2.2. Molecular dynamics simulation

To analyse atomic mechanisms of instability and failure of defect-free nanosized crystal under uniform triaxial (hydrostatic) tension, MD simulation was employed. Simulation was performed using the software (program package) XMD [http://xmd.sourceforge.net/].

Extended Finnis–Sinclair semiempirical potential [30] (Tables 1 and 2) was utilised for MD simulation of hydrostatic tension of tungsten. This potential overcomes the "soft" behaviour of the original Finnis–Sinclair potential [31].

Atomic Mechanisms Governing Strength of Metallic Nanosized Crystals http://dx.doi.org/10.5772/intechopen.75159 39

Figure 2. The schemes of the nanoneedle tip and loading (r is the protrusion radius) before failure (a) and FIM microfractograms of tungsten nanotip acquired at 5.60 kV immediately after failure (b).


Table 1. Potential parameters for W.

the hemispherical cap in the conical part. So, the failure of needle-shaped specimens was usually initiated at the nanosized cap in the region of the ultimate field strength. An electric

<sup>σ</sup><sup>s</sup> <sup>¼</sup> <sup>ε</sup>0F<sup>2</sup>

where ε<sup>0</sup> is the electric constant. This stress acts normal to the surface element at all points. The stress state of a needle-shaped specimen is possible to estimate to any plane in the specimen apex section by integration of the stress components at each point on the surface. The Maxwell

where dS is an element of surface area, β is the angle between the axis and the normal to that surface facet. The integral is taken over the whole specimen surface. The local Maxwell stress is calculated by integration of the field-induced stress and dividing by the area of the section concerned. The near surface Maxwell stress under the hemispherical envelope of the specimen is virtually hydrostatic and equals to the local surface stress determined by Eq. (3). The stress state in the conical shank region is corresponded to the tensile test with uniaxial tension. It enables both to deform nanocrystal plastically and to break it under uniaxial tension (Figure 1b). For nanocrystal failure under multiaxial uniform (hydrostatic) tension, a protrusion was formed on the spherical part of specimen. It was created by selection of special schedules for the high-field evaporation of atoms from the nanospecimen surface [19]. This is shown schematically in Figure 2a. Electric field strength and, accordingly, mechanical stresses increase with decrease in the curvature radius of a surface. So, maximal stresses act in this protrusion. This makes it possible to fracture the metal exactly in this volume. It is done by a steep increase in the electric field strength. Appearance of the crater on a nanoneedle surface is the direct evidence that fracture occurs in the local region, where hydrostatic tension acts (Figure 2b).

The Maxwell stress was determined with accuracy of �4%. The systematic error due to uncertainties in the field calibration is �3% (see [28, 29]), and, correspondingly, the systematic error in determination of the Maxwell stress is �6%. The total error in determination of the

To analyse atomic mechanisms of instability and failure of defect-free nanosized crystal under uniform triaxial (hydrostatic) tension, MD simulation was employed. Simulation was

Extended Finnis–Sinclair semiempirical potential [30] (Tables 1 and 2) was utilised for MD simulation of hydrostatic tension of tungsten. This potential overcomes the "soft" behaviour of

performed using the software (program package) XMD [http://xmd.sourceforge.net/].

Maxwell stress is �10%.

38 Molecular Dynamics

2.2. Molecular dynamics simulation

the original Finnis–Sinclair potential [31].

stress acting over the metal surface produces the force in the axis direction is given by:

<sup>f</sup> <sup>z</sup> <sup>¼</sup> <sup>1</sup> 2 ðð ε0F<sup>2</sup>

<sup>2</sup> (2)

cosβdS, (3)

field applied to a metal specimen induces the Maxwell surface stress


Table 2. Potential parameters for Mo.

According to EAM or FS formalism, the total energy of a system is given by the expression:

$$\mathcal{U}\_{\text{tot}} = \frac{1}{2} \sum\_{\vec{\eta}} V(\mathbf{r}\_{\vec{\eta}}) + \sum\_{i} f(\rho\_i) \tag{4}$$

where the first term is the conventional central pair-potential summation:

$$V(r) = \begin{cases} (r-c)^2(c\_0 + c\_1r + c\_2r^2 + c\_3r^3 + c\_4r^4), & r \le c\\ 0, & r > c \end{cases}$$

and the second term is the n-body term:

$$f(\rho\_i) = \sqrt{\rho\_i} \quad \rho\_i = \sum\_{i \neq j} A^2 \wp(r\_{ij})$$

$$\wp(r) = \begin{cases} (r-d)^2, & r \le d \\ 0, & r > d \end{cases}$$

Uniform triaxial tension of ball-shaped nanosized tungsten crystal was also simulated. Initial radius of nanocrystal was 10.8 nm. Such a size enabled to eliminate the influence of rigid surface shell on the behaviour of atoms in the inner region of specimen. To describe the distribution of atomic velocities for given temperature, the Boltzmann distribution was used. To hold simulated temperatures, velocity of atoms was scaled by ð Þ Tt=Tc <sup>1</sup>=<sup>33</sup> (Tt is the value of temperature kept during simulation; Tc is the instantaneous system temperature at each time step). То avoid а shock wave when loading, gradient velocity is added to interior atoms [32]. Thermal motion of the atoms of the surface layer of a thickness 0.5 nm was "frozen." These atoms had only the radial component of the velocity, which ensured uniform triaxial expansion of the ball shell with the rate 5 m/s. Accordingly, the strain rate was 4.63 � 108 <sup>s</sup> �1 . Tension of nanowires of molybdenum and tungsten in three crystallographic directions [100], [110] and [111] was modelled also. Cylinder with the axis oriented along a corresponding crystallographic direction was cut from the set of atoms placed exactly at a lattice site. Relation between the diameter and the length of cylinder is one-quarter. After that, boundary conditions are applied. The central part (gauge length) of the cylinder with length of 3 diameters is still free, but atoms of border parts are made "frozen." Time step is 10�<sup>15</sup> sec. The diameter of nanospecimens varied within the range 1÷13 nm. In addition, to interpret the experimental data on the destruction of nanopillars under uniaxial compression, MD simulation of the compression of cylindrical Mo specimens in the [100] direction was performed.

For the quantitative analysis of the driving forces of atomic rearrangements, in addition to global stresses σij (average over the specimen), local ξij stresses (acting within the volume per atom) were estimated. The local stresses ξij were determined as:

$$\xi\_{ij}^{k} = \frac{1}{2\Omega^{k}} \sum\_{m(\neq k)}^{n} f\_{i}^{km} r\_{j}^{km} \,. \tag{5}$$

True strain value was employed as a measure of longitudinal strain of the nanowire as a whole:

Δlxx l0

(8)

http://dx.doi.org/10.5772/intechopen.75159

Atomic Mechanisms Governing Strength of Metallic Nanosized Crystals

(9)

41

exx ¼ ln 1 þ

where l<sup>0</sup> and Δlxx are the initial gauge length of a specimen and its increment, respectively.

value was determined as:

3. Results and discussion

in shape and orientation of an ideal crystal [33–35].

