Tensile strain-induced softening of iron at high temperature

Tensile strain-induced softening of iron at high temperature
received: 27 September 2015
accepted: 16 October 2015
Published: 10 November 2015
Tensile strain-induced softening of
iron at high temperature
Xiaoqing Li1, Stephan Schönecker1, Eszter Simon2, Lars Bergqvist3, Hualei Zhang1,4,
László Szunyogh2,5, Jijun Zhao6, Börje Johansson1,7 & Levente Vitos1,7,8
In weakly ferromagnetic materials, already small changes in the atomic configuration triggered by
temperature or chemistry can alter the magnetic interactions responsible for the non-random atomicspin orientation. Different magnetic states, in turn, can give rise to substantially different
macroscopic properties. A classical example is iron, which exhibits a great variety of properties as
one gradually removes the magnetic long-range order by raising the temperature towards its Curie
point of TC0 = 1043 K. Using first-principles theory, here we demonstrate that uniaxial tensile strain
can also destabilise the magnetic order in iron and eventually lead to a ferromagnetic to
paramagnetic transition at temperatures far below TC0. In consequence, the intrinsic strength of the
ideal single-crystal body-centred cubic iron dramatically weakens above a critical temperature of
~500 K. The discovered strain-induced magneto-mechanical softening provides a plausible atomiclevel mechanism behind the observed drop of the measured strength of Fe whiskers around
300–500 K. Alloying additions which have the capability to partially restore the magnetic order in the
strained Fe lattice, push the critical temperature for the strength-softening scenario towards the
magnetic transition temperature of the undeformed lattice. This can result in a surprisingly large
alloying-driven strengthening effect at high temperature as illustrated here in the case of Fe-Co alloy.
Iron is one of the most abundant elements in the Milky Way and constitutes the major component of
earth’s core1. Processed iron is the main ingredient in steels and other ferrous materials, whose technology is a central pillar in today’s industrial world. Beyond the obvious practical interests, the physical
properties of Fe have attracted great attention in many fields of sciences. Probably the most conspicuous
phenomenon is the marvellous coupling between magnetism and mechanical properties. Although recognised long time ago2, it is only recently that atomic-level tools can give insight into the mechanisms
behind the magneto-mechanical effects in Fe and its alloys. Magnetic order in solid iron emerges with
the formation of a crystalline lattice. The stable body-centred cubic (bcc) phase (α-Fe) is characterised
by long-range ferromagnetic (FM) order below T C0. The face-centred cubic (fcc) structure appears in the
phase diagram at temperatures above 1189 K in the paramagnetic (PM) state (γ-Fe). At low temperature,
metastable fcc Fe has a complex antiferromagnetic state associated with a small tetragonal deformation3,4.
This diversity corroborates the strong interplay between structural and magnetic degrees of freedom in
Department of Materials Science and Engineering, KTH - Royal Institute of Technology, 10044, Stockholm,
Sweden. 2Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8.,
HU-1111, Budapest, Hungary. 3Department of Materials and Nano Physics, KTH Royal Institute of Technology,
Electrum 229, SE-16440, Kista, Sweden. 4Center of Microstructure Science, Frontier Institute of Science and
Technology, Xi’an Jiaotong University, 710054, Xi’an, China. 5MTA-BME Condensed Matter Research Group,
Budafoki út 8., HU-1111, Budapest, Hungary. 6Key Laboratory of Materials Modification by Laser, Ion and Electron
Beams (Dalian University of Technology), Ministry of Education, 116024, Dalian, China. 7Department of Physics
and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-75120, Uppsala, Sweden. 8Research
Institute for Solid State Physics and Optics, Wigner Research Center for Physics, P.O. Box 49, HU-1525, Budapest,
Hungary. Correspondence and requests for materials should be addressed to X.L. (email: [email protected]) or S.S.
(email: [email protected]) or J.Z. (email: [email protected])
Scientific Reports | 5:16654 | DOI: 10.1038/srep16654
Figure 1. The ITS of bcc Fe in tension along the [001] direction as a function of temperature (σm(T),
solid line and circles); the ITS taking into account only magnetic disorder (dashed line, open
diamonds); the ITS taking into account magnetic disorder and neglecting the change of TC with
structural deformation (σmc , dotted line, open triangles). The Fe0.9Co0.1 alloy (stars) possesses a slightly
larger ITS at 0 K, but compared to pure Fe the ITS drastically increases at high temperatures. All data are
normalised to the ITS of Fe at 0 K (12.6 GPa). (Inset) Experimentally determined tensile strength of [001]
oriented Fe whiskers as a function of temperature from ref. 17. The two data sets give the average tensile
strength and the maximum tensile strength for whiskers possessing nearly equal diameter (5.1–5.4 μm).
