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.