# Twinning and amorphization as plasticity mechanisms in ultra small tungsten nanoparticles

Laurent Pizzagalli, Julien Godet, Julien Durinck, Sandrine Brochard, Jean

Furstoss, Felipe J. Valencia

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Laurent Pizzagalli, Julien Godet, Julien Durinck, Sandrine Brochard, Jean Furstoss, et al. Twinning and amorphization as plasticity mechanisms in ultra small tungsten nanoparticles. Computational Materials Science, 2025, 258, pp.114077. 10.1016/j.commatsci.2025.114077. hal-05172063

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# Twinning and amorphization as plasticity mechanisms in ultra small tungsten nanoparticles

L. Pizzagalli<sup>a,†</sup>, J. Godet<sup>a,‡</sup>, J. Durinck<sup>a,§</sup>, S. Brochard<sup>a,¶</sup>, J. Furstoss<sup>a,¶</sup>, F. Valencia<sup>b</sup>

## ARTICLE INFO

**Keywords:**

Compression

Nanoparticles

Tungsten

First-principle calculations

Molecular dynamics

## ABSTRACT

First principles molecular dynamics calculations are carried out to simulate the uniaxial compression of ultra small tungsten nanoparticles, along two crystallographic directions. We find that the predicted elastic moduli and strengths are larger than bulk values, up to 45% more for the former and 69% more for the latter. Most of nanoparticles yield by amorphization, which is a plasticity mechanism known to be favored at small sizes. However, twinning is found to occur for the two largest nanoparticles compressed along (110). Our analyses reveal the homogeneous nucleation of a two-layers {112} twin embryo thanks to the simultaneous glide of two opposite partial dislocations.

## 1. Introduction

The mechanical properties of materials can be modified and improved in a number of ways, for instance through alloying, impurities addition, or structural engineering at various scales. Another key factor is the characteristic dimension of materials, due to the intrinsic multi-scale nature of mechanical properties. This aspect has been extensively studied at low scales by investigating sub-micrometer samples such as nanopillars, nanowires and nanoparticles subjected to a mechanical solicitation [1–3]. A general result is that the decrease of the available volume as well as the increased influence of surfaces profoundly change the plasticity properties of material, by triggering processes like dislocation starvation, heterogeneous nucleation of dislocations [4], or the activation of single-arm sources [5]. It overall leads to an increase of strength when the dimension is reduced, as well as an increase of toughness and ductility of materials normally brittle at larger sizes [6–9].

Nanoparticles (NP) made of various materials have been investigated both experimentally and theoretically, in order to determine their mechanical properties. Most studies concern Face Centered Cubic (FCC) metals [10–20] and related alloys [21–25]. Oxides [26–32], semiconductors [33–44], and carbon-based materials [45–48] are other classes of materials which have also been the focus of numerous studies. Finally, several groups have studied NP with core–shell geometries [18, 49–56]. Such systems are especially interesting since they allow for more flexibility to tune a specific property. In stark contrast to this abundant literature, experiments and calculations of metallic NP with a

Body Centered Cubic (BCC) structure are scarce, with a few studies on Fe [57], Nb [58], and Mo [59], despite the technological importance of this class of materials. This clearly calls for additional investigations.

Most of the investigations listed above concern NP with dimensions ranging from a few tens to several hundreds of nanometers. Considering that even smaller NPs, of only a few nanometers, can be synthesized, there is a clear incentive to investigate mechanical properties at smaller scales. However, measuring the plasticity properties of these ultra small systems is extremely challenging due to technical limitations [60]. In our group we have recently developed an ab initio molecular dynamics (AIMD) approach to predict at the density functional theory (DFT) level of accuracy the properties of 1–2 nm NP subjected to uniaxial compression [61]. This approach has been successfully applied to carbon, silicon, silicon carbide, and aluminum NP [44,62–64]. One of the main results is the prediction that the strength of these ultrasmall NP is equal or even seemingly greater than the bulk theoretical value. Another interesting feature is that amorphization becomes a competing plasticity mechanism at this scale due to high local stresses.

