Thermal surface free energy and stress of iron

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received: 18 June 2015

accepted: 07 September 2015

Published: 06 October 2015

# Thermal surface free energy and stress of iron

Stephan Schönecker 1 , Xiaoqing Li 1 , Börje Johansson 1,2 , Se Kyun Kwon 3 & Levente Vitos 1,2,4

Absolute values of surface energy and surface stress of solids are hardly accessible by experiment.
Here, we investigate the temperature dependence of both parameters for the (001) and (110) surface
facets of body-centered cubic Fe from first-principles modeling taking into account vibrational,
electronic, and magnetic degrees of freedom. The monotonic decrease of the surface energies of
both facets with increasing temperature is mostly due to lattice vibrations and magnetic disorder.
The surface stresses exhibit nonmonotonic behaviors resulting in a strongly temperature dependent
excess surface stress and surface stress anisotropy.

Introduced by Gibbs, the surface free energy ( _γ_ ) and the surface stress ( _σ_ ), are the two essential macroscopic parameters that characterize the thermodynamic properties of crystalline surfaces 1,2 . Today, there
is growing appreciation that both parameters play an important role in understanding various surface
phenomena, e.g., faceting, roughening, segregation, surface reconstruction, crystal growth, adsorption,
bottom-up self-organization, and surface melting 1–6 . Surface stress also bears enormous potential for
prospective molecular sensing and actuation devices exploiting its sensitivity to changes of chemical
bonding and morphology in the surface region 7,8 .

Although Langmiur 9 pointed out already a century ago that rising thermal agitation of atoms in solids
would lead to a reduced work of creation of surface (the surface energy), its temperature dependence
and that of the surface stress are still largely unknown in the absence of direct measurements. Today,
we face the situation that the polar dependence of _γ_ and _σ_ is accessible experimentally—the anisotropy
of the surface energy is encoded in the equilibrium shape of 3D crystals—, but the determination of
their absolute values often lacks reliability and accuracy hampered by experimental difficulties 1–3,10,11 .
Available absolute surface energies were in most cases either obtained at temperatures slightly below the
melting point ( _T_ m) where plastic flow occurs or were derived from the surface tension in the liquid phase
( _γ_ LV) 12–14 . Absolute surface stress has proved elusive for direct measurements so far, merely few data for
noble metals were derived from surface stress induced lattice strain in nanometer-sized particles 11 .

In view of the experimental difficulties, the determination of surface energy and surface stress from
theory has become a convenient alternative route 15–18, although modeling has been essentially limited to
the ground state ( _T_ = 0 K) hitherto. Semi-empirical approaches 12,19 estimated configurational and vibrational contributions to the surface excess entropy for an “average” high-index surface facet to access
_γ_ ( _T_ ). These estimations rely, however, on dependable surface-tension data and carry errors of unknown
magnitude at low-temperatures due to the uncertainty in the extrapolation procedure based on certain
assumptions for the temperature dependence, e.g., a linear dependence resembling the behavior of the
surface tension (Eötvös rule) 19 .

Various thermodynamic integration (TI) methods in combination with off-lattice Monte Carlo or
molecular dynamics (MD) simulations have been employed to compute the vibrational contribution to
the surface parameters, both potential-based 20–22 and by means of _ab inito_ simulations 23 . Unfortunately,

1Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology,
Stockholm SE-10044, Sweden. 2 Department of Physics and Astronomy, Division of Materials Theory, Uppsala
University, Box 516, SE-75120, Uppsala, Sweden. 3 Graduate Institute of Ferrous Technology, Pohang University of
Science and Technology, Pohang 790-784, Korea. 4 Research Institute for Solid State Physics and Optics, Wigner
Research Center for Physics, Budapest H-1525, P.O. Box 49, Hungary. Correspondence and requests for materials
[should be addressed to S.S. (email: stesch@kth.se)](mailto:stesch@kth.se)

the applicability of most TI variants is limited to solid systems that do not undergo an allotropic or
magnetic phase transition in the temperature interval of interest.

The phase stability and mechanical properties of Fe are strongly influenced by magnetism and temperature 24–26 . Magnetic long range order and magnetic correlations in Fe are responsible for the stability of the body-centered cubic (bcc) ( _α_ ) phase up to 1189 K significantly above the Curie temperature
( _T_ C = 1043 K). The face-centered cubic (fcc) phase is stable in the temperature interval 1189 K–1662 K,
but iron eventually melts from the reoccurring bcc ( _δ_ ) phase at 1807 K. Surface magnetism in Fe was
shown to have a pronounced impact on the magnitudes of surface energy and surface stress in the
ground state 17,27 .

