Ab initio study of the theoretical strength and magnetism of the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites

Ab initio study of the theoretical strength and magnetism of the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites

Accepted Manuscript Ab initio study of the theoretical strength and magnetism of the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites Tomá š Ká ňa, Martin Zouhar...

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Accepted Manuscript Ab initio study of the theoretical strength and magnetism of the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites Tomá š Ká ňa, Martin Zouhar, Miroslav Černý, Mojmír Šob PII: DOI: Reference:

S0304-8853(18)30124-0 https://doi.org/10.1016/j.jmmm.2018.08.027 MAGMA 64224

To appear in:

Journal of Magnetism and Magnetic Materials

Received Date: Revised Date: Accepted Date:

16 January 2018 29 May 2018 11 August 2018

Please cite this article as: T. Ká ňa, M. Zouhar, M. Černý, M. Šob, Ab initio study of the theoretical strength and magnetism of the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites, Journal of Magnetism and Magnetic Materials (2018), doi: https://doi.org/10.1016/j.jmmm.2018.08.027

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Ab initio study of the theoretical strength and magnetism of the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites Tomáš Káňaa, Martin Zouharb,c, Miroslav Černýd,e,f, Mojmír Šob b,f,g,* a

Central European Institute of Technology, CEITEC IPM, Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic b Central European Institute of Technology, CEITEC MU, Masaryk University, Kamenice 5, CZ-625 00 Brno, Czech Republic c Institute of Scientific Instruments, Academy of Sciences of the Czech Republic, Královopolská 147, CZ-612 64 Brno, Czech Republic d Central European Institute of Technology, CEITEC BUT, Brno University of Technology, Technická 2896/2, CZ-616 00 Brno, Czech Republic e Faculty of Mechanical Engineering, Brno University of Technology, Technická 2, CZ-616 00 Brno, Czech Republic f Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Žižkova 22, CZ-616 62 Brno, Czech Republic g Department of Chemistry, Faculty of Science, Masaryk University, Kotlářská 2, CZ-611 37 Brno, Czech Republic * Corresponding author Email: [email protected]

Keywords: nanocomposites, ab initio calculations, theoretical strength, magnetism Abstract We studied the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites formed by Fe nanowires embedded in the fcc Pd, Pt or Cu matrix. The Fe atoms form nanowires oriented along the [001] crystallographic direction. They replace second nearest neighbor atoms in the matrix. By means of varying the distance between the nanowires we arrived to the chemical compositions X15Fe, X8Fe and X7Fe where X stands for Pd, Pt and Cu. The mechanical and magnetic properties of the nanocomposites were obtained by ab initio simulations. We performed tensile and compressive tests along the [001] direction and compared the results with the deformation behavior of the fcc matrix and the known intermetallic compounds FePd3 and FePt3. It turned out that the maximum attainable stress for the Fe−Pd and Fe−Pt nanocomposites is higher than the stress attainable for the Pd and Pt matrices. The maximum stress increased with the increasing Fe content. The increase was due to the enhanced stability in the nanocomposites described by the C11 − C12 > 0 condition. This effect was particularly pronounced in the Fe−Pt nanocomposites. On the contrary, the Fe nanowires in the Fe−Cu nanocomposites do not enhance the stability and strength of the Cu matrix. They even make the Cu matrix more compliant to compression. Regarding the magnetic ground states, the Fe−Pd and Fe−Pt nanocomposites prefer a ferromagnetic configuration where the spins of all Fe atoms are oriented in parallel manner. On the other hand, the Fe−Cu nanocomposites exhibit an antiferromagnetic configuration where the spins of all Fe atoms assigned to a particular nanowire are oriented parallel, but antiparallel to the spins of a neighboring Fe nanowire. The Young modulus E001 along the [001] crystallographic direction increases linearly with the Fe content in both the Fe−Pd and Fe−Pt nanocomposites. 1

