Ideal strength of nanoscale materials induced by elastic instability

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Ideal strength of nanoscale materials induced by elastic instability Duc Tam Ho , Soon Kim , Soon-Yong Kwon , Sung Youb Kim PII: DOI: Reference:

S0167-6636(19)30396-5 https://doi.org/10.1016/j.mechmat.2019.103241 MECMAT 103241

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Mechanics of Materials

Received date: Revised date: Accepted date:

12 May 2019 18 September 2019 6 November 2019

Please cite this article as: Duc Tam Ho , Soon Kim , Soon-Yong Kwon , Sung Youb Kim , Ideal strength of nanoscale materials induced by elastic instability, Mechanics of Materials (2019), doi: https://doi.org/10.1016/j.mechmat.2019.103241

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Ideal strength of nanoscale materials induced by elastic instability Duc Tam Ho1, Soon Kim1, Soon-Yong Kwon2, and Sung Youb Kim1, * 1

Department of Mechanical Engineering, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea 2

School of Material Science and Engineering, Ulsan National Institute of Science and Technology, Ulsan 44919, South Korea (2019.09.18) *

To whom correspondence should be addressed, e-mail: [email protected]

Abstract The ideal strength of a defect-free material, which is the stress causing a material failure, is one of the fundamental mechanical properties. In this study, we investigate ideal strengths of some facecenter cubic nanostructures using molecular statics simulations and an elastic stability criterion. The simulation results show that ideal strength depends strongly on loading direction, loading mode (tension or compression), side surface orientation, shape of cross-section, and size. Consequently, nanostructures can exhibit the “smaller is stronger” trend, the “smaller is weaker” trend, and the “size-independent strength plateau” trend. Our semi-analytic model for prediction of ideal strengths of nanostructures is in good agreement with molecular statics simulation results. Key words: ideal strength; elastic instability; molecular statics simulation; nanowires; and nanoplates.

Introduction Strength is the stress that causes some forms of material failures such as phase transformation, initiation and movement of dislocations, and fracture. For macroscale metals, material failures are governed by lattice defects such as dislocations and grain boundaries; the corresponding strength is usually called the yield strength. For extremely small structures such as whiskers, nanowires, and nanoparticles yield strengths are significantly large (Han et al., 2015; Richter et al., 2009; Yue et al., 2011). For example, the yield strength of a Au nanowire (NW) is about 100 times larger than that of macroscale Au (Wu et al., 2005). Here, we focus on ideal strength, i.e., the strength of defect-free materials. In practice, many nanostructures are in fact defect-free materials (Krishnamachari et al., 2004; Lin et al., 2018; Wu et al., 2016; Zhang et al., 2015). The concept of elastic stability is initially introduced by Born (Born, 1940; Born and Fürth, 1940) and modified later by Hill and Milstein to take into account the contribution of external loading (Hill, 1975; Hill and Milstein, 1977). For a given crystalline material under a given loading, the elastic stability theory provides a set of elastic stability conditions, and the ideal strength for this loading condition is the stress right before the onset of elastic instability (when at least one elastic stability condition is violated). Many previous works use atomistic simulations and density functional theory calculations with the support of the elastic stability theory to measure ideal strengths of single crystals under various loading conditions (Breidi et al., 2016; Chantasiriwan and Milstein, 1998; Pokluda et al., 2004; Wang and Li, 2009; Zhang et al., 2008).

1

It is well-known that due to the effects of surface stress, the core of an extremely thin nanostructure is subjected to multi-axial loading although the nanostructure is under uniaxial stress condition (Diao et al., 2004; Ho et al., 2016, 2017a). Therefore, if the failure of a nanostructure is governed by the elastic instability rather than conventional plasticity, it is expected that the ideal strength will be different from that of its bulk counterpart because the ideal strength depends on the loading conditions (Černý and Pokluda, 2007, 2010; Ho et al., 2015b; Krenn et al., 2002; Šob et al., 1997). A previous molecular dynamics simulations study shows that the stress drops prior to initiation of dislocation at low temperatures, i.e., the material fails with defect-free deformation (Zepeda-Ruiz et al., 2007); elastic instability is deemed to be the origin of this defect-free failure (Ho et al., 2017b, 2015a). In addition, the failure of some ductile alloys governed by elastic instability was observed experimentally (Saito et al., 2003). It was shown that ideal tensile strength in the [100]-direction of some face-centered cubic (FCC) (001) nanoplates (NPs) decreases when the NP thickness is lower (Ho et al., 2017b). However, in the case of nanostructures, other issues such as direction, loading mode (tension or compression), side surface orientation, shape of cross-section, and size might influence the ideal strength. It should be noted that these issues have been shown to strongly effect the mechanical properties of nanostructures, such as failure mode and Poisson’s ratio (Ho et al., 2016; Park et al., 2006). In this work, we study the ideal strength of nanostructures 100 / 100 NWs, 110 / 110 , 100 NWs, 100 NPs, and {110} NPs in different directions using molecular statics (MS) simulations and the elastic stability criterion. Before studying ideal strength of nanostructures, effect of superimposed transverse compressive stresses (STCSs) on the ideal strength of bulk FCC materials, which is convenient for the study of nanostructures (will be shown later), is investigated. This study emphasizes that ideal strengths of nanostructures depend on the loading direction, side surface orientation, shape of cross-section, and size, and that ideal strengths have three different trends “smaller is stronger”, “smaller is weaker”, and “size-independent strength plateau” rather than the well-known “smaller is stronger” trend. Our semi-analytic model for prediction of ideal strengths is in good agreement with MS results.

Simulation methods Table 1: A list of NWs and NPs with different surface orientations and different loading directions. For all cases, the load is applied in the x-direction. x

y

z

100 / 100 NW

[100]

[010]

[001]

110 / 110 , 100 NW

[110]

110

[001]

100 NP

[100]

[010]

[001]

100 NP

[110]

110

[001]

2

{110} NP

[110]

110

[001]

We employ MS simulations to predict mechanical responses of the bulk materials and nanostructures. The six FCC metals Au, Ag, Cu, Ni, Pd, and Pt are modeled using the embedded-atom method (EAM) potential model (Foiles et al., 1986). Different considered nanostructures and their crystallographic directions are presented in Table 1. To model the bulk materials, we apply periodic boundary conditions (PBCs) in all directions. To model the NPs with the surface in the z-direction, we apply PBCs along the x- and y-directions (not in the z-direction). To model the NWs, PBC is apply only along the x-direction. A length along a periodic direction is about 6a0 where a0 is the lattice parameter. Nanostructures are initially under a full relaxation process where the positions of atoms and the periodic lengths of the simulation box are allowed to change to eliminate the pressures along the periodic directions. For bulk metals, the stress along the x-direction is set to be zero, whereas the stresses along the y- and z-directions are set to be target values. To perform a loading test, we stretch or compress a structure from its initial state along the x-direction with an incremental strain of 0.001; the simulation boxes in the transverse directions with periodic boundary conditions are adjusted so that the stresses along that direction are zero ( for the uniaxial stress condition) or target values (for a multi-axial loading condition). A conjugate gradient (CG) energy minimization technique is employed, and the minimization is terminated when the relative change of the magnitude of total energy between successive iterations was less than 10−16. All atomistic simulations are conducted using the publicly-available simulation code LAMMPS (Plimpton, 1995), and all visualizations are conducted using the open visualization tool OVITO (Stukowski, 2010).

