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,
NanoSIMS, FIB-ATEM, e-probe, micro-FTIR)
Major and trace element
zoning in Plag
Stable isotope zoning
18O 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
t0, T0, P0, X0
distance
tt, Tt, Pt, Xt
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 ‘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
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 56Fe 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 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)
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
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
core
Examples
* Fe-Mg, Ca, Ni, Mn in olivine
(Costa and Chakraborty, 2004;
Costa and Dungan, 2005)
Blue Creek flow,
Post-HRT
core
6
0.1 ky
airabraded
5 ky
18 O ‰ VSMOW
3. Using a homogeneous
concentration profile
2 ky
4
2
10 ky
* O in zircon (Bindeman and
Valley, 2001)
25 ky
0
0
20
Radius, m
40
rim
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)
t1
t2
distance
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)
CC 1960 plag1_t1
50
An mol %
45
40
35
30
0
100
200
300
400
distance from rim in micrometers
500
600
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
X-Ray Map of Mg
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 2D effects
Not OK
OK
Neglecting multidimensional effects tends to
overestimate time scales
Effects of geometry and multiple dimensions
Example of plagioclase
1D, t = 225 y
800
ini
700
600
Mg ppm
500
400
calc
300
200
equi
100
Costa et al (2003), GCA 67
0
0
50
100
microns
150
200
Effects of geometry and multiple dimensions
700
600
500
400
300
200
100
0
calc
Mg ppm
Mg ppm
ini 800
800
700
600
500
400
300
200
100
0
1D, t = 225 y ------> 2D, t = 60 y
almost a factor of 4!
Fe-Mg zoning in olivine: diffusion anisotropy and 2
dimensional effects
Mg zoning
Model
Initial
c
b
Fo zoning
150 µm
Anisotropy of diffusion
EBSD
T6: , , ??
~0, ~, ~
Dt6 ??
Dt6 ~ Da
Anisotropy of diffusion
a
2D
c
Fe-Mg diffusion in olivine : Dc ~ 6 Da ~ 6 Db
This can be used as a test for diffusion-controlled zoning
Errors and uncertainties on the parameters
1. T uncertainties from geothermometers are ~ 30 oC
E (kJmol-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
800
Long Valley system
Rb-Sr (felds)
U-Pb (Zrc)
U-Th dis. (Zrc)
BT
U-Th dis. (other)
700
Residence time in ky
dOD
600
500
400
dOL
dYA
dYG
300 dY
dOC
dDM
dMK
200
100
WMC
0
0
500
1000
40
1500
39
14
Eruption age ( Ar/ Ar, K-Ar, C) in ka
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?
Radioactive decay vs. chemical diffusion time scales
t1 = 100000 y
t2 = 100 y
Radioactive time =
t1 + t2
Cooling and
crystallization;
Small intrusions
or eruptions
Time recorded by
isotopes is
100 000 y
Diffusion time =
t2
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,
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.