1. Introduction The ongoing search for nanomaterials with desired properties has been vastly supported by ab initio modeling. In this way, one can predict properties of proposed materials without the need of synthesizing them and thus spare a lot of laboratory work. A recent experimental work [1] studied the deformation behavior of a widely existing nanocomposite pearlite (lamellar structure of alternating Fe and Fe3C layers). It turned out that a stable columnar nanosized grain structure impedes dislocation motion and enables to achieve an extremely high tensile strength of 7 GPa, making this alloy the strongest ductile bulk material known. The columnar structure can be regarded as a form of a nanocomposite at a mesoscopic level. Using molecular beam epitaxy, Fe nanoparticles with a diameter of 2 nm embedded in a Pd matrix were prepared [2]. Alloying of Fe and Pd atoms has formed an FexPd1−x shell surrounding a reduced pure Fe core. Magnetic measurements of these nanocomposite samples pointed to a magnetized cloud of Pd atoms surrounding the embedded nanoparticles. Recently, a novel [email protected](Fe) nanocomposite has been successfully fabricated via facile alcohol reduction approach [3]. It exhibited superior photoactivity toward degradation of pharmaceutical and personal care products under visible light irradiation. Nanocomposite Fe100−xPtx (x = 40, 50, and 60) powders were prepared by mechanical alloying followed by annealing [4]. They showed various combinations of phases, away from thermodynamic equilibrium and a strong intergrain coupling. The Fe−Pt nanocomposites were also created in the form of thin films [5]. A Fe−Cu nanocomposite was prepared through a powder metallurgy route [6,7]. It was found that the composite specimens exhibit enhanced microhardness as compared with rule-of-mixtures [8] predictions. This enhancement was attributed to the interface strengthening at the fcc-bcc interphase boundaries. Similarly in the field of technology of production of integrated circuits, one is interested to what scale one can arrive and still keep the desired mechanical properties in the field of nanocomposite fabrication. At the atomic level, one arrives to the idea of metallic nanowires embedded in the surrounding matrix. A type of a nanocomposite at the atomic level is formed in such a way that some of the matrix atoms along selected crystallographic directions are substituted by the guest atoms, forming thus a nanowire. In this work, we focus on Fe nanowires [9]. The Fe−Cu nanocomposites at this level have been already modeled [10] in order to obtain the magnetic anisotropy. However, it is tempting to model their mechanical behavior by simulating a tensile/compressive test. If there is some strengthening or weakening in tension and compression, it should be assigned to the net forces between individual atoms. A part of the forces should come from the magnetic interactions between atoms. Such magnetic forces can even stabilize a particular structure, as is, for example, the case of bcc Fe [11]. From the modeling point of view, it is most easy to put the Fe nanowires along the [001] crystallographic direction of the embedding matrix. In this way, an ordered structure arises where the Fe atoms mutually stand for second nearest neighbors. There is a question whether the Fe second nearest neighbors still form a nanowire embedded in the fcc matrix or the whole structure will behave more like a solid solution. This will very probably depend on the interatomic bonding. An attempt to study the ordering mechanism in FePt via the vacancy mechanism in atomic migration has been made very recently [12]. 2

