Embedded-atom potential for hcp and fcc cobalt

PHYSICAL REVIEW B 86, 134116 (2012)

Embedded-atom potential for hcp and fcc cobalt

G. P. Purja Pun* and Y. Mishin†
Department of Physics and Astronomy, MSN 3F3, George Mason University, Fairfax, Virginia 22030, USA

(Received 14 August 2012; published 22 October 2012)

We report on the development of an embedded-atom interatomic potential representing basic properties of both
the hcp and the fcc phases of cobalt with nearly equal accuracy. The potential also reproduces the structural phase
transformation between the two phases at a temperature close to the experimental value. The proposed potential
can be used for large-scale atomistic simulations of cobalt microstructures over a wide range of temperatures. In
a more general context, it offers a model for studying thermodynamic and kinetic properties of hcp/fcc interfaces
and microstructure evolution in two-phase materials.

DOI: 10.1103/PhysRevB.86.134116 PACS number(s): 61.50.Ah, 61.50.Ks, 64.70.K−, 64.30.Ef

I. INTRODUCTION Ti, Ru, Hf, Zn, Mg, and Be.31 The overwhelming majority of
these potentials were designed to reproduce only properties of
Cobalt is a 3d transition metal, which is widely used the hcp phase. The attention was usually focused on the lattice
constant, cohesive energy, elastic constants, and the vacancy
in many magnetic applications, including magnetic shape formation energy. Only in a few cases did the authors examine
memory alloys.1–7 It belongs to the class of allotropic solids the wider set of properties of the hcp phase.29,30 Besides the
hcp phase, it was usually only checked that the fcc and bcc
along with other technologically important metals, such as structures were less stable than hcp at 0 K. The only exception
known to us is the EAM potential for Zr, which correctly
Ti, Fe, Zr, Sn, and U. The equilibrium crystalline structure predicts the hcp-bcc phase transformation at a temperature
close to the experiment and reproduces a number of physical
of Co is hexagonal close packed (hcp) below 700 K and properties of both phases.30 For Co, however, the existing
potentials22,23,28,31 were only fitted to properties of the hcp
becomes face-centered cubic (fcc) above this temperature. This phase with limited or no testing for properties of other phases.
These potentials do not reproduce the hcp-fcc transformation
structural phase transformation has a very small transformation with the temperature, which prevents them from being useful
enthalpy H hcp→fcc ≈ 0.004 eV/atom (Refs. 8 and 9) and for studying the mechanisms of this transformation and its
occurs by a martensitic (diffusionless) mechanism. Relatively effect on microstructure and properties of Co.

thick films of metastable fcc Co can be obtained even at In this paper, we develop an EAM potential for cobalt,
room temperature by epitaxial growth on a Cu(110) surface.10 which reproduces not only properties of both the hcp and the
fcc phases, but also the structural transformation between them
Furthermore, ferromagnetic body-centered cubic (bcc) Co at a temperature, which is very consistent with the experiment.
layers as thick as 357 A˚ can be grown epitaxially on a GaAs The results of testing this potential suggest that it can be
substrate.11 A recent paper12 reports on yet another metastable employed in future MD simulations of thermal and mechanical
processing as well as the microstructure development in cobalt.
phase, called , with a complex cubic structure (space group
II. METHODOLOGY OF POTENTIAL DEVELOPMENT
P 4132) containing 20 atoms in a unit cell. Its cohesive energy
is reported to be slightly larger than that of the magnetic fcc In the EAM formalism, the total energy of a collection of
atoms is represented by20
Co, indicating possible transformations into the hcp or fcc
E= 1 φ(rij ) + F (ρ¯i), (1)
structures. 2
The microscopic origin and mechanisms of the marten- i,j i

sitic hcp-fcc phase transformation have been studied both where φ(rij ) is a pair-interaction potential as a function of
theoretically13,14 and by inelastic neutron scattering15,16 and distance rij between atoms i and j and F (ρ¯i) is the embedding
other experimental methods.17 Frey et al.15 and Strauss energy as a function of the host electron density ρ¯i on atom i.
et al.16 did not find any softening of the T [ζ ζ 0] and T [ζ ζ ζ ] The host electron density,
branches of the phonon dispersion that often accompanies such
transformations. In fact, no precursor effect was observed ρ¯i = ρ(rik) (2)
during the transition.15 In another interesting experiment, it
was shown that the hcp-fcc transformation can be suppressed in k=i
small particles of fcc Co synthesized at ambient temperatures
during slow heating and cooling between the temperatures of is the sum of atomic electron densities created at site i by all
28 and 293 K.18 other atoms of the system.