3

Elastic instability of materials b

9 6

primary path

unstable

xx (GPa)

12 stable

unstable

secondary path

0

A

-3 -0.2

Transverse strain

B

3

0.2

Stability term (GPa)

a

0

xx

0.2

c

150

100

50 A

B

0

0.4

-0.1

zz

B22+B23-2B122/B11 B22-B23 B44 B55

-2.5

-0.05

0

xx

0.1

-2.7

yy=zz

-0.1

secondary path

-2.8

-0.2 -0.3

primary path secondary path

-0.2

0

xx

0.1

primary path

-2.6

0.0

0.05

d

Energy (eV/atom)

15

yy

-2.9 0.2

0.4

-0.2

0

xx

0.2

0.4

Figure 1: Mechanical response of the bulk Ag under the uniaxial stress in the [100]-direction: (a) stress-strain curves; (b) Elastic stability condition terms as functions of the applied strain; (c) Lateral strains as functions of the applied strain; (d) Energies as functions of the applied strain. In the compression, B22  B23  2B122 B11  0 is first violated at point A with a strain of 0.088 ; in the tension, B22  B23 becomes zero at point B with a strain of 0.096. A solid is said to be mechanically stable at the current state if the following condition is met for all possible and arbitrary virtual infinitesimal deformations (Hill and Milstein, 1977):

U  W  0

(3.1)

where U and W are the internal energy and work done by external loadings, respectively, and  indicates the incremental change of U and W due to the virtual infinitesimal deformation. The inequality (3.1) is equivalent to

det B  0

(3.2)

where B  (B + BT) and B is the elastic stiffness tensor expressed by following Voigt notation. The expression of the coefficient of the tensor B following by tensor notation is (Wallace, 1967; Wang et al., 1995):

4

Bijkl  Cijkl 

1 ik jl   jk il  il jk   jl ik  2 kl ij  . 2



(3.3)



Here, Cijkl  1 V  2U Eij Ekl are components of the elastic constant tensor C where V is the volume of the system, Eij are components of the Lagrangian strain tensor E,  ij are components of the external stress tensor  , and  ij is the Kronecker delta. For the uniaxial stress in the [100]direction, the FCC crystal has six independent elastic stiffness components B11 ; B 22  B33 ;

B12  B13 , B 23 ; B 44 ; B55  B66 in Voigt notation. Consequently, the elastic stability condition in (3.2) is simplified to the conditions:

B22  B23  2 B122 B11  0

(3.4)

B22  B23  0

(3.5)

B44  0

(3.6)

B55  0

(3.7)

As mentioned by Hill and Milstein, mechanical response of a FCC material under tensile stress in the [100]-direction is special because the crystal can follow either a primary path or secondary path. In the primary path, the deformation always follows the relation  yy   zz . On the other hand, the deformation can also go astray from the primary path to a secondary path due to perturbations and fluctuations at a critical strain when the stability condition B22  B23  0 is violated. Fig. 1 shows the mechanical response of the bulk Ag under uniaxial stress in the [100]-direction. In the tension case, the stress increases with the applied strain and reaches a maximum value of 14.44 GPa at a critical strain of 0.316, and it then decreases gradually with the applied strain for the primary path as shown in Fig. 1a. With perturbations, at a critical strain of 0.096, the deformation drifts from the primary path to the secondary path, and the stress drops significantly right after this point. The stability condition term B22  B23 decreases from 38.3 GPa at the initial state to zero at the critical strain of 0.096 as shown in Fig. 1b, confirming the elastic stability loss of the material. While the  yy   zz relation is always seen the primary path, there is a large change in the crystal at the onset of the instability in the secondary path in Fig. 1c i.e., the lateral strain component  yy  zz decreases/increases significantly with small increase of the applied strain  xx . The deformation mode





in the secondary path can be approximated as  xx ;  yy ;  zz ;  yz ;  xz ;  xy   0; 1;1;0;0;0  . In other words, there is a tetragonal to orthorhombic phase transformation. This phase transformation is predicted theoretically and observed computationally in some FCC materials in previous studies (Hill and Milstein, 1977; Ho et al., 2016; Milstein et al., 2004; Wang et al., 1995). It is worth mentioning that at a particular applied strain, the energy of the secondary path is much smaller than that of the primary path (Fig. 1d). Our simulation results show that the crystal always follows the primary path when it is compressed in the [100]-direction. Magnitude of the compressive stress first increases with magnitude of the compressive strain, then decreases after reaching a maximum value at a strain of 0.088 (Fig. 1a) 5

when one of the stability conditions B22  B23  2 B12 2 B11  0 is violated first. We note that since the effective Young’s modulus along *100+-direction of a material with tetragonal symmetry is

E[100]   B11  B22  B23   2 B122   B22  B23 , the violation of B22  B23  2 B122 B11  0 is 4

a

Stability term (normalized)

equivalent to the vanishing of the effective Young’s modulus at the critical strain (Fig. 1a). A

xx (GPa)

3

2

1

0 0

0.03

0.06

0.09

xx

0.12

0.15

Dn1

b

Dn2 Dn3 Dn4

3

Dn5

2

Dn6

1 A

0 0

0.05

xx

0.1

0.15

Figure 2: Mechanical response of the bulk Ag under uniaxial tensile loading in the [110]-direction: (a) Stress-strain curve; (b) Elastic stability terms as functions of the applied strain.

Din (i  1,6)  Di Di  xx  0  . We now present the mechanical response of FCC bulk materials under the uniaxial tensile loading in the [110]-direction. We also chose the bulk Ag as an example. The [110]-, 110  -, and [001]directions are assigned to the x-, y-, and z-directions, respectively. Under the uniaxial stress in the [110]-direction, FCC materials keep their orthogonal symmetry, i.e., the [110]-, 110  -, and [001]axes remain mutually perpendicular (Djohari et al., 2006) resulting in nine independent elastic

stiffness components B11 , B22 , B33 , B12 , B13 , B23 , B44 , B55 , and B66 . Consequently, the explicit elastic stability conditions for the materials for this loading condition are:

D1  B11  0 ,

(3.8)

B11 B12

(3.9)

D2 

B11 D3  B12 B13

B12 0, B22

B12 B22 B23

B13 B23  0 , B33

(3.10)

D4  B44  0 ,

(3.11)

D5  B55  0 ,

(3.12)

D6  B66  0 .

