geological materials are slow, minerals record high T events (as opposed to room T) distance

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

2. Diffusion, flux, and Fick’s law GAS LIQUID SOLID

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 ‘distance 2/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 Di*= tracer diffusion coefficient w = frequency of a jump to an adjacent site l = distance of the jump f = related to symmetry, coordination number e. g. , diffusion of 56 Fe in homogenous olivine

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 Fe. Mg olivine

3. Diffusion coefficient Perform experiments at controlled conditions to determine D* or DFe. Mg 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, f. O 2, 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 8 h 30’

4. Solving the diffusion equation Initial and boundary distribution (conditions)

4. Initial distribution (conditions) 4 strategies for initial distribution Shape of profile may retain info about 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 X-Ray Map of P OLIVINE 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) 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)

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

Conc. Initial conditions Conc. Equil % 100 0 time distance

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

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 X-Ray Map of Fe 1. Open: the crystal exchanges with the surrounding (e. g. , Fe-Mg in melt-olivine interface)

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 e. g. , Ca in olivine-plag contact Olivine Plag

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) X-Ray Map of Mg Olivine Plag Kahl et al. (2008)

Boundary conditions Conc. Open boundary Conc. Equil % 100 0 time Closed boundary distance

Boundary conditions: effects on time scales 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 of crystal growth or dissolution (a) Neglecting crystal growth tends to overestimate time scales (b) Neglecting crystal dissolution tends to underestimate time scales (e. g. , shortening of diffusion profiles) Conc. t 1 Conc. t 2 distance

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)

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, 13 h 40’, Tuesday,

5. Potential pitfalls and errors

Errors and uncertainties associated with time determinations Two types: (1) those associated with how well we understand 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; 2 D effects on sectioning X-Ray Map of Mg Kahl et al. (2008)

Effects of geometry and multiple dimensions Data acquisition Careful with the orientation of the diffusion front with respect to the traverse- one can create unnecessary and artificial 2 D effects Not OK OK Neglecting multidimensional effects tends to overestimate time scales

Effects of geometry and multiple dimensions Example of plagioclase 1 D, t = 225 y ini calc equi Costa et al (2003), GCA 67

Effects of geometry and multiple dimensions ini calc 1 D, t = 225 y ------> 2 D, t = 60 y almost a factor of 4!

Fe-Mg zoning in olivine: diffusion anisotropy and 2 dimensional effects Mg zoning Initial Model c b Fo zoning 150 µm

Anisotropy of diffusion EBSD T 6: , , ? ? ~0, ~ Dt 6 ? ? Dt 6 ~ Da

Anisotropy of diffusion a 2 D c � Fe-Mg diffusion in olivine 6 Da ~D 6 Db c ~ �: � This can be used as a test for diffusion-controlled zoning

Errors and uncertainties on the parameters 1. T uncertainties from geothermometers are ~ 30 o. C E (k. Jmol-1) 500 Uncer. Factor At magmatic T 4 200 -300 100 2 1. 3 Lower uncertainties if T is determined experimentally 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

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?

Radioactive decay vs. chemical diffusion time scales t 1 = 100000 y t 2 = 100 y Radioactive time = t 1 + t 2 Cooling and crystallization; Small intrusions or eruptions Time recorded by isotopes is 100 000 y Diffusion time = t 2 Major intrusion; entrainment of old cumulates. Crystal overgrowth drive diffusion

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, Nano. SIMS, 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.