Many technologically important properties of cobalt de-
pend not only on its crystalline structure, but also on the
microstructure, which includes dislocations, grain bound-
aries, phase boundaries, and other defects. One of the
most efficient approaches for studying the microstructure
is offered by large-scale molecular dynamics (MD) simu-
lations with semiempirical interatomic potentials.19 Several
embedded-atom methods20 (EAMs) and modified embedded-
atom methods21 potentials have been developed for hcp
metals.22–30 In addition, Finnis-Sinclair-type many-body po-
tentials have been developed for hcp metals, such as Co, Zr,

1098-0121/2012/86(13)/134116(9) 134116-1 ©2012 American Physical Society

G. P. PURJA PUN AND Y. MISHIN PHYSICAL REVIEW B 86, 134116 (2012)

In this paper, the pair interactions were described by the 0.20
generalized Lennard-Jones-type function,
0.10
φ(r) = ψ r − rc V0 b2 − b1 +δ ,
h b2 − b1 (r/r1)b1 (r/r1)b2 (3) 0.00

where rc, h, V0, b1, b2, r1, and δ are fitting parameters. This φ (eV) −0.10
function is smoothly truncated at the cutoff distance r = rc
−0.20
using the truncation function,
−0.30
ψ(x) = x4 , x < 0, (4) 1.5 2.5 3.5 4.5 5.5 6.5
1+x4
r (Å)
0, x 0.
0.04
The analytical form of the electron-density function was 0.03
chosen to be 0.03
0.02
ρ(r) = ψ r − rc [A0zy exp−γ z(1 + B0 exp−γ z) + C0], (5) 0.01
h 0.01
0.01
where z = r − r0 with the parameters A0, y, γ , B0, C0, and 0.00ρ (arb. units)
r0. These parameters are not independent because the electron
density can be scaled by an arbitrary factor without changing 1.5 2.5 3.5 4.5 5.5 6.5
the total energy or forces.32,33 To remove this arbitrariness
during the fitting, A0 was chosen to give ρ¯ = 1 in equilibrium r (Å)
hcp Co with the ideal c/a ratio.
0.0
The embedding function of the potential was not defined
−0.5
analytically. Instead, it was constructed numerically from the
−1.0
requirement that the energy per atom of hcp Co with the ideal
c/a should follow Rose’s universal equation of state,34 −1.5

E(a) = Ec(1 + αx) exp−αx , (6) −2.0

where x ≡ a/a0 − 1 and α ≡ (−9 0B/Ec)1/2. The param- −2.5F (eV)
eters a0, 0, Ec, and B represent, respectively, the lattice
constant, atomic volume, cohesive energy, and bulk modulus 0.0 0.2 0.4 0.6 0.8 1.0 1.2

of equilibrium hcp Co with the ideal c/a ratio at 0 K. ⎯ρ (arb. units)
FIG. 1. (Color online) Potential functions of the EAM Co
The above parametrization describes the Co potential with potential developed in this paper. (a) Pair interaction potential,
(b) Electron-density function, and (c) Embedding energy function.
a total of 12 adjustable parameters. They were determined
fitted data included a mix of experimental and ab initio
by minimizing the weighted mean-square deviation of a set properties of hcp Co as detailed in the tables below. When
several target values were available, the fit was to their average.
of properties from their target values using the simulated Note that properties of fcc Co were not directly included
annealing algorithm applied in our recent papers.35–40 The

TABLE I. Optimized fitting parameters of the proposed EAM
potential for cobalt.

Parameter Value

rc (A˚ ) 0.649 9539 × 101
h (A˚ ) 0.142 7612 × 101
V0 (eV) −0.615 0127 × 100
r1 (A˚ ) 0.243 6873 × 101
b1 0.117 3323 × 102
b2 0.349 8429 × 101
δ (eV) −0.261 8860 × 101
A0 0.127 4313 × 102
y 0.110 0594 × 100
γ (A˚ −1)
B0 0.191 3083 × 101
C0 0.358 5129 × 101
r0 (A˚ ) 0.205 0327 × 10−1
−0.216 0589 × 101

134116-2

EMBEDDED-ATOM POTENTIAL FOR hcp AND fcc COBALT PHYSICAL REVIEW B 86, 134116 (2012)

in the fit. Instead, several potentials were generated, which strains and fitting the energy change with a parabolic function.
reproduced hcp Co with about the same accuracy. They were After each application of strain, the structure was relaxed with
then tested for fcc-Co properties, the hcp-fcc transformation respect to the internal degrees of freedom of the unit cell.44
temperature, and the potential, which demonstrated the best As indicated in Table II, the elastic constants predicted by the
accuracy for both phases was selected as final. The best two potential are in accurate agreement with the existing databases.
to three potentials gave about the same quality of fit, and the The surface energy γs for the basal-plane orientation, the
selection of the final version was based on the accuracy of the intrinsic stacking fault energy γI2 , the vacancy formation (Evf ),
phase transformation temperature. and migration (Evm) energies are in reasonable agreement with
available experimental and first-principles data. This is not
The optimized fitting parameters are listed in Table I, and very surprising given that this information was included in the
the plots of the potential functions are shown in Fig. 1. The fitting process. What is more interesting is that the properties
tabulated forms of the functions can be downloaded from of fcc Co, which were not fitted to directly, came out to be in
the National Institute of Standards and Technology (NIST) equally good, if not better, agreement with the experimental
Interatomic Potentials Repository.41,42 and first-principles data (Table III). This agreement attests
to a good transferability of the proposed potential to various
III. RESULTS AND DISCUSSIONS alternate structures, an important property, which is further
examined below.
A. Properties of hcp and fcc Co
Phonon-dispersion curves at 0 K were computed for both
Table II summarizes the physical properties of hcp Co hcp (Fig. 2) and fcc Co (Fig. 3) by Fourier transformation of the
predicted by the potential in comparison with experimental dynamical matrix in several high-symmetry directions. For hcp
and first-principles data. All these properties were calculated Co, the potential accurately reproduces the low- and medium-
after full relaxation at 0-K temperature. The vacancy migration frequency regions of the phonon spectrum but overestimates
barrier was computed for a jump in the basal plane using the frequencies of the optical branches in comparison with
the nudged elastic-band method.43 The elastic constants cij room-temperature neutron-scattering experiments45 (Fig. 2).
were computed by applying a set of small volume-conserving