(3.13)

As shown in Fig. 2a, the stress becomes maximum at a critical strain of 0.075, which is also the point when D3  0 is violated first among the stability conditions (Fig. 2b). We note that the slope of the stress-strain curve is also 0 at the critical strain, similar to the case of uniaxial compressive loading in 6

the [100]-direction presented above, because the effective Young’s modulus of the crystal is

D3

B

22

6

  .

B33  B 23

2

a

A2

xx (GPa)

E2 4 A1 E1 P2

2

P1 [110]-direction [100]-direction

0 0

0.03

0.06

xx

0.09

0.12

Figure 3: (a) Stress-strain curves of the Ag (001) NP with a thickness of 5 nm under uniaxial stress conditions in the [100]-direction and [110]-direction. Configurations of Ag NPs at different important points marked in Fig. 3a in for (b) the [110]-direction case and (c) [100]-direction case. Mechanical responses of the Ag (001) NP with a thickness of 5nm under two uniaxial stresses in the [100]- and [110]-directions are presented in Figure 3. Under loading in the [110]-direction the stressstrain curve is similar to that of the bulk counterpart presented in Fig. 2a, e.g., the stress increases non-linearly with the applied strain, becomes maximum at a critical strain of 0.081 (larger than that of the bulk counterpart), and gradually decreases beyond the critical strain. On the other hand, under loading in the [100]-direction the stress-strain curve of the Ag (001) NP is different from that of the bulk counterpart presented in Fig. 1a, e.g., it decreases smoothly after the maximum stress (whereas the stress-strain curve of bulk Ag, shown in Fig. 1a, follows the secondary path and drops drastically after the maximum stress point). In addition, the critical strain at the maximum stress is 0.086, which is lower than that of bulk Ag. When a FCC (001) NP is under uniaxial stress in the [100]-direction, there are nine independent components of the effective elastic moduli matrix B11 ; B22 ; B33 ; B12 ; B13 ; B23 ; B44 ; B55 and B66 (compared to six for a bulk FCC material). Therefore, the elastic stability conditions for an FCC (001) NP under uniaxial loading in the [100]-direction are the same as those for a bulk FCC material under 7

uniaxial loading in the [110]-direction, i.e., inequalities (3.8)-(3.13) (Ho et al., 2017b). Similarly, when an FCC (001) NP is under uniaxial stress in the [110]-direction, the stiffness matrix also has nine independent components. Consequently, the elastic stability conditions for the case of an FCC (001) NP under uniaxial loading in the [100]-direction are represented by inequalities (3.8)-(3.13). We confirmed that the stability condition D3  0 is also the first violated condition for Ag (001) NPs in the both loading conditions. Since the failure of Ag (001) NPs is governed by elastic instability rather than conventional plasticity in both loading conditions, the NP fails with dislocation-free deformation. Figs. 3b shows the configurations of a Ag (001) NP at different important points marked in Fig. 3a for a [110] loading direction. We show only atoms with a centro-symmetry parameter larger than 1 to detect any planar defects such as dislocations. At point E1, although the applied strain is 0.090, which is larger than the critical strain for the elastic instability (0.081), no dislocation is observed. Dislocation can be observed only when the applied strain is sufficiently large, e.g., at point P1 with an applied strain of 0.096. Similarly, for the [100] loading direction, shown in Fig. 3c, dislocation-free deformation is observed when the applied strain is slightly above the critical strain, whereas dislocation deformation can be observed when the applied strain is sufficiently large. We note that the failure mode of metals caused by the elastic instability, i.e., dislocation-free deformation, has been observed computationally and experimentally in previous studies (Ho et al., 2017b, 2015a; Saito et al., 2003). In short, in this section, we studied the mechanical response of bulk Ag and Ag (001) NPs under uniaxial loading conditions in the [100]- and [110]-directions. Both the bulk Ag and Ag NPs fail due to elastic instability; hence, the failure deformation is homogeneous and dislocation-free. We note that ideal strength of Ag (001) NPs in the [100]-direction is 5.14 GPa, which is lower than the ideal strength of bulk Ag in the [100]-direction, 6.39 GPa. On the other hand, the ideal strength of Ag (001) NPs in the [110]-direction is 4.07 GPa, which is higher than the ideal strength of bulk Ag in the [110]-direction, 3.54 GPa. The ideal strength of bulk materials under different loading conditions is presented in Section 4 and the ideal tensile strength of nanostructures will be presented and discussed in Section 5 and Section 6.

Effect of transverse loading We study the effects of STCSs on ideal strengths of bulk materials in the [100]- and [110]-directions. A multi-axial loading for bulk materials is constituted by a loading in a direction ([100]- or [110]direction), which we call the primary loading direction, and one STCS is applied in another direction or two STCSs are applied in the other two directions. In the case of the [100] primary loading direction, we assigned [100]-, [010]-, and [001]-directions as the x-, y-, and z-directions, respectively, and in the case of the [110] primary loading direction, we assigned [110]-, 110  -, and [001]directions as the x-, y-, and z-directions, respectively, as indicated in Table 1.

8

Effect of STCSs on ideal tensile strength in the [110]-direction

xx (GPa)

4

uniaxial stress

[110]-direction

a[001]= - 0.4 GPa

3

a[001]= - 0.8 GPa

2 a[001]= - 1.2 GPa

1 [001]= - 1.6 GPa a

0 0

0.03

0.06

0.09

xx

0.12

Figure 4: Stress-strain curves of the bulk Cu under the uniaxial stress in the [110]-direction and multiaxial loading conditions. A multiaxial loading condition is the combination of the tensile loading in the [110]-direction and a STCS in the [001]-direction. Tension in [110];  = [xx;0;[001];0;0;0]

I-0I (GPa)

a

0.0

a

-1.5 -3.0

MS F1(x) = 1.395x

0.0

-0.5

-1.0

a[001] (GPa)

-1.5

Tension in [110];  = [x;[110];0;0;0;0]

I-0I (GPa)

a

0.8

b

0.4

MS F2(x) = -0.365x

0.0 0.0

-0.5

-1.0

a[110] (GPa)

-1.5

Figure 5: Influence of STCS on the ideal tensile strength of bulk Cu in the 110 -direction. The material is under a multiaxial loading condition, which is a combination of a tensile stress in the a a [110]-direction and an STCS  [001] in the [001]-direction (a), and an  [110] in the 110 -direction (b). a a The ideal tensile strength decreases with  [001] , whereas it increases with  [110] .  I and  I0

(4.20 GPa) are the ideal tensile strengths of the material under a multiaxial loading and a uniaxial tensile loading, respectively.