Further, it is instructive to compare the behavior of Fe nanowires in the Cu matrix, as Cu does not form any intermetallic compounds with Fe, [13] to the properties of Fe nanowires embedded in the Pd and Pt matrices which do form intermetallic compounds with Fe [14−16]. Another motivation to model nanocomposites at the atomic level [10,17−21] are the experimentally observed magnetic [22−30] or catalytic [31−35] properties of nanocomposites at the mesoscopic level. The paper is organized as follows. After introducing the structure of the X15Fe, X8Fe and X7Fe (X = Pd, Pt, Cu) nanocomposites in Section 2 we provide a brief description of calculation methods. Section 3 presents the results of ab initio calculations, namely the ground state magnetic configurations of the proposed Fe−Pd, Fe−Pt and Fe−Cu nanocomposites and the results of tensile and compressive tests. A discussion of the obtained results is also provided in this Section. At last, the outcomes are summarized in Section 4. 2. Materials and methods The studied nanocomposites are presented in Table 1 and illustrated in Fig. 1 on the example of Fe−Pd system. The Fe atoms form linear nanowires equidistantly distributed in the matrix. The perpendicular distance of the Fe nearest neighbor (nn) nanowires determines Fe concentration in the nanocomposite. If the nn nanowires are set 2a apart with a being the lattice constant of the Pd matrix (see Fig. 1a) then the chemical composition is Pd 15Fe. If the perpendicular distance of the Fe nanowires reduces to a√2 (Fig. 1b), then the chemical composition becomes Pd7Fe. The last nanostructure Pd8Fe (Fig. 1c) has the perpendicular distance of the nn nanowires 1.5a. It is formed by a 3×3×1 Pd supercell with the centers of all faces occupied by Fe atoms. It further turns out that all three considered nanocomposites exhibit a tetragonal structure. As it may be seen from Table 2, the calculated lattice constant c (measured along the [001] direction) is very close to the lattice constant a (measured along the [100] or [010] direction) so that the c/a ratio is not very far from 1 (maximum deviation is 1.4 %). Table 1 Lattice parameters of the fcc supercells needed to build the Fe−X nanocomposites, the perpendicular distance d of the nn Fe nanowires, Fe concentration cFe in atomic %, chemical formula used for the nanocomposites and their structure together with the reference to the panels of Fig. 1. The elemental fcc matrix completes the table. fcc supercell d cFe (%) XnFe structure

2a×2a×c

2a

6.25

3a×3a×c

1.5a 11.11 X8Fe Fig. 1c

2a×2a×c

√2a 12.5

a×a×a

a

25

FeX3 Fig. 1d (L12)

a×a×a



0

X

3

X15Fe Fig. 1a X7Fe Fig. 1b

fcc

Fig. 1. The Pd15Fe (a), Pd7Fe (b) and Pd8Fe (c) nanocomposites and the FePd3 intermetallic compound (d) with the L12 structure. The Fe nanowires follow the [001] crystallographic direction.

The FePd3 and FePt3 intermetallics crystallize in the L12 structure with the corner positions of the fcc Pd unit cell occupied by Fe atoms and can be regarded as natural nanocomposites formed by the Fe nanowires that are just one Pd lattice constant a apart (see Fig. 1d). Performing tensile or compressive test simulation consists in incremental stretching or compressing the ideal crystal along the considered loading axis and relaxing the structure so that the perpendicular biaxial stresses vanish. This corresponds to the modeling of Poisson contraction during the test. The total energy E is minimized with respect to structure cross section A for every strain increment. The investigated structure is represented within its unit cell where the positions of atoms are relaxed. The strain ε in the [001] direction is given by ε = c/c0 – 1 where c0 is the equilibrium lattice parameter of the structure in the [001] direction. The stress σ is evaluated as [36−41] σ = 1/(Ac0) ∂E/∂ε. The total energy E is calculated using the full-potential linearized augmented plane wave method (FLAPW) as implemented in the WIEN2k code [42]. For exchange-correlation energy, we used the generalized gradient approximation (GGA-PBE) [43]. The parameters of calculations of the total energy are as follows. The radii of muffin-tin spheres have fixed values of 2.21 a.u. for Fe, 2.23 a.u. for Cu, 2.32 a.u. for Pd and 2.40 a.u. for Pt. The energy dividing the valence and (semi)core electrons has been set to −9.0 Ry and the product of muffin-tin radius and the maximum reciprocal space vector, RMTKmax, was equal to 8.8. The maximum value ℓ for the waves inside the atomic spheres, ℓmax, was set to 10 and the 4