TABLE II. Properties of hcp Co computed with the present EAM potential in comparison with experimental data and first-principles
calculations. The asterisks mark properties included in the potential fit.

Property Ab initio Experiment EAM
−4.72,a −5.49,b −3.66b
Ec (eV/atom)∗ −4.39c −4.3910
a (A˚ )∗ 2.476,d 2.5007e 2.507f 2.5187
c/a∗ 1.642,b 1.614,d 1.622e 1.623f 1.6103
B (GPa)∗ 191.4c
c11 (GPa)∗ 353,g 331,h 297h 293,h 319.50,i 307.1j 195.1
c12 (GPa)∗ 188,g 161,h 131h 143,h 166.09,i 165.0j 311.9
c13 (GPa)∗ 116,g 104.0,h 82h 90,h 102.09i 102.7j 146.9
c33 (GPa)∗ 443,g 369,h 342h 339,h 373.60,i 358.1j 119.6
c44 (GPa)∗ 78,h 82.41,i 75.5j 359.4
γs (0001) (mJ/m2) 63,g 94,h 96h 2550k 91.7
γI2 (mJ/m2) 2315
Evf (eV)∗ −0.038d 27l 39.8
Evm (eV) 1.4,k 1.38m
T hcp→fcc (K) 1.49
690,n 695,o 700p 0.98
H fcc→hcp (eV) −0.004,o −0.0048p 717
−0.0061
aReference 61.
bReference 12.
cReference 50.
dReference 62.
eReference 63.
fReference 64.
gReference 65.
hReference 66.
iReference 67.
jReference 68.
kReference 69.
lReference 70.
mReference 71.
nReference 55.
oReference 9.
pReference 8.

134116-3

G. P. PURJA PUN AND Y. MISHIN PHYSICAL REVIEW B 86, 134116 (2012)

TABLE III. Properties of fcc Co computed with the present EAM potential in comparison with experimental data and first-principles
calculations. These properties were not included in the potential fit.

Property Ab initio Experiment EAM
−3.5446e
Ec (eV/atom) −4.35,a −5.48,b −3.79b 3.5447,f 3.568g −4.3849
a (A˚ ) 3.4806,c 3.54,d 3.518b 260,i 225,j 223k 3.5642
160,i 160,j 186k
c11 (GPa) 325h 110,i 92,j 110k 275.7
189h 1.34-1.91l 158.9
c12 (GPa) 156h 108.2
2.34,d 1.71d 1770,m 1768n
c44 (GPa) 1.19,d 0.98d 1.56
Evf (eV) 0.95
Evm (eV) 2470
γs (100) (mJ/m2) 2604
γs (110) (mJ/m2) 2333
γs (111) (mJ/m2) 1898

Tm (K)

aComputed from Ecfcc − Echcp = 0.038 eV (Ref. 62).
bReference 12.
cReference 62.
dReference 72.
eReference 73.
fReference 74.
gReference 75.
hReference 65.
iReference 76.
jReference 77.
kReference 16.
lReference 78.
mReference 50.
nReferences 9 and 8.

It should be noted that quite similar discrepancies were found 4000-atom periodic cell. The agreement with experimental
in all previous calculations for hcp metals.31,46–48 A similar data is reasonable (Fig. 4), especially given that thermal
comparison was found for fcc Co with accurate agreement expansion factors were not in the potential fit.
at relatively low frequencies and significant discrepancies at
high frequencies (Fig. 3). In this case, however, part of the The melting temperature Tm of fcc Co was calculated by
discrepancies could be attributed to the fact that the neutron- the solid-liquid interface method used in our previous paper.36
scattering experiment was performed for a Co0.92Fe0.08 not a A periodic simulation block containing 20 000 atoms with
pure Co.49 approximately equal amounts of the solid and liquid phases
separated by a (001) interface was prepared by melting half of
Thermal expansion factors of both phases were computed the initially solid phase. The system was brought to mechanical
by zero-pressure canonical (NPT) MD simulations of a