In this section, we present the effects of STCS on the ideal tensile strength of bulk Cu in the [110]direction. Fig. 4 presents the stress-strain curves of the material under the loading conditions where

9

a the primary loading axis is the [110]-direction and an STCS  [001] is applied in [001]-direction, i.e., the





a stress components are  x ; y ; z ; yz ; xz ; xy    [110] ;0; [001] ;0;0;0 . The ideal tensile strength a linearly decreases with magnitude of  [001] (Fig. 5a). In contrast, when an STCS is applied in the 110





a -direction, i.e., the stress components are  x ; y ; x ; yz ; xz ; xy    [110] ; [110] ;0;0;0;0 , the a ideal strength increases with magnitude of  [110] (Fig. 5b)

Effect of STCSs on ideal tensile strength in the [100]-direction Tension in [100];  = [x;[010];[001];0;0;0]

I-0I (GPa)

a

0

[010]=[001]=

a

-1

a

a

a

a

MS F3(x) = 0.969x

-2 0.0

-0.5

-1.0

 (GPa) a

-1.5

Tension in [100];  = [x;[010];0;0;0;0]

I-0I (GPa)

a

0 -2 -4 -6

b

MS F4(x) = 3.300x

0.0

-0.5



a [010]

-1.0

-1.5

(GPa)

Figure 6: Influence of STCS(s) on the ideal tensile strength of bulk Cu in the 100 -direction. The material is under a multiaxial loading condition which is a combination of a tensile stress in the [100]-direction and STCSs in both the [010]- and [001]-directions (a) or an STCS in the [010]-direction (b). In both cases, the STCS reduces the ideal tensile strength of the material. However, the reduction is more significant when STCS is applied only in one transverse direction.  I and  I0 (7.22 GPa) are the ideal tensile strengths of the material under a multiaxial loading and a uniaxial tensile loading, respectively. In this section, we investigate the effects of STCS on the ideal tensile strength in the 100 -direction of the bulk Cu. One case with an STCS applied in the [001]-direction and another case with two STCSs with the same value applied in both the [010]- and [001]-directions are considered. For the case with two STCSs the ideal tensile strength decreases linearly with magnitude of the STCSs (Fig. 6a). For the case with one STCS, the ideal tensile strength decreases more significantly with magnitude of the STCS. For example, for STCSs with a value of 1.0 GPa in both [010]- and [001]-directions, the ideal tensile strength is 6.19 GPa, whereas when 1.0 GPa is applied only in the [001]-direction , the ideal tensile strength is reduced to 3.72 GPa.

10

Effect of STCSs on ideal compressive strength in the [100]-direction Compression in [100];  = [x;[010];[010];0;0;0]

I-0I (GPa)

a

-2

a

-1

a

MS F3(x) = 1.055x

a[010]=a[001]=a

0 0.0

-0.5

-1.0

-1.5

 (GPa) a

Compression in [100];  = [x;[010];0;0;0;0]

I-0I (GPa)

a

-1.0

b

-0.5

MS F4(x) = 0.547x

0.0 0.0

-0.5

-1.0

a[010] (GPa)

-1.5

Figure 7: Influences of STCS(s) on the ideal compressive strength of bulk Cu in the 100 -direction. The material is under a multiaxial loading condition, which is a combination of a compressive stress in the [100]-direction and (a) two STCSs in the [010]- and [001]-directions (b) or an STCS in the [001]direction. In both cases, STCSs enhance the ideal compressive strength of the material but the enhancement is more significant when two STCSs are applied in both transverse directions.  I and

 I0 ( 2.73 GPa) are the ideal compressive strengths of the material under a multiaxial loading and the uniaxial compressive loading in the [100]-direction, respectively. We also studied effect of STCS on the ideal compressive strength in the [100]-direction of the bulk Cu. As in Section 4.2, we also considered two cases, i.e., an STCS applied in the [001]-direction or two STCSs with the same value applied in the [010]- and [001]-directions. How STCS affects the ideal compressive strength in the [100]-direction is different from how it does on the ideal tensile strength. First, as shown in Fig. 7, the STCSs results in enhancement of the ideal compressive strength rather than a reduction. For example, the ideal compressive strength of bulk Cu in the [100]-direction is 2.73 GPa for a uniaxial compressive loading; it becomes 3.79 GPa for the case of two STCSs with a value of 1.00 GPa. In addition, Figs. 7a and 7b show that contrary to the trend observed for the ideal tensile strength, the influence of one SCTS on the ideal compressive strength is more significant when two STCSs are applied in both transverse directions.

Discussion on the effect of transverse loadings on ideal strength of materials STCSs can either reduce or enhance the magnitude of the ideal strength of bulk materials. As shown in Fig. 5a, the ideal tensile strength in the [110]-direction linearly decreases as the magnitude of the STCS in the [100]-direction increases. In addition, as shown in Figs. 6a and 6b, STCSs applied in the lateral directions also reduce the ideal tensile strength of bulk Cu in the [100]-direction. The reduction behavior agrees with previous observation (Černý and Pokluda, 2010; Ho et al., 2015b). However, Fig. 5b shows that the STCS in the 110 -direction enhances the ideal tensile strength in the [110]-direction. Therefore, by simply changing the loading direction of the STCS, the ideal

11

strength of a material in a particular direction can be either increased or decreased. We also note that the difference between the effect of STCS on ideal strengths of the defect-free bulk metals and yield strength of macroscale metals is that the STCS always decreases the yield strength in the loading direction (e.g., in von Mises yield criterion), whereas it can increase ideal tensile strength (Fig. 5b).

200

0

0.03

0.06

xx

d

uniaxial stress a a [110] = -1.6 GPa; [001] = 0 a[110] = 0; a[001] = -1.6 GPa

120 90 60

0

0.03

100

0.09

0.06

150

xx

0

0.03

0.06

0.09

xx

e

uniaxial stress a a [110] = -1.6 GPa; [001] = 0 a a [110] = 0; [001] = -1.6 GPa

250

100

150

0

[110] = 0; [001] = -1.6 GPa a

0

a

0.03

0.06

0.09

xx

0.06

xx

0.09

uniaxial stress a a [110] = -1.6 GPa; [001] = 0 a a [110] = 0; [001] = -1.6 GPa

90 60

0.03

f

120

uniaxial stress a[110] = -1.6 GPa; a[001] = 0

90 60

c

150

120

0.09

300

200

150

B13 (GPa)

B12 (GPa)

150

250

uniaxial stress a[110] = -1.6 GPa; a[001] = 0 a a [110] = 0; [001] = -1.6 GPa

200

150 100

b

B33 (GPa)

250

300

B23 (GPa)

uniaxial stress a[110] = -1.6 GPa; a[001] = 0 a a [110] = 0; [001] = -1.6 GPa

B22 (GPa)

a

B11 (GPa)

300

0

0.03

0.06

xx

0.09

Figure 8: Elastic stiffness components of the bulk Cu as functions of applied strain under three different loading conditions: uniaxial tensile stress in the [110]-direction (solid blue lines), the tensile stress in the [110]-direction combined with an STCS in the [001]-direction  a001  1.6 GPa (dotted pink lines), and tensile stress in the [110]-direction combined with an STCS in the 110  direction  110  1.6 GPa (dashed red lines). a