This paper brings forwards an density-functional theory (DFT) description of _γ_ ( _T_ ) and _σ_ ( _T_ ) for the
two most stable surface facets of bcc Fe [(001) and (110)] 18,28 . Free energies [ _F_ = _F_ ( _T_, _V_ )] were computed
for a surface subsystem and for a bulk subsystem, from which the surface energies, _γ_ = _A_ −1 ( _F_ surf - _F_ bulk ),
and the surface stresses, _σ_ = (2 _A_ ) −1 ∂( _F_ surf - _F_ bulk )/∂ _ε_ | _ε_ = 0, were obtained 2 . Here, _ε_ specifies an isotropic
and elastic in-plane deformation of the surface with unit area _A_ and _σ_ represents half the trace of the
surface stress tensor.

Theoretical modeling
We rely on the adiabatic approximation shown to be applicable to Fe 29,30 and model the individual vibrational, electronic, and magnetic excitations to the free energy, _F_ = _E_ 0 + _F_ vib + _F_ el + _F_ mag. _E_ 0 is the total
energy of the ferromagnetic ground state. The vibrational free energy contribution ( _F_ vib) and the bulk
thermal expansion were derived within the quasiharmonic approximation (QHA) 31 in the ferromagnetically ordered state using a gradient-corrected exchange-correlation parameterisation (PBE) 32 . The
force-constant matrix was obtained within the framework of density-functional perturbation theory 31,33
as implemented in the projector-augmented wave method (VASP) 34 and employing Phonopy 35 to compute the phonon density of states (DOS) and _F_ vib. The electronic ( _F_ el) and magnetic ( _F_ mag = _E_ mag - _TS_ mag)
contributions to the free energy were obtained with the exact muffin-tin orbitals (EMTO) method and
the full charge-density technique 36–38 using PBE in conjunction with the coherent-potential approximation (CPA) 39,40 . _F_ el takes into account electronic excitations due to smearing of the Fermi-Dirac distribution 41,42 . The magnetic disorder effect on the total energy ( _E_ mag) was approximated by the partially
disordered-local moment (PDLM) model described within the framework of CPA 43,44 . The PDLM model
connects the ordered ferromagnetic phase with the disordered paramagnetic (DLM) phase without taking into account short range order 45 . Accordingly, the gradual loss of magnetic long-range order, characterised by the staggered magnetisation ( _m_, 1 ≥ _m_ ≥ 0), is captured by a gradual concentration change
( _x_, 0 ≤ _x_ ≤ 0.5) of a random binary alloy with anti-parallel moment orientation, Fe1↑− _x_ Fe↓ _x_, where _x_ = 0
and _x_ = 0.5 correspond to the ferromagnetic phase ( _m_ = 1) and the paramagnetic (DLM) phase ( _m_ = 0),
respectively. To describe the loss of magnetisation as a function of temperature, _m_ was mapped to _T_ via
an analytic representation ( _m_ = [1 − 0.35 _τ_ 3/2 - 0.65 _τ_ 4 ] 1/3, _τ_ = _T_ / _T_ C) 46 of the experimental magnetisation
curve of Fe 47 . Magnetic entropy ( _S_ mag) within the PDLM model was considered by the mean-field expression, _S_ mag/ _k_ B = 2 _x_ ln( _μ_ + 1), where _k_ B and _μ_ denote the Boltzmann constant and the local magnetic
moment in units of Bohr magneton, respectively. We notice that the measured magnetic entropy of
paramagnetic bcc iron is in agreement with this mean-field estimate 48 . Thermal lattice expansion was
incorporated into the magnetic and electronic contributions via the lattice parameters obtained from the
QHA for _F_ vib.

For the surface energy _γ_ full explicit phononic contributions and thermal expansion were considered.
For the surface stress _σ_, we included the thermal expansion effect only. That is because previous investigations found that the dominating phononic contribution to the surface stress (of atomically ordered
surfaces) arises from thermal expansion rather than from a temperature-induced change of the vibrational amplitude, even for open surface facets 21 . The computational details can be found in the Method’s
section.