magnitude of the largest reciprocal vector G in the charge Fourier expansion, Gmax, was equal to 16. We employed 20 000 k-points in the first Brillouin zone in the mesh generated by the Monkhorst-Pack scheme [44] for an fcc structure and used a similar density of k-points for all other structures. The energy convergence criterion was 2 × 10 -5 eV/atom and on the basis of the convergence tests with respect to the number of k-points and RMT Kmax, the error in calculated total energies may be estimated to be less than 8 × 10-4 eV/atom. If symmetry conditions did not dictate the exact atomic coordinates in the unit cells as was the case of the elements Pd, Pt and Cu and the intermetallics FePd3 and FePt3, then we relaxed them so that the remaining forces between the atoms in the nanocomposites were less than 1mRy/bohr (≈ 26 meV/Å). 3. Results and discussion 3.1. Ground-state parameters The magnetic ground states, equilibrium lattice parameters, bulk moduli, Young moduli, average magnetic moments in the cell and the energy difference between the ferromagnetic and antiferromagnetic configuration of the Fe--Pd, Fe--Pt and Fe--Cu nanocomposites are presented in Table 2. Table 2 The equilibrium magnetic state, lattice parameter a and tetragonal ratio c/a, bulk modulus B, Young modulus E001 in the [001] direction, the total magnetic moment µ in the cell and the ferromagnetic energy gain EAFM − EFM (in meV/atom) with respect to the antiferromagnetic configuration of the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites and intermetallics. The experimental values of B and E001 were calculated from the elastic constants given in Ref. [45]. magnetic a (Å) c/a B (GPa) E001 (GPa) µ (µB/atom) EAFM − EFM state comp. expt. [45] comp. comp. expt. comp. expt. comp. expt. Pd NM 3.945 3.890 1 165 193 75 73 0 0 Pd15Fe FM 3.937 0.996 167 91 0.503 3.0 Pd8Fe FM 3.924 0.996 168 104 0.650 7.9 Pd7Fe FM 3.934 0.988 168 106 0.671 8.1 a FePd3 FM 3.891 3.848 1 169 123 1.076 1.09 13.0 Pt NM 3.972 3.924 1 249 283 143 136 0 0 Pt15Fe FM 3.964 0.996 241 156 0.362 1.2 Pt8Fe FM 3.953 0.995 238 165 0.583 1.7 Pt7Fe FM 3.951 0.995 235 168 0.599 2.2 FePt3 AFM (110) 3.910 3.872a 1 227 202 0 −19.4 Cu NM 3.632 3.615 1 143 138 82 66 0 0 Cu15Fe AFM (100) 3.640 0.996 143 90 0 −3.2 Cu8Fe AFM (100) 3.655 0.986 143 75 0 −6.1 Cu7Fe AFM (100) 3.656 0.986 143 85 0 −9.1 bcc Fe FM 2.836 2.867 1 199 169 131 119 2.178 2.2 fcc Fe FM low spin 3.491 1 217 264 1.057 fcc Fe NMb 3.44b 3.647, 3.572c 1 202d 274b 0 a)

Ref. [46]

b)

Ref. [47]

c)

Ref. [48]

d)

Ref. [49]

All these values were determined from data obtained during calculating the energy-volume and energy-strain curves. Atomic positions inside the corresponding unit cells were relaxed at 5

each volume or strain magnitude. For the considered elements and the FePd3 and FePt3 intermetallic compounds, we also provide a comparison with available experimental values [45]. The default proposed antiferromagnetic configuration is the AFM (100) where the Fe atoms lying in the (100) atomic planes distant by a flip their spin orientation. The a stands for the lattice parameter of the embedding Pd, Pt or Cu matrix. The magnetic ground state structure for the Fe−Pd and Fe−Pt structures is ferromagnetic in most cases with the only exception of FePt3 that prefers antiferromagnetic ordering AFM (110). There, the Fe atoms in the (110) atomic planes distant by a√2 flip their spin orientation. The antiferromagnetic energy gain of 19.4 meV/atom compared to the ferromagnetic state is quite unusual in the Pt-Fe part in Table 2. Let us note that the antiferromagnetic ordering AFM (100) is by 3.8 meV/atom more energetically demanding than the AFM (110). The employed exchange-correlation energy functional within the GGA slightly overestimates the equilibrium atomic volume of Pd (by 4.3 %), Pt (by 3.7 %) and Cu (by 1.4 %). On the other hand, it underestimates the atomic volume of bcc Fe by 3.2 %. Overall, the equilibrium volumes are reasonably close to available experimental values. In the Fe–Pd and Fe–Pt ferromagnetic nanocomposites, the Fe atoms induce magnetic moments on the surrounding Pd (see Ref. [2]) or Pt atoms. These induced moments are presented in Fig. 2. The Fe atoms in the Fe−Pd nanocomposites exhibit magnetic moments between 3.355 µB and 3.367 µB and the values of the moments induced on the Pd atoms vary between 0.180 µB and 0.355 µB. All these moments are calculated for the electrons inside the muffin-tin spheres. The electrons in the interstitial region show net negative magnetic moments between −0.179 µB and −0.267µB. In particular, in the FePd 3 intermetallic compound, the Fe atoms exhibit magnetic moments of 3.326 µB and all Pd atoms exhibit quite high induced moment of 0.354 µB. The electrons in the interstitial region exhibit magnetic net moment of −0.083 µB.