When modelling hydrostatic tension of a nanosized ball-shape specimen, engineering strain

<sup>e</sup> <sup>¼</sup> <sup>Δ</sup><sup>r</sup> r0

As it is known, microscopic defects (dislocations and twins) are the main reason for decrease in strength of materials. Modern nanotechnologies enable us to create materials, which, in most cases, are defect-free. Therefore, initially, it was believed that their strength must tend to the ideal strength and be the intrinsic material trait much like the elastic constants. However, it was appeared that the strength of nanosized crystals changes within the wide range depending on their sizes, temperature, loading condition, etc. However, the concept of "ideal strength" can be used as a starting point for the analysis of atomic mechanisms governing the strength of nanosized crystals. The ideal strength of a material is defined to be the maximal homogenous stress that an ideal crystal can withstand. Reaching of the limit state of such crystal is related to uniform and simultaneous break of atomic bonds or its reformation. In general case, two instability modes exist for metals, namely (i) instability of crystal under the tensile stresses (instability on the Bain path for bcc crystals) and (ii) shear instability. The first mode of instability of an ideal crystal results in its disintegration. Shear instability gives rise to change

A fundamental difference of nanosized crystals from the ideal ones is that the nanocrystals can be defect-free but not ideal since the ideal position of atoms in the crystal lattice is disturbed by both (i) action of surface tension forces and (ii) thermal vibrations of atoms. Action of the surface tension forces leads to inhomogeneous distribution of local stresses. Thermal vibration of atoms causes local stress fluctuations. These two factors are the main reasons for localisation of process of breaking and reformation of atomic bonds in nanosized crystals [36, 37]. As a result, the above two modes of an ideal lattice instability occur in a nanosized crystal, but this instability is realised in a limited local region of the crystal. This gives rise to the fact that defects form as a result of above instability. For instance, formation of dislocations and twins is a direct consequence of this localised shear instability. The key difference of these defects from those in macrosized single crystals lies in the fact that they are highly non-equilibrium, since they are formed in a defect-free crystal at stresses much exceeding the critical stress of their

where r<sup>0</sup> and Δr are the initial values of radius of ball and its increment, respectively.

where k refers to the considered atom; m refers to the neighbouring atom; f km <sup>i</sup> is the force vector between atoms k and m determined as the gradient of energy functional; rkm <sup>j</sup> is the position vector between atoms k and m; n is the number of the nearest neighbouring atoms; Ω<sup>k</sup> is the atomic volume; and i, j are the stress tensor indexes.

An expression for the global stresses σij is the following:

$$
\sigma\_{i\dot{\jmath}} = \frac{1}{N} \sum\_{k=1}^{N} \xi\_{i\dot{\jmath}}^{k}. \tag{6}
$$

where N is the total number of atoms in the nanowire.

For calculation of the local shear stresses, the following formula was used:

$$\mathcal{L}\_{ns} = \sum\_{i,j} \mathcal{L}\_{ij} n\_i s\_j \tag{7}$$

where ni is the normal vector for the glide plane, sj is the slip direction.

True strain value was employed as a measure of longitudinal strain of the nanowire as a whole:

$$\sigma\_{\text{xx}} = \ln\left(1 + \frac{\Delta l\_{\text{xx}}}{l\_0}\right) \tag{8}$$

where l<sup>0</sup> and Δlxx are the initial gauge length of a specimen and its increment, respectively.

When modelling hydrostatic tension of a nanosized ball-shape specimen, engineering strain value was determined as:

$$e = \frac{\Delta r}{r\_0} \tag{9}$$

where r<sup>0</sup> and Δr are the initial values of radius of ball and its increment, respectively.

### 3. Results and discussion

Uniform triaxial tension of ball-shaped nanosized tungsten crystal was also simulated. Initial radius of nanocrystal was 10.8 nm. Such a size enabled to eliminate the influence of rigid surface shell on the behaviour of atoms in the inner region of specimen. To describe the distribution of atomic velocities for given temperature, the Boltzmann distribution was used. To hold simulated temperatures, velocity of atoms was scaled by ð Þ Tt=Tc <sup>1</sup>=<sup>33</sup> (Tt is the value of temperature kept during simulation; Tc is the instantaneous system temperature at each time step). То avoid а shock wave when loading, gradient velocity is added to interior atoms [32]. Thermal motion of the atoms of the surface layer of a thickness 0.5 nm was "frozen." These atoms had only the radial component of the velocity, which ensured uniform triaxial expan-

�1

<sup>j</sup> : (5)

km

ij: (6)

ξijnisj (7)

<sup>i</sup> is the force vector

<sup>j</sup> is the position

. Tension

sion of the ball shell with the rate 5 m/s. Accordingly, the strain rate was 4.63 � 108 <sup>s</sup>

compression of cylindrical Mo specimens in the [100] direction was performed.

ξk ij <sup>¼</sup> <sup>1</sup> 2Ω<sup>k</sup>

where k refers to the considered atom; m refers to the neighbouring atom; f

For calculation of the local shear stresses, the following formula was used:

where ni is the normal vector for the glide plane, sj is the slip direction.

between atoms k and m determined as the gradient of energy functional; rkm

atom) were estimated. The local stresses ξij were determined as:

atomic volume; and i, j are the stress tensor indexes.

40 Molecular Dynamics

An expression for the global stresses σij is the following:

where N is the total number of atoms in the nanowire.

of nanowires of molybdenum and tungsten in three crystallographic directions [100], [110] and [111] was modelled also. Cylinder with the axis oriented along a corresponding crystallographic direction was cut from the set of atoms placed exactly at a lattice site. Relation between the diameter and the length of cylinder is one-quarter. After that, boundary conditions are applied. The central part (gauge length) of the cylinder with length of 3 diameters is still free, but atoms of border parts are made "frozen." Time step is 10�<sup>15</sup> sec. The diameter of nanospecimens varied within the range 1÷13 nm. In addition, to interpret the experimental data on the destruction of nanopillars under uniaxial compression, MD simulation of the

For the quantitative analysis of the driving forces of atomic rearrangements, in addition to global stresses σij (average over the specimen), local ξij stresses (acting within the volume per

> Xn mð Þ 6¼k f km <sup>i</sup> r km

vector between atoms k and m; n is the number of the nearest neighbouring atoms; Ω<sup>k</sup> is the

k¼1 ξk

<sup>σ</sup>ij <sup>¼</sup> <sup>1</sup> N X N

<sup>ξ</sup>ns <sup>¼</sup> <sup>X</sup> i, j

As it is known, microscopic defects (dislocations and twins) are the main reason for decrease in strength of materials. Modern nanotechnologies enable us to create materials, which, in most cases, are defect-free. Therefore, initially, it was believed that their strength must tend to the ideal strength and be the intrinsic material trait much like the elastic constants. However, it was appeared that the strength of nanosized crystals changes within the wide range depending on their sizes, temperature, loading condition, etc. However, the concept of "ideal strength" can be used as a starting point for the analysis of atomic mechanisms governing the strength of nanosized crystals. The ideal strength of a material is defined to be the maximal homogenous stress that an ideal crystal can withstand. Reaching of the limit state of such crystal is related to uniform and simultaneous break of atomic bonds or its reformation. In general case, two instability modes exist for metals, namely (i) instability of crystal under the tensile stresses (instability on the Bain path for bcc crystals) and (ii) shear instability. The first mode of instability of an ideal crystal results in its disintegration. Shear instability gives rise to change in shape and orientation of an ideal crystal [33–35].

A fundamental difference of nanosized crystals from the ideal ones is that the nanocrystals can be defect-free but not ideal since the ideal position of atoms in the crystal lattice is disturbed by both (i) action of surface tension forces and (ii) thermal vibrations of atoms. Action of the surface tension forces leads to inhomogeneous distribution of local stresses. Thermal vibration of atoms causes local stress fluctuations. These two factors are the main reasons for localisation of process of breaking and reformation of atomic bonds in nanosized crystals [36, 37]. As a result, the above two modes of an ideal lattice instability occur in a nanosized crystal, but this instability is realised in a limited local region of the crystal. This gives rise to the fact that defects form as a result of above instability. For instance, formation of dislocations and twins is a direct consequence of this localised shear instability. The key difference of these defects from those in macrosized single crystals lies in the fact that they are highly non-equilibrium, since they are formed in a defect-free crystal at stresses much exceeding the critical stress of their

Figure 3. Images of Fe-nanowires and diagrams of their strain at temperature 300К for three crystallographic orientations: arrows indicate instability regions; RC is the instability stress (strength of nanowire).

propagation in a crystal (Figure 3). As shown in Figure 3, the stress of formation for these defects predetermines the level of strength of nanosized crystals [37].