Results and Discussion
Figure 1 shows the computed temperature dependence of surface free energy and surface stress for the
(001) and (110) surface facets of bcc Fe. Rising temperature from 0 K to _T_ m leads to a monotonically
decreasing surface energy for both faces, i.e., by 28% and 29% for (110) and (001), respectively. The
close-packed (110) surface facet maintains the lower surface energy in the entire stability range of the
bcc phase, but the anisotropy _γ_ (110)/ _γ_ (001) slightly decreases with increasing temperature.

The surface stress of the (001) and (110) facets exhibits a nonmonotonic dependence on _T_ (Fig. 1).
While _σ_ (001) is significantly lower than _σ_ (110) in the fully-ordered magnetic state at 0 K, they approach each
other with increasing temperature mainly due to a significant increase of the surface stress of the (001)
facet. The surface stresses of the two facets are nearly of the same magnitude in the paramagnetic phase.

Figure 2 shows how the individual magnetic, electronic, and vibrational contributions add up to
the surface free energy change as a function of temperature for the (110) surface facet. Accordingly,
lattice vibrations reduce the surface energy most significantly followed by magnetic disorder. The magnetic disorder contribution to _γ_ below _T_ C originates mainly from the continuous loss of magnetic long
range order, while the magnetic entropy determines the magnetic contribution above _T_ C. The overall

2.6

2.4

2.2

2.0

1.8

1.6

2.5

2.0

1.5

1.0

0.5

0.0

0 500 1000 1500 _T_ m

_T_ C

(001)
(110)
γ(0K), γ( _T_ m) Tyson _et al._
γ(0K) Tyson _et al._ + γmag
γ(0K) de Boer _et al._
γ(0K) de Boer _et al._ + γmag
γ(δ Fe) Price _et al._
γLV(Tm) Allen

fcc phase

(110)
(001)

_T_ C

0 500 1000 1500 _T_ m
temperature _T_ (K)

Figure 1. The calculated temperature effect on the surface energy and surface stress of the (110) and
(001) facets of bcc Fe. Experimental surface energies of _δ_ Fe were reported by Price _et al._ 14, and the
available surface tension measurements ( _γ_ LV) of liquid Fe were reviewed by Allen 13 . The surface energy at _T_ m
was estimated by Tyson _et al._ 12 and at 0 K by Tyson _et al._ 12 and de Boer _et al._ 19 . The latter two 0 K estimates
amended by the missing magnetic contribution ( _~~γ~~_ mag) are also plotted and discussed in the text. Data from
refs 12,14,19 should be understood as to represent an average surface facet. Lines guide the eye. We estimate
the error of the computed surface energy (surface stress) at 0 K and _T_ m to < 0.1 J/m 2 (< 0.2 J/m 2 ) and
0.1–0.2 J/m 2 (0.3–0.4 J/m 2 ), respectively.

2.6

2.5

2.4

2.3

2.2

2.1

2.0

1.9

1.8

1.7

1.6

fcc phase

mag.
mag.+el.
mag.+el.+vib.

bulk
(001)
(110)

00 2 4 6 8 10
frequency (THz)

_T_ C

0 500 1000 1500 _T_ m
temperature _T_ (K)

Figure 2. Decomposition of magnetic, electronic, and vibrational contributions to _γ_ (110)( _T_ ) for the
(110) surface facet of bcc Fe. The inset shows the phonon DOSs for the (110) and (001) surface facets in
comparison to the bulk phonon DOS.

small electronic disorder effect becomes significant only at temperatures higher than several hundreds of
Kelvin. We note that the relative magnitudes of the individual contributions to _γ_ are similar for the (001)
surface facet. The absolute vibrational (magnetic) contribution to the surface free energy is, however,
larger (smaller) for the open (001) facet than for the close-packed (110) facet. The different contributions
due to phonons can be understood by comparing the phonon DOS for the two surface atomic layers
with the bulk one as shown in the inset of Fig. 2 (DOSs computed for the equilibrium geometry). The

1043

994
1807

921

0
293

921 623

1043 722

1807

513

(110) [T(K)]
(001) [T(K)]

0 1 2 3 4
moment enhancement ∆µ2 (µ2B )

2 (µ2B )

Figure 3. Correlation between moment enhancement and magnetic surface stress as a function of
temperature (indicated by labels to the data points) on bcc (110) and (001) surface facets of Fe. The
proportionality constants obtained from a fit to the data are C(110) = 1 42. and C(001) = 0 57. .

reduced coordination number at both surface leads to an enhanced DOS at low phonon frequencies and
a lower Debye temperature compared to the bulk counterpart. This softening of the vibrational modes is
more pronounced for the (001) surface of Fe than for the (110) one 49 resulting in a larger excess entropy
at the more open surface facet.