The objective of the present work is to determine the mechanical properties of ultrasmall BCC tungsten NP by using this numerical approach. Tungsten is one of the strongest metals, and it is considered as a first wall material in fusion reactor divertors like ITER. It is reported that the interaction of plasma during operation leads to the formation of a tungsten NP dust [65]. Tungsten NP are also potentially interesting for applications in energy storage, catalysis, biomedicine or as mechanical strengtheners. It appears then scientifically and technologically relevant to determine their mechanical properties.

<footnote><sup>a</sup> Institut P', CNRS UPR 3346, Université de Poitiers, SP2MI, Boulevard Marie et Pierre Curie, TSA 41123, 86073 Poitiers Cedex 9, France</footnote>

<footnote><sup>b</sup> Departamento de Computación e Industrias, Facultad de Ciencias de la Ingeniería, Universidad Católica del Maule, Talca 3480112, Chile</footnote>

<footnote>* Corresponding author.
E-mail address: Laurent.Pizzagalli@univ-poitiers.fr (L. Pizzagalli).</footnote>
<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Variation of contact force, nN (a), energy per atom, eV/at (b), and contact stress, GPa (c) as a function of the engineering strain $\epsilon$ for the investigated systems (blue: W$_{59}$, orange: W$_{65}$, green: W$_{78}$, khaki: W$_{113}$, red: W$_{137}$, indigo: W$_{169}$). Full lines show results for (110) compressions, and dashed lines results for (100) compressions. The force and stress curves are time-averaged to smooth out fluctuations for a better clarity. The studied NP models are represented on the right part of the figure, for the (110) (d) and (100) (e) orientations.</figcaption></figure>

<caption>**Table 1**</caption>

Calculated lattice parameter $a_0$ (Å) and elastic constants (GPa). The Young (E) and shear (G) moduli are calculated using the Voigt average of elastic constants. Experimental values for lattice parameter [66] and elastic constants [67] are reported for comparison.

<table><thead><tr><th></th><th>a<sub>0</sub></th><th>C<sub>11</sub></th><th>C<sub>12</sub></th><th>C<sub>44</sub></th><th>E</th><th>G</th></tr></thead><tbody><tr><td>This work</td><td>3.1753</td><td>546</td><td>193</td><td>158</td><td>421</td><td>165</td></tr><tr><td>Exp.</td><td>3.165</td><td>523</td><td>204</td><td>160</td><td>410</td><td>160</td></tr></tbody></table>

In the following of this paper we first describe the NP models and the computational approach used to simulate the compression. We then report in Section 3 calculated mechanical data such as elastic modulus, critical strain and strength. It is revealed that for each property there is no clear relation with the size or shape of the NP. The most striking feature is the high values for elastic modulus and strength, which are both greater than bulk theoretical values. The next section is dedicated to the description of the plasticity mechanisms. The NP are found to yield by amorphization and twinning for (110) compression, depending on the NP size. For the (100) compression, only amorphization is obtained. These results are discussed in Section 5.

## 2. Methods

All calculations are made in the framework of DFT, as implemented in the Quantum Espresso code [68]. The electronic structure is calculated using the Perdew-Burke-Ernzerhof functional for describing exchange-correlation contributions [69], the Projector Augmented-Wave method [70] for ion-electron interactions with the valence electron configuration $5s^25p^65d^46s^2$, and the cold smearing method [71]. The convergence is obtained by using a plane wave energy cutoff of 40 Ry and a charge density cutoff of 320 Ry. The lattice parameter and the elastic constants of bulk tungsten, computed using $\gamma$-centered k-points meshes of 20 × 20 × 20 and a minimum force criterion of

$10^{-4}$ eV Å$^{-1}$, are reported in Table 1. The comparison with experiments is satisfactory, with deviations in the usual range of DFT calculations.