Fig. 2. The magnetic moment of the Fe atoms and the induced magnetic moments on the surrounding Pd (or Pt) atoms in the Fe−Pd (Fe−Pt) nanocomposites.

The Fe atoms in the Fe−Pt nanocomposites exhibit slightly lower magnetic moments between 3.281 µB and 3.317 µB and the values of the moments induced on the Pt atoms are also lower than in the Fe−Pd nanocomposites. Their values are within the range from 0.098 µB to 0.264 µB. The electrons in the interstitial region show net negative magnetic moments between −0.085 µB and −0.139 µB. Further, in the FePt3 intermetallic compound, the Fe atoms in the AFM(110) magnetic configuration exhibit the moment of ±3.297 µB and there is essentially 6

zero induced moment on the Pt atoms. This is a difference compared to the case of FePd 3 and an interesting property indicating that the magnetic configuration changes if the [001] Fe nanowires are arranged very densely in the Pt matrix. A similar study for the Mn−Pt compounds and nanocomposites [21] revealed that also the superstructure MnPt7 should exhibit antiferromagnetic ordering. Comparison of the values of magnetic moments in Fig. 2 indicates that the values of induced magnetic moments on the matrix atoms are most evenly distributed in the Pd8Fe and Pt8Fe nanocomposites. In contrast, Fe nanowires embedded in the Cu matrix prefer an antiferromagnetic ordering. In general, we found that all Fe atoms in a particular nanowire have parallel orientation of their spins. The total energy calculations reveal that the spins of individual Fe atoms in neighboring nanowires prefer mutual antiparallel orientation. The values of magnetic moments of electrons inside the Fe muffin-tin spheres in the Fe−Cu nanocomposites are between ±2.771 and ±2.789 µB. The magnetic moments induced on the Cu atoms attain rather low values limited to the range from ±0.010 to ±0.043 µB. The electrons in the interstitial region show zero net magnetic moment. For low Fe concentration in the Fe−Pd and Fe−Pt nanocomposites, it can be said that the embedded Fe nanowires increase linearly their Young modulus (see Fig. 3). For the Fe−Pd nanocomposites, the increase is linear up to the Pd8Fe. On the other hand, E001 of the antiferromagnetic FePt3 intermetallic compound deviates from the linear trend valid for the ferromagnetic Fe−Pt nanocomposites.

Fig. 3. The dependence of the Young modulus in the [001] crystallographic direction E001 on the Fe content in the Fe−Pd and Fe−Pt nanocomposites and the FePd3 and FePt3 intermetallic compounds. Linear fits of E001 in the range of 0 < cFe < 25 % are displayed for both Fe−Pd and Fe−Pt. For illustration, the values of E001 for the bcc and fcc Fe are denoted by horizontal lines.