In contrast to the monotonic trends seen for _γ_ and the weakly temperature dependent _σ_ (110), _σ_ (001)
increases significantly between 0 K and _T_ C (Fig. 1) resulting in a strongly temperature-dependent surface
stress anisotropy ( _σ_ (110)/ _σ_ (001)). We analyzed this striking behavior by considering the magnetic contribution ( _σ_ mag) to the surface stress, where _σ_ mag is defined as the difference between the nonmagnetic value
of the surface stress (no spin-polarization considered) and the surface stress from Fig. 1 (spin-polarization
considered). The geometry of the lattice was kept fixed in this additional study. According to Punkkinen
_et al._ ’s 17 analysis for magnetic metals, _σ_ mag was found to be proportional to the surface moment enhancement (Δ _μ_ 2 ), viz., _σ_ mag ∝ C( _hkl_ )Δ _μ_ 2, where Δ _μ_ 2 ≡ _μ_ surf2 - _μ_ bulk2 . The surface magnetic moment ( _μ_ surf)

ment (Δ _μ_ 2 ), viz., _σ_ mag ∝ C( _hkl_ )Δ _μ_ 2, where Δ _μ_ 2 ≡ _μ_ surf2 - _μ_ bulk2 . The surface magnetic moment ( _μ_ surf)

of Fe is larger than the bulk one ( _μ_ bulk) leading to a compressive _σ_ mag, i.e., _σ_ mag tends to expand the lattice
and to reduce the total tensile surface stress 17 . Here, we introduced the facet-dependent positive proportionality constant (C( _hkl_ )) to reflect the influence of atomic coordination and surface electronic structure
on _σ_ mag. Δ _μ_ 2 for the open (001) facet was obtained by adding up the contributions from both the surface
layer and the first subsurface layer, since _μ_ differed significantly from _μ_ bulk in these two layers.

The proportionality between _σ_ mag and Δ _μ_ 2 shown in Fig. 3 is excellent for the (110) surface facet and
describes somewhat less satisfactorily the data of the (001) facet. _σ_ mag is nearly constant above _T_ C due to
the small influence of thermal expansion on _σ_ mag and Δ _μ_ 2 in the DLM state. We find that a temperature
increase from 0 K to _T_ m has a more significant effect on Δ _μ_ 2 for the (001) facet than for the (110) facet.
This is plausible since the magnetic moments in the surface region of the densely packed (110) facet are
smaller and converge more rapidly towards the bulk value compared to the situation at the open (001)
facet 18 . We also observe that C(110) - C(001) indicating that the enhanced surface magnetic moment for
the (110) surface facet generates a larger surface stress than in the case of (001). Taking into account that
the unit area per surface atom for (110) is a factor of 2 smaller than for (001), the magnetic surface
stress per atom for (001) is still ≃ 43% smaller than that for (110). A possible explanation lies in the lower
coordination number, and hence the smaller _d_ -bandwidth, for an Fe atom situated at the (001) surface
compared to when it is located at the (110) surface. Since C( _hkl_ ) is inversely proportional to the height of
the single-particle DOS 17,50, C( _hkl_ ) should roughly scale with the width of the _d_ -band. Thus, the difference
in the surface atom coordination number confirms our finding C(110) - C(001) .
Available experimental data for _γ_ of _δ_ Fe allows for a comparison with the present _ab initio_ surface
energies. Price reported _γ_ = 1.95 J/m 2 ± 10% in the temperature range 1673 K – _T_ m14. Our prediction for

2, where Δ _μ_ 2 ≡ _μ_ surf2 - _μ_

surf2 - _μ_ bulk2

energies. Price reported _γ_ = 1.95 J/m 2 ± 10% in the temperature range 1673 K – _T_ m14. Our prediction for