Six models of BCC tungsten NP are built either by carving spheres from bulk, or by using the Wulff construction procedure together with calculated surface energies [72]. They include 59 to 169 W atoms, with characteristic dimensions in a 1–2 nm range. Previous calculations show that for this size range a BCC crystalline structure is energetically favored compared to a disordered phase [72]. Larger systems are not considered because of the huge computational effort needed to perform long AIMD calculations for tungsten NP. These models are next introduced in a supercell large enough to allow for a minimal separation of 10 Å between periodic replicas. They are oriented with either the (110) or the (100) crystallographic directions, or both, along the compression axis (Fig. 1-d, e). After relaxation, the uniaxial compression of the NP is achieved by performing AIMD calculations with the Born–Oppenheimer scheme. The electronic structure is calculated with a $\gamma$ point sampling of the Brillouin zone, as is customary for isolated systems. A constant temperature of 1 K is obtained using the Bussi-Donadio-Parrinello thermostat [73]. Such a low temperature combined with the high thermostat stability allow for a timestep of 5 fs without appreciable energy drift in the timescale of the simulation, and for largely reducing the influence of high strain rates typical of MD simulations. The uniaxial compression of the NP is obtained thanks to two mobile planar repulsive classical force fields [61–63], moving towards each other along the compression axis with a velocity of $0.05$ Å s$^{-1}$. It corresponds to strain rates ranging from 5.8 to 8.3 × $10^9$ s$^{-1}$. Classical molecular dynamics tests have been performed with lower values ($10^7–10^9$ s$^{-1}$) to make sure that the calculated strength and deformation mechanisms did not depend on the strain rate. A force field strength of 1836 eV Å$^{-3}$ is used, which simulates incompressible flat-punch indenters. The typical duration of an AIMD run is about 30 ps, what makes it possible to achieve deformations between 0.18 and 0.34 depending on the NP. In total, nine different compression tests are considered in this study.

The variations of the potential energy, of the forces applied on the NP by classical force fields, and of the atomic structure are monitored during each run. The contact force with the flat punch indenters is determined as the average of these applied forces. The engineering strain $\epsilon$ is calculated relatively to the difference between classical force field positions, with an origin corresponding to applied forces different from zero. With this strain definition, the reference energy is obtained by averaging the potential energy of configurations for $\epsilon < 0$. This reference is then subtracted to the energy variation. Finally the contact stress is calculated as the ratio of the contact force over the contact surface. The definition of the latter is ambiguous and critical at the nanoscale, as shown in the literature [74–77]. For instance, the area $S_0$ equal to the convex Hull of surface atoms is the lowest value of the true surface (assuming an homogeneous contact), yielding unrealistic contact stresses at the smallest sizes. An upper limit of the contact surface area can be obtained by assuming that a surface atom contribution is equal to $\Sigma$, the area per atom of a perfect surface. Approximating the contact surface as a regular polygon, an estimation of its area is $S = S_0 + (m/2 + 1)\Sigma$, with m the number of surface atoms. Here $\Sigma = a_0^2/\sqrt{2}$ for (110) compression, and $\Sigma = a_0^2$ for (100) compression. Atoms less than 0.5 Å from the force field position are considered to be part of the contact surface.

## 3. Elasticity and strength

The variations of contact force, energy, and contact stress as a function of strain are represented in Fig. 1-(a-c) for all compression runs. Force and stress curves exhibit an initial linear increase up to a first maximum, generally rounded, followed by a gentle decrease and finally a seemingly constant level with significant fluctuations. The energy initially increases quadratically as expected for an elastic deformation. In most cases, the first energy maximum occurs at a slightly larger
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Elastic modulus (a), critical strain $\epsilon_c$ (b) and critical contact stress $\sigma_c$ (c) for all compression runs, as a function of the shape factor. The latter is defined as the area ratio of the initial contact surface over the initial maximum section (normal to the compression axis). The size of the markers is proportional to the NP size (in nm), and the color indicates the compression crystallographic direction (blue for (100) and orange for (110)). The blue (orange) dashed line in (a) indicates the bulk (100) ((110)) Young modulus. The dark gray strip in (c) corresponds to a theoretical bulk strength range of 35.7–36.4 GPa (see text for details).</figcaption></figure>

