35
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between electrons, we have decided the GGA functional, parameterized by Perdew, Burke, and Ernzerhof44, which has been utilized extensively for solid state calculations. The interaction between core and valence electrons were dealt with the projector augmented wave (PAW) pseudo-potential45. The electrons of wavefunctions were extended up to 500 eV of cutoff energy in a plane-wave basis set.
All calculations were completely relaxed until force between ions becomes less than 0.01 eV/ Å. To sample the Brillouin zone, we have utilized 23231 and 2311 Monkhorst-Pack scheme46 for the nanoplate and nanowire, respectively. The periodic boundary condition is applied along the x- and y- directions for nanoplate (vacuum is applied along the z-direction) and x-direction for nanowire (vacuum is applied along the y- and z-directions). Here, we consider Cu(001) nanoplate and Cu(100) nanowire.
4.2.2 Molecular static and dynamics calculations
All molecular statics (MS) and molecular dynamics (MD) calculations have performed using the large- scale atomic/molecular massively parallel simulator (LAMMPS)48. The periodic boundary conditions are applied to the x- and y-directions to build up the nanoplate. The size of lattice for nanoplate structure in the x- and y-directions is 5a0 5a0 with different thickness in z-direction, where a0 is the lattice constant at equilibrium state. For FCC metal nanoplate, we employ the interatomic potential proposed by Foiles et al.49 to exclude any artificial effect that may arise from the usage of a single potential.
4.3 Relaxation and surface energy of nanoplate
In this Chapter, we have focused on the relaxation of Cu(001) nanoplate with different boundary conditions. The relaxation methods are categorized in un-, surface-, and full-relaxation. For nanoplate during un-relaxation, the atomic configuration and structural lattice along the in-plane directions, which are referred from its bulk counterpart, is not relaxed, while that the charge distribution around ions is only optimized. And, for surface-relaxation, the atomic position of nanoplate is relaxed in thickness direction ([001]-direction) as well as the relaxation of charge distribution around ions. Finally, all structural parameters are fully optimized during the calculation, so-called full-relaxation, which is optimizing the charge distribution, ionic position, and box size in lateral directions ([010]- and [001]- directions, respectively). As shown in Figure 4.1a, the cohesive energy (eV/atom) for Cu(001) nanoplate, obtained by the difference between nanoplate and its corresponding bulk, is calculated with different thickness under un-relaxation. Here, thickness is obtained as the difference between top and bottom layer of nanoplate model shown in Figure 4.1b. With the increase of thickness, the cohesive energy of Cu(001) nanoplate decreases approaching to its bulk counterpart. Under surface-relaxation, the
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properties of Cu(001) nanoplate is investigated in Figure 4.2. First, we have observed the change in relaxed thickness of nanoplate during optimization in atomic position with different thickness shown in Figure 4.2a. In thickness direction, the surface-relaxed length of nanoplate is shrunk as previously reported84. As increasing the number of nanoplate layers, the amount of relaxation compared to bulk configuration decreases. The amount of change in thickness length during surface-relaxation is obtained as the difference between surface-relaxed nanoplate and its corresponding bulk. Second, the cohesive energy during surface-relaxation is shown in Figure 4.2b. As the number of layers in nanoplate increases, the cohesive energy decreases, which is relatively smaller than un-relaxed nanoplate. We have observed that during surface-relaxation the atoms in first layer of nanoplate is moved inside, so-called in-ward84 shown in Figure 4.2c. We may say that the introduction of surface is able to affect the atomic configuration. To investigate whether structure of Cu(001) nanoplate can be relaxed more, the transverse directions (x- and y-directions shown in Figure 4.1b) are relaxed to be relatively stable configuration. Figure 4.3a shows the tendency of applied compressive strain along the [010]- and [001]- directions with different layers of nanoplate. Here, compressive strain is estimated as a relation of
bulk bulk
(a−a ) /a , where a is the lattice constant along the transverse directions from fully-relaxed nanoplate, abulkis the lattice constant from bulk configuration. In Figure 4.3b, the cohesive energy of full-relaxed nanoplate is calculated with increase of nanoplate layers, which it is slightly more stable than un- and surface-relaxed nanoplate. For comparison to surface energy (in meV/Å) under surface- (black solid line) and full-relaxation (red solid line), which surface energy is calculated as
nanoplate bulk / 2
(E −nE ) A, where Enanoplateis the total energy of plate model, n is the used number of atom for plate, Ebulk is the total energy of bulk model, A is the cross-sectional are to create surface, the results are displayed in Figure 4.3c. During full-relaxation, there is the decrease in cross-sectional area of plate (A), resulting in that surface energy by full-relaxation is smaller than that of surface-relaxation.