Γ [00ζ] X [ζζ0] Γ [ζζζ] L
10
11 Γ [ζζ0] K M [ζ00] Γ [00ζ] A
10 0.1 0.2
Phonon frequency (THz) 8
Phonon frequency (THz)9
8 6
7
6 4
5
4 2
3
2 0 0.5
1 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 0.0
0 0.3 0.4 0.5 0.3 0.10.0 0.5
0.0 Reduced wave vector ζ Reduced wave vector ζ

FIG. 2. (Color online) Phonon-dispersion curves of hcp Co FIG. 3. (Color online) Phonon-dispersion curves of fcc Co
predicted by the present EAM potential at 0 K (lines). The points
predicted by the present EAM potential at 0 K (lines) and obtained represent experimental neutron-scattering measurements for the
by neutron-scattering measurements at room temperature45 (points). Co0.92Fe0.08 alloy at room temperature.49

134116-4

EMBEDDED-ATOM POTENTIAL FOR hcp AND fcc COBALT PHYSICAL REVIEW B 86, 134116 (2012)

2.00 data were found in the literature that could be used for
comparison with the potential predictions. The latter are,
1.75 hcp-Co, a-axis therefore, reported for the purpose of comparison with future
hcp-Co, c-axis first-principles calculations. Nevertheless, the overall ranking
of the structural energies correlates well with the known
Linear thermal expansion (%) 1.50 hcp-Co, a-axis ranking for other fcc and hcp metals.38,51–53 The bcc and
hcp-Co, c-axis diamond structures are predicted to be mechanically unstable,
whereas, all other structures are mechanically stable.
1.25 fcc-Co
The critical test of the proposed potential was its ability
fcc-Co to reproduce the hcp-fcc transformation. This test required
1.00 calculation of the free-energy difference between the phases as
a function of temperature. The methodology of this calculation
0.75 was similar to the one employed in our previous papers.38,40
Two separate hcp and fcc simulation blocks were created, each
0.50 with approximately 4000 atoms with all-periodic boundary
conditions. The blocks were equilibrated at several tempera-
0.25 tures ranging from 50 to 1200 K by running MD simulations in
the zero-pressure NPT ensemble with independent variations
0.00 in the block sizes in all three directions. The average enthalpy
H per atom was recorded at each temperature, and the enthalpy
-0.25 difference between the phases H (T ) = H fcc(T ) − H hcp(T )
was fitted by the function,
-0.50 200 400 600 800 1000 1200
0 Temperature (K)

FIG. 4. (Color online) Linear thermal expansion factors of hcp H (T ) = H0 + AT + BT 2, (7)
and fcc Co relative to room temperature (293 K) predicted by
the present EAM potential (points) in comparison with experiment with adjustable constants H0, A, and B. The enthalpy is related
(lines).55 The vertical dashed line marks the experimental hcp-fcc to the Gibbs free energy G(T ) through the Gibbs-Helmholtz
transformation temperature.
equation,
equilibrium by a short NPT MD run at zero pressure and a
temperature close to the expected Tm. The simulation ensemble ∂[G(T )/T ] =− H (T ) . (8)
was then switched to microcanonical (NVE), and the system
was allowed to reach full equilibrium. The equilibrium was ∂T p T2
typically reached at some temperature Tm different from
the initial guess and some nonzero pressure p. This NVE- Integration of this equation gives the free-energy difference
equilibration process was generally accompanied by partial between the phases,
melting or crystallization of the solid phase. This NPT-NVE
cycle was repeated a number of times, producing a set of G(T ) = T + H0 T − BT (T − T0)
temperatures and corresponding positive or negative pressures G0 T0 1−
representing different two-phase equilibrium states. The plot of
Tm versus p thus obtained was interpolated to p = 0 to predict T0
the zero-pressure melting point of the material. The obtained
Tm = 1898 K overestimates the experimental melting point − AT ln T . (9)
of fcc Co (1770 K).50 It should be considered, however, that
semiempirical potentials rarely predict the melting point more T0
accurately, unless the latter is directly included in the potential
fit (which was not the case in this paper). Attempts to estimate Here, G0 is the integration constant representing the free
the melting temperature of hcp Co were unsuccessful due to the energy at a chosen reference temperature T0. If T0 is chosen low
thermodynamic instability of this phase at high temperatures enough, then G0 can be obtained by a harmonic calculation.54
(see below). In this paper, we used T0 = 300 K.

B. Structural stability and phase transformation The computed functions H (T ) and G(T ) are shown in

The cohesive energies and equilibrium lattice constants Fig. 5. The harmonic calculations for hcp and fcc Co were
of several alternate structures of Co are summarized in
Table IV. The hcp, fcc, bcc, and structures were observed performed on periodic supercells with 800 and 864 atoms,
experimentally (Sec. I), whereas, the rest of the structures were
imaginary and were only included for testing the potential. respectively. The calculation of G0 was based on classical
Note the significant scatter of the available experimental and statistics. The plot clearly shows a change in sign for G(T )
first-principles data for the experimentally observed phases. at the temperature of T hcp→fcc = 717 K. Thus, the hcp phase
The cohesive energies and lattice constants of these phases is predicted to be stable below T hcp→fcc, whereas, the fcc
computed with the EAM potential are very consistent with the
existing data. For the imaginary structures, no first-principles phase is above this temperature. The experimental phase
transformation temperature is around 700 K.8,9,55

It should be mentioned that the calculated transformation

temperature slightly depends on the choice of the reference

temperature T0. Variations in T0 within the physically reason-
able limits of ±100 K produce variations in T hcp→fcc by about
±3 K. Furthermore, purely harmonic calculations of G over
a wide temperature range above and below T0 give practically
the same values as the thermodynamic integration by Eq. (9).