Lines guide the eye.
The weak FM nature of α-Fe makes its magnetic interactions, and hence its characteristic properties,
susceptible to chemical perturbations by, e.g., transition metal impurities. The effect is reflected in the
well-known Slater-Pauling curves for the magnetic moment and Curie temperature of dilute Fe-based
binaries5,6. The magneto-elastic coupling is manifested in the observed softening of the elastic parameters
near the magnetic phase transition in bcc Fe7. The role of magnetism in the phase stability of hydrostatically pressurised bcc Fe has also been underlined8,9. However, explorations of the interplay among
magnetism, lattice strain, and micro-mechanical properties are missing hitherto.
Using computer simulations, we reveal the impact of the magneto-structural coupling on the
temperature-dependent mechanical strength of bcc Fe under large anisotropic strain. We apply uniaxial
tensile load to a perfect bcc Fe single-crystal and investigate the non-linear stress-strain response up to
the point of mechanical failure of the lattice (the maximum stress that a perfect lattice can sustain under
homogeneous strain). The stress at the failure point under tension defines the ideal tensile strength (ITS,
denoted by σm), which is a fundamental intrinsic mechanical parameter of theoretical and practical
importance10–15. We model the temperature effects in terms of a first-principles theoretical approach by
taking into account contributions arising from phononic, electronic and magnetic degrees of freedom.
The simulation details can be found in the method section.
Results and Discussion
Our predicted maximum tensile strength of Fe as a function of temperature T [σm(T)] is shown in
Fig. 1 (solid line and circles). At 0 K, the ITS amounts to 12.6 GPa. For comparison, the measured
room-temperature ultimate tensile strengths of bulk bcc iron ranges between 0.1 GPa and 0.3 GPa16,
and the largest reported one is ~6 GPa for [001] oriented Fe whiskers17. Temperature is found to have a
severe impact on the ITS of Fe. Namely, σm(T) remains nearly constant up to ~350 K, decreases by ~8%
between ~350 K and ~500 K compared to the 0 K strength and then drops by ~90% between ~500 K
and ~920 K. At temperatures above ~1000 K, Fe can resist a maximum tensile strength of ~1.0 GPa. For
all investigated temperatures, we found that the ideal Fe lattice maintains body-centred tetragonal (bct,
lattice parameters a and c) symmetry during the tension process and eventually fails by cleavage of the
(001) planes18.
Figure 1 also shows the ITS of Fe considering merely the thermal magnetic disorder effect (dashed
line, open diamonds). Comparing these data to the full ITS (solid line and circles), it can be inferred that
the main trend of σm(T) is governed by the magnetic disorder term. The pronounced drop of the ITS at
~500 K is related to the magnetic properties of strained Fe, more precisely to its magnetic transition temperature. The Curie temperature is shown as a function of the bct lattice parameters in Fig. 2. Compared
to bcc Fe, TC of bct Fe is strongly reduced. The main trend is that TC decreases as the tetragonality (c/a)
increases. A drop in TC with increasing c/a was reported in previous fixed-volume calculations19, which
turns out to be the case for the present uniaxial tensile deformations as well.
Scientific Reports | 5:16654 | DOI: 10.1038/srep16654
Figure 2. Contour plot of TC of bct Fe as a function of the lattice parameters a and c. The region where
TC is shown is confined by the bcc ground state (black asterisk) and the failure points (εm) at different T
(stars, numbers denoting T). The Curie temperature corresponding to each failure point (TCεm ) can be read
from the legend. The ground state bcc structure possesses the highest calculated TC in the region shown. The
dashed line represents the hyperbola of constant volume equal to the bcc equilibrium volume. The insets
sketch local spins for the unstrained parent lattice and for the strained lattice at high temperature,
illustrating the magnetic disorder increase with tensile strain.
In order to illustrate the significance of the lattice strain-induced reduction of TC on the strength of
Fe, we computed an auxiliary ITS [σmc (T ) ] assuming a constant TC for the strained lattice. We chose
without loss of generality the present theoretical T C0 of bcc Fe (1066 K) (see Table S1 in the Supplementary
information). The magnetic disorder effect on σmc (T ) is shown in Fig. 1 (dotted line, open triangles). The
drop of the ITS σmc (T ) is shifted to higher temperatures compared to σm(T). We conclude that the
strongly reduced Curie temperatures of the distorted bct Fe lattices compared to bcc Fe are primarily
responsible for the drop of the ITS at ~500 K. In other words, lattice deformation destabilises the magnetic order (shown schematically in insets of Fig. 2) which leads to an unexpected softening of the lattice
already at temperatures far below T C0.