#### 3.1. The effect of surface

A characteristic feature of nanosized objects is the significant influence of the surface on their properties. For mechanical properties, the surface effect is realised through the surface tension. As it is known, in nanosized specimens, the level of stresses created by surface tension may reach significant values even without loading [8]. Surface tension forces induce tensile stresses in the surface layer of the crystal, which are balanced by compressive stresses in the bulk. It gives rise to inhomogeneous local stress distribution within interior of the nanosized specimen. Figure 4 presents the cross-sectional distribution of local shear stresses ξns, which act in shear systems, where instability of crystal is observed. According to these data, even in the case of the unloaded nanowire, the value of shear stresses in a surface layer of Mo nanowire may reach values of the order of 4 GPa. This is only 3 times less than theoretical shear stress for Mo. At loading of such specimen, these stresses cause the lattice instability in sub-surface layer (Figure 4d). Surface tension not only gives rise to localization of the instability region in surface layer, but it is also the reason for size effect, which manifests itself in growth or decrease in strength of nanosized crystals (Figure 5). Recently, two approaches exist to explain

Figure 4. Distribution of the local shear stresses ξns in the slip systems <211>{111}, <11-2>{111} and <121>{1-11} for unloaded Mo nanowires in directions [100] - (а), [110] - (b), and [111] - (c) at T = 0 К. The cross-section diameter is 2.1 nm;

Atomic Mechanisms Governing Strength of Metallic Nanosized Crystals

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43

(d) localisation of lattice instability in surface layers.

propagation in a crystal (Figure 3). As shown in Figure 3, the stress of formation for these

Figure 3. Images of Fe-nanowires and diagrams of their strain at temperature 300К for three crystallographic orienta-

A characteristic feature of nanosized objects is the significant influence of the surface on their properties. For mechanical properties, the surface effect is realised through the surface tension. As it is known, in nanosized specimens, the level of stresses created by surface tension may reach significant values even without loading [8]. Surface tension forces induce tensile stresses in the surface layer of the crystal, which are balanced by compressive stresses in the bulk. It gives rise to inhomogeneous local stress distribution within interior of the nanosized specimen. Figure 4 presents the cross-sectional distribution of local shear stresses ξns, which act in shear systems, where instability of crystal is observed. According to these data, even in the case of the unloaded nanowire, the value of shear stresses in a surface layer of Mo nanowire may reach values of the order of 4 GPa. This is only 3 times less than theoretical shear stress for Mo. At loading of such specimen, these stresses cause the lattice instability in sub-surface layer (Figure 4d). Surface tension not only gives rise to localization of the instability region in surface layer, but it is also the reason for size effect, which manifests itself in growth or decrease in strength of nanosized crystals (Figure 5). Recently, two approaches exist to explain

defects predetermines the level of strength of nanosized crystals [37].

tions: arrows indicate instability regions; RC is the instability stress (strength of nanowire).

3.1. The effect of surface

42 Molecular Dynamics

Figure 4. Distribution of the local shear stresses ξns in the slip systems <211>{111}, <11-2>{111} and <121>{1-11} for unloaded Mo nanowires in directions [100] - (а), [110] - (b), and [111] - (c) at T = 0 К. The cross-section diameter is 2.1 nm; (d) localisation of lattice instability in surface layers.

Figure 5. Dependence of strength of Mo nanowire on its diameter at T = 77К and T = 300 K for three crystallographic orientations: points are the results of MD simulation; solid lines are calculations by (12) for the following orientations (with corresponding values of parameters):[110] (f xx = 1.777 n/m, h ¼ 0.29 nm, m ¼ 0.470, τ<sup>c</sup> ¼ 15:9 GPa (T = 77 K), τ<sup>c</sup> ¼ 13:9 GPa (T = 300 K), α ¼ 1);[111] (f xx = 2.345 n/m, h ¼ 0.40 nm, m ¼ 0.314, τ<sup>c</sup> ¼ 15:9 GPa (T = 77 K), τ<sup>c</sup> ¼ 13:9 GPa (<sup>T</sup> = 300 K), <sup>α</sup> <sup>¼</sup> 1);[100] (<sup>f</sup> xx = 2.070 n/m, <sup>h</sup> <sup>¼</sup> 0.10 nm, <sup>h</sup><sup>∗</sup> <sup>¼</sup> 0.40 nm, <sup>m</sup> <sup>¼</sup> 0.47, <sup>τ</sup><sup>c</sup> <sup>¼</sup> <sup>9</sup>:7 GPa (<sup>T</sup> = 77 K), <sup>τ</sup><sup>c</sup> <sup>¼</sup> <sup>8</sup>:3 GPa (<sup>T</sup> <sup>=</sup> 300 K), α ¼ 1).

size effect. The first of them is based on a "global" criterion of limit state [8, 10, 16], and the second one employs the "local" criterion of nanocrystal instability [14, 15, 18, 19].

Because of the surface tension, an inner part of a nanosized crystal is compressed, and the thin surface region is stretched. If thickness of this stretched layer is neglected, then compressive stresses σin xx in the inner part of specimen may be estimated as:

$$
\sigma\_{\text{xx}}^{in} = -\frac{4f\_{\text{xx}}}{d} \tag{10}
$$

increase with reduction of a specimen diameter. At tension, the effect of a specimen diameter on the strength must be inverse, i.e., the value of RC must increase with the diameter growth. This agrees with findings of MD simulation of tension and compression of nanowires with fcc lattice [7–10, 36]. According to the MD simulation finding on tension of nanowires made of fcc metals (Cu, Ni, Au, Ag), at a cross-sectional dimension less than 4.0 nm, there is an increase by 10–20% of their strength level with a decrease in the cross-sectional dimension. It should be emphasised that this effect is observed for orientations [100], [110], and [111] in fcc nanowires and in nanowires of bcc metals at orientations [100] [7–10, 12, 16, 36, 37]. At the same time, it was shown in [37] that an opposite effect is observed under tension in the [110] direction of Mo nanowires. In this case, the strength of nanowire decreases with decreasing its diameter. A similar effect was ascertained for Fe nanowires [17]. MD simulation of the Mo nanowire carried out in this paper shows that for Mo, analogous regularities are also observed for [111] orientation (Figure 5). This difference in the behaviour of fcc and bcc nanowires cannot be explained within the framework of existing global approach. Based on the results of MD simulation and direct experimental data, it was shown in [4, 14] that under uniaxial tension, the failure of metal nanowires is governed by shear instability, the result of which is the formation of non-equilibrium dislocations or twins in the surface layer. Such a mechanism of failure is observed even in bcc nanowires at low temperatures. Therefore, in the proposed local

approach, the instability criterion can be written in terms of local shear stresses ξns, as:

To derive the criterion of nanocrystals instability within the framework of continuum approach, the concept of "effective" thickness h of surface layer can be used. This is the thickness of a layer

where d is the diameter of specimen; m is the orientation factor for system where shear instability occurs; α is the coefficient taking into account biaxial stress state on the nanospecimen surface

xx ¼ � <sup>4</sup><sup>f</sup> xx<sup>d</sup>

<sup>m</sup> � <sup>f</sup> xx<sup>d</sup>

According to this dependence, at d >> h, the surface tension must give rise to decrease in strength RC, but the degree of this decrease does not depend on specimen diameter. At small

xx <sup>¼</sup> mf xx<sup>d</sup>

xx is the effective tensile stress, which acts in the subsurface layer. Therefore, the

xx acting in internal volume of specimen is:

where τ<sup>c</sup> is the critical stress of local instability of crystal in a surface layer.

is the same as in the case of real inhomogeneous stress distribution. In this case:

σin

Using the criterion (10) with (11) gives the first approximation for strength RC:

RC <sup>≈</sup> <sup>τ</sup><sup>c</sup>

ξsf ns <sup>¼</sup> <sup>m</sup>σsf

where the distribution of effective shear stresses ξsf

value of effective compressive stress σin

(α ≥ 1); and σsf

ξns ≥ τ<sup>c</sup> (11)

Atomic Mechanisms Governing Strength of Metallic Nanosized Crystals

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45

ns is homogeneous, but their effect on strength

<sup>α</sup>h dð Þ � <sup>h</sup> (12)

ð Þ <sup>d</sup> � <sup>2</sup><sup>h</sup> <sup>2</sup> (13)

<sup>α</sup>h dð Þ � <sup>h</sup> (14)

where f xx is the surface tension along the specimen axis, and d is the diameter of specimen.