Our results show that with different boundary conditions for relaxation as well as the number of layers for nanoplate the properties of material such as cohesive energy, length in periodic boundary, and surface energy could be affected.
4.4 Mechanical response of nanoplate under tensile loading
In Section 3.3, we have considered full-relaxed configuration as the equilibrium state of Cu(001) nanoplate. Here, to investigate the mechanical response of Cu(001) nanoplate, uniaxial tensile loading is applied to equilibrium state of Cu(001) nanoplate. As shown in Figure 4.4, under uniaxial loading
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along the [100]-direction, the change of transverse strain and the applied stress (GPa) are summarized for 6-, 10-layer nanoplate, and its corresponding bulk. Interestingly, as increasing the mechanical strain to bulk and Cu(100) nanoplate, other behavior of change in transverse lattice is observed as shown in Figure 4.4a. In case of Cu bulk, at instability point around 0.11 of tensile strain, there is sudden change in structural transformation as expected by Hill’s theory. While, for nanoplate configuration, the smooth change in transverse strain is observed. As we discussed in Chapter 2.4, for bulk metal, the lengths in transverse directions undergo equal reduction by uniaxial tensile loading. However, we have observed the negative Poisson’s ratio in metal nanoplate, before critical strain corresponding to ideal tensile strength, indicating the expand in one of length ([010]- or [001]-direction) during tensile loading along the x-direction. These change in transverse lattice is dependent on the thickness of nanoplate. For 6- layer plate, the transverse strain is significantly applied compared to 10-layer plate and its bulk counterpart shown in Figure 4.4a. Also, Ho et al.31 reported same phenomenon (negative Poisson’s ratio) for the mechanical response on metal nanoplate using DFT and MD simulations. Under uniaxial loading to Cu, the strain-stress curve is investigated in Figure 4.4b for 6-, 10-layer, and bulk structure. Compared to bulk, the maximum stress (ideal tensile strength) of nanoplate is significantly decreased. In addition, the computed ideal strength of Cu is 2.78, 4.37, 9.49 GPa for 6- and 10-layer nanoplate, and its corresponding bulk, respectively, indicating the dependence of ideal strength with different thickness of nanoplate. These findings show that the nanoplate is weaker than its bulk counterpart in terms of ideal tensile strength. In order words, we have thought of that the surface effect of nanoplate system is important resulting in smooth bifurcation and the drop of ideal strength as the size of thickness decreases.
4.5 Origin of surface effect
To elucidate the reason of surface effect in nanomaterial, we have performed MD simulation on the property of Cu(001) nanoplate at extremely low temperature (0.01K) to investigate the stress distribution for each layer of nanoplate. As we consider the relaxation methods (surface- and full- relaxation) in Chapter 3.3, they are utilized during calculation of nanoplate. Figure 4.5a exhibits the stress profile of nanoplate for each layers (normalized) under surface- and full-relaxation along the thickness direction. Both relaxation methods show same tendency in shape of stress distribution along the number of layers of nanoplate (normalized). Interestingly, the tensile stress along the z-direction is observed in top and bottom layer while compressive stress is applied in next layer of top and bottom.
For inside of nanoplate, stress, z, would be not detected. However, the lateral stresses (along the x- and y-directions) are dependent on boundary condition of relaxation, as shown in Figure 4.5b. In top and bottom layer, the stress is tensile during surface- and full-relaxation. Mostly interesting, for most
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of inside layers, the compressive stress is applied to compensate for the high tensile stress induced in top and bottom layers during full-relaxation. As a result, the stress profile in the nanoplate along the in- plane directions is not homogeneously distributed, unlike bulk. This is a unique characteristic of nanoplates and is ascribable to the so-called surface effect.
To portray the surface effect in bulk configuration, we have calculated the property of Cu bulk under uniaxial tensile loading (x-direction) with additional compressive stress (y-direction around 500MPa and 1000MPa), and relaxation along the z-direction to be almost zero stress using DFT calculations.
First, the lattice parameters of Cu bulk under loading condition, are introduced in Figure 4.6a. Here, the compressive stress of 0 GPa along the y-direction could be thought of that the original bulk. As increasing the compressive stress to bulk, the trend of lattice change is like nanoplate as discussed in Chapter 3.4. Figure 4.6b shows that the applied stress along the [100]-direction is varied with different compressive stress, approximately 9.49 (0MPa along the [010]-direction), 6.63 (500MPa), and 5.05 GPa (1000MPa), like strain-stress tendency of nanoplate. We have confirmed that the compressive stress is induced along the normal direction with respect to surface. In other words, these behaviors induced by surface effect such as negative Poisson’ ratio and decrease of ideal tensile strength could be explained by compressive stress.