400



a

b

100

E(GPa)

D3(104GPa3)

300

150

uniaxial stress a a [110] = -1.6 GPa; [001] = 0 a[110] = 0; a[001] = -1.6 GPa

200 100

uniaxial stress a a [110] = -1.6 GPa; [001] = 0 a a [110] = 0; [001] = -1.6 GPa

50 0 -50

0 0

0.03

xx

0.06

-100

0.09

0

0.03

xx

0.06

0.09

Figure 9: (a) The elastic stability term D3 and (b) the effective Young’s modulus E of bulk Cu under three different loading conditions: uniaxial tensile stress in the [110]-direction (solid blue lines), the

12

tensile stress in the [110]-direction combined with an SCTS in the [001]-direction  a001  1.6 GPa (dotted pink lines), and tensile stress in the [110]-direction combined with an SCTS in the 110  -





direction  110  1.6 GPa (dashed red lines). STCS applied in the 110 -direction enhances both D3 a





and E so that the elastic instability occurs at higher strains. On the other hand, STCS applied in the [001]-direction reduces both D3 and E so that the material become elastically unstable at lower strains, causing a decrease in the ideal tensile strength. To better understand the difference between the ideal strength behaviors observed in Figs. 5a and 5b, we focus on the elastic instability behavior of the bulk Cu under three loading conditions: the uniaxial tensile stress in the [110]-direction, the loading condition presented in Fig. 5a with a STCS a a  [001]  1.6 GPa, and the loading condition presented in Fig. 5b with a STCS  [110]  1.6 GPa. Among

the elastic stability conditions, D3  0 is the first stability condition violated for all three cases. Fig. 8 plots the six elastic stiffness components B11 , B22 , B33 , B12 , B13 , and B 23 for bulk Cu under the three loading conditions. For the uniaxial stress B11 , B22 , and B33 decrease with the applied strain, whereas B12 , B13 , and B23 increase with the applied strain. D3 (a combination of the six components) decreases with the applied strain and becomes zero at a critical strain of 0.072 (Fig. 9a) which is the maximum point of the stress in Fig. 4a. For the other two loading conditions, the six elastic stiffness components follow the same trend as for the uniaxial stress. However, at a given a strain value, the STCS  [001] B11 , B22 and B33 for the case of is smaller than those of the uniaxial

stress, whereas it enhances the values of B12 , B13 and B23 . The opposite trend is observed for STCS in the 110 -direction; the values of B11 , B22 , and B33 are lower whereas the values of B12 , B13 , and B23 are higher. Consequently, as shown in Fig. 9a, the elastic instability occurs at lower strain values for Fig. 5a loading conditions and at higher strain values for Fig. 5b loading conditions. These results help us understand why STCS in the 100 -direction reduces the ideal strength of a material in the [110]-direction while STCS in the 110 -direction enhances it.

13

Ideal strength of nanostructures Formulation f11y

f 22y f

z 22

 22R  

 33R  

b a

f11z

z

z 22

2f b

 11R  

2 f 22y a

y x

2  bf11y  af11z  ab

Figure 10: An RNW model. Surface stresses on the RNW (above) and stresses induced by the surface stresses in the interior of the RNW (below) are presented. The surface energy density of a material  is the energy needed to create a unit area of the material. The surface stress of the material is related to the surface energy density and the applied strain via the following equation (Shenoy, 2005):

f 

1 E S     A  

(5.1)

where f are components of the surface stress tensor f, A is the area of the surface, E S is the total surface energy of A,   are components of the surface strain tensor  , and  is the Kronecker delta. For an infinitely long RNW with a cross-sectional area of a × b under mechanical tot

loading, its total energy E is the summation of the total bulk energy E B and total surface energy

E S (Dingreville and Qu, 2007):



E tot  E B  E S  E B  2 E yS  EzS



(5.2)

where E yS and EzS are the total surface energies of the y- and z-plane surfaces of the RNW, respectively. We suppose that the strain components in the x-direction of all atoms are the same, i.e.,  11 . Taking the first derivative of Eq. (5.2) with respect to  xx and dividing both sides by the volume V of the RNW yields: S 1 E tot 1 E B 2  E y EzS      . V  xx V  xx V   xx  xx 

(5.3)

where V  Ay b  Az a , where Ay and Az are the areas of the free surfaces of the y- and z-planes, respectively. Eq. (5.3) becomes:

 1 E yS 1 E tot 1 E B 1 EzS    2  . V  xx V  xx  aAy  xx bAz  xx 

14

(5.4)

We note that  N 

1 E tot 1 E B   and B are the x-components of the effective stresses of V  xx V  xx

the RNW and the bulk crystal, respectively. In addition, according to Eq. (5.1),

S 1 E y and Ay  xx

1 EzS are the x-components of the surface stress tensors of the y-plane surface ( f xxy ) and z-plane Az  xx surface ( f xxz ) of the material, respectively. As a result, Eq. (5.3) can be rewritten as:

N B

bf 2

y xx

 af xxz ab

.

(5.5)

The first term on the right hand side of Eq. (5.5) is the contribution from the core and the second term is the contribution from the surface stresses. Suppose that the elastic instability occurs homogeneously in the RNW, at the onset of elastic instability  N becomes the ideal strength of the RNW, noted  NI , and  B becomes the ideal strength of the bulk material, noted  BI . In addition, we suppose that the changes due to applied strain of the surface stresses are negligible. Consequently, the ideal strength can be written as:

 NI   BI  2

bf

y xx

 af xxz ab

.

(5.6)

Eq. (5.6) implies that the ideal strength of the nanostructure is the combination of the ideal strength of the bulk material and the modification due to surface stress. The core of a nanostructure is under a multi-axial loading condition even when the nanostructure is under uniaxial stress conditions. The reason can be explained as follows. The number of bonds between the atoms on the surface is smaller than the number bonds between the atoms in the core. Therefore, the equilibrium interatomic distance of the atoms at the surface are not the same as that of atoms in the core, and thus the atoms in the core can be regarded as they are subjected to an additional load by the atoms on the surface (Cammarata, 1994). With a tensile surface stress, e.g., in the case of FCC {100} and FCC {110} nanostructures, the atoms in the core of the nanostructures rearrange themselves to balance the forces, thereby causing compressive stress in the core (Diao et al., 2003). In the case of RNWs, the induced stress component in the x-direction, i.e., the axial direction is:



a x

bf  2

y xx

 af xxz ab

;

(5.7)

and the induced stress components in the y-direction (  ya ) and z-direction (  za ) can be approximated by (Yang et al., 2009):

  2 a y

f yyz

(5.8)

b

15

 za  2

f 22y a

(5.9)

Since the core under a multi-axial loading condition, the ideal strength of the core is a function of the induced transverse stresses,  ya and  za in Eqs. (5.8) and (5.9):

 BI   BI ,0  FT  ya , za 

(5.10)

where FT is the additional term that takes into account the influence of the induced transverse stress components. Eq. (5.6) can now be written as:

 NI   BI ,0  FT  ya ,  za   FA ,



where FA  2 bf xxy  af xxz



(5.11)

ab . FA is actually the amount of external stress needed to overcome

the residual compressive stress along the loading. Therefore, FT represents the contribution of the transverse surface stress components and FA represents the contribution of the axial surface stress components. In the following section, we investigate the ideal strength behavior of some selected nanostructures.