Global approach supposes that at compression of a nanosized specimen, stress σin xx is "added" to applied stress, which gives rise to decrease in the global stress required for transition from elastic to plastic strains, i.e., to decrease its strength. According to (10), the value of σin xx should increase with reduction of a specimen diameter. At tension, the effect of a specimen diameter on the strength must be inverse, i.e., the value of RC must increase with the diameter growth. This agrees with findings of MD simulation of tension and compression of nanowires with fcc lattice [7–10, 36]. According to the MD simulation finding on tension of nanowires made of fcc metals (Cu, Ni, Au, Ag), at a cross-sectional dimension less than 4.0 nm, there is an increase by 10–20% of their strength level with a decrease in the cross-sectional dimension. It should be emphasised that this effect is observed for orientations [100], [110], and [111] in fcc nanowires and in nanowires of bcc metals at orientations [100] [7–10, 12, 16, 36, 37]. At the same time, it was shown in [37] that an opposite effect is observed under tension in the [110] direction of Mo nanowires. In this case, the strength of nanowire decreases with decreasing its diameter. A similar effect was ascertained for Fe nanowires [17]. MD simulation of the Mo nanowire carried out in this paper shows that for Mo, analogous regularities are also observed for [111] orientation (Figure 5). This difference in the behaviour of fcc and bcc nanowires cannot be explained within the framework of existing global approach. Based on the results of MD simulation and direct experimental data, it was shown in [4, 14] that under uniaxial tension, the failure of metal nanowires is governed by shear instability, the result of which is the formation of non-equilibrium dislocations or twins in the surface layer. Such a mechanism of failure is observed even in bcc nanowires at low temperatures. Therefore, in the proposed local approach, the instability criterion can be written in terms of local shear stresses ξns, as:

$$
\xi\_{ns} \ge \mathfrak{r}\_c \tag{11}
$$

where τ<sup>c</sup> is the critical stress of local instability of crystal in a surface layer.

size effect. The first of them is based on a "global" criterion of limit state [8, 10, 16], and the

Figure 5. Dependence of strength of Mo nanowire on its diameter at T = 77К and T = 300 K for three crystallographic orientations: points are the results of MD simulation; solid lines are calculations by (12) for the following orientations (with corresponding values of parameters):[110] (f xx = 1.777 n/m, h ¼ 0.29 nm, m ¼ 0.470, τ<sup>c</sup> ¼ 15:9 GPa (T = 77 K), τ<sup>c</sup> ¼ 13:9 GPa (T = 300 K), α ¼ 1);[111] (f xx = 2.345 n/m, h ¼ 0.40 nm, m ¼ 0.314, τ<sup>c</sup> ¼ 15:9 GPa (T = 77 K), τ<sup>c</sup> ¼ 13:9 GPa (<sup>T</sup> = 300 K), <sup>α</sup> <sup>¼</sup> 1);[100] (<sup>f</sup> xx = 2.070 n/m, <sup>h</sup> <sup>¼</sup> 0.10 nm, <sup>h</sup><sup>∗</sup> <sup>¼</sup> 0.40 nm, <sup>m</sup> <sup>¼</sup> 0.47, <sup>τ</sup><sup>c</sup> <sup>¼</sup> <sup>9</sup>:7 GPa (<sup>T</sup> = 77 K), <sup>τ</sup><sup>c</sup> <sup>¼</sup> <sup>8</sup>:3 GPa (<sup>T</sup> <sup>=</sup>

Because of the surface tension, an inner part of a nanosized crystal is compressed, and the thin surface region is stretched. If thickness of this stretched layer is neglected, then compressive

xx ¼ � <sup>4</sup><sup>f</sup> xx

where f xx is the surface tension along the specimen axis, and d is the diameter of specimen.

to applied stress, which gives rise to decrease in the global stress required for transition from

<sup>d</sup> (10)

xx is "added"

xx should

second one employs the "local" criterion of nanocrystal instability [14, 15, 18, 19].

σin

Global approach supposes that at compression of a nanosized specimen, stress σin

elastic to plastic strains, i.e., to decrease its strength. According to (10), the value of σin

xx in the inner part of specimen may be estimated as:

stresses σin

300 K), α ¼ 1).

44 Molecular Dynamics

To derive the criterion of nanocrystals instability within the framework of continuum approach, the concept of "effective" thickness h of surface layer can be used. This is the thickness of a layer where the distribution of effective shear stresses ξsf ns is homogeneous, but their effect on strength is the same as in the case of real inhomogeneous stress distribution. In this case:

$$
\xi\_{ns}^{sf} = m \sigma\_{xx}^{sf} = \frac{m f\_{xx} d}{\alpha h (d - h)} \tag{12}
$$

where d is the diameter of specimen; m is the orientation factor for system where shear instability occurs; α is the coefficient taking into account biaxial stress state on the nanospecimen surface (α ≥ 1); and σsf xx is the effective tensile stress, which acts in the subsurface layer. Therefore, the value of effective compressive stress σin xx acting in internal volume of specimen is:

$$
\sigma\_{\rm xx}^{\rm in} = -\frac{4f\_{\rm xx}d}{\left(d - 2h\right)^2} \tag{13}
$$

Using the criterion (10) with (11) gives the first approximation for strength RC:

$$R\_{\mathbb{C}} \approx \frac{\mathbf{r}\_{\mathfrak{c}}}{m} - \frac{f\_{\text{xx}}d}{\alpha h(d-h)}\tag{14}$$

According to this dependence, at d >> h, the surface tension must give rise to decrease in strength RC, but the degree of this decrease does not depend on specimen diameter. At small values of d (in our case at d ≤ 4÷5 nm), reduction of d must result in growth of ξsf ns, and, respectively, decrease in RC. This agrees well with the results of MD simulation of tension in directions [110] and [111] (Figure 5). At tension in these crystallographic directions, local instability results in formation of non-equilibrium dislocation. Layer width, necessary to form this dislocation, is of the order of the Burgers vector. This is comparable with the width of stretched sub-surface layer of nanospecimen, and so, surface tension facilitates its formation. At tension of bcc nanosized crystals in the direction [100], the local instability is due to formation of non-equilibrium twin [16]. To form this kind of defect, the greater crystal volume is needed; so, the region of the crystal, where twin forms, is located in the site where the compressive stresses act. This reduces the magnitude of the resulting shear stress acting at the time of defect formation. As a result of this, the level of applied stresses necessary for the instability of a nanospecimen, i.e., its strength, increases. In this case, the upper estimate for the nanowire strength is:

$$R\_{\mathbb{C}} \approx \frac{\pi\_{\varepsilon}}{m} - \frac{f\_{xx}d}{a\hbar\left(1+\beta\right)(d-h)} + \frac{4\beta f\_{xx}d}{a\left(1+\beta\right)(d-2h)^{2}}\tag{15}$$

$$\text{where } \beta \approx \frac{h^\*}{h} \left[ 1 - \frac{h + h^\*}{d - h} \right] \tag{16}$$

dislocation (bcc nanowires: orientations <110> and < 111>) gives rise to the opposite effect—a

