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First-principles prediction of
diffusion coefficients in non-dilute,
multi-component solids
Anton Van der VenDepartment of Materials Science and Engineering
University of Michigan
Ann Arbor, MI
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Coarse graining time
Diffusion in a crystal
Two levels of time coarse graining
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Coarse graining time
Diffusion in a crystal
Two levels of time coarse graining
kT
EBexp*
Short-time coarse graining:
transition state theory
- MD simulations
- Harmonic approximation
Vineyard, J. Phys. Chem. Solids 3, 121 (1957).
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Coarse graining time
Diffusion in a crystal
Two levels of time coarse graining
kT
EBexp*
Short-time coarse graining:
transition state theory
- MD simulations
- Harmonic approximation
A second level of coarse graining
that leads to Fick’s law
J D C
Green-Kubo
Kinetic coefficients derived from
fluctuations at equilibrium
Vineyard, J. Phys. Chem. Solids 3, 121 (1957).Zwanzig, Annu. Rev. Phys. Chem. 16, 67 (1965).
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Diffusion
• Distinction between:
– interstitial diffusion
– substitutional diffusion
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Interstitial diffusion
• C diffusion in bcc Iron (steel)
• Li diffusion in transition metal oxide host
• O diffusion on Pt-(111) surface
In all examples, diffusion occurs on a rigid lattice
which is externally imposed by a host or substrate
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Example of interstitial diffusion
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Irreversible thermodynamics:
interstitial diffusion of one component
LJ
CDJ
dC
dLD
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Notation
M = number of lattice sites
N = number of diffusing atoms
vs = volume per lattice site
x = N/M
C=x/vs
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Interstitial diffusion:
one component
LD
C
Kubo-Green relations
(linear response statistical mechanics)
L1
(2d)tMvskT
R i t
i 1
N2
Thermodynamic factor
Kinetic coefficient
A. Van der Ven, G. Ceder, Handbook of Materials Modeling, chapt. 1.17, Ed. S. Yip, Springer (2005).
R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/
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More familiar form
~JDD
˜ kT
ln x
DJ
1
2d t
1
N
R i t
i 1
N2
Thermodynamic factor
Self diffusion coefficient
R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/
J D C
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R i t
DJ
1
2d t
1
N
R i t
i 1
N2
Trajectories can be sampled with
kinetic Monte Carlo simulations
R j t
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Common approximation
~*DD
x
kT
ln
~
D*
Ri t2
2d t
Thermodynamic factor
Tracer diffusion coefficient
R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/
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Intercalation Oxide as Cathode in
Rechargeable Lithium Battery
LixCoO2
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Example:
LixCoO2
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Configurational Variables
Binary system alloy (A, B atoms)
Assign occupation variables to each position in crystal
1i
1i
if atom A occupies site i
if atom B occupies site i
Total of 2N configurations
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Polynomials of occupation variables
form a basis in configuration space
ji
m n o
kl
J. M. Sanchez et al Physica A128, 334, 1984
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Polynomials of occupation variables
form a basis in configuration space
ji
m n o
kl
E
Vo Vi i
i
Vi, j i j
i, j
Vi, j,k i j k
i, j,k
...
J. M. Sanchez et al Physica A128, 334, 1984
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First-principles energies of
a “few” excitations
Lattice Model Hamiltonian
Monte Carlo
Thermodynamics
E
Vo Vi i
i
Vi, j i j
i, j
Vi, j,k i j k
i, j,k
...
Z exp
E
kBT
F kBT ln Z
2
1
1 1
2
eN
i nuc i nuci ji i j
H V r E Rr r
Extrapolate to
all other excitations
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First principles energies (LDA)
of different lithium-vacancy configurations
A. Van der Ven, et al, Phys. Rev. B 58 (6), p. 2975-87 (1998).
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Cluster expansion for
LixCoO2
, , ,
, , ,
...o i i i j i j i j k i j k
i i j i j k
E V V V V
A. Van der Ven, et al, Phys. Rev. B 58 (6), p. 2975-87 (1998).
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Calculated LixCoO2 phase diagram