4.6 Relaxation and edge energy of nanowire
In this Chapter 3.6, we have focused on the relaxation of Cu(100) nanowire and its edge energy. As mentioned previously in Chapter 3.3, we have considered the un-relaxation, surface-relaxation, and full- relaxation. In order to study the effect of size of nanowire, we change the mode size of nanowire in thickness direction (y- and z-directions shown in Figure 4.7b) from 3a03a0 (36 atoms) to 7a07a0
(196 atoms)and plot the changes of the cohesive energy with different radii of nanowire under un- relaxation condition with its bulk counterpart in Figure 4.7. Here, the radius (in Å) is expressed as the relation of R= A/ , where A is the cross-sectional area of nanowire after simulation shown in Figure 4.7b. One can see the changes of cohesive energy are different with increase of radius of nanowire. For further allowing the relaxation in atomic position along the thickness directions (or surface direction), the cohesive is slightly decreased compared to un-relaxation case. Also, we can see that with increasing the radius of nanowire, the cohesive energy is approaching to its bulk counterpart shown in Figure 4.8a. In Figure 4.8b, during surface-relaxation, the atoms nearby edge of nanowire is moved far from inside of nanowire compared corresponding bulk configuration, so-called out-ward.
Unlike nanoplate, the atoms located in surface move to out-ward direction along the surface directions.
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We have thought that the introduction of edge which have a very low atomic coordination affect the property of materials. Figure 4.9a indicates the relationship between the applied compressive strain along the length direction of nanowire and the radius of nanowire. When nanowire is small in surface direction, the amount of compressive deformation is significantly applied along the length direction to be energetically stable configuration. In this study, for the smallest nanowire using 3a03a0 of lattice along the surface directions, we have obtained the compressive strain around 0.07 for Cu(100) nanowire.
With increase of radius of nanowire, the amount of applied compression decreases. As shown in Figure 4.9, during full-relaxation, there is the influence by the edge. To identify the contributions of edge in nanowire, in this Chapter, we have defined the edge energy (eV/atom) as
nanowire edge
U A Vβ
E n
− −
= . (3.1)
where Unanowire is the total energy of nanowire, A is the cross-sectional area of nanowire, is the surface energy per area calculated from its nanoplate counterpart, β is the bulk energy per volume obtained from its bulk counterpart, n is the number of atoms located to edge of nanowire. As shown in equation (3.1), to estimate the edge energy of nanowire per atom (eV/atom), the references of corresponding bulk and nanoplate are essentially required. Here, using our designed methods, we have tried to select the bulk and nanoplate as references with different relaxation conditions such as un- (nanoplate is dealt with bulk configuration), surface-, and full-relaxation (all references are stress-free).
Also, the references could be extracted from nanowire without calculation of bulk and nanoplate, indicating that some specific regions in nanowire could be chosen to represent the bulk (inside of nanowire) and nanoplate (few layers of nanowire and far from edge). By dividing three regions in nanowire, the computational cost could be significantly saved. As shown in Figure 4.9c, the edge energy is obtained by full-relaxation method. Although the value of edge energy with different definition of references is diverged, they show almost same tendency with different radius of nanowire using DFT and MD simulations. In addition, to investigate the strain effect to edge energy, we have utilized upper defined references. However, among those definitions, we have failed to explain the consistent trend of edge energy with the increase of mechanical deformation. Later, it would be our next topic for study of edge energy. Figure 4.9c shows the behavior of edge energy obtained from DFT and MD at 0.01K with different radius of nanowire. As can be seen, the edge energy is nearly constant especially with different size of nanowire in surface directions, which is also reported by atomic calculation based on EAM approximation for Al and Ni very thin nanowires85.
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4.7 Mechanical response of nanowire under tensile loading
To observe the mechanical response of metal nanowire under uniaxial loading, we have modeled copper (Cu) nanowire with 3a03a0 of size along the surface directions (y- and z-directions as shown in Figure 4.7b) for square shape of nanowire (SNW). Figure 4.10a shows the results of strain-stress curve for Cu(100) nanowire with comparison to bulk’s tendency. In case of SNW, the stiffness of material shows linear and relatively smaller behavior compared to its corresponding bulk. As increasing the mechanical deformation to structure, the stress in loading directions increase linearly and reach to maximum value with 10.17 GPa (9.49 GPa for bulk) at 0.17 (0.10 for bulk) of strain. We have shown that the critical strain of instability and its ideal tensile strength of SNW is higher than bulk configuration. As shown in Figure 4.10b, at instability point (0.18 of strain), similar structural change in lattice induces to structure like bulk configuration. Figure 4.10c shows the comparison of atomic configuration for before (left) and after (right) instability. After failure in nanowire, the shape of cross- sectional area of SNW is not exactly rectangular like bifurcation (or branching) at bulk. However, the increase and decrease in the out-plane direction of SNW could be found, as shown in Figure 4.10b-c.