Atomic Mechanisms Governing Strength of Metallic Nanosized Crystals

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As noted above, there are two main reasons for localisation of the lattice instability in nanocrystals, namely: (i) the effect of surface tension and (ii) the local stress fluctuation due to oscillations of atoms in the lattice. Statistical distribution of local shear stresses in nanosized crystal of iron at two temperatures is exhibited on Figure 6. Limit state of a nanosized crystal is reached when the local shear stresses ξns attain the critical stress of local instability τc. In general case, τ<sup>c</sup> is the critical stress, at reaching of which the highly non-equilibrium dislocation or twin forms in crystal. The value of a fluctuation component of local shear stresses δξns depends on temperature. As shown in Figure 6, it leads to temperature dependence of the

δξns ¼ t

where D<sup>ξ</sup> is the value of variance of local shear stresses in a subsurface layer, where dislocation or twin forms; t is the dimensionless characteristic of the fluctuation value. Its magnitude depends on the probability of occurrence of such fluctuation. According to the criterion of

Figure 6. Distributions of local shear stresses ξns (in the slip system ([1 1 1](211)) for instability point RC in Fe–nanowire

ffiffiffiffiffiffiffiffiffiffiffiffiffi Dξð Þ T q

(17)

47

reduction in strength with a reduction in the diameter of the nanowire.

The fluctuation component δξns can be presented as:

under uniaxial tension in [100] direction at T = 3 K and T = 300 K [37].

3.2. The temperature effect

strength RC.

h<sup>∗</sup> is the thickness of the layer within which the defect is affected by oppositely directed shear stresses; β characterises the ratio of the slip plane areas located within the regions of compressive and tensile stresses.

The third term in this dependence accounts for the effect of decrease in total value of shear stresses influencing the twin, which is due to the fact that part of twin is located in a compressed region. As it is exhibited in Figure 5, dependence (15) enables to describe the effect of increase in strength at tension in the direction [100]. Thus, this rather simple model enables to explain not only the anomaly of the size effect at tension of nanowires of bcc metals in the <100> directions but also the difference between the regularities of size effect in fcc and bcc metals. For all three orientations <100>, <110>, and <111>, local instability in fcc metals is due to formation of a non-equilibrium stacking fault, the critical size of which exceeds a thickness of a tensile surface layer. As a result, for these three crystallographic orientations, the effect of increase in RC at decrease in a specimen diameter is observed. In bcc metals, the same effect is observed only at tension in directions <100>, when twin forms. At tension in directions <110> and <111>, the inverse effect is observed, because local instability results in formation of nonequilibrium dislocations. Therefore, differences in regularities of manifestation of size effect in bcc and fcc metals are due to different kinds of defects, which form as a result of local instability of a crystal. Thus, dependence of the magnitude of the local shear stresses in the surface layer of nanowire on its diameter is the reason for the existence of a size effect for the strength of nanowires. The "sign" of this effect (increase or decrease in strength with decreasing diameter) is determined by the kind of non-equilibrium defect, which is formed as a result of local instability of the nanocrystal. When a non-equilibrium twin (bcc nanowire: orientation <100>) or a stacking fault (fcc nanowires: orientations <100>, <110>, and < 111>) is formed, the strength of the nanowire increases with decreasing its diameter. Formation of a non-equilibrium dislocation (bcc nanowires: orientations <110> and < 111>) gives rise to the opposite effect—a reduction in strength with a reduction in the diameter of the nanowire.

#### 3.2. The temperature effect

values of d (in our case at d ≤ 4÷5 nm), reduction of d must result in growth of ξsf

nanowire strength is:

46 Molecular Dynamics

sive and tensile stresses.

RC <sup>≈</sup> <sup>τ</sup><sup>c</sup>

<sup>m</sup> � <sup>f</sup> xx<sup>d</sup>

<sup>α</sup><sup>h</sup> <sup>1</sup> <sup>þ</sup> <sup>β</sup> ð Þ <sup>d</sup> � <sup>h</sup> <sup>þ</sup>

<sup>h</sup> <sup>1</sup> � <sup>h</sup> <sup>þ</sup> <sup>h</sup><sup>∗</sup> d � h 

h<sup>∗</sup> is the thickness of the layer within which the defect is affected by oppositely directed shear stresses; β characterises the ratio of the slip plane areas located within the regions of compres-

The third term in this dependence accounts for the effect of decrease in total value of shear stresses influencing the twin, which is due to the fact that part of twin is located in a compressed region. As it is exhibited in Figure 5, dependence (15) enables to describe the effect of increase in strength at tension in the direction [100]. Thus, this rather simple model enables to explain not only the anomaly of the size effect at tension of nanowires of bcc metals in the <100> directions but also the difference between the regularities of size effect in fcc and bcc metals. For all three orientations <100>, <110>, and <111>, local instability in fcc metals is due to formation of a non-equilibrium stacking fault, the critical size of which exceeds a thickness of a tensile surface layer. As a result, for these three crystallographic orientations, the effect of increase in RC at decrease in a specimen diameter is observed. In bcc metals, the same effect is observed only at tension in directions <100>, when twin forms. At tension in directions <110> and <111>, the inverse effect is observed, because local instability results in formation of nonequilibrium dislocations. Therefore, differences in regularities of manifestation of size effect in bcc and fcc metals are due to different kinds of defects, which form as a result of local instability of a crystal. Thus, dependence of the magnitude of the local shear stresses in the surface layer of nanowire on its diameter is the reason for the existence of a size effect for the strength of nanowires. The "sign" of this effect (increase or decrease in strength with decreasing diameter) is determined by the kind of non-equilibrium defect, which is formed as a result of local instability of the nanocrystal. When a non-equilibrium twin (bcc nanowire: orientation <100>) or a stacking fault (fcc nanowires: orientations <100>, <110>, and < 111>) is formed, the strength of the nanowire increases with decreasing its diameter. Formation of a non-equilibrium

where <sup>β</sup> <sup>≈</sup> <sup>h</sup><sup>∗</sup>

4βf xxd

<sup>α</sup> <sup>1</sup> <sup>þ</sup> <sup>β</sup> ð Þ <sup>d</sup> � <sup>2</sup><sup>h</sup> <sup>2</sup> (15)

respectively, decrease in RC. This agrees well with the results of MD simulation of tension in directions [110] and [111] (Figure 5). At tension in these crystallographic directions, local instability results in formation of non-equilibrium dislocation. Layer width, necessary to form this dislocation, is of the order of the Burgers vector. This is comparable with the width of stretched sub-surface layer of nanospecimen, and so, surface tension facilitates its formation. At tension of bcc nanosized crystals in the direction [100], the local instability is due to formation of non-equilibrium twin [16]. To form this kind of defect, the greater crystal volume is needed; so, the region of the crystal, where twin forms, is located in the site where the compressive stresses act. This reduces the magnitude of the resulting shear stress acting at the time of defect formation. As a result of this, the level of applied stresses necessary for the instability of a nanospecimen, i.e., its strength, increases. In this case, the upper estimate for the

ns, and,

(16)

As noted above, there are two main reasons for localisation of the lattice instability in nanocrystals, namely: (i) the effect of surface tension and (ii) the local stress fluctuation due to oscillations of atoms in the lattice. Statistical distribution of local shear stresses in nanosized crystal of iron at two temperatures is exhibited on Figure 6. Limit state of a nanosized crystal is reached when the local shear stresses ξns attain the critical stress of local instability τc. In general case, τ<sup>c</sup> is the critical stress, at reaching of which the highly non-equilibrium dislocation or twin forms in crystal. The value of a fluctuation component of local shear stresses δξns depends on temperature. As shown in Figure 6, it leads to temperature dependence of the strength RC.