A. Van der Ven, et al, Phys. Rev. B 58 (6), p. 2975-87 (1998).
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Individual hops:
Transition state theory
kT
EBexp*
BE
Can be calculated with
First-principles
Density functional theory
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Migration mechanism in LixCoO2
Single vacancy hop
mechanism
Divacancy hop
Mechanism
Oxygen
Cobalt
Lithium
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Single vacancy hop Divacancy hop
Oxygen
Cobalt
Lithium
A. Van der Ven, G. Ceder, M. Asta, P.D. Tepesch, Phys Rev. B 64 (2001) 064112
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Many types of hop possibilities
in the lithium plane
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Migration barriers depend
configuration and concentration
Divacancy
mechanism
Single-vacancy
mechanism
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Barrier dependence on concentration
Co3+ Co4+
as lithium is removed:
increased electrostatic
repulsion for
Li+ in the activated state
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Calculated diffusion coefficient(First Principles cluster expansion + kinetic Monte Carlo)
JDDDiffusion coefficient
at 300 KThermodynamic
factor
A. Van der Ven, G. Ceder, M. Asta, P.D. Tepesch, Phys Rev. B 64 (2001) 064112
Due to Li-ordering
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Divacancy diffusion mechanism
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Divacancy diffusion mechanism
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Divacancy diffusion mechanism
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Divacancy diffusion mechanism
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Divacancy diffusion mechanism
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Divacancy diffusion mechanism
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Divacancy diffusion mechanism
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Divacancy diffusion mechanism
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Divacancy diffusion mechanism
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Divacancy separation
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Divacancy separation
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Diffusion in substitutional solids
Connecting atomistic hop mechanisms with macroscopic
diffusion coefficients
Real solids contain grain boundaries
and dislocations which serve as vacancy
sources and sinks
Diffusion within
a perfect crystal
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Individual hops:
Transition state theory
kT
EBexp*
BE
B
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Thermodynamic driving forces for
substitutional diffusion
BABAAAA LLJ ~~
BBBABAB LLJ ~~
VAA~
VBB~
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LAA
R A t
2
2d tVkT LBB
R B t
2
2d tVkT
LAB
R A t
R B t
2d tVkTLBA
Kubo Green formalism
BABAAAA LLJ ~~
BBBABAB LLJ ~~
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Kinetic coefficients (fcc lattice in dilute vacancy limit, ideal solution)
tMd
tRtR
L
ji
ij2
~
Kubo-Green
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Diffusion in Al-Li alloys
Dark field TEM
A. Kalogeridis, J. Pesieka, E. Nembach, Acta Mater 47 (1999) 1953
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fcc Al-Li alloy
Binary cluster expansion
, , ,
, , ,
...o i i i j i j i j k i j k
i i j i j k
E V V V V
1i
1i
Li at site i
Al at site i
Fit to LDA energies of
70 different Al-Li
arrangements on fcc
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Calculated thermodynamic and kinetic
properties of Al-Li alloy
First principles cluster expansion + Monte Carlo
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Equilibrium vacancy concentration(Monte Carlo applied to cluster expansion)
A. Van der Ven, G. Ceder, Phys. Rev. B71, 054102(2005)
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Vacancy surrounds itself by Al
Short range order around a vacancy
750 Kelvin
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Vacancies reside on lithium sublattice
in L12
Al
Li
600 Kelvin
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Migration barriers for lithium and
aluminum differ by ~150 meV
Li barriers
Al barriers
vAl*
4.5 1013
Hz
vLi*
7 1013
Hz
Calculated (LDA) in
107 atom supercells
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L-coefficients for Li-Al
contains an ordered phase
JLi LLiLi˜
Li LLiAl˜
Al
JAl LAlLi˜
Li LAlAl˜
Al
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Frequency of hop angles between
successive hops
Al
Li
A. Van der Ven, G. Ceder, Phys. Rev. Lett. 94, 045901 (2005).
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BABAAAA CDCDJ
BBBABAB CDCDJ
BABAAAA LLJ ~~
BBBABAB LLJ ~~
Perfect crystalFickian flux expressions
DAA DAB
DBA DBB
˜ L AA˜ L AB
˜ L BA˜ L BB
˜ A
kT
xA
˜ A
kT
xB
˜ B
kT
xA
˜ B
kT
xB
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Diagonalize the D-matrix
Yields a mode corresponding to
(a) density relaxation (vacancy mobility)
(b) interdiffusion
1
0
0EE
DD
DD
BBBA
ABAA
K. W. Kehr, et al, Phys. Rev. B 39, 4891 (1989)
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Physical meaning of modes + and -
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Physical meaning of modes + and -
Density fluctuations relax with a
time constant of +
K. W. Kehr, et al, Phys. Rev. B 39, 4891 (1989)
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Physical meaning of modes + and -
Density fluctuations relax with a
time constant of +
Compositional inhomogeneities
decay with a time constant of
K. W. Kehr, et al, Phys. Rev. B 39, 4891 (1989)
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Vacancy mobility and
interdiffusion
Random alloy
Thermodynamically ideal
AlxLi1-x alloy
Non-ideal thermodynamics
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The role of vacancy sources and
sinks
Real solids contain grain boundaries
and dislocations which serve as vacancy
sources and sinks
Diffusion within
a perfect crystal
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Uniform source sink
approximation
BABAAAA LLJ ~~
BBBABAB LLJ ~~
0V
AAA CDJ
BBB CDJ
0Vd
Gibbs-Duhem
Assume vacancy concentration
in equilibrium everywhere
0BBAA dxdx
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Lattice frame and laboratory frame of
reference
BAmVmlattice JJVJVv
VAAA JxJJ~
Fluxes in the laboratory frame
VBBB JxJJ~
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Lattice frame and laboratory frame of
reference
BAmVmlattice JJVJVv
VAAA JxJJ~
Fluxes in the laboratory frame
VBBB JxJJ~
BVBB CDJxJ~
BV CWJ~
BAAB DxDxD~
BA DDW~
Drift
Interdiffusion
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Comparisons of different treatments
Random alloy
B=100x AB=10x A
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Physical difference between D and
Perfect crystalPlenty of vacancy
sources and sinks
~
D~
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Al
Li
Calculated interdiffusion
coefficient
A. Van der Ven, G. Ceder, Phys. Rev. Lett. 94, 045901 (2005).
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Conclusion
• First-principles methods can be used to
comprehensively characterize diffusion
in interstitial and substitutional alloys
• The role of vacancy sources and sinks
needs to be further investigated (both
theoretically as well as experimentally)
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Available migration mechanisms for
each lithium ion
Channels into a
divacancy
Number of vacancies
around lithium
Channels into
isolated vacancies
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Diffusion occurs with a divacancy
mechanism