Case studies

16

Table 2: Surface stress values (unit J/m2) of a Cu (100) surface and a Cu (110) surface. For the Cu (100) surface, the x-direction corresponds to [010]-direction and the y-direction corresponds to [001]-direction. For the Cu (110) surface, the x-direction corresponds to the 110 -direction and the y-direction corresponds to [001]-direction.

Surface

(100)

(110)

a

( Shenoy, 2005)

b

(Dingreville and Qu, 2007)

f xx

f yy

1.38

1.38

(1.10)a

(1.10) a

(1.40)b

(1.40)b

0.99

1.12 a

(0.99) a

(0.99)b

(1.13)b

(0.60)

We evaluate the semi-analytic model for the ideal strength of nanostructures by simplifying Eq. (5.11) for some nanostructures and compared the prediction of the model with MS simulation results. We study the ideal tensile strengths in the 110 -direction of FCC {100} NPs and FCC {110} NPs, and both the ideal tensile and compressive strengths in the 100 -direction of FCC

100 /{100} RNWs and FCC {100} NPs. The relevant surface stresses are measured using a previous method (Ho et al., 2014). Table 2 provides a list of the surface stress values for the {100} and {110} surfaces of a Cu sample, and those obtained in previous studies (Dingreville and Qu, 2007; Shenoy, 2005). Ideal tensile strength in the 110 -direction

17

Table 3: Values of transverse compressive stresses induced by surface stresses, and expressions of FT and FA for tensile strengths of Cu {110} NPs and {100} NPs. The y-direction and z-direction correspond to the 110 -direction and [001]-directions, respectively.

{110} NP

 ya

{100} NP

  2

0

 za

  2 a

{110} f 100 , 100

FA

2

t

0

t

0.365 a

1.395 a

FT

{100} f 110 , 110

a

{110} f 110 , 110

2

t

{100} f 110 , 110

t

We calculate the ideal tensile strength in the [110]-direction of Cu {110} NPs and Cu {100} NPs. For Cu {110} NPs, we assign [110]-, [001]-, and 110 -directions as the x-, y-, and z-directions,

respectively, and for Cu {100} NPs, we assigned [110]-, 110 - and [001]-directions as the x-, y-, and zdirections, respectively. Details of the MS simulations for uniaxial stress loading for nanostructures can be found in Section 2. The induced transverse compressive stresses in the y- and z-directions of the core of the Cu {110} NP can be approximated using Eqs. (5.8) and (5.9), respectively:  y   [1 10]  0 , and a

a

a {110}  za   [001]  2 f 100 t where t is the thickness of the NP. Therefore, the core can be assimilated , 100

to a bulk material under a multiaxial loading condition, which is a combination of a stress in the a {110} [110]-direction and an STCS  [001]  2 f 100 t in the [001]-direction. As a result, the contribution , 100

of the transverse surface stress components for Cu {110} NPs FT can be approximated by the function F1 presented in Fig. 5a:

FT  2.790

{110} f 100 , 100

t

.

(5.12)

{110} In addition, FA can be simplified to 2 f 110 t (Table 3). Consequently, the expression of the ideal , 110

strength of the Cu {110} NP in the 110 -direction is:

  I N

I ,0 B

 2.790

{110} f 100 , 100

t



2



110 f 110 , 110

t

.

(5.13)

Similarly, for the FCC {100} NP stretched in the 110 -direction, the induced transverse compressive a stresses in the core are  y   [110]   2 f 110 , 110 t (the y-component) and  za   [001]  0 (the z-

a

a

{100}

18

component). Therefore, the core can be assimilated to the bulk material under a combination of a {100} tensile stress in the [110]-direction and an STCS  ya  2 f 110 t in the 1 10 -direction (y, 110

direction). As a result, FT can be approximated by the function F2 presented in Fig. 5b:

FT  0.730

{100} f 110 , 110

t

.

(5.14)

{100} The third term FA on the right hand side of Eq. (5.11) can be simplified to 2 f 110 t . As a result, , 110

we obtain the expression of the ideal strength of the Cu {100} NP in the 110 -direction:

 

I ,0 B

 0.730

Ideal strength (GPa)

I N

{100} f 110 , 110

t



2

t

.

(5.15)

<110> loading

7

Cu {110} NP (MS) Cu {110} NP (model) Cu {100} NP (MS) Cu {100} NP (model)

6 5



100 f 110 , 110

bulk

4 3 0

7

14

21

28

35

Thickness (nm) Figure 11: Size-dependent ideal tensile strength in the 110 -direction of Cu {110} NPs and Cu {100} NPs. The ideal strength of the Cu ,110- NP shows the “smaller is weaker” trend, whereas the ideal strength of the Cu {100} NP shows the “smaller is stronger” trend. The prediction of our model is in good agreement with MS results. Fig. 11 shows the change of the ideal tensile strength of the Cu {110} NP and Cu {100} NP in the 110 -direction as functions of the NP thickness. As the Cu {110} NP thickness is smaller, the ideal strength becomes smaller, i.e., the ideal strength of the Cu {110} NP shows the “smaller is weaker” trend. In contrast, ideal strength of the Cu {100} NP in the 110 -direction shows the “smaller is stronger” trend. The predictions of our models (Eq. (5.13) for Cu {110} NPs and Eq. (5.15) for Cu {100} NPs) are in good agreement with the MS simulation results. This section shows that although the two NPs are under the same loading conditions, the ideal strength behaviors are entirely different. Ideal tensile strength in the 100 -direction

19

Table 4: The values of transverse compressive stresses induced by surface stresses and expressions of FT and FA for ideal tensile strengths of Cu 100 / 100 SNWs and Cu {100} NPs. The y-direction and z-direction correspond to the  010 -direction and [001]-directions, respectively. 100 / 100 } SNW

 ya

 a  2

 za

  2 a

FT FA

{100} NP

{100} f 100 , 100

{100} f 100 , 100

 a  2

t {100} f 100 , 100

t

0

t

0.969 a

3.300 a

{100} f 100 , 100

{100} f 100 , 100

4

2

t

t

10

Ideal strength (GPa)

<100> loading 8 6 bulk 4

SNW (MS) SNW (model) NP (MS) NP (model)