_γ_ slightly below _T_ m lies within the uncertainty of the experiment, see Fig. 1. The jump of _γ_ across the
liquid-solid phase transition may further be used to estimate _γ_ ( _T_ m) from _γ_ LV adding to it the latent heat
of fusion per surface area, viz., _γ_ ( _T_ m) ~ Δ _H_ f/ _A_ + _γ_ LV51. Using available values for Fe, Δ _H_ f = 13.81 kJ/mol52

of fusion per surface area, viz., _γ_ ( _T_ m) ~ Δ _H_ f/ _A_ + _γ_ LV51. Using available values for Fe, Δ _H_ f = 13.81 kJ/mol52

and _γ_ LV = 1.78 ± 0.08 J/m 2 13, we arrive at 2.04 ± 0.08 J/m 2 for the (001) facet and at 2.15 ± 0.08 J/m 2 for the
(110) facet. These values are somewhat larger than our predicted surface energies at _T_ m, _γ_ (110) = 1.76 J/m 2
and _γ_ (001) = 1.79 J/m 2 . A possible reason for the underestimation of _γ_ near _T_ m could be that the vibrational
contribution to _γ_ for paramagnetic Fe is somewhat overestimated by the present theory. An estimation
of the influence of the magnetic state of Fe on the vibrational contributions to the surface energy can be
found in the Supplementary Information online. For the (110) surface facet, we found that the surface

vibrational excess is nearly identical in the two magnetic phases. This indicates that the here reported
vibrational contribution to _γ_ (110) is not expected to be significantly altered by the magnetic state of Fe.
For the (001) surface facet, we estimated that the surface vibrational excess to _γ_ in the paramagnetic
state is ≃ 40% smaller than in the ferromagnetic state. This corresponds to an underestimation of _γ_ (001)
by 0.1 J/m 2 at _T_ m. The updated estimate for _γ_ (001)( _T_ m), accounting for the reduced vibration excess in the
paramagnetic state of Fe, is 1.89 J/m 2 in closer agreement with the experimental data from Price 14 and
the above value derived from _γ_ LV.

Semi-empirical estimates of _γ_ (0 K) for an average surface facet, derived from experimental
surface-tension data and estimated configurational and vibrational contributions, are available for comparison: _γ_ (0 K) = 2.42 J/m 2 from ref. 12 based on a nonlinear extrapolation scheme, and _γ_ (0 K) = 2.48 J/m 2
from ref. 19 assuming that _γ_ is a linear function of _T_ . Magnetic disorder was not considered in either
approach, although the decomposition of _γ_ ( _T_ ) (Fig. 2) suggests that the magnetic part should be taken
into account for a more accurate low temperature estimate. We quantified the magnetic part for an average surface facet on the basis of the presently calculated magnetic contribution to the surface energy. To
this end, the magnetic contributions to _γ_ (001) and _γ_ (110) in the range 0 K to _T_ m were weighted by their
fractional areal contributions to the equilibrium shape of a 3D crystallite according to the Wulff construction 10 (other crystal faces possessing larger surface energies were not considered, as they contribute
only by small fractions to the equilibrium shape 18 ). Using the present 0 K values, _γ_ (110) = 2.46 J/m 2 and
_γ_ (001) = 2.53 J/m 2, the fractions of {001} facets and of {110} facets to the equilibrium crystal shape are 0.23
and 0.77, respectively. This gives an average magnetic contribution _~~γ~~_ mag = 0 14. J/m 2, which when added
to the aforementioned estimated figures results in 2.56 J/m 2 (with the value from ref. 12) and 2.62 J/m 2
(ref. 19) respectively. The former estimate is in very good agreement with the present values.