3.2. Uniaxial loading All the structures were then subjected to simulated uniaxial tensile and compressive tests along the [001] direction. The obtained stress-strain curves are illustrated in Figs 4, 5 and 6. Let us note that in our tensile tests we calculate ideal tensile strength (of material without defects, also without dislocations) and look after the first elastic instability, without any attempt to include plastic deformation and motion of dislocations. Theoretical tensile strength represents the upper limit of the materials strength and provides valuable information about the strength and character of interatomic bonding, as discussed in details e.g. in Refs. [39-41]. 7

Under tension, the stresses in most of the studied systems reach their maxima σ+max at strains ε+max = 0.3−0.4. These stress maxima are associated with vanishing of the Young modulus. However, as it has been shown in numerous studies [50−54] fcc crystals loaded in tension along the [001] direction tend to fail by shear at strains lower than ε+max. In such cases, analyses of elastic stability typically predict violation of other stability condition C11 − C12 > 0. For this reason we evaluated the difference C11 − C12 (proportional to the tetragonal shear modulus) as a function of axial strain ε and identified strains ε+ and stresses σ+ related to its vanishing. The maximum stress in tension σ+max and the stress σ+ related to occurrence of the elastic instability due to violation of the condition C11 − C12 > 0 are presented for the Pd, Pt and Cu elements and the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites in Table 3 along with the corresponding strains ε+max and σ+.

Table 3 Computed values of the maximum stress σ+max during tensile loading and the corresponding strain ε+ max for the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites and intermetallics. The next two columns present values of the stress σ+ and strain ε+ at which the stability condition C11 − C12 > 0 is violated. The last two columns present maximum compressive stress σ−max and the corresponding strain ε−max. All stresses are in GPa. magn. state σ+max ε+max σ+ ε+ σ−max ε−max Pd Pd15Fe Pd8Fe Pd7Fe FePd3 Pt

NM FM FM FM FM NM

Pt15Fe Pt8Fe Pt7Fe FePt3 Cu

FM FM FM AFM(110) NM

Cu15Fe AFM(100) Cu8Fe AFM(100) Cu7Fe AFM(100) bcc Fe FM fcc Fe NM a)

Ref. [21] (LDA) d) Ref. [37]

23.3 0.355 9.0 23.5 0.367 14.4 24.0 0.362 15.4 24.4 0.362 15.0 25.5 0.328 18.7 27.9 0.314 16.0 34.7a 0.324a 34.1b 0.34b 28.9 0.313 26.7 29.2 0.308 27.6 29.2 0.311 27.6 30.6 0.300 29.7 24.5 0.358 9.9 24.3c 0.33c 9.4c 24.1b 0.36b 24.7 0.369 9.5 24.8 0.378 9.3 24.8 0.382 8.4 12.1 0.145 12.1 12.7d 0.150d 48.1e 0.275e b)

0.099 0.151 0.156 0.141 0.161 0.114

−3.2 −4.7 −5.4 −5.2 −7.7 −6.5 −7.8a

0.228 0.232 0.234 0.242 0.099 0.10c

−8.8 −0.107 −10.5 −0.111 −10.1 −0.111 −12.3 −0.108 −3.6 −0.087 −3.5c −0.09c

0.100 0.108 0.096 0.145

−3.5 −0.090 −2.9 −0.086 −2.8 −0.077 −26.2 −0.112

Ref. [55] (LDA) e) Ref. [47]

8

c)

−0.107 −0.097 −0.106 −0.096 −0.114 −0.115 −0.114a

Ref. [52]

Fig. 4. Top: The stress-strain curves for the Pd matrix, the Fe−Pd nanocomposites and the intermetallic compound FePd3. Bottom: The change of material constant C11 − C12 during the compressive and tensile loading.

9

Fig. 5. Top: The stress-strain curves for the Pt matrix, the Fe--Pt nanocomposites and the intermetallic compound FePt3. Bottom: The change of material constant C11 − C12 during the compressive and tensile loading.

10

Fig. 6. Top: The stress-strain curves for the Cu matrix and the Fe−Cu nanocomposites. Bottom: The change of material constant C11 − C12 during the compressive and tensile loading.