This indicates that the proposed potential is highly harmonic

134116-5

G. P. PURJA PUN AND Y. MISHIN PHYSICAL REVIEW B 86, 134116 (2012)

TABLE IV. Summary of cohesive energies and lattice constants of stable, metastable, and unstable phases of Co. The properties included
in the potential fit are marked by asterisks.

Structure Property Ab initio Experiment EAM

hcp Ec (eV)∗ −4.72,a −5.49,b −3.66b −4.39c −4.3910
a (A˚ )∗ 2.476,d 2.5007e 2.507f 2.5187
fcc c/a∗ 1.623f 1.6103
1.614,d 1.642,b 1.622e −3.5446g
bcc Ec (eV) −4.35,d −5.48,b −3.79b 3.5447,i 3.568j −4.3849
a (A˚ ) 3.4806,d 3.54,h 3.518b 3.5642
phase 2.82,m 2.827,n 2.825o
Ec (eV) −4.23b −4.3273
A15 a (A˚ ) 2.74,k 2.82l 6.097p 2.815
−5.45,b −3.70b
Simple cubic Ec (eV) 6.057,b 6.128b −4.294
a (A˚ ) 6.128
Diamond
Ec (eV) −4.2271
aReference 61. a (A˚ ) 4.5481
bReference 12.
cReference 50. Ec (eV) −3.7608
dReference 62. a (A˚ ) 2.3711
eReference 63.
fReference 64. Ec (eV) −2.7275
gReference 73. a (A˚ ) 5.3436
hReference 72.
iReference 74.
jReference 75.
kReference 79.
lReference 65.
mReference 80.
nReference 11.
oReference 81.
pReference 82.

and provides additional reassurance in the correctness of the amounts of stacking faults near the solid-liquid interface
predicted transformation temperature. until the solid structure adjacent to the interface eventually
became predominantly fcc. Although this rendered further
To further demonstrate that fcc Co becomes the ther- melting point calculations of hcp Co meaningless, this local
modynamically most stable phase at high temperatures, we transformation to the fcc structure was very consistent with the
created a fully periodic liquid block with dimensions 35.6 A˚ × expected greater stability of this phase at high temperatures.
178.2 A˚ × 35.6 A˚ containing about 20 000 atoms. After a
long anneal above the melting point intended to erase any IV. DISCUSSION AND CONCLUSIONS
memory of the previous history, the block was cooled down
to a temperature of 1500 K, which was below Tm. The melt Many-body interatomic potentials20,56 have been widely
crystallized forming two large grains, one of which eventually used for materials’ simulations for nearly three decades.19
disappeared (Fig. 6). Structural analysis shows that the solid One of the existing challenges in this field is the development
phase predominantly has a fcc structure with a large amount of reliable potentials capable of reproducing phase equilibria
of stacking faults. The fact that the melt crystallizes to the and phase transformations. Despite the availability of a large
fcc structure confirms its larger stability at this temperature in number of potentials for single-component metals, few of
comparison with the hcp and other phases. them are capable of reproducing phase transformations with
temperature or pressure.30,57 This limits the applicability
Additional evidence comes from our failed attempts to of atomistic simulations to multiphase materials and, in
calculate the melting point of hcp Co. Following the same particular, to phase boundaries.
methodology as described in Sec. III A, a simulation block was
prepared with approximately equal amounts of hcp Co and the In this paper, we developed an EAM potential, which reli-
melt with the basal-plane orientation of the solid-liquid inter- ably reproduces some of the basic physical properties of hcp
face. During the subsequent equilibrating anneals in the NVE and fcc cobalt, including the lattice constants, cohesive ener-
ensemble, the interface position spontaneously fluctuated up gies, elastic constants, vacancy formation and migration ener-
and down by melting and resolidification of thin solid layers. gies, stacking faults, surface energies, and thermal expansion.
These fluctuations resulted in the accumulation of increasing

134116-6

EMBEDDED-ATOM POTENTIAL FOR hcp AND fcc COBALT PHYSICAL REVIEW B 86, 134116 (2012)