strain than the first maximum in force/stress curves, and is followed by a small energy decrease. As an example, we describe the results obtained for the (110) compression of $W_{78}$ (full green lines in Fig. 1). Distinct force and stress maxima of 31 nN and 49 GPa, respectively, are achieved when a 0.13 strain is reached. The first energy maximum, equal to 0.16 eV/at, occurs when the strain is about 0.16. For three systems (the (100) compression of $W_{113}$, $W_{137}$ and $W_{169}$), no clear energy maxima can be identified. At large strains the energy continues to increase according to an approximately linear variation. Although NP are comparable in size and shape, force, stress, and energy curves are remarkably diverse. It is also noteworthy that it is not possible to draw a clear distinction between (100) and (110) compression curves.

The elastic modulus is determined by linear regression of the stress in the linear elastic regime. We note that for some (110) compressed NP, an initial low slope can be identified up to strains equal to 0.02–0.05 depending on the NP (see $W_{59}$ for instance, full blue curve in Fig. 1). The contact forces show similar variations, indicating that this effect originates from the intrinsic mechanical response of the NP, and is not a consequence of the contact surface definition. A careful analysis of atomic displacements was unsuccessful to identify the cause of this phenomenon, which does not occur in case of (100) compression. For a coherent and meaningful comparison of elastic moduli, this initial soft slope is not taken into account for the linear regression.

Fig. 2-(a) shows elastic moduli as a function of a NP shape factor, which is the ratio of the number of atoms belonging to a contact surface over the number of atoms belonging to the central cross-section layer. The shape factor is then equal to 1 for a cubic NP, while it would be close to 0 for spherical geometries. Similar shape factors are proposed by various authors [20,77,78]. Fig. 2-(a) also displays information about the NP size and the compression orientation. We notice that there is no clear correlation between the elastic moduli and all these characteristics. The values range from 457 to 646 GPa,

except for $W_{137}$ compressed along (100) for which a much lower value of 295 GPa is obtained. The likely origin of the outlier is the singular contact surface geometry. In fact four of the contact surface atoms have only three first neighbors in the subsurface layer (see light gray atoms in Fig. 4). It is in stark contrast with all other NP, for which all contact surface atoms have four first neighbors. This peculiar configuration is less stable mechanically, and these four atoms make little contribution to the elastic response. Discounting these atoms in the calculation of the contact surface area, an elastic modulus of 531 GPa is obtained, in the same range than the other systems.

The comparison of our data to the Young modulus for bulk tungsten along both orientations, determined from the elastic constants reported in Table 1, indicates that the NP elastic moduli are systematically higher, up to 45% more. This finding is at odds with a recent study of gold NP finding greater or lower elastic moduli compared to the bulk depending on the shape and compression orientation [77]. We emphasize that the evaluation of the elastic modulus for such small sizes is critically linked to the contact surface definition. To prove the point, we computed the elastic modulus of W NP using the convex Hull surface $S_0$. The values now range from 830 to 2380 GPa, which are too high to be physically realistic.

We also report in Fig. 2-(b, c) the critical strain $\epsilon_c$ and stress $\sigma_c$ corresponding to the first maximum in stress–strain curves. We note that $\epsilon_c$ is typically a few percents smaller than the strain associated with the first energy maximum. The difference decreases as a function of the NP size, and its origin is not clear. We find that $\epsilon_c$ ranges from 0.096 to 0.153, which is comparable to values predicted for aluminum NP of similar sizes [64]. There is no obvious correlation between $\epsilon_c$ and shape, or size, or compression orientation. A similar conclusion can be made regarding $\sigma_c$, the later defining the strength of the NP. Values ranging from 37.8 GPa ($W_{65}$ (110)) to 62.4 GPa ($W_{169}$ (100)) are obtained.