Our results show that edge could delay the instability of material and increase the ideal strength of material. For Cu SNW, it can be thought of that the equal compressive stress is applied to material.
Therefore, as expected by elastic stability condition term (T3 ), the failure mode of SNW can be predicted like ideal bulk as shown in Figure 4.11c. These results are in good agreement with previous atomic simulation32 for gold SNW under 0.01K.
To observe the contribution of the edge effect to material, we have added the four hydrogen atoms at edge of nanowire as the hydrogenation. After full relaxation (to reach almost zero stress for all directions), the mechanical loading along the [100]-direction is applied to hydrogenated structure as shown in Figure 4.11a. Mostly interesting, the behavior of hydrogenated SNW shows almost similar properties of ideal bulk in terms of critical strain for instability, stiffness (the gradient of strain-stress curve), and ideal strength (maximum stress). Figure 4.11b is the stress tendency of hydrogenated SNW under uniaxial tensile loading. Ideal strength of hydrogenated SNW is higher than its corresponding bulk counterpart around 10.62 GPa (9.49 for bulk). In Figure 4.11c, the atomic configurations at 0.11 and 0.12 of tensile strain are compared. At 0.12 of tensile strain shown in right side of Figure 4.11c, the failure of hydrogenated SNW starts with stress drop which happens shortly later than bulk case. The phase transformation in terms of transverse strain is observed like bare SNW but earlier. After hydrogenation at edge of nanowire, the stiffness of hydrogenated SWN show almost similar value of bulk. These findings say that edge effect could be dominant and change the mechanical properties of nanowire, especially in stiffness and critical strain for instability. In addition, the proper design of materials such as low dimension and hydrogenation could produce the high stiffness of materials like
42 bulk.
As same calculation, we have modeled the rectangular shape of nanowire (RNW) using 3a05a0 of lattice parameter along the y- and z-directions, respectively. Here, the pure and hydrogenated RNW are investigated. As shown in Figure 4.12, the mechanical properties of RNW are summarized for clean edge of nanowire. The stress tendency under uniaxial tensile loading is non-linear unlike SNW with 5.98 GPa of ideal tensile strength (9.49 and 10.17 GPa for bulk and SNW, respectively). At small region from 0.08 and 0.09 of tensile strain in Figure 4.12b, the negative Poisson’s ratio is detected as observed in metal nanoplate (See Figure 4.4a). In addition, between before and after instability, the configuration of RNW with different loading is smoothly transited as shown in Figure 4.12c. In fact, the RNW can be thought of that the compressive stress is not equally induced along the normal direction with respect to surface. The mechanical property of RNW could be understood like nanoplate (See Figure 4.4) due to non-homogeneous distribution of applied stress in structure. Mostly interestingly, even adding hydrogen atoms to edge of nanowire, the remarkable change in mechanical properties would be found in terms of critical strain for instability, ideal strength, and structural change as shown in Figure 4.13. After hydrogenation, the ideal strength of RNW is around 5.79 GPa, indicating the edge effect is not dominant factor in RNW.
4.8 Summary
In this Chapter, we have investigated the mechanical properties of Cu nanoplate and nanowire. Unlike bulk material, as introducing the surface to structure, the interesting behavior of nanoplate is observed such as atomic relaxation (in-ward), gradual transformation under uniaxial loading, and the reduction of ideal strength by controlling the number of thickness. To understand the relaxation process in nanoplate, three kinds of approaches are utilized such as un-, surface-, and full-relaxation. In addition, the relationship between properties (cohesive energy, atomic configuration, and transverse strain) and thickness of nanoplate have been simulated. With different thickness of nanoplate, the amount of change in applied transverse strain under uniaxial loading is diverged. Compared bulk (9.49 GPa), the ideal strength (2.78 and 4.37 GPa for 6- and 10-layer nanoplate, respectively) is also dependent on the thickness of nanoplate. These unique characteristics can be explained by the generation of compressive stress in middle of nanoplate due to surface. As well as nanoplate, as applying compressive stress, these phenomena could be found in bulk materials.
Next, we have calculated the square (SNW) and rectangular (RNW) shape of nanowire for their mechanical properties using DFT method. The common calculations are first performed to obtain the equilibrium state of SNW and RNW under un-, surface-, and full-relaxation (The optimized RNW