A possible measure to indicate surface reconstruction is based on a bulk continuum elastic model,
viz., _κ_ ≡( _γ_ - _σ_ )/ _Gb_  0.1–0.2, where the upper limit corresponds to more corrugated surface facets 1,53 .
Accordingly, the larger is the excess surface stress ( _γ_ - _σ_ ), or the smaller is the shear modulus ( _G_ ) [the
length of the Burger’s vector ( _b_ ) is nearly constant], the higher is the tendency to reconstruct. The predicted trends for _γ_ and _σ_ (Fig. 1) indicate that the excess surface stress and hence the propensity to
reconstruct would decrease with rising temperature for both surface facets. However, the observed strong
softening of the shear elastic constants with increasing temperature 24,25 works against the excess surface
stress. Expressing _G_ through the Hill average 54 of the experimental single-crystalline elastic constants
from refs 24,25 and evaluating _κ_ for _α_ Fe in the ferromagnetic and paramagnetic phases, we found that
_κ_ decreases as _T_ increases. The largest _κ_ values (at 0 K) are 0.09 and 0.02 for the (001) facet and the (110)
facet, respectively, indicating stable surfaces. This prediction is in line with observations, namely, no
surface reconstruction was observed for the (110) and (001) facets of Fe in the temperature range
300–500 K 55 .
To conclude, temperature has a pronounced effect on the surface energy and the surface stress of bcc
Fe. The surface energies of the (001) and (110) facets decrease nonlinearly by approximately 30% from
0 K up to the melting point with slight reduction of their anisotropy. This reduction of the work of separation is primarily due to lattice vibrations and magnetic disorder. There is good agreement between the
present surface energies and the 0 K values from Tyson _et al._ ’s and de Boer _et al._ ’s databases 12,19 especially
if the latter values are corrected for the missing magnetic term. The surface stresses are nonmonotonic
functions of temperature exhibiting a strongly temperature dependent surface stress anisotropy. Surface
stress originating from the thermal magnetic disorder at the (001) surface is more strongly affected by
temperature than the one at (110) and is mainly responsible for the pronounced increase of the stress
at (001) below the Curie temperature. We predict that the propensity for surface reconstruction at (001)
and (110) lowers as temperature increases.

Today, the theoretical description of metal surfaces at elevated temperature is a very challenging task.
In order to make such a study feasible in the case of complex systems, one has to adopt several simplifications and approximations. Despite of that, the present theoretical results turn out to follow closely the
previous semi-empirical estimates, demonstrating the predictive power of our _ab initio_ tools in the case
of high-temperature surface properties. These finding offers a solid platform for extending the existing
theoretical methodology to study the surface properties of technological alloys.

Methods
Convergence of all numerical parameters was carefully checked. Bulk and surface reference systems were
of identical size to ensure numerical error cancellation. _F_ vib for the surface subsystem was obtained from
32–34 Å thick slabs separated by 10–14 Å vacuum. The relaxation at 0 K included all interlayer distances
perpendicular to the surfaces until the residual forces on each layer were smaller than 0.05 meV/Å. To
compute the force constant matrix (VASP), we used a plane wave cut-off of 500 eV and a _k_ -point mesh
equivalent to a 24 × 24 × 2(1) Monkhorst-Pack mesh of the primitive bulk (surface) reference system.
The phonon properties were sampled on a 3 × 3 × 1 mesh. All derived vibrational quantities that involve
an integration of the phonon density of states (i.e., free energies) are converged to within ± 2% against a
denser 4 × 4 × 1 sampling mesh. We checked the temperature-dependent surface inter-layer relaxation of
Fe within the QHA according to ref. 56. We found that relaxation at the (110) facet, determined at 1043 K
and at 1805 K, change _γ_ and _σ_ by < 2% and by < 4%, respectively. Since this relaxation effect represents

only a small correction to both _γ_ and _σ_ at significantly higher computational cost, finite temperature
surface inter-layer relaxation was not included in the present _F_ vib, i.e., the relaxed _T_ = 0 K interatomic
distances were rigidly rescaled according to the bulk thermal expansion.

The magnetic and electronic disorder effects were modelled by slabs with a converged thicknesses
of 13 atomic layers and 9 atomic layers for the (001) surface and the (110) surface, respectively, separated by ~10 Å vacuum. Surface energies and surface stresses calculated with EMTO were sampled on a
15 × 15 × 2(1) _k_ -point mesh in the case of the bulk (surface) reference system.

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Acknowledgements
The Swedish Research Council, the Swedish Foundation for Strategic Research, the China Scholarship
Council, the Basic Science Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Science, ICT and future Planning (NRF-2014R1A2A1A12067579), and the
Hungarian Scientific Research Fund (research projects OTKA 84078 and 109570) are acknowledged for
financial support. The computations were performed using resources provided by the Swedish National
Infrastructure for Computing (SNIC) at the National Supercomputer Centre in Linköping.

Author Contributions
S.S. and X.L. performed the calculations. S.S. analysed the results. S.S. and L.V. wrote the manuscript.
All authors reviewed the manuscript.

Additional Information
Supplementary information [accompanies this paper at http://www.nature.com/srep](http://www.nature.com/srep)

Competing financial interests: The authors declare no competing financial interests.

How to cite this article : Schönecker, S. _et al._ Thermal surface free energy and stress of iron. _Sci. Rep._
5, 14860; doi: 10.1038/srep14860 (2015).

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