2 0

0

7

14

21

28

35

Thickness (nm) Figure 12: Ideal tensile strength of Cu 100 /{100} NWs and Cu {100} NPs in the 100 -direction. Although the two nanostructures are under the same loading conditions, the SNW shows the “smaller is stronger” trend, whereas the NP shows the “smaller is weaker” trend. In this section, we focus on the ideal tensile strength in the 100 -direction of Cu 100 /{100} SNWs and Cu {100} NPs by following the approach in the section 5.2.1. We assign [100]-, [010]-, and [001]-directions as the x-, y-, and z-directions, respectively. For the case of Cu 100 /{100} SNWs, the core is under a combination of tensile strain in the [100]-direction, an STCS in the [010]direction, and another STCS in the [001]-direction, and thus the function FT for this case can be approximated as the function F3 in Fig. 6a. For Cu {100} NPs, the core is under a combination of tensile strain in the [100]-direction and an STCS in the [010]-direction, so the function FT can be approximated as the function F4 in Fig. 6b. The STCSs, FT, and FA for the two cases are presented in

20

Table 4. Consequently, the expression for the ideal tensile strength in the 100 -direction of a Cu

100 / {100} SNW with thickness of t is:

  I N

I ,0 B

 2.062

{100} f 100 , 100

t

,

(5.16)

and that for a Cu {100} NP with thickness of t is:

 NI   BI ,0  4.600

{100} f 100 , 100

t

.

(5.17)

Fig. 12 presents the ideal tensile strengths of the Cu 100 /{100} SNWs and Cu {100} NPs as a function of the thickness. The ideal strength of the Cu 100 /{100} SNW in the 100 -direction with a thickness of 1.8 nm is 8.67 GPa, which is more than two times larger than that of the Cu {100} NP with the same thickness, 3.73 GPa. As the thickness increases, the ideal tensile strength of the SNW decreases while that of the NP increases; both values are close to that of bulk Cu (7.22 GPa) when thickness becomes larger 30 nm. Fig. 12 also shows that the predictions of our model are in excellent agreement with the MS simulation results. Although the two nanostructures have the same loading direction and the same surface, the RNW exhibits the “smaller is stronger” trend whereas the NP shows the “smaller is weaker” trend. Ideal compressive strength in the 100 -direction Table 5: The values of transverse compressive stresses induced by surface stresses and expressions of FT and FA for ideal compressive strengths of Cu 100 / 100 RNWs and Cu {100} NPs. The ydirection and z-direction correspond to the  010 -direction and [001]-direction, respectively. 100 / 100 NW

 ya



a z

FT FA

  2 a

  2 a

{100} NP

{100} f 100 , 100

0

t {100} f 100 , 100

t

  2 a

{100} f 100 , 100

t

1.055 a

0.547 a

{100} f 100 , 100

{100} f 100 , 100

4

t

21

2

t

Ideal strength (GPa)

-3

bulk

<100> loading

-2 SNW (MS) SNW (model) NP (MS) NP (model)

-1

0

0

7

14

21

28

35

Thickness (nm) Figure 13: Ideal compressive strength of Cu {100} NPs and Cu 100 / 100 SNWs in the [100]direction. Both nanostructures show “smaller is weaker” trend. In this section, the ideal compressive strengths in the [100]-direction of 100 /{100} SNWs and {100} NPs are considered. The induced transverse stresses in the two cases are the same as those in the tension presented in Section 5.2.2. Consequently, the contribution of the transverse surface stress components for the Cu 100 /{100} SNWs FT can be approximated by F5 in Fig. 7a and that for the Cu {100} NPs can be approximated by F6 in Fig. 7b. Details of the induced transverse stresses, contributions of the transverse surface stress components, contributions of the axial surface stress components, and ideal strength functions are listed in Table 5. As shown in Fig. 13, the ideal compressive strengths of both nanostructures exhibit the “smaller is weaker” trend, and the degree of reduction of strength with size for the SNWs is more significant than that for the NP.

Discussions We have shown that the ideal strengths of the nanostructures depend strongly on loading direction and side surface orientation. For example, for Cu {100} NPs tensile strength in the [110]-direction shows “smaller is stronger” trend (Fig. 11), whereas that in the [100]-direction shows “smaller is weaker” trend (Fig. 12). In addition, the size effect of the ideal tensile strengths in the 110 direction of Cu {110} NPs and Cu {100} NPs have completely different from each other (Fig. 11). This indicates importance of the interplay between the loading direction and side surface orientations. We show that the stress induced in the core of nanostructure by the surface stress changes the elastic instability behavior of the nanostructures, thus causing a change in its ideal strength. For defect-free metallic nanostructures, the surfaces and edges are usually regarded as the main sites for dislocation nucleation, which in turn strongly influence their yield strength (Cao and Ma, 2008). Therefore, how the surface influences ideal strength is thoroughly different from how it influences the yield strength of nanostructures, i.e., it changes the ideal strength by influencing the elastic instability behavior whereas in the case of the yield strength it acts as a local source for dislocation nucleation. We note that the expressions for the ideal strengths of the nanostructures presented in

22

Tables 3-5 follow a generic scaling law of properties of nanostructures by Wang et al. (Wang J et al., 2006)

Effect of loading direction The change of ideal strength with size depends strongly on loading direction. One reason is that loading direction effects elastic instability behavior resulting in modification of FT in Eq. (5.11). For example, in the case of Cu {100} NPs FT for the expression of ideal tensile strength in 110 -direction is enhanced with an induced transverse compressive stress (Table 3) whereas the ideal tensile strength in the 100 -direction is reduced with an induced transverse compressive stress (Table 3). As a result, ideal strength in the 110 -direction shows “smaller is stronger” (Fig. 11) but ideal strength for the [100]-direction shows “smaller is weaker” trend (Fig. 12). Another reason is that loading direction might modify FA in Eq. (5.11) due to the change of the axial surface stress components in the anisotropic surface stress tensors (see the case of Cu (110) surface in Table 2).

Ideal tensile strength versus ideal compressive strength The effects of both axial and transverse surface stresses for ideal tensile strength case is opposite to those for ideal compressive strength case. As mentioned early, for tensile surface stresses FA in Eq. (11) is the external stress needed to overcome the residual compressive stress along the loading direction, and thus the axial surface stresses enhances the ideal tensile strength but it reduces ideal compressive strength. In addition, contribution of the lateral surface stress components via FT, is different for the ideal tensile strength and the ideal compressive strength. We discuss the Cu 100 /{100} SNWs and Cu {100} NPs cases. For Cu 100 /{100} SNWs, the enhancement/reduction of the ideal tensile/compressive strength due to the axial surface stress component is approximately two times larger than the reduction/reduction due to the effect of the lateral stress components (Table 4) causing the “smaller is stronger” trend for the ideal tensile strength case (Fig. 12), and “smaller is weaker” for ideal compressive strength case. For Cu {100} NPs although they show “smaller is weaker” trend for both tension and compression cases, the main contribution of the trend for each case is different, i.e, the contribution of the axial stress component is dominant for the ideal tensile strength while the contribution of the lateral stress component is more important (see Tables 3 and 5).

