However, before the material reaches the theoretical strength, a failure of the material occurs, which causes a limitation of the strength of the material, such as instabilities (based on phonon and elastic criteria) or other reasons, the so-called ideal strength. More interestingly, for different loading conditions and other configurations such as hexagonal close packing (HCP) and body-centered cubic (BCC), the stabilities with respect to elastic and phonon measures for element Cu, Ag and Au show consistent results. Other atomic configurations of bulk materials such as diamond carbon (C), silicon (Si) and germanium (Ge) are also investigated based on DFT calculation.
Furthermore, we focused on the mechanical response under uniaxial tensile loading by DFT calculation and characterized the edge quantity, such as edge energy with different size and shape of nanowires, by molecular statics and dynamic simulations. In addition, we discuss the change in band gap energy and critical stresses for the direct-to-indirect transition under different types of stresses, such as uniaxial, biaxial, and pure shear. 112 2.25 Elastic stability under symmetric transverse loading on Cu, Ag and Au bulk after initiation.
Introduction
Background of density functional theory
The first two terms on the right-hand side explain the kinetic energies of electrons and nuclei, respectively. The solution to the electronic Schrödinger equation is the electron wave function Ψelec and the. Although the Hohenberg-Kohn theorem is quite useful in principle, it cannot provide a method to predict the ground-state electron density of a system in practice, because the exact form of the function Eee[ ( )] r is unclear. .
As previously discussed, the problem that arises in Hohenberg and Kohn's theorem is the vagueness of Eee[ ( )] r term. Variation of the total energy functional E[ ( )] r with respect to i*( )r leads to an effective one-electron equation for the. GGA considers one additional term to the LDA with the gradient of the electron density at specific region.
Stability criterion for crystal structure
Hill and Milstein8 showed that condition (1.20) is a necessary and sufficient condition for the stability of a material and is coordinate invariant. In this dissertation, several loading conditions are considered to investigate the stability of the structure according to Eq. The determinant of B reaches zero when and only when one of the stability conditions vanishes.
A lattice wave in a solid is defined by the frequency ( , )ωq s , where q is the wave vector in reciprocal space and s denotes the polarization (i.e. the mode of the wave for longitudinal and two transverse modes, in the case of a single mode atom in unit cell) and phonon branches (i.e. acoustic and optical branches in case of more than two atoms in unit cell for optical phonon at least). As the energies of the phonon become negative, the crystal will grow in the increase of amplitude to structure, which is the driving force to transform into another stable configuration. Generally, the phonon scattering is accepted as a more reliable stability criterion than elastic stability ratio (designed by Hill) as shown by Clatterbuck et al.22 which is one of the common examples.
Study aim and constitution of dissertation
So far we have discussed the theory of stability, based on elastic and phonon criteria. However, in this study we used both criteria to precious metals (Cu, Ag and Au) to confirm which criterion shows better performance. Therefore, we tried to obtain the correct ideal strength of material for simple (uniaxial tension) and complicated (multiaxial or introduction of surface/edge) loading condition.
By DFT calculation, we tried to understand the influence of surfaces and edges on the structures and their origin. The relationship between the edge effect and the onset of instability in nanowire metals was also discussed in section 4. In the appendix, we calculated the properties of very thin materials such as graphene and monolayer MoS2.
Bulk Metals
- Introduction
- Computational method
- Equilibrium state of FCC structures
- Mechanical response under uniaxial loading
- Stability of structure under uniaxial loading
- Phase transformation by instability
- Stability for HCP and BCC configuration
- Stability of tensioned structure with additional transverse deformation
- Stability of compressed structure with additional transverse deformation
- Summary
Next, we have focused on the elastic stability under uniaxial compressive loading as shown in figure 2.7a-c. We have compared the primary and secondary pathways in terms of lattice parameter in Figure 2.10a-c for Cu, Ag and Au respectively. It is related to the elastic state T4 shown in Figure 2.7 with 19%, 17% and 9% of compressive strain for Cu, Ag and Au respectively.
First, for HCP Cu, Ag and Au, the elastic stability state terms (in GPa) are summarized in Table 2.5. In contrast, as the tensile transverse strain increases, the elastic steady state term (T2) increases for Cu, Ag and Au. As shown in Figure 2.26, the same trend is observed in elastic steady state term T2 with the critical strains of and 0.0484 (Also in the elastic steady state term T4 for Au) along the transverse directions for Cu, Ag and Au, respectively .
Covalent Materials
- Introduction
- Computational method
- Equilibrium state of covalent materials
- Mechanical response under uniaxial loading and elastic stability
- Phase transformation at unstable state
- Effect of symmetrical transverse loading
- Summary
Based on these configurations, the elastic steady state conditions of tetragonally deformed covalent materials are evaluated as shown in Figure 3.5. In the case of Ge-bulk, we have found another instability around 0.26 of the tensile stress based on the term of the elastic stability condition of T5 as shown in Figure 3.8c. The trend of elastic stability conditions with different transverse loads is shown in Figure 3.9a.
For most Ges with 23% strain, the elastic steady state conditions are largely improved by the application of additional tensile transverse stress as shown in Figure 3.10a. However, for bulk Si and Ge, the homogeneous deformation can be well detected by the term of the elastic stability condition of T3. Also, we have seen that the tensile strength can be increased by transverse loading for bulk Si and Ge.