0.007

0.006

0.005 ΔH
0.004

ΔH, ΔG (eV) 0.003 Reference
0.002
0.001 ΔG

0.000

-0.001 FIG. 6. (Color online) Typical snapshots of MD simulations of
Co crystallization at 1500 K. (a) Initial melt. (b)–(d) Evolution of
-0.002 the structure every 10 ps. The visualization is based on the common-
neighbor analysis83,84 with fcc-type atomic environments shown in
-0.003 200 400 600 800 1000 1200 the dark colors and the stacking faults shown in the bright color.
0 T (K)
transformation near the solid-liquid interface. This inability to
FIG. 5. (Color online) Differences in the enthalpy H (T ) = compute the melting point of the metastable hcp phase can be
H fcc − H hcp and Gibbs free energy G(T ) = Gfcc − Ghcp between explained by the relative ease of the hcp-fcc transformation
the fcc and the hcp phases of Co as functions of temperature. The by the formation of stacking faults in comparison with the
square symbols represent the G values obtained by harmonic hcp-bcc transformation in Zr. The latter occurs by a different
calculations, whereas, the solid line is the result of thermodynamic mechanism and apparently has a larger barrier. The fact that the
integration by Eq. (9). The close agreement between the two fcc Co phase is more stable than hcp Co at high temperatures
calculations of G indicates that the proposed interatomic potential was also confirmed by examining the solidification structures.
is very harmonic.
We suggest that the Co potential developed in this paper
The experimental phonon-dispersion curves, without fitting, can be used for atomistic simulations of two-phase structures
are reproduced with reasonable accuracy. The melting tem- as a simple model of Co-based alloys, particularly, those for
perature of Co, without fitting, is also in reasonable agreement magnetic shape memory and other magnetic applications.1–7
with the experiment. Overall, the potential describes properties Although the EAM model does not capture magnetism
of hcp Co more accurately than the previous potentials22,23,28,31 directly, understanding of microstructure development through
(discussions of specific properties of the previous potentials atomistic simulations can be an important component in
can be found in the literature22,23,28,31 and are not repeated magnetic alloy design. A further step towards more realistic
here). In contrast to the existing potentials, the potential modeling would be to cross this Co potential with, e.g., Ni, Al,
proposed here reproduces properties of both hcp and fcc phases and/or other elements to obtain alloy potentials for binary and
with nearly equal accuracy. Furthermore, it demonstrates a multicomponent Co-based systems.
good transferability to various structural environments as
evidenced by the comparison of its structural energies and Finally, it should be mentioned that the current under-
the tests of a wide range of structure-sensitive properties. standing of the atomic structure, thermodynamics, and kinetic
Transferability is probably the most important property of properties of phase boundaries remains rather incomplete.58
atomistic potentials, which ultimately determines their utility Earlier MD studies of the motion of the hcp-fcc martensitic
and the range of possible applications.33 transformation front relied on highly simplistic pairwise
potentials.59,60 The proposed Co potential offers the op-
The proposed potential reproduces the hcp-fcc phase portunity to study the dynamics and mechanisms of the
transformation at a temperature of 717 K, which is close to the transformation front motion in realistic materials that could
experimental value of 700 K. This transformation temperature be compared with experimental observations.
was computed by thermodynamic integration using harmonic
reference free energies of the phases at 300 K. An alternative ACKNOWLEDGMENTS
approach would be to use the crystal-melt coexistence points
for the two phases as reference states as was performed for This work was supported by the National Aeronautics and
the hcp-bcc transformation in Zr.30 In our case, however, Space Administration through the Langley Research Center
this approach was precluded by the spontaneous hcp-fcc (Grant No. NRA # NNX08AC07A).

*[email protected] 2Z. H. Liu et al., J. Phys. D 37, 2643 (2004).
†[email protected] 3P. J. Brown et al., J. Phys.: Condens. Matter. 17, 1301 (2005).
1Y. Tanaka, T. Ohmori, K. Oikawa, R. Kainuma, and K. Ishida, 4T. Omori, Y. Sutou, K. Oikawa, R. Kainuma, and K. Ishida, Mater.

Mater. Trans. 45, 427 (2004). Sci. Eng. A 438–440, 1045 (2006).