Under uniaxial compression, the stresses reach their (negative) maxima σ−max already at strains ε−max that are remarkably lower (around −0.1) than those in tension. The reason is that, under the [001] uniaxial compression, fcc crystals undergo a structural transformation along tetragonal Bain's path to the bcc structure and, therefore, due to existence of symmetrydictated energy maximum at ε− ≈ −0.206 (corresponding to the bcc structure obtained for c/a = 1/√2, if we consider the fcc structure as a tetragonal structure with c/a = 1 − see e.g. Ref. [5658], where the bcc and fcc structures are treated at rescaled values of c/a = 1 and √2 for bcc and fcc structures, respectively). This explains also occurrence of tensile (positive) stresses in the region of negative strains (particularly visible in Fig. 5 for Pt and Fe−Pt nanocomposites). One can thus understand increasing the amplitude of compressive strain as a sequence of loading of fcc structure in compression and unloading of bcc structure in tension with its subsequent loading in compression. Former analyses of elastic stability [51-53] of fcc crystals under [001] compression reported no shear instability prior vanishing of Young modulus (at ε−max). Also our results (strain-dependent values of C11 − C12) do not indicate occurrence of any instability 11

related to vanishing of the tetragonal shear modulus within the range of studied compressive strains. The values of maximum compressive stress σ−max can be, therefore, considered to be the ideal compressive strength and were added (together with the corresponding critical compressive strains ε−max) to Table 3. Pd attains maximum stress in tension σ+max = 23.3 GPa at the strain ε+max = 0.355. However, this value is greatly reduced to σ+ = 9.0 GPa due to the violation of the C11 − C12 > 0 condition that occurs already at the strain ε+ = 0.099 (see Fig. 4). Table 3 and Fig. 4 show that the embedded Fe nanowires in the Pd 15Fe, Pd8Fe, Pd7Fe nanocomposites affect the σ+max value only marginally but substantially increase their attainable stress σ+ along the [001] direction by shifting the ε+ at which the stability condition is violated to higher values. This trend is also illustrated in the top-left panel of Fig. 7.

Fig. 7. Top left: The difference between the maximum stress σ+ attainable in tension for a particular FeXn nanocomposite and for the elemental matrix X (X = Cu, Pd, Pt) prior to the violation of the C11 − C12 > 0 condition. The difference ∆σ+ is displayed as a function of the Fe concentration cFe in the nanocomposite. Bottom left: The same for the corresponding deformation ε+. Top right: The difference between the maximum stress σ− max attainable in compression for a particular FeXn nanocomposites and for the elemental matrix X. Bottom right: The same for the corresponding deformation ε−max.

The maximum tensile stress in Pt σ+max reaches the value of 27.9 GPa at the strain ε+max = 0.314 but the maximum attainable stress σ+ is reduced to 16.0 GPa by the shear instability occurring at ε+ = 0.114. The effect of increasing σ+ and ε+ by the embedded nanowires in the Pt15Fe, Pt8Fe and Pt7Fe nanocomposites is here even more pronounced, shifting the σ+ close to the σ+max (see top-left panel of Fig. 7). In the case of Cu, the maximum stress in tension σ+max = 24.5 GPa attainable at the strain ε+max = 0.358 is reduced to only 9.9 GPa due to the violation of the C11 − C12 > 0 stability condition at the strain ε+ = 0.099 (coincidentally the same strain as was in the case of Pd). This result is in a good agreement with that of former analysis of elastic stability of Cu [52]. In tension, the 12

