Supplementary Information for “First-Principles Studies on
Vacancy-modified Interstitial Diffusion Mechanism of Oxygen in
Nickel, Associated with Large-Scale Atomic Simulation Techniques”
H. Z. Fang 1,2 S. L. Shang 1.2, Y. Wang 1,2 and Z. K. Liu 1,2, D. Alfonso 1,3 and D. E. Alman 1,3, Y.
K. Shin 1,4, C. Y. Zou 1,4 and A. C. T. van Duin 1,4, Y. K. Lei 1,5 and G. F. Wang 1,5
1
National Energy Technology Laboratory Regional University Alliance
2
Department of Materials Science and Engineering, The Pennsylvania State University,
University Park, PA 16802, USA
3
National Energy Technology Laboratory, U.S. Department of Energy, Pittsburgh, PA 15236,
USA
4
Department of Mechanical and Nuclear Engineering, The Pennsylvania State University,
University Park, PA 16802, USA
5
Department of Mechanical Engineering and Materials Science, University of Pittsburgh, PA
15261, USA
1. Kinetic Monte Carlo Method
The KMC simulation is based on Bortz-Kalos-Lebowitz (BKL) algorithm
1-3
.In this
scheme, the entire system is scanned to order to inventory all the possible diffusion processes at
the current system configuration. The rate of a particular diffusion process ki is calculated and
N
the overall rate K is obtained by adding up the rates of all N possible processes ( K   k i ). A
i 1
1
random number  1 , with a random distribution in the range [0,1], is generated and a process j
that satisfies the following condition
j 1
k
i 1
j
i
  1 K   ki
(S1)
i 1
is chosen. After the system is updated, the time is advanced according to
t   ln(  2 ) / K
(S2)
where  2 is another random number in the range [0, 1]. From the updated configuration, a new
process table is determined, and the whole procedure starts again. Following this scheme, the
system can be propagated step by step, starting from an initial configuration at time t0 to the next
one at time t1 and so on.
A periodic Ni supercell consisting of 10×10×10 fcc cubes containing one O and one
vacancy is employed. Diffusion occurs by two mechanisms – a vacancy hop and a interstitial hop
for O atom in the fcc Ni. The microscopic rate constant for a hop ki associated with a process i is
calculated according to the transition state theory:
k i   i exp(
 EiTS
k BT
)
(S3)
where k B is the Boltzmann constant, i is the pre-exponential factor, and EiTS denotes the energy
barrier for process i. The EiTS depends on the immediate environment of the diffusing particles.
This effect is accounted for by calculating the effective interaction energy J between O and
vacancy up to a certain distance R in the initial and final configurations and then calculating the
energy barrier using a Kinetic Ising based model 4.
EiTS  Eio,TS  ( J R final  J R init )
(S4)
2
where E io ,TS is the DFT-LDA calculated activation energy of O or Ni diffusion when O and
vacancy are very far from each other. J R final and J R init are the effective interaction energy in the
final and initial states. The effective interaction energy between O and vacancy is extracted from
DFT-LDA calculations using
J R n  E tot ( Nibulk [O  vac] n )  E tot ( Nibulk [0  vac]  )
(S5)
where the two terms in the right hand side are the energies of Ni matrix containing O and
vacancy in the nth nearest neighbor shell and two infinitely separated O atom and vacancy. The
results are summarized in Figure 7 in the manuscript. A 3×3×4 periodic fcc supercell was used to
obtain these values. Based on Figure 7, up to third neighbor interaction was included in the
model. The interactions beyond this are ignored since they are very small. Using the theory of
random walks, the O diffusion parameter is determined through the mean square displacement
(MSD) as:
1
D  lim
t  6t


2
r(t  t )  r(t )
(S6)
t 0


where t is the simulation time, the vector r (t  t )  r (t ) represents the change in position after a
time interval t , and the angular brackets indicates the average over all steps.
The vacancy concentration in the simulation cell is generally much higher than the
equilibrium concentration at T calculated using Eq. 9 in the manuscript. To obtain diffusion
parameters at vacancy concentration comparable with the experimental data, we follow the
correction scheme in Ref. 5. For each T, additional calculations were done using different
supercell sizes. Linear fit of the predicted O diffusion parameters and supercell vacancy
concentrations was then performed. From this linear fit, the O diffusion parameter can be
determined at the vacancy concentration obtained from Eq. 9.
3
2. MEAM potential
6-7
A second nearest neighbor (2NN) modified embedded atomic method (MEAM)
potential has been developed to describe the interaction between Ni and O. The parameter of this
potential is given in Table S1 and Table S2. The properties of the Ni-O system predict by this
potential is shown in Table S3 comparing to density functional theory (DFT) calculations. Both
the MEAM potential and the DFT calculations indicate that the vacancy increased the migration
energy of O in Ni.