The fluctuation component δξns can be presented as:

$$
\delta \xi\_{ns} = t \sqrt{D\_{\xi}(T)} \tag{17}
$$

where D<sup>ξ</sup> is the value of variance of local shear stresses in a subsurface layer, where dislocation or twin forms; t is the dimensionless characteristic of the fluctuation value. Its magnitude depends on the probability of occurrence of such fluctuation. According to the criterion of

Figure 6. Distributions of local shear stresses ξns (in the slip system ([1 1 1](211)) for instability point RC in Fe–nanowire under uniaxial tension in [100] direction at T = 3 K and T = 300 K [37].

local instability (10) and expression (16), the first approximation for the temperature dependence of nanosized strength Run may be presented as:

$$R\_{un} \approx \frac{\left[\pi\_c - \mathcal{E}\_{ns}^{\circ f} - t\sqrt{D\_\xi(T)}\right]}{m} \tag{18}$$

causes local instability in defect-free structure and nucleation of defects. In the second case, the lattice vibration facilitates mobility of already existent defects. It is necessary to emphasise the fundamental difference in the mechanisms and structural levels of realisation of these processes. In the first case, we are talking about the thermal activation of instability of the crystal lattice (atomic-scale event). In the second case, the oscillation of the dislocation line (a microscopic object) plays a key role, leading to the formation of kinks pairs, by means of which a thermally activated overcoming by dislocation of the potential relief occurs. As a result, the laws of temperature dependence differ as well as change in the absolute strength value. For instance, over the temperature range 77К…300К, the value of yield strength of typical bcc transition metals decreases 3–5 times, while the changes of a critical stress of NSC instability do not exceed

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As noted above, there are two modes of crystal instability, namely shear instability and instability under the action of tensile stresses. In ideal bcc crystals, this is manifested in the existence of instability on the orthorhombic deformation path and instability on the Bain path. As mentioned above, a shear instability in nanowires is usually realised at tension in directions <110> and <111>. In the general case, when the bcc crystals are stretched in the <100> direction, ab-initio computation predicts the possibility of instability, both on the orthorhombic and Bain paths. As a result, the mode of instability is realised, for which a lower value of strains is required. So, for example, for an ideal niobium crystal, the value of critical strain of instability on the orthorhombic path is almost 1.5 times less than the corresponding strain on the Bain path [34]. This means that the strength of Nb nanowires at tension in the direction <100> should be governed by local shear instability. The situation with Mo, W, and Fe is more complicated. According to the results of ab-initio calculations, for ideal crystals of these metals, the values of critical strains and stresses on both paths are very close [33–35]. However, the results of the MD simulation on tension of nanowires of these metals indicate the realisation of local shear instability. One of the results of such instability is the reorientation of the crystal lattice from [100] to [110]. According to the ab-initio calculations, such a reorientation should be a consequence of the instability on the orthorhombic path. The only difference is that in an ideal crystal, this reorientation must occur simultaneously in the entire crystal, and in a nanosized specimen, this occurs locally with subsequent propagation to the entire volume of nanospecimen (Figure 8). These regularities were considered in the works [14, 40]. Later, similar results were obtained on nanowires with a square cross-section [16]. As a result of realisation of the shear instability mechanism at tension of Mo, W, and α-Fe nanowires in the direction [100], brittle fracture (Bain instability) does not occur, but on the contrary, the plastic strain is larger as compared with that for orientations [110] and [111] [14, 40]. In this connection, the question arises whether it is possible to realise instability on the Bain path in nanosized specimens made of the bcc metals. Ваіn instability was found in [14] at MD-simulation of hydrostatic tension of ball-shaped specimen of iron. Under hydrostatic tension, the value of the average shear stresses is zero. This contributes to the realisation of the Bain instability, since there is no driving force for the shear instability of the crystal. However, the bcc!fcc transition requires the realisation of a certain relationship between the strains of lattice along the

3.3. Nanosized crystal instability under hydrostatic tension

15–25%.

Thus, temperature dependence of the local stress fluctuations δξns (16) gives the main contribution to the temperature dependence of the strength of nanosized crystals. From the Debye model [38] and results of computer simulation [39], it follows that within the range of temperatures, greater than the Debye temperature (θD), the variance of atomic displacements is linear function of temperature, and for lower temperatures, susceptibility D<sup>ξ</sup> to change in temperature decreases. In [18], experimental evidence was first obtained for the temperature dependence of Mo nanosized specimens (Figure 7). From these data, it follows that at temperatures higher than θD, the expression (18) gives the correct qualitative description of the temperature dependence of the strength of nanosized crystals, if assume the linear dependence of D<sup>ξ</sup> on T. It should be noted that the nature of the strength temperature dependence for nanosized crystals and macrosized single crystals differs essentially because, in first case, thermal vibration of atoms

Figure 7. Dependence of strength of nanosized W crystals on the temperature and crystallographic direction: , , are the results of MD-simulation, are the experimental evidence [18].

causes local instability in defect-free structure and nucleation of defects. In the second case, the lattice vibration facilitates mobility of already existent defects. It is necessary to emphasise the fundamental difference in the mechanisms and structural levels of realisation of these processes. In the first case, we are talking about the thermal activation of instability of the crystal lattice (atomic-scale event). In the second case, the oscillation of the dislocation line (a microscopic object) plays a key role, leading to the formation of kinks pairs, by means of which a thermally activated overcoming by dislocation of the potential relief occurs. As a result, the laws of temperature dependence differ as well as change in the absolute strength value. For instance, over the temperature range 77К…300К, the value of yield strength of typical bcc transition metals decreases 3–5 times, while the changes of a critical stress of NSC instability do not exceed 15–25%.

## 3.3. Nanosized crystal instability under hydrostatic tension

local instability (10) and expression (16), the first approximation for the temperature depen-

Thus, temperature dependence of the local stress fluctuations δξns (16) gives the main contribution to the temperature dependence of the strength of nanosized crystals. From the Debye model [38] and results of computer simulation [39], it follows that within the range of temperatures, greater than the Debye temperature (θD), the variance of atomic displacements is linear function of temperature, and for lower temperatures, susceptibility D<sup>ξ</sup> to change in temperature decreases. In [18], experimental evidence was first obtained for the temperature dependence of Mo nanosized specimens (Figure 7). From these data, it follows that at temperatures higher than θD, the expression (18) gives the correct qualitative description of the temperature dependence of the strength of nanosized crystals, if assume the linear dependence of D<sup>ξ</sup> on T. It should be noted that the nature of the strength temperature dependence for nanosized crystals and macrosized single crystals differs essentially because, in first case, thermal vibration of atoms

Figure 7. Dependence of strength of nanosized W crystals on the temperature and crystallographic direction: , , are

the results of MD-simulation, are the experimental evidence [18].

ns � <sup>t</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>D</sup>ξð Þ <sup>T</sup> � � <sup>p</sup>

<sup>m</sup> (18)

Run <sup>≈</sup> <sup>τ</sup><sup>c</sup> � <sup>ξ</sup>sf

dence of nanosized strength Run may be presented as:

