Inter-Atomic Potential

In this work, we present calculated SFE using the LKKR-CPA method for Al-Cu and Al-Mg which are of interest from the point of view of superplasticity. We use the SFE to validate the rigid band model which allows a deeper insight into the electronic structure and its implication on the nature of inter-atomic potentials. 

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

A similar calculation can be done for ionic crystals. In this case the Coulomb interaction is taken into account, in addition to the van der Waals attraction and the Pauli repulsion. Although the van der Waals attraction contributes little to the three-dimensional lattice energy, its contribution to the surface energy is significant and typically 20-30%. The calculated surface energy depends sensitively on the particular choice of the inter-atomic potential. 

In the case of the recrystallization progress of the amorphous structure, the determination of the a-Si microstructure is of great importance as an initial