It is interesting to compare the NP strength with bulk-like quantities to estimate finite size effects. Using DFT calculations, Černý et al. determined that the theoretical strength of tungsten uniaxially compressed along (110) is 36.4 GPa [79]. This value is reached at a critical strain of 0.11, above which a 45° rotation of the BCC crystal is observed. For the (100) orientation, calculations by the same authors yield a maximum strength of 58 GPa [80]. The theoretical shear stress (TSS) is another measure of the strength of a material, corresponding to yield by dislocation nucleation [81]. Černý et al. reported TSS values of 17.1 GPa and 16.8 GPa for the (111){110} and (111){112} slip systems, respectively [79,80], in close agreement with an earlier investigation [82]. Noting that the Schmid factor is at most 0.408 for (111){110}, and at most 0.471 for (111){112}, for both (110) and (100) compression, these values correspond to bulk compressive strengths of 41.9 GPa for (111){110} and 35.7 GPa for (111){112}. Considering all these values, we can estimate that the bulk tungsten subjected to a (110) uniaxial compression should yield at a stress of 35.7 GPa by dislocation nucleation along {112} planes, or at 36.4 GPa by structural transformation. For the (100) compression, yield should preferentially occur by dislocation nucleation along {112} planes at 35.7 GPa. Our computed NP strengths are all larger than these values, up to 69% more for $W_{169}$ (100). This is in agreement with our previous investigations of ultrasmall aluminum NP [64]. The meaning and implication of this result are discussed in Section 5.

## 4. Plasticity mechanisms

In this section we focus on the stress relaxation mechanisms associated with the first stress maxima in Fig. 1. The analyses of atomic displacements reveal two different behaviors that are twinning along {112} planes, and amorphization. The former occurs during the (110) compression of $W_{169}$ and $W_{137}$. Fig. 3 describes the main steps of the twinning mechanism for $W_{169}$ in relation with the stress and energy variations (Fig. 3-(c)). Up to a strain $\epsilon = 0.117$, at which the stress is
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Results obtained for the (110) compression of W₁₆₉. (a) Ball-stick representation of W₁₆₉ oriented along ⟨110⟩. (b) [110]-projection of W₁₆₉, with selected atoms in yellow to highlight a {112} slip plane. (c) Stress in GPa (blue curves) and energy in eV (orange curve) as a function of strain. (d)–(g) [110]-projection of W₁₆₉ at ε = 0.117, 0.135, 0.152, 0.185. Arrows centered on atomic positions are proportional to cumulative atomic displacements, relatively to ε = 0.117. The arrow color depends on the sign of the displacement along the compression axis [110]. The orange dashed line in (e) shows the region where twinning seems to initiate. The one in (f) is a guide to visualize the displacement between the two sheared regions of the NP.</figcaption></figure>

maximum, the deformation is purely elastic (Fig. 3-(d)). This configu-
ration is used as the reference to calculate the atomic displacements at
greater strains. At ε = 0.135, we observe that the energy is maximum
and that the largest displacements occur in {112} layers adjacent to the
central {112} plane (with atoms colored in yellow in Fig. 3-(b, d-g)).
These displacements also belong to a {110} plane, and they are mainly
oriented along a ⟨111⟩ direction, with opposed directions on both side
of the central layer (Fig. 3-(e-g)). It seems that these displacements are
initiated from the center of the NP (orange circle in Fig. 3-(e)), which
would suggest that the twin nucleation is homogeneous. At ε = 0.152,
the NP is sheared with two halves displaced in opposite directions
(Fig. 3-(f)). However the central {112} layer is only slightly affected.
Note that due to compression the sheared NP slightly tilts in the [001]
direction. In fact, the initial angle between the central {112} layer
and the compression axis is arccos(1/√3) = 54.73°, which decreases
to 48.4° at ε = 0.185. Increasing further the compression achieves the
process, with perfectly symmetric displacements in the two halves (Fig.
3-(g)). The formation of a two-layers {112} twin (orange line in Fig. 3-
(f)) is then obtained by the simultaneous glide of two opposite partial
dislocations along ⟨111⟩ directions in two adjacent {112} slip planes.
We find that a similar mechanism occurs during the ⟨110⟩ compression
of W₁₃₇.