48 Molecular Dynamics

As noted above, there are two modes of crystal instability, namely shear instability and instability under the action of tensile stresses. In ideal bcc crystals, this is manifested in the existence of instability on the orthorhombic deformation path and instability on the Bain path. As mentioned above, a shear instability in nanowires is usually realised at tension in directions <110> and <111>. In the general case, when the bcc crystals are stretched in the <100> direction, ab-initio computation predicts the possibility of instability, both on the orthorhombic and Bain paths. As a result, the mode of instability is realised, for which a lower value of strains is required. So, for example, for an ideal niobium crystal, the value of critical strain of instability on the orthorhombic path is almost 1.5 times less than the corresponding strain on the Bain path [34]. This means that the strength of Nb nanowires at tension in the direction <100> should be governed by local shear instability. The situation with Mo, W, and Fe is more complicated. According to the results of ab-initio calculations, for ideal crystals of these metals, the values of critical strains and stresses on both paths are very close [33–35]. However, the results of the MD simulation on tension of nanowires of these metals indicate the realisation of local shear instability. One of the results of such instability is the reorientation of the crystal lattice from [100] to [110]. According to the ab-initio calculations, such a reorientation should be a consequence of the instability on the orthorhombic path. The only difference is that in an ideal crystal, this reorientation must occur simultaneously in the entire crystal, and in a nanosized specimen, this occurs locally with subsequent propagation to the entire volume of nanospecimen (Figure 8). These regularities were considered in the works [14, 40]. Later, similar results were obtained on nanowires with a square cross-section [16]. As a result of realisation of the shear instability mechanism at tension of Mo, W, and α-Fe nanowires in the direction [100], brittle fracture (Bain instability) does not occur, but on the contrary, the plastic strain is larger as compared with that for orientations [110] and [111] [14, 40]. In this connection, the question arises whether it is possible to realise instability on the Bain path in nanosized specimens made of the bcc metals. Ваіn instability was found in [14] at MD-simulation of hydrostatic tension of ball-shaped specimen of iron. Under hydrostatic tension, the value of the average shear stresses is zero. This contributes to the realisation of the Bain instability, since there is no driving force for the shear instability of the crystal. However, the bcc!fcc transition requires the realisation of a certain relationship between the strains of lattice along the

Figure 8. Stress-strain dependence for Mo nanowire. Uniaxial tension in the direction [100] at T = 30 K, аnd schemes of the lattice re-arrangement.

transition within the local regions, i.e., instability on the Bain path is realised in these regions. At further loading of a specimen, the regions with such type of a lattice expand and new regions with fcc lattice may form as well. Figure 9 demonstrates by different colours the atoms belonging to bcc and fcc lattices, atoms forming the boundaries between these regions, and atoms from the

Figure 9. Diagram of deformation of the ball-shaped Mo nanosized specimen under hydrostatic tension at T = 30 K and crystallographic structure of the ball-shaped nanosized specimen at the moment of its instability and at the moment of crack initiation. The colour range indicates the number of atoms in the first coordination sphere for the selected atom.

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To analyse quantitatively the instability initiation for the temperature 77К, distribution of local tensile stresses ξf g <sup>100</sup> , acting on planes {100} at the moment of instability of nanosized specimen, was built (Figure 10). According to these data, significant fluctuation of the local stresses ξf g <sup>100</sup> is observed. This gives rise to change in local stresses over the wide range of values from 10.6 GPa to 40.8 GPa at the average value of 28 GPa. The local maximum on the left branch of distribution appears due to rigid surface shell, which is necessary to realise uniform tension of specimen. According to the evidence obtained, Bain instability is initiated at the value of local stresses ξcm ≈ 40 GPа. This stress exceeds significantly that for bcc!fcc transition at purely uniaxial tension σct ¼ 28.9 GPа [41]. Action of tensile stresses in the direction normal to that, in which Bain instability realises, is the reason for above excess. Findings of ab-initio calculations for iron presented in [42] show that when applying a tensile stresses in the orthogonal direction, the Bain instability stress increases as compared to purely uniaxial tension. In our case, the Bain transition occurs in one of the directions <100> as a result of fluctuation of local tensile stress in this direction. This is the feature of the atomic mechanism of

regions with strongly non-uniform distribution of atomic displacements (read colour).

directions OX, OY, and OZ [34]. This is impossible to be realised under conditions of uniform triaxial tension. To analyse the details of this mechanism of bcc!fcc transition in nanosized crystals, hydrostatic tension of a molybdenum nanoball with a diameter of 10.8 nm was simulated [19]. MD simulations were performed for T = 77 K (the temperature for which the experimental data were obtained) and also for a lower temperature of 30 K. Simulation at lower temperatures makes it possible to obtain a clearer picture of atomic rearrangements, since thermal "smearing" of atom locations is reduced. Based on the results of this simulation, the diagram of deformation of the ball was built (Figure 9). To analyse local bcc-fcc rearrangements, for each atom, the number of neighbouring atoms in the first coordination sphere was calculated. This permits to determine the regions in crystal with both initial bcc lattice and formed unstable fcc lattice, as well as heavily deformed regions (Figure 9).

According to the MD-simulation findings, when critical stress of instability of a specimen under uniform triaxial tension, RC, is reached, displacements of atoms in the specimen are essentially non-uniform (Figure 9). Bands form in the specimen, and deformation of the lattice in them differs from the deformation in the other part of the specimen. Detailed analysis of atomic rearrangements showed that global instability of entire specimen is initiated by bcc!fcc

Atomic Mechanisms Governing Strength of Metallic Nanosized Crystals http://dx.doi.org/10.5772/intechopen.75159 51

Figure 9. Diagram of deformation of the ball-shaped Mo nanosized specimen under hydrostatic tension at T = 30 K and crystallographic structure of the ball-shaped nanosized specimen at the moment of its instability and at the moment of crack initiation. The colour range indicates the number of atoms in the first coordination sphere for the selected atom.

transition within the local regions, i.e., instability on the Bain path is realised in these regions. At further loading of a specimen, the regions with such type of a lattice expand and new regions with fcc lattice may form as well. Figure 9 demonstrates by different colours the atoms belonging to bcc and fcc lattices, atoms forming the boundaries between these regions, and atoms from the regions with strongly non-uniform distribution of atomic displacements (read colour).

directions OX, OY, and OZ [34]. This is impossible to be realised under conditions of uniform triaxial tension. To analyse the details of this mechanism of bcc!fcc transition in nanosized crystals, hydrostatic tension of a molybdenum nanoball with a diameter of 10.8 nm was simulated [19]. MD simulations were performed for T = 77 K (the temperature for which the experimental data were obtained) and also for a lower temperature of 30 K. Simulation at lower temperatures makes it possible to obtain a clearer picture of atomic rearrangements, since thermal "smearing" of atom locations is reduced. Based on the results of this simulation, the diagram of deformation of the ball was built (Figure 9). To analyse local bcc-fcc rearrangements, for each atom, the number of neighbouring atoms in the first coordination sphere was calculated. This permits to determine the regions in crystal with both initial bcc

Figure 8. Stress-strain dependence for Mo nanowire. Uniaxial tension in the direction [100] at T = 30 K, аnd schemes of

the lattice re-arrangement.

50 Molecular Dynamics

lattice and formed unstable fcc lattice, as well as heavily deformed regions (Figure 9).

According to the MD-simulation findings, when critical stress of instability of a specimen under uniform triaxial tension, RC, is reached, displacements of atoms in the specimen are essentially non-uniform (Figure 9). Bands form in the specimen, and deformation of the lattice in them differs from the deformation in the other part of the specimen. Detailed analysis of atomic rearrangements showed that global instability of entire specimen is initiated by bcc!fcc To analyse quantitatively the instability initiation for the temperature 77К, distribution of local tensile stresses ξf g <sup>100</sup> , acting on planes {100} at the moment of instability of nanosized specimen, was built (Figure 10). According to these data, significant fluctuation of the local stresses ξf g <sup>100</sup> is observed. This gives rise to change in local stresses over the wide range of values from 10.6 GPa to 40.8 GPa at the average value of 28 GPa. The local maximum on the left branch of distribution appears due to rigid surface shell, which is necessary to realise uniform tension of specimen. According to the evidence obtained, Bain instability is initiated at the value of local stresses ξcm ≈ 40 GPа. This stress exceeds significantly that for bcc!fcc transition at purely uniaxial tension σct ¼ 28.9 GPа [41]. Action of tensile stresses in the direction normal to that, in which Bain instability realises, is the reason for above excess. Findings of ab-initio calculations for iron presented in [42] show that when applying a tensile stresses in the orthogonal direction, the Bain instability stress increases as compared to purely uniaxial tension. In our case, the Bain transition occurs in one of the directions <100> as a result of fluctuation of local tensile stress in this direction. This is the feature of the atomic mechanism of

Figure 10. Distribution of local normal stresses ξf g <sup>100</sup> <sup>11</sup> in the ball-shaped nanosized specimen at the moment of it instability at T = 77 K: RC is the global stress of instability of the ball-shaped nanosized specimen; ξcm is local critical stress of bcc ! fcc transition; δξf g <sup>100</sup> <sup>11</sup> is local stress fluctuation on {100} planes, required for realisation of the local instability of crystal.

instability initiation in nanosized specimen under global uniform triaxial tension. It should be emphasised that in this case, the appearance of fluctuations gives rise not only to excess of the local stresses over the average ones but also to deviation from uniform triaxial tension within the local region where the Bain transition is realised.