At temperatures T  650 K, Fe reaches the maximum strength (fails) very close to the magnetic instability (T / T Cεm >0.9). The strength of these magnetically barely ordered systems approaches the very low
strength of the PM state. The emerging question is why the ideal tensile strength of PM Fe is so much
lower than that of FM Fe. We recall that bcc Fe is susceptible to the formation of sizeable moments in
both FM and PM states due to its peculiar electronic structure20,21. The nonmagnetic (NM) bcc Fe is
thermodynamically and dynamically unstable as illustrated in Fig. 3(a), where the total energy along the
constant-volume Bain path is shown. The pronounced E2g peak in the NM density of states (DOS) at the
Fermi level (EF) [Fig. 3(b)] drives the onset of FM order, that is, the majority and the minority spin
channels populate and depopulate, respectively, causing an exchange split and moving the peak away
from EF22. When the FM equilibrium state is reached (at 0 K), EF is located near the bottom of the pseudo
gap at approximately half band filling in the minority channel which explains the pronounced stability
of FM bcc Fe. This is reflected by a deep minimum in the energy versus a and c map (see Fig. S2 in the
Supplementary information). The depth of this energy minimum in connection to the ITS is best characterised by the total energy cost to reach the ideal strain (εm) from the unstrained bcc phase (strain
energy), which amounts to 0.426 mRy/atom per % strain in the FM state.
The formation of local magnetic moments in the PM state also stabilises the bcc structure. This is
illustrated in Fig. 3(a), where we show the constant-volume Bain path of PM bct Fe for various fixed values of the local magnetic moment (μ). It is found that increasing μ gradually stabilises the bcc structure
and for the equilibrium value of the local magnetic moment in the PM state (2.1μB), a shallow minimum
is formed on the tetragonal energy curve. Figure 3(a) (inset) displays the total energy change of the bcc
structure corresponding to a small volume-preserving tetragonal deformation [Δ E(δ) = E(δ) − E(0)] for
the same μ values as in the main figure. Δ E(δ) is negative for NM Fe, but increasing μ turns Δ E(δ) positive indicating the mechanical stabilisation of PM bcc Fe upon PM moment formation. It is found that
the mechanical stabilisation due to local magnetic moment formation, embodied in the trend of Δ E(δ)
as a function of μ, is in fact determined by the kinetic energy (solid symbols) and thus by the details of
the electronic structure. In the PM state, the local magnetic moment formation is due to a split-band
mechanism for which the average exchange field is zero20,21. The disorder-broadened DOSs of PM Fe
[Fig. 3(b)] reveal that the formation of local magnetic moments effectively removes the peak at EF seen
for NM Fe. However, this mechanism does not yield a comparably large cohesive energy increase as for
the FM case. The strain energy cost to reach εm starting from the shallow energy minimum equals 0.024
mRy/atom per % strain in the PM phase, which is approximately 18 times smaller than in the FM case.
Scientific Reports | 5:16654 | DOI: 10.1038/srep16654
Figure 3. (a) Total energy of NM bct Fe and PM bct Fe for various values of the local magnetic moment.
The energies are plotted with respect to the energy of the PM bcc state with equilibrium local magnetic
moment (2.1μB), and the volume is fixed to that of PM bcc Fe. The inset displays the total energy change
(open symbols) and the kinetic energy (solid symbols) change corresponding to a small volume-preserving
tetragonal shear (arbitrary units). (b) Total DOS of PM bcc Fe for different local magnetic moments. (c,d)
The exchange interactions J1 and J2 of bcc Fe and Fe0.9Co0.1 and J1–J3 of one representative bct structure
(a = 2.732 Å, c = 3.109 Å).
Hence, the significantly lower ITS of PM Fe compared to that of FM Fe arises from the differences in
the magnetism-driven stabilisation mechanisms and the corresponding energy minima within the Bain
configurational space (see Fig. S2 in the Supplementary information).
The 0 K ideal strength of Fe can be sensitively altered already by dilute alloying with, e.g., V or Co23.
Apart from the intrinsic chemical effect of the solute atom, there is a magnetic effect due to the interaction between the solute atom and the Fe host. Here we show that the strength of Fe can be significantly
enhanced at high temperature by alloying with Co.