Twinning in BCC metals is a well documented mechanism [85]. Wang et al. demonstrated that twinning would prevail over dislocation slip in tungsten nanowires where large stresses can be reached, for sizes below 40 nm [86,87]. It is also assumed to be favored by low temperatures and high strain rates [85], as is the case in the present work. Important unresolved issues concern the nucleation of twins, and in particular the minimum size of embryos and the involved mechanisms. The earliest theoretical investigations based on interatomic potentials calculations indicate that twin embryos need to be at least three layers wide to be stable [85]. DFT calculations of twinning generalized stacking fault energies rather suggest a minimum stable width of two layers [87,88]. Experimentally, the thinnest observed twin embryos are six-layers wide, but it is extremely difficult to observe the primary formation steps due to fast dynamics [85,87]. Recent theoretical investigations confirm that a two-layers twin embryo is stable [89,90], and highlight the decisive influence of compressive stress [89]. In the present work we indeed observe the formation of

a two-layers twin in presence of a significant normal stress component. A crucial information missing in literature is the formation mechanism of such a twin embryo. Our calculations reveal an original formation path for the homogeneous nucleation of the twin, which is based on the simultaneous glide of two opposite partial dislocations in two adjacent slip planes. We note that this mechanism was proposed by Xiao et al. to hand-build an input for their twin calculations (Fig. 1d in Ref. [91]). Our calculations suggest that it can be active at the nanoscale.

The other identified stress relaxation mechanism is amorphization. As an example, Fig. 4 shows the evolution of W₁₃₇ subjected to ⟨100⟩ compression (Fig. 4-(a)), and variations of stress and energy (Fig. 4-(b)). We also monitor the number of BCC atoms as determined by the polyhedral template matching method with a RMSD threshold of 0.4 [83], and the average local entropy with a cutoff distance of 5.76 Å and a smoothing parameter of 0.2 [84], as indicators of the level of disorder (Fig. 4-(c)). Both quantities are computed using the mdapy package [92]. At ε = 0.0 the number of identified BCC atoms is 51, excluding NP surface atoms (Fig. 4-(c, d)). Under compression, this number tends to initially decrease with large fluctuations, while the average local entropy increases. The stress maximum is reached for ε = 0.148 (Fig. 4-(b)). Lattice distortions are overall delocalized, with larger amplitudes at surfaces, and with a high level of symmetry. The NP tends to become oblate with a rounder shape. This phenomenon is accentuated until ε = 0.188 (Fig. 4-(f)), for which a local minimum of stress is obtained. Note that the energy continues to grow but at a lower rate. Both the number of identified BCC atoms and the local entropy become constant on average. At larger deformation (ε = 0.257), a new stress maximum is reached. The NP structure is now clearly fully disordered, with no symmetry left (Fig. 4-(g)). Increasing ε up to 0.3, the stress decreases and the structure evolves but remains largely disordered.

We find that all the NP compressed along ⟨100⟩ yield by amor-
phization. It is also the case for the smallest NP compressed along
⟨110⟩, i.e. W<sub>59</sub>, W<sub>65</sub>, and W<sub>78</sub>. Amorphization is then the favored stress
relaxation mechanism for ultrasmall tungsten nanoparticles. We remark
that in the case of the ⟨110⟩ compression, the NP become disordered
thanks to uncorrelated atomic displacements mostly located in {112}
planes. It is different from the ⟨100⟩ compression, for which atomic
displacements appear to be fully random.
<figure><img src="image_5.png" /><figcaption>Fig. 4. Results obtained for the (100) compression of W₁₃₇. (a) Ball-stick representation of W₁₃₇ oriented along (100). Light gray spheres show the W atoms discussed in Section 3. These atoms have only three first neighbors, which could explain the low elastic modulus for this specific system (Fig. 2). (b) Stress in GPa (blue curves) and energy in eV (orange curve) as a function of strain. (c) Number of BCC atoms as identified with the polyhedral template matching method [83] (green curve), and average local entropy [84] (red curve), as a function of strain. (d)–(h) [010]-projection of W₁₃₇ at ε = 0.0, 0.148, 0.188, 0.257, 0.300.</figcaption></figure>