Side surface orientation The side surface orientation has a strong influence on the ideal strength of nanostructures. Since the surface stress tensors depend on the side surface, the contribution due to the axial surface stress components, i.e., FA, and the stress state of the core also vary with the change of the side surfaces. {100} For example, for the ideal tensile strength of Cu {100} NP in the 110 -direction, FA is 2 f 110 t, , 110

which is larger than FA for the ideal tensile strength of Cu {110} NP in the 110 -direction, {110} {100} {110} is smaller than the value of f 110 (Table 2). In 2 f 110 t ; this is because the value of f 110 , 110 , 110 , 110

addition, when the Cu (100) NP is stretched in the 110 -direction, the core has one non-zero lateral stress component in the 110 -direction; hence, the loading conditions are similar to that of bulk Cu presented in Fig. 5b. On the other hand, when the Cu {110} NP is stretched in the 110 -direction, the core has one non-zero lateral stress component in the 100 -direction; the loading conditions are similar to that of bulk Cu presented in Fig. 5a. 23

Shape of cross-section

bulk 6 =1.00 =0.83 =0.50 =0(Cu(001) NP)

4 0

6

12

18

24

30

ty(nm)

7

b

<110> loading

6

=0(Cu(001) NP) =1 =2 =4 = (Cu(110) NP)

+

8

<100> loading

Ideal strength (GPa)

Ideal strength (GPa)

a 8

5

+

bulk

+

+

+

+

+

4 0

6

12

18

24

30

ty (nm)

Figure 14: Effect of the shape of cross-section on the ideal strength. (a): in the 100 -direction of Cu [100]/(001) RNWs; (b): in the 110 -direction of Cu 110 /{110}, {100} RNWs. We now discuss on effects of the shape of cross-section on ideal tensile strength in the 100 direction of Cu 100 /{100} RNWs. As discussed previously, there is a competition between the enhancement due to the contribution of the axial surface stress FA and the reduction due to the contribution of the lateral surface stress components FT. For SNWs, FA is the dominant factor and the ideal tensile strength has a “smaller is stronger” trend. For NPs, FT is the dominant factor and the ideal tensile strength has a “smaller is weaker” trend. Since the main difference between SNWs and NPs is the shape of cross-section, it can be deduced that the ideal tensile strength of a nanostructure can be controlled by changing its shape. The aspect ratio of the cross-section is defined as   t z t y where t y and t z are the thickness of the nanostructure in the y- and z-directions, respectively. Fig. 14a presents the ideal tensile strength as a function of thickness for different cross-sections. The ideal strength of the SNW forms the upper boundary whereas the ideal strength of the NP forms the lower boundary of the curves. Therefore, the RNW initially shows a “smaller is stronger” trend but is transformed into a “smaller is weaker” trend when the aspect ratio is sufficiently small. This change occurs for an aspect ratio of about 0.83, where the ideal strength shows a “size-independent strength plateau”. We also discuss on effects of the shape of cross-section on ideal tensile strength in the in the 110 direction of Cu 110 / 110 , 100 RNWs. The contribution of the axial surface stress components FA also enhances the ideal tensile strength as in the case of the Cu 100 /{100}RNWs, but the contribution of the transverse surface stress components FT is different. When it is applied on the {100} surface, it tends to enhance the ideal tensile strength (via the induced compressive stress in

the core in the 110 -direction). However, when it is applied on the {110} surface, it tends to reduce the ideal tensile strength (via the induced compressive stress in the core in the [100]-direction). Consequently, the ideal tensile strength of the RNW has a lower boundary that is defined by the ideal tensile strength of the Cu {110} NP and an upper boundary that is defined by the ideal tensile strength of the Cu {100} NP. The ideal tensile strength of the RNW also shows a “size-independent strength plateau” for an aspect ratio of about 4.0. 24

Ideal strength (normalized)

Other fcc metals 1.2

SNW

1.0 0.8

Nanoplate

0.6 0.4 Ag

Ni

Cu

Au

Pd

Pt

Figure 15: Ideal strengths of different 100 / 100 SNWs and {100} NPs, normalized with values of the bulk metal. All values were measured in the 100 -direction. All SNWs and nanoplates have the same thickness of 12a0. For all metals, the ideal strength of the nanowires is always larger whereas the ideal strength of the nanoplates is always smaller than that of the bulk counterpart. So far, we have mainly focused on the ideal strength behavior of various Cu nanostructures. We verify that ideal strength of nanostructures of different metals namely Ag, Au, Ni, Pd, and Pt show a similar behavior. For the metals stated above, Fig. 15 shows that the ideal tensile strength of the bulk material in the 100 -direction is larger than that of the corresponding {100} NP but smaller than that of the 100 / 100 SNW. In general, the ideal strength values of the SNWS with a thickness of 12a0 are about 5-10% larger compared to the values of the bulk material. On the other hand, ideal strength of the metal NPs are much smaller than of that of the bulk material; the lowest difference is observed for the Ag NP (22% lower than bulk Ag) and largest for the Pt NP (58% lower than bulk Pt). For the Au NP, the difference 40% and 33% is observed in this study and a previous study (Ho et al., 2017b), respectively.

Conclusions In conclusion, we study ideal tensile and compressive strengths of various FCC nanostructures with different orientations. This study shows that ideal strength of nanostructures depends on loading direction, model of loading (tension or compression), side surface orientation, shape of crosssection, and size. Our MS simulation results show that ideal strengths can show “smaller is stronger” (the tensile strengths in the 100 -direction of 100 / 100 SNWs, the 110 -direction of

110 / 110 , 100 SNWs and 100 NPs), “smaller is weaker” (both tensile and compressive strengths in the 100 -direction of {100} NPs, tensile strength in the 110 -direction of {110} NPs, and the compressive strength in the 100 -direction of 100 / 100 SNWs), and a “size-independent strength plateau” (ideal tensile strengths in 110 -direction of some 100 / 100 RNWs and in

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110 -direction of some 110 / 110 , 100 RNWs). We also developed a simple semi-analytic model to predict the ideal strength of nanostructures. Our model is based on the fact that the surface influences the ideal strength of nanostructures by changing the elastic instability behavior of the entire nanostructure rather than acting as local resources for dislocation nucleation, which is the case of yield strength. The prediction of our model is in good agreement with MS simulation results. Acknowledgement We gratefully acknowledge the support from the High-speed Manufacturing and Commercialization of Ultra-lightweight Composites Research Fund (Project Number: 1.180035.01) of UNIST and the supercomputing resources of the UNIST Supercomputing Center.

Conflicts of Interest Statement We have no conflict of interest to declare.

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