embedded nanowires practically do not affect the strength of the nanocomposites. This indicates that the model of a solid solution of Fe atoms in Cu matrix is more suitable for the description of the Fe−Cu nanocomposites. On the other hand, the Fe nanowires weaken the Fe−Cu nanocomposites in compression. The value σ−max = −3.6 GPa is effectively reduced for the Cu8Fe and Cu7Fe nanocomposites. The weakening in compression can be explained realizing that the resistance of a material against compression is proportional to its hardness. A chemically pure single crystal iron is very soft (3 MPa HB) [59]. On the contrary, for the Fe−Pd and Fe−Pt nanocomposites, the embedded iron nanowires significantly increase the resistance against compression as can be seen from Table 3, Fig. 4 and 5 and top-right panel in Fig. 7. Namely, Pd attains maximum stress in compression σ−max = −3.2 GPa at ε−max = −0.107. The addition of Fe nanowires into the Pd matrix strengthens the resulting Fe−Pd nanocomposites in compression by about 45−70 %. The maximum attainable compressive stress in Pt σ− max = −6.5 GPa is reached at the strain ε−max = −0.115. Similarly to the Fe--Pd case, σ−max calculated for the Fe−Pt nanocomposites is also increased by the addition of Fe nanowires by about 35−60 %. For completeness, the bcc Fe attains maximum tensile stress σ+max = 12.1 GPa at the strain ε+max = 0.145. These values also correspond to σ+ and ε+ since the stability condition C11 − C12 > 0 is violated at a relatively high strain ε = 0.270. Our calculations for bcc Fe also yield the maximum stress in compression σ+max = −26.2 GPa attained at the strain ε−max = −0.112. 4. Conclusions In summary, we have studied the theoretical strength and magnetic properties of the Fe−Pd, Fe−Pt and Fe−Cu nanocomposites by means of ab initio calculations. It turns out that the Fe−Pd and Fe−Pt nanocomposites prefer ferromagnetic arrangement whereas the Fe−Cu nanocomposites prefer an antiferromagnetic configuration. There, the Fe atoms corresponding to one particular Fe nanowire have the same spin orientation and the Fe atoms of the neighboring nanowire have antiparallel spin orientation. The Pd and Pt atoms exhibit magnetic moments induced by the magnetic moment of the Fe atoms. The induced magnetic moments on the matrix atoms are most evenly distributed in the Pd8Fe and Pt8Fe nanocomposites. The studied Fe nanowires are formed by the next nearest neighbor atoms. As a result, they do not significantly affect the tensile strength of the Fe−Cu nanocomposites along the [001] crystallographic direction. On the other hand, the Fe nanowires soften the Cu matrix in compression along the [001] direction. In contrast, the Fe nanowires improve the mechanical properties of the Fe−Pt and Fe−Pt nanocomposites during both tensile and compressive tests along the [001] direction. Both compressive and tensile strength of the nanocomposites are improved, the latter mainly due to the increase of the maximum attainable strain ε+ at which the stability condition C11 − C12 > 0 is violated. In the case of Fe−Pt nanocomposites, the embedded Fe nanowires almost allow to reach the maximum stress σ+max corresponding to vanishing of the Young modulus.

13

In the Fe−Pd nanocomposites, the Young modulus E001 in the [001] crystallographic direction increases approximately linearly with the Fe content. The trend is also linear for the Fe−Pt nanocomposites. We note that the present results will motivate experimentalists to investigate these interesting nanocomposites. Acknowledgements This research has been financially supported by the Czech Science Foundation (Project No. GA16-24711S), by the Ministry of Education, Youth and Sports of the Czech Republic under the Project CEITEC 2020 (Project No. LQ1601), and by the Academy of Sciences of the Czech Republic (Institutional Project No. RVO:68081723). Computational resources were provided by the Ministry of Education, Youth and Sports of the Czech Republic under the Projects CESNET (Project No. LM2015042) and CERIT-Scientific Cloud (Project No. LM2015085) within the program Projects of Large Research, Development and Innovations Infrastructures. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

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Highlights: •

Fe–Pd, Fe–Pt and Fe–Cu nanocomposites formed by Fe nanowires embedded in the fcc Pd, Pt or Cu matrix are studied with the help of ab initio calculations.



Tensile and compressive tests along the [001] direction are simulated and the deformation behaviour of the nanocomposites is compared with that of fcc matrix and selected intermetallics.



Maximum attainable stresses for the Fe–Pd and Fe–Pt nanocomposites are higher than the stresses attainable for the Pd and Pt matrices but the Fe nanowires in the Fe–Cu nanocomposites make the Cu matrix more compliant to compression.



Fe–Pd and Fe–Pt nanocomposites prefer a ferromagnetic configuration. On the other hand, the Fe–Cu nanocomposites exhibit an antiferromagnetic arrangement.

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Studied nanocomposites. The Fe nanowires follow the [001] crystallographic direction.

Young modulus as a function of iron concentration.

The stress-strain curves for the Pd matrix, the Fe−Pd nanocomposites and the intermetallic compound FePd3. Bottom: The change of material constant C11 − C12 during the compressive and tensile loading.