5. Discussion

One main result of the present work is the high NP strengths (Fig. 1-(c)), which are substantially greater than the theoretical value for bulk tungsten (Fig. 2-(c)). We point out that a similar outcome is reported for NP of comparable sizes and made of different materials [44,64]. The extrapolation towards small sizes using the predicted relations between strength and size for FCC NP by Feruz and Mordehai also yields values much larger than the theoretical bulk strength [93]. The experimental compression of Mo and Ni NP with sizes of several hundreds of nm also revealed high strength values, close to the theoretical limit [17,59], despite much larger sizes than in the present work. High strength values could be unique of nanoparticles. In fact Wang et al. reported a resolved shear stress of 9 GPa for twinning on {112} planes, from in-situ nanomechanical compression of tungsten nanowires 15 nm in diameter [86]. This is equivalent to a [110] compression strength of 19 GPa, which is much lower than the values reported here.

It may be legitimately surprising that the NP strength is greater than the bulk theoretical value, the latter being assumed to be the largest sustainable stress by the bulk material at 0 K. A potential source of error in NP strength calculations is the determination of the contact surface. Underestimating this quantity leads to an overestimation of the strength. In atomistic investigations the contact surface is often computed using the convex Hull, i.e. S₀ in the present work. S₀ is an underestimation of the contact surface which becomes an issue for NP sizes of a few nm. It is also likely that using S₀ affects the predicted stress-size relations in previous studies [93]. We find that the NP strengths calculated using S₀ range from 71 to 256 GPa, i.e. two to seven times higher than the theoretical bulk value. To tackle this issue, we use S as defined in Section 2. S can be considered as a good approximation of the upper limit of the contact surface, and leads to strength values ranging from 37.8 to 62.4 GPa. These high but reasonable values are then likely not caused by the determination of the contact surface.

A possible explanation is the presence of an inhomogeneous stress
distribution inside the compressed NP. It is in fact well known that
stress concentrations occur at surface edges for a contact angle different
from 90° [94], i.e. for NP with a non-constant cross section along
the compression axis, in presence of an applied stress σ. The local
stress distribution includes all stress concentrations in addition to the

projected stress mσ, with m the Schmid factor. Conversely, for a cubic
NP with no stress concentrations, the stress distribution inside the
NP is homogeneous with a value equal to mσ. and the crystal yields
when mσ becomes equal to the TSS. In presence of stress distributions,
the relation between σ and the local stresses distribution, and its
dependence on the NP shape and size, is not known. If we assume that
the local stress increase rate in a compressed NP could be lower than
m, due to elastic screening between the different stress concentration
regions, a larger stress σ would be necessary for a local stress to be
equal to the TSS. And then the strength of the NP would appear to be
larger than the bulk strength, as shown in Section 3. This hypothesis
calls for further investigations, but those are out of the scope of the
present work.

Our investigations reveal that tungsten NP yield mostly by amor-
phization, except for W₁₃₇ and W₁₆₉ compressed along (110) for which
twinning occurs. Amorphization is a general plasticity mechanism ob-
served in different materials [95]. It is also favored by low temperatures
and high strain rates conditions, which are typically present in our
simulations [96]. One rationale for the dominance of amorphization
plasticity at small sizes is that the amorphous state becomes increas-
ingly stable relatively to crystal phases [97]. This is also the case for
tungsten NP [72]. It seems reasonable to assume that the energy barrier
between crystalline and amorphous phases will also decrease when the
NP size is reduced. This would explain why here all the smallest NP
yield by amorphization. It is noteworthy that we previously reported
amorphization as the main plasticity mechanism for Si and SiC NP with
similar dimensions [44].

The second observed mechanism is the homogeneous nucleation of a two-layers twin. The BCC crystallography is characterized by a twinning/anti-twinning asymmetry along {112} planes [98]. Twinning is obtained by shearing the crystal by 1/6⟨111⟩ in the twinning direction. In the opposite so-called anti-twinning direction, it is necessary to shear the crystal by 1/3⟨111⟩ to obtain a stable stacking fault. The energy barrier to create such a fault is much larger than for twinning [98], although a recent investigation suggests that it could be reduced with a shuffling mechanism [99]. There are also experimental evidences of anti-twinning in presence of high stresses [100].