Table S1 Parameters of the MEAM potentials for the Ni-O system. The reference structures of Ni, O and
Ni-O systems are face centered cubic, dimer and NiO B1 crystal, respectively. The cohesive energy 𝑬𝒄
(eV/atom), the equilibrium nearest-neighbor distance 𝒓𝒆𝒒 (Å), the decaying exponential factor of the
universal energy 𝜶, the scaling factor of the embedding function 𝑨, four decaying exponential factors of
the atomic charge density 𝜷(𝒊) , four weighting factors of the atomic charge density 𝒕(𝒊), and the scaling
factor of the atomic charge density 𝝆𝟎 .
𝜶
𝒓𝒆𝒒
𝑬𝒄
𝑨
𝜷(𝟎)
𝜷(𝟏)
𝜷(𝟐)
𝜷(𝟑)
𝒕(𝟎)
𝒕(𝟏)
𝒕(𝟐)
𝒕(𝟑)
𝝆𝟎
Ni
4.99 2.49 4.45
1.10 2.45
1.50
6.00
1.50
1.00 5.79
1.60
3.70
1.10
O
3.60 1.21 2.56
2.30 2.20
2.20
5.00
7.00
1.00 -3.00
6.00
1.50
13.6
Ni-O
5.60 2.10 4.18
-
-
-
-
-
-
-
-
-
-
Table S2 Angular screening factors of the atomic charge density in Ni-O system.
Ni-Ni-NI
Ni-O-Ni
O-Ni-Ni
O-O-Ni
O-Ni-O
O-O-O
𝑪𝒎𝒂𝒙
2.80
2.06
3.70
2.80
2.80
2.80
𝑪𝒎𝒊𝒏
0.80
0.46
2.00
0.80
0.80
0.80
4
mig
Table S3 Solution energy EOsol and migration energy EO of O in Ni. The units of all the energy are eV.
mig
The definition of EOsol and EO is given in the results section.
MEAM
DFT
EOsol at octahedral site
-3.61
-3.63
EOsol at tetrahedral site
-3.26
-3.28
EOsol at substitutional site
-2.98
-2.31
EOmig for O-O path
1.52
1.40
EOmig for O-T path
1.21
0.93
EOmig for O-O path with Vacancy
1.95
2.30
EOmig for T-O path with Vacancy
1.70
1.36
Molecular dynamic (MD) simulations are used to determine the diffusion coefficient of O
in Ni. Two models have been built to simulate the O diffusion in Ni with and without vacancy.
For the O diffusion in Ni without vacancy, a supercell of 4000 Ni atoms has been built as the
matrix, and an O atom has been put into an octahedral site. For the O diffusion in Ni with
vacancy, a vacancy has been created next to the O atom. The MD simulations are performed on
these two models at constant temperature for 1 ns. The MSD of the O atom is calculated for
different diffusion time intervals. And a linear fitting between the MSD and the diffusion time
has been made. The diffusion coefficient is obtained from the slope of the line divided by 6. All
the simulation and the MSD analysis have been done from 1200K to 1700K.
5
3. ReaxFF MD simulations 8-9
For the purpose of investigating the effect of vacancy on the diffusivity of oxygen in Ni,
two systems are prepared: system I contains 255 Ni atoms, 1 vacancy and 1 oxygen atom
initially placed at the octahedral site near the vacancy; system II contains 256 Ni atoms (without
vacancy) and 2 oxygen atoms initially placed at two randomly chosen octahedral sites within the
lattice. Both systems are first equilibrated in NPT ensemble (where particle number, pressure and
temperature are kept constant) at targeted temperature for 2.5 ps allowing free volume
adjustment and system equilibration at that temperature; after the system has been fully
equilibrated, the simulation was restarted from the end point of the NPT stage and was switched
to NVE ensemble (where particle number, volume and system energy are conserved). The MSD
of the oxygen atoms was calculated and averaged over all the oxygen atoms in the system. The
diffusion coefficient was calculated according to Eq. S6.
4. Complementary first-principles results
When the oxygen jumps onto the 2nd nearest-neighboring octahedral site (O2) with respect
to the vacancy, the binding effect of vacancy on the migration barriers for oxygen to diffuse
away becomes insignificant. As shown in Figure S1, the migration barrier of oxygen jumping
from O2 site to the 4th nearest-neighboring octahedral site (O4) is about 0.92 - 1.17 eV, which is
comparable to the barrier of perfect O-T-O jump in Ni without vacancy (0.93 eV).
6
1.4
1.4
TS
1.2
TS
1.0
1.0
0.8
1.17 eV
0.8
0.6
0.92 eV
Migration barrier, eV
1.2
0.4
0.4
T3
O4
0.2
0.0
0.6
O2
0.2
0.0
Reaction coordinate
Figure S1 Migration energy profile as the oxygen jumping from O2 to O4 site by passing the T3 site. TS
denotes the transition state.
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2
3
4
5
6
7
8
9
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