For quantitative estimates, the expression for value of local stress of Bain transition, ξcm, under multiaxial stress state, can be written as follows:

$$
\xi\_{cm} = \overline{E}\mathfrak{e}\_{cm} + \nu(\xi\_{YY} + \xi\_{ZZ}) \tag{19}
$$

Respectively:

ν = 0.27, one obtains ξcm = 40.3 GPa.

ξcm ≈ Eecm þ 2νRC (21)

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Dependence (20) enables to estimate the value of local stress of initiation of local instability, ξcm, at a given value of global stress RC. In this case, the experimental value of the global stress of instability of a ball-shape nanospecimen RC ¼ 28 � 3 GPa [19] (Figure 11). Assuming that ecm is approximately equal to critical strain of the Bain transition at uniaxial tension (ecm ≈ 0.12 [41]); and E ≈ 0:5Ef g <sup>100</sup> [19], where Ef g <sup>100</sup> = 409 GPa is elasticity modulus of tungsten and

To estimate ξcm, the value of critical strain of initiation of Bain transition at pure uniaxial tension was used. According to the data of ab-initio calculations, the value of critical strain of the lattice instability increases at transition from uniaxial to triaxial tension, and at uniform triaxial tension, it reaches maximum value of 0.15 [42]. It means that obtained value ξcm = 40.3 GPa should be considered as the low estimation of critical stress of Bain transition of tungsten at multiaxial tension. In spite of this, the calculated value of ξcm agrees sufficiently with the value of ξcm obtained from the analysis of distribution of local stresses in ball-shaped specimen (Figure 10). According ab-initio calculation findings, the instability stress of an ideal Mo crystal under uniform triaxial tension is 50.0–52.0 GPa [42]. However, such a value of strength can be reached only in an ideal crystal but not in nanosized specimens. Fluctuations in local tensile stresses are the reason for this. They do not enable the uniform triaxial tension in local regions of the crystal to be

Figure 11. Hydrostatic tensile strength of tungsten nanocrystals at 77 K; d is the diameter of nanocrystal region fractured

under hydrostatic tension; RC is the experimental evidence of strength.

where ecm is the critical strain of the Bain transition under multiaxial tension; E is the secant modulus; ν is the Poisson's ratio; ξYY and ξZZ are the local orthogonal tensile stresses.

The probability of simultaneous fluctuations of local stresses in three mutually orthogonal directions is many orders of magnitude less than probability of fluctuation in only one direction, so, in the first approximation, at the moment of local instability:

$$
\xi\_{\Upsilon\Upsilon} \approx \xi\_{ZZ} \approx \mathsf{R}\_{\mathbb{C}} \tag{20}
$$

where RC is the global (average) tensile stress at the moment of initiation of nanosized crystal instability.

Respectively:

instability initiation in nanosized specimen under global uniform triaxial tension. It should be emphasised that in this case, the appearance of fluctuations gives rise not only to excess of the local stresses over the average ones but also to deviation from uniform triaxial tension within

instability at T = 77 K: RC is the global stress of instability of the ball-shaped nanosized specimen; ξcm is local critical stress

<sup>11</sup> is local stress fluctuation on {100} planes, required for realisation of the local instability of

For quantitative estimates, the expression for value of local stress of Bain transition, ξcm, under

where ecm is the critical strain of the Bain transition under multiaxial tension; E is the secant

The probability of simultaneous fluctuations of local stresses in three mutually orthogonal directions is many orders of magnitude less than probability of fluctuation in only one direc-

where RC is the global (average) tensile stress at the moment of initiation of nanosized crystal

modulus; ν is the Poisson's ratio; ξYY and ξZZ are the local orthogonal tensile stresses.

tion, so, in the first approximation, at the moment of local instability:

ξcm ¼ Eecm þ ν ξð Þ YY þ ξZZ (19)

<sup>11</sup> in the ball-shaped nanosized specimen at the moment of it

ξYY ≈ ξZZ ≈ RC (20)

the local region where the Bain transition is realised.

Figure 10. Distribution of local normal stresses ξf g <sup>100</sup>

of bcc ! fcc transition; δξf g <sup>100</sup>

crystal.

52 Molecular Dynamics

multiaxial stress state, can be written as follows:

instability.

$$
\xi\_{cm} \approx \overline{E}e\_{cm} + 2\nu \mathcal{R}\_{\mathbb{C}} \tag{21}
$$

Dependence (20) enables to estimate the value of local stress of initiation of local instability, ξcm, at a given value of global stress RC. In this case, the experimental value of the global stress of instability of a ball-shape nanospecimen RC ¼ 28 � 3 GPa [19] (Figure 11). Assuming that ecm is approximately equal to critical strain of the Bain transition at uniaxial tension (ecm ≈ 0.12 [41]); and E ≈ 0:5Ef g <sup>100</sup> [19], where Ef g <sup>100</sup> = 409 GPa is elasticity modulus of tungsten and ν = 0.27, one obtains ξcm = 40.3 GPa.

To estimate ξcm, the value of critical strain of initiation of Bain transition at pure uniaxial tension was used. According to the data of ab-initio calculations, the value of critical strain of the lattice instability increases at transition from uniaxial to triaxial tension, and at uniform triaxial tension, it reaches maximum value of 0.15 [42]. It means that obtained value ξcm = 40.3 GPa should be considered as the low estimation of critical stress of Bain transition of tungsten at multiaxial tension. In spite of this, the calculated value of ξcm agrees sufficiently with the value of ξcm obtained from the analysis of distribution of local stresses in ball-shaped specimen (Figure 10).

According ab-initio calculation findings, the instability stress of an ideal Mo crystal under uniform triaxial tension is 50.0–52.0 GPa [42]. However, such a value of strength can be reached only in an ideal crystal but not in nanosized specimens. Fluctuations in local tensile stresses are the reason for this. They do not enable the uniform triaxial tension in local regions of the crystal to be

Figure 11. Hydrostatic tensile strength of tungsten nanocrystals at 77 K; d is the diameter of nanocrystal region fractured under hydrostatic tension; RC is the experimental evidence of strength.

realised at a global (on average) uniform triaxial tension of the entire specimen. Probability of the fluctuation of local stresses in three directions simultaneously is much lower than the probability of fluctuation along one of the directions <100>; so, initiation of instability of nanosized tungsten crystal occurs at local stresses ξcm ≈ 40 GPa. This means that in local regions, the instability occurs at non-uniform triaxial tension (ξ<sup>11</sup> ≈ 40 GPa; ξ<sup>22</sup> ≈ ξ<sup>33</sup> ≈ 28 GPa).

Thus, the strength of nanosized crystals under hydrostatic tension will always be below the strength of an ideal crystal under the same conditions. In this case, the strength of tungsten nanosized crystals under hydrostatic tension is 28 GPa. This is approximately 1.90 times less than the strength of an ideal tungsten crystal. The reason for this is fluctuations of local tensile stresses, which leads to: (i) exceeding of the value of local stresses over global (fluctuations themselves) and (ii) to deviation from triaxial uniform tension within the local region where this transition realises. For nanosized Mo specimens, the first factor results in a decrease in strength by 1.43 times, while the second one—by 1.25–1.30 times.