The (110) compression induces a crystal shear in the twinning
direction [85,101,102]. This explains why twinning is obtained for
W<sub>169</sub> and W<sub>137</sub> compressed along (110) (Fig. 5-(a)). We also compute
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** (a) [110]-projection of $W_{169}$. The inset shows the 1/6(111) displacements in a {112} plane (red arrows), corresponding to the twinning direction (T). (b) [011]-projection of $W_{137}$. The inset shows the 1/6(111) displacements in a {112} plane (red arrows), corresponding to the anti-twinning direction (AT). The full and dashed blue and orange lines highlight the central {112} plane and two of its neighbors in (a) and (b). (c) Variation of the average Schmid factor as a function of strain for $W_{169}$ compressed along (110) (blue lines) and $W_{137}$ compressed along (100) (orange lines), for the three central planes represented in colors in (a) and (b).</figcaption></figure>

the variation of the Schmid factor for a (111)(112) slip in the central planes, where the twin nucleates. Fig. 5-(c) shows that the Schmid factor continuously increases during compression, which further stimulates twinning. Conversely, the (100) compression favors shear in the anti-twinning direction (Fig. 5-(b)). This mechanism is however not observed in our simulations, likely because the stress needed for the anti-twin formation is greater than the one required to trigger amorphization. In addition, the Schmid factor decreases during the (100) compression, which further penalizes anti-twinning (Fig. 5-(c)).

Finally it is noteworthy that the main plasticity mechanism in BCC materials is the formation and propagation of dislocations, even in nanostructures. For instance Sainath et al. reported dislocation slip in (100) compressed Fe nanowires [101]. However, dislocation-mediated plasticity is not observed in our calculations. Several explanations can be brought forward, such as high strain rate and low temperature which favor twinning [85]. The low dimensions of our NP might also hinder the nucleation of dislocations and promotes twinning, as shown for tungsten nanowires [87].

## 6. Conclusions

This study focus on the mechanical properties of 1–2 nm tungsten NP determined by performing AIMD uniaxial compression calculations. We find that there are no clear relations between mechanical properties like elastic moduli, critical strain and strength, and the size or shape of the NP. Both the NP elastic moduli and strengths are overall greater than the corresponding bulk theoretical values, with increases up to 45% for the former and to 69% for the latter. We propose that these surprisingly high values originate from the inhomogeneous distribution of local stress, a clear consequence of their shape. It is found that all NP compressed along (100) yield by amorphization. Along (110), amorphization is observed for the smallest NP, and twinning for the two largest ones. The latter occurs thanks to the homogeneous nucleation of a two-layers twin according to an original mechanism.

In this work we identify the initiation of the plastic deformation in relation with the presence of a maximum in stress strain curves. In most cases, it precedes an energy maximum at a slightly larger deformation,

followed by a substantial energy drop. Such a drop suggests that the new NP structure is stable, and would be preserved upon compression stress release. However there are simulations where no drops are present as for instance the case of the (100) compression of $W_{137}$. We observe that the structure of the NP at the first stress maximum remains distinctly crystalline, despite noticeable lattice distortions. One could then expect a complete recovery in case of compression reversal, as observed for a fullerene [62]. Recent studies also questioned the stability of twins [103]. Future investigations will therefore be welcome to provide answers to these issues.

## CRediT authorship contribution statement

L. Pizzagalli: Writing – review & editing, Writing – original draft, Investigation, Conceptualization. J. Godet: Writing – review & editing. J. Durinck: Writing – review & editing. S. Brochard: Writing – review & editing. J. Furstoss: Writing – review & editing. F. Valencia: Writing – review & editing.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

Computer time for this study was provided by the MCIA (Mésocentre de Calcul Intensif Aquitain). This work pertains to the French Government program "Investissements d'Avenir" (EUR INTREE, reference ANR-18-EURE-0010, and LABEX INTERACTIFS, reference ANR-11-LABX-0017-01).

## Data availability

Data will be made available on request.

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