We assessed the magnetic properties of the random Fe0.9Co0.1 solid solution using the same methodology as for Fe. The alloying effect of Co on TC is mainly connected to the strengthening of the first
nearest neighbour exchange interactions (J1)5,24. We show the influence of Co on the eight first [J1(8)]
and six second [J2(6)] nearest neighbour exchange interactions in Fig. 3(c) for the bcc phase. J1 between
both similar and dissimilar atomic species increases significantly in Fe0.9Co0.1 compared to J1 in pure Fe.
At the same time, alloying with Co weakens J2, but this effect is of lesser importance for TC than the
strengthening of J1.
We computed the ITS of the Fe0.9Co0.1 alloy at 0 K23 and in the high-temperature interval between
500 K and 1000 K. As shown in Fig. 1, Fe0.9Co0.1 exhibits ~10% larger ITS than Fe at temperatures
below 500 K. However, the strengthening impact of alloying on the ITS becomes more dramatic in the
high-temperature region. The physical origin underlying this effect is the higher Curie temperature of bct
Fe0.9Co0.1 compared to bct Fe, which is a consequence of both the enhanced nearest neighbour exchange
interactions and the stronger ferromagnetism in Fe0.9Co0.1, i.e., both the magnetic moment and exchange
interactions are larger and more stable in the Fe-Co alloy than in pure Fe upon structural perturbation (tetragonalisation)6,23,24. The alloying effect of Co on J1(8), J2(4), and J3(2) (the two third nearest
neighbour interactions) is shown in Fig. 3(d) for one representative bct structure (with a = 2.732 Å and
c = 3.109 Å). The addition of Co leads to a strong increase of J1 and a weakening of J2 and J3 similar to the
Scientific Reports | 5:16654 | DOI: 10.1038/srep16654
bcc phase. For the chosen example, the TCs of Fe and Fe0.9Co0.1 amount to 707 K and 1053 K, respectively,
i.e., Co addition results in an increase of TC which is in magnitude much more pronounced for the bct
structure than for the bcc one.
The predicted strong magneto-mechanical softening of Fe above ~500 K should be observable in flawless systems. Indeed, for single-crystalline Fe whiskers tensioned along [001]17,25,26, Brenner reported a
pronounced temperature dependence of the average and maximum tensile strengths as shown in Fig. 1
(inset). It seems plausible to assume that the maximum strength corresponds to whiskers with the lowest
defect density and highest surface perfection. The failure of whiskers with diameter < 6μm was reported
to occur without appreciable plastic deformation. It is important to realise that the observed temperature
gradient of the measured strength is much stronger than the one assuming only the thermally-activated
nucleation of dislocations at local defects17. Hence, the observed softening of Fe whiskers requires additional intrinsic mechanisms emerging from the atomic-level interactions. The present results obtained for
the tensile strength of ideal Fe crystal show a strong temperature gradient due to tensile-strain induced
magnetic softening which actually nicely follows the experimental trend. The qualitative difference in the
theoretical and experimental temperature dependence may be ascribed to the low strain-rate inherent in
experiment, which can allow for the activation of dislocation mechanisms27,28.
We have discovered a strong magneto-mechanical softening of iron single-crystals upon tensile loading.
We showed that the strength along the [001] direction persists up to ~500 K, however, diminishes most
strongly in the temperature interval ~500–900 K due to the loss of the net magnetisation upon uniaxial
strain. The strength in the paramagnetic phase is more than 10 times lower than that in the ferromagnetic phase. We found that Fe fails by cleavage at all investigated temperatures.
We have demonstrated that the intrinsic strength of Fe at high temperatures is significantly enhanced
by alloying with Co. This finding opens a way to carefully scrutinise the proposed connection29,30 between
the tensile strength of a defect-free ideal crystal and the measured tensile strength of single-crystal whiskers or nanoscale systems (e.g., nanopillars). Measuring the tensile strength of Fe-Co whiskers at elevated
temperature could verify the here predicted differences in the high-temperature intrinsic mechanical
properties of Fe and Fe-Co alloy.
Extending the above mechanism to other systems, we expect that the large magnetic effect on the
strength exists for all dilute Fe alloys and that the magneto-mechanical softening temperature can be
sensitively tuned by proper selection of the alloying elements. Solutes that strengthen (weaken) the ferromagnetic order in the tetragonally distorted Fe increase (decrease) the intrinsic strength of defect-free
Fe crystals at temperatures above (below) ~600 K.