134116-7

G. P. PURJA PUN AND Y. MISHIN PHYSICAL REVIEW B 86, 134116 (2012)

5J. Liu and J. G. Li, Mater. Sci. Eng. A 454-455, 423 (2007). 39Y. Mishin, M. J. Mehl, and D. A. Papaconstantopoulos, Acta Mater.
6F. Chen, B. Tian, Y. Tong, and Y. Zheng, Int. J. Mod. Phys. B 23,
53, 4029 (2005).
1931 (2009). 40Y. Mishin, Acta Mater. 52, 1451 (2004).
7H. Morito et al., Scr. Mater. 63, 379 (2010). 41C. A. Becker, Models, Databases and Simulation Tools Developed
8R. Hultgren, R. L. Orr, P. D. Anderson, and K. K. Kelley, Selected
and Needed to Realize the Vision of ICME (ASM International,
Values of the Thermodynamic Properties of Metals and Alloys
Materials Park, OH, 2011).
(Wiley, New York, 1963). 42NIST Interatomic Potentials Repository: http://www.ctcms.nist.
9A. F. Guillermet, Int. J. Thermophys. 8, 481 (1987).
10G. R. Harp, R. F. C. Farrow, D. Weller, T. A. Rabedeau, and R. F. gov/potentials/.
43H. Jo´nsson, G. Mills, and K. W. Jacobsen, in Classical and Quantum
Marks, Phys. Rev. B 48, 17538 (1993).
11G. A. Prinz, Phys. Rev. Lett. 54, 1051 (1985). Dynamics in Condensed Phase Simulations, edited by B. J. Berne,
12V. A. de la Pen˜a O’Shea, I. de P. R. Moreira, A. Rolda´n, and F. Illas,
G. Ciccotti, and D. F. Coker (World Scientific, Singapore, 1998),
J. Chem. Phys. 133, 024701 (2010).
13Y. M. Mishin and I. M. Razumovskii, Acta Metall. Mater. 40, 2707 p. 1.
44R. Pasianot and E. J. Savino, Phys. Status Solidi B 176, 327
(1992).
14P. Toledano, G. Krexner, M. Prem, H. P. Weber, and V. P. Dmitriev, (1993).
45N. Wakabayashi, R. H. Scherm, and H. G. Smith, Phys. Rev. B 25,
Phys. Rev. B 64, 144104 (2001).
15F. Frey, W. Prandl, J. Schneider, C. Zeyen, and K. Ziebeck, J. Phys. 5122 (1982).
46F. Cleri and V. Rosato, Phys. Rev. B 48, 22 (1993).
F 9, 603 (1979). 47R. R. Rao and A. Ramanand, J. Low Temp. Phys. 30, 127
16B. Strauss, F. Frey, W. Petry, J. Trampenau, K. Nicolaus, S. M.
(1978).
Shapiro, and J. Bossy, Phys. Rev. B 54, 6035 (1996). 48O. N. Singh and S. S. Kushwaha, Nuovo Cimento Soc. Ital. Fis., B
17R. Nicula et al., Thermochim. Acta 526, 137 (2011).
18H. Sato, O. Kitakami, T. Sakurai, and Y. Shimada, J. Appl. Phys. 21, 354 (1974).
49S. M. Shapiro and S. C. Moss, Phys. Rev. B 15, 2726 (1977).
81, 1858 (1997). 50C. Kittel, Introduction to Sold State Physics (Wiley-Interscience,
19Handbook of Materials Modeling, edited by S. Yip (Springer,
New York, 1986).
Dordrecht, The Netherlands, 2005). 51Y. Mishin, D. Farkas, M. J. Mehl, and D. A. Papaconstantopoulos,
20M. S. Daw and M. I. Baskes, Phys. Rev. B 29, 6443 (1984).
21M. I. Baskes, J. S. Nelson, and A. F. Wright, Phys. Rev. B 40, 6085 Phys. Rev. B 59, 3393 (1999).
52Y. Mishin, M. J. Mehl, D. A. Papaconstantopoulos, A. F. Voter, and
(1989).
22D. J. Oh and R. A. Johnson, J. Nucl. Mater. 169, 5 (1989). J. D. Kress, Phys. Rev. B 63, 224106 (2001).
23R. Pasianot and E. J. Savino, Phys. Rev. B 45, 12704 (1992). 53R. R. Zope and Y. Mishin, Phys. Rev. B 68, 024102 (2003).
24M. I. Baskes and R. A. Johnson, Modelling Simul. Mater. Sci. Eng. 54S. M. Foiles, Phys. Rev. B 49, 14930 (1994).
55Thermal Expansion: Metallic Elements and Alloys, edited by Y. S.
2, 147 (1994).
25A. S. Goldstein and H. Jo´nsson, Philos. Mag. B 71, 1041 (1995). Touloukian, R. K. Kirby, R. E. Taylor, and P. D. Desai, Vol. 12
26W. Hu, H. Deng, X. Yuan, and M. Fukumoto, Eur. Phys. J. B 34,
(Plenum, New York, 1975).
429 (2003). 56M. W. Finnis and J. E. Sinclair, Philos. Mag. A 50, 45 (1984).
27S. Chen, J. Xu, and H. Zhang, Comput. Mater. Sci. 29, 428 (2004). 57T. Lee, M. I. Baskes, S. M. Valone, and J. D. Doll, J. Phys.: Condens.
28M. Jiang, K. Oikawa, and T. Ikeshoji, Metall. Mater. Trans. A 36,
Matter 24, 225404 (2012).
2307 (2005). 58Y. Mishin, M. Asta, and J. Li, Acta Mater. 58, 1117 (2010).
29D. Y. Sun, M. I. Mendelev, C. A. Becker, K. Kudin, T. Haxhimali, 59J. V. Lill and J. Q. Broughton, Phys. Rev. Lett. 84, 5784 (2000).
60J. V. Lill and J. Q. Broughton, Phys. Rev. B 63, 144102 (2001).
M. Asta, J. J. Hoyt, A. Karma, and D. J. Srolovitz, Phys. Rev. B 73, 61C. M. Singal and T. P. Das, Phys. Rev. B 16, 5093 (1977).
62Alloy database: http://alloy.phys.cmu.edu/.
024116 (2006). 63C. M. Singal and T. P. Das, Phys. Rev. B 16, 5068 (1977).
30M. I. Mendelev and G. J. Ackland, Philos. Mag. Lett. 87, 349 64W. B. Pearson, A Handbook of Lattice Spacings and Structure of