Nanoplate and Nanowire Metals
- Introduction
- Computational method
- Relaxation and surface energy of nanoplate
- Mechanical response of nanoplate under tensile loading
- Origin of surface effect
- Relaxation and edge energy of nanowire
- Mechanical response of nanowire under tensile loading
- Summary
Here the thickness is obtained as the difference between the top and bottom layers of the nanoplate model, shown in Figure 4.1b. Firstly, we observed the change in the relaxed thickness of the nanoplate during atomic position optimization with different thicknesses, shown in Figure 4.2a. However, the lateral stresses (along the x and y directions) depend on the relaxation boundary condition, as shown in Figure 4.5b.
Here, the radius (in Å) is expressed as the ratio R= A/ , where A is the cross-sectional area of the nanowire after the simulation, shown in Figure 4.7b. After complete relaxation (to achieve near-zero stress for all directions), a mechanical strain along the [100] direction is applied to the hydrogenated structure, as shown in Figure 4.11a. As shown in Figure 4.12, the mechanical properties of the RNW are summarized for the clean edge of the nanowire.
Conclusion
Also, the ideal tensile strength and structural change of nanoplates are observed under uniaxial tensile deformation. It can be thought that the strength of materials is determined by the thickness and introduction of the surface. Interestingly, the introduction of surface, as opposed to bulk, induces the compressive stress toward the interior of the nanoplate along the normal direction to the surface.
It is unique for material to undergo negative Poisson's ratio under uniaxial tensile loading. We have shown that the compressive stress along the in-plane directions is the dominant factor. It is proved by using ideal bulk with compressive stress, which shows similar trend in lattice change under uniaxial tensile loading.
For nanowire materials, it is conceivable that biaxial compressive stress is applied to material. Here we have simulated the mechanical response of square (SNW) and rectangular (RNW) nanowire under uniaxial tensile loading and the contribution of edge effect in material. For SNW, since applied compressive stress is homogeneous inside the material, the ideal strength is higher than its corresponding bulk due to high critical stress due to edge effect.
We have shown that by hydrogenation the overall mechanical properties of SNW are analogous to its bulk counterpart. Unlike SNW, the hydrogenated and pure RNW show similar trend, indicating that the edge effect in RNW is not dominant in affecting the mechanical property of material.
Appendix
- Lateral diffusion of single carbon atom onto graphene and Ni(111) surface
- Lateral diffusion of adsorbate onto Cu(100) and Cu(111) surface
- Mechanism of oxidation of graphene-coated copper
- Engineering band gap energy of MoS 2 monolayer by strain
The available adsorption position of the Cadatom in a 4×4 free-standing graphene sheet is shown schematically in Figure 6.3a. For example, we performed spin-polarized DFT calculations on the configuration of the H2O molecule on a graphene MV, as shown in Figures 6.12 and 6.13. The H atom is bonded to the top of the C atom and the O atom is stabilized in the middle of the C atoms.
The initial configuration of the optimized carboxyl group at graphene MV is as shown in the middle configuration of Figure 6.15a. In addition, for the application of optoelectronic devices, it is essential to maintain the band gap of the MoS2. The changes of the band gap energy of the MoS2 monolayer due to the corresponding mechanical deformations are summarized in Figure 6.19.
It should also be noted that the transitions occur near or at the maximum values of the band gap energy when compressive stresses are applied, as shown in Figure 6.20. The dependence of the band gap changes on the mechanical strain directions can be explained by the crystal structures shown in Figure 6.19. Even at a biaxial strain of 0.7% (either compressive or tensile), the bandgap in the MoS2 monolayer becomes indirect.
Up to a compressive pure shear strain of 2%, the band gap of the MoS2 monolayer remains direct. As the tensile load increases further, the energy of the VBM at the K point becomes lower than that. However, the energy shifts of the VBM at the K point based on the strains are large enough to initiate the bandgap-type changes.
We also found that the band structure of the MoS2 monolayer changes from direct to.
The effect of transverse loading on the ideal tensile strength of in-plane-centered cubic materials. Influence of superimposed biaxial stress on the tensile strength of perfect crystals from first principles. Cohesive properties of noble metals by van der Waals-corrected density functional theory: Au, Ag and Cu as case studies.
Elastic stability, bifurcation and ideal strength of gold under hydrostatic stress: An ab initio calculation. Atomic scale investigation of graphene grown on Cu foil and thermal annealing effects. Atoms, molecules, solids and surfaces: Applications of the generalized gradient approximation to exchange and correlation.
Enhanced reduction of graphene oxide by charge and electric fields applied to hydroxyl groups.
List of Tables
Optimized lattice constant a 0 (Å), Ecohesive cohesive energy (eV/atom), obtained by the difference of the total energy per atom between the ideal bulk material and its corresponding isolated individual atom, elastic constant C 11. Optimized lattice constant a 0 (Å), bond lengths dS-Mo and d S-S, bond angle S-Mo-S, cohesion energy Ecohesive (eV/atom), obtained by the difference of the total energy per atom between the ideal bulk material and its corresponding isolated single atom, gap energy E g (eV).
Acknowledgement