The adopted first-principles method is based on density-functional theory as implemented in the exact
muffin-tin orbitals (EMTO) method31–33 with exchange-correlation parameterised by Perdew, Burke,
and Ernzerhof (PBE)34,35. EMTO features the coherent-potential approximation (CPA), which allows to
describe the disordered PM state by the disordered local moment (DLM) approach21.
The ITS (σm) with corresponding strain (εm) is the first maximum of the stress-strain curve,
1 + ε ∂G
σ (ε) = V (ε) ∂ε [V(ε) and G(ε) are the relaxed volume and the calculated free energy at strain ε], upon
uniaxial loading. The tensile stress was determined by incrementally straining the crystal along the [001]
direction (the weakest direction for bcc crystals) and taking the derivative of the free energy [G(ε)] with
respect to ε. The two lattice vectors perpendicular to the [001] direction were relaxed at each value of
the strain allowing for a possible symmetry lowering deformation with respect to the initial body-centred
tetragonal symmetry23.
The temperature induced contribution from electronic excitations to the ITS was considered by
smearing the density of states with the Fermi-Dirac distribution36. To measure the effect of explicit lattice
vibrations, we computed the vibrational free energy as a function of strain in the vicinity of the stress
maximum within the Debye model37 employing an effective Debye temperature which was determined
from bulk parameters38.
In order to take into account the effect of thermal expansion, the ground-state bcc lattice parameter
at temperature T was obtained by rescaling the theoretical equilibrium lattice parameter with the experimentally determined thermal expansion39,40. The expanded volume was stabilised by a hydrostatic stress
pT derived from the partial derivative of the bcc total energy with respect to the volume taken at the
expanded lattice parameter corresponding to T. pT was accounted for in the relaxation during straining
the lattice. To this end, the relaxed lattice maintained an isotropic normal stress pT in the (001) plane
for each value of the strain.
We modelled the effect of thermal magnetic disorder on the total energy by means of the partially
disordered local moment (PDLM) approximation41,42. Within PDLM, the magnetic state of Fe and
Fe0.9Co0.1 alloy are described as a binary Fe1↑− x Fe ↓x and a quaternary (Fe ↑0.9Co ↑0.1)1− x (Fe 0↓.9Co 0↓.1)x alloy
with concentration x varying from 0 to 0.5 and anti-parallel spin orientation of the two alloy components. Case x = 0 corresponds to the completely ordered FM state with magnetisation m = 1. As x is
gradually increased to 0.5, the magnetically random PM state lacking magnetic long and short range
Scientific Reports | 5:16654 | DOI: 10.1038/srep16654
order (m = 0) is obtained (DLM state). The PDLM approach describes the energetics underlying the loss
of magnetic order. Connection to the temperature is provided through the magnetisation curve [m(τ),
τ ≡ T/TC being the reduced temperature] which maps the computed m to T. It is important to realise
that for Fe and Fe0.9Co0.1 the thermal spin dynamics embodied in the shape of m(τ) and in the TC value
change under the presently applied uniaxial loading. The computational details for TC and m(τ) are given
in the Supplementary information.
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Scientific Reports | 5:16654 | DOI: 10.1038/srep16654
We thank E. K. Delczeg-Czirjak, M. D. Kuz’min, and L. Udvardi for helpful discussions. The Swedish
Research Council, the Swedish Foundation for Strategic Research, the European Research Council, the
China Scholarship Council and the Hungarian Scientific Research Fund (research project OTKA 84078
and 109570), and the National Magnetic Confinement Fusion Program of China (2015GB118001) are
acknowledged for financial support. L.B. acknowledges financial support from the Swedish e-Science
Research Centre (SeRC) and Göran Gustafsson Foundation. The computations were performed using
resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National
Supercomputer Centre in Linköping. L.S. was supported by the European Union, cofinanced by the
European Social Fund, in the framework of the TÁMOP 4.2.4.A/2-11-1-2012-0001 National Excellence
Author Contributions
L.V., J.Z., S.S. and X.L. designed research. X.L. and S.S. performed research and analysed the results.
E.S., L.B., H.Z. and L.S. contributed to the calculations. X.L., S.S., and L.V. wrote the paper. All authors
reviewed the paper.
Additional Information
Supplementary information accompanies this paper at http://www.nature.com/srep
Competing financial interests: The authors declare no competing financial interests.
How to cite this article: Li, X. et al. Tensile strain-induced softening of iron at high temperature. Sci.
Rep. 5, 16654; doi: 10.1038/srep16654 (2015).
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Scientific Reports | 5:16654 | DOI: 10.1038/srep16654
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