(2007). Metals and Alloys, Vol. 2 (Pergamon Press Ltd., Oxford, England,
31M. Igarashi, M. Khantha, and V. Vitek, Philos. Mag. B 63, 603
1967).
(1991). 65G. Y. Guo and H. H. Wang, Chinese J. Phys. 38, 949 (2000),
32M. S. Daw, in Atomistic Simulation of Materials: Beyond Pair
http://psroc.phys.ntu.edu.tw/cjp/.
Potentials, edited by V. Vitek and D. J. Srolovitz (Plenum, 66D. Antonangeli, M. Krisch, G. Fiquet, D. L. Farber, C. M. Aracne,

New York, 1989), pp. 181–191. J. Badro, F. Occelli, and H. Requardt, Phys. Rev. Lett. 93, 215505
33Y. Mishin, in Handbook of Materials Modeling, edited by S. Yip,
(2004).
Chap. 2.2 (Springer, Dordrecht, The Netherlands, 2005), pp. 459– 67G. Simmons and H. Wang, Single Crystal Elastic Constants and

478. Calculated Aggregate Properties (MIT Press, Cambridge, MA,
34J. H. Rose, J. R. Smith, F. Guinea, and J. Ferrante, Phys. Rev. B 29,
1977).
2963 (1984). 68H. J. McSkimin, J. Appl. Phys. 26, 406 (1955).
35F. Apostol and Y. Mishin, Phys. Rev. B 83, 054116 (2011). 69F. R. de Boer, R. Boom, W. C. M. Mattens, A. R. Miedema, and A. K.
36G. P. P. Pun and Y. Mishin, Philos. Mag. 89, 3245 (2009).
37H. Chamati, N. Papanicolaou, Y. Mishin, and D. A. Niessen, Cohesion in Metals: Transition Metal Alloys, Landolt-

Papaconstantopoulos, Surf. Sci. 600, 1793 (2006). Bo¨rnstein New Series, Vol. 1 (Elsevier, Amsterdam, 1988).
38P. L. Williams, Y. Mishin, and J. C. Hamilton, Modelling Simul. 70A. Korner and H. P. Karnthaler, Philos. Mag. A 48, 469 (1983).
71H. J. Wollenberger, Physical Metallurgy, 3rd. ed. (Elsevier,
Mater. Sci. Eng. 14, 817 (2006).
Amsterdam, 1983).

134116-8

EMBEDDED-ATOM POTENTIAL FOR hcp AND fcc COBALT PHYSICAL REVIEW B 86, 134116 (2012)

72M. R. LaBrosse, L. Chen, and J. K. Johnson, Modelling Simul. 77J. Gump et al., J. Appl. Phys. 86, 6005 (1999).
Mater. Sci. Eng. 18, 015008 (2010). 78P. Ehrhart, P. Jung, H. Schultz, and H. Ullmaier, Atomic Defects

73T. Nishizawa and K. Ishida, J. Phase Equilibria (formerly Bull. in Metals, Landolt-Bo¨rnstein New Series, Group III, Crys-
Alloy Phase Diagrams) 4, 387 (1983).
tal and Solid State Physics, Vol. 25 (Springer-Verlag, Berlin,
74CRC Handbook of Chemistry and Physics, edited by D. R. Lide and
W. M. M. Haynes, 90th ed. (CRC, Boca Raton, FL, 2009). 1991).
79A. Y. Liu and D. J. Singh, Phys. Rev. B 47, 8515 (1993).
75Smithells Metals Reference Book, 8th ed., edited by W. F. Gale and 80Y. U. Idzerda, W. T. Elam, B. T. Jonker, and G. A. Prinz, Phys. Rev.
T. C. Totemeir (Elsevier Butterworth-Heinemann, Burlington, MA,
2004). Lett. 62, 2480 (1989).
81P. C. Riedi, T. Dumelow, M. Rubinstein, G. A. Prinz, and S. B.
76Elastic, Piezoelectric, Pyroelectric, Pieqooptic, Electrooptic Con-
stants and Nonlinear Dielectric Susceptibilities of Crystals, edited Qadri, Phys. Rev. B 36, 4595 (1987).
by K.-H. Hellwege and A. M. Hellwege, Landolt-Bo¨rnstein Numer- 82D. P. Dinega and M. G. Bawendi, Angew. Chem., Int. Ed. 38, 1788
ical Data and Functional Relationships in Science and Technology,
New Series, Group III, Crystal and Solid State Physics, Vol. 11 (1999).
(Springer-Verlag, Berlin, 1979). 83J. D. Honeycutt and H. C. Andersen, J. Phys. Chem. 91, 4950

(1987).
84D. R. Clarke, J. Am. Ceram. Soc. 70, 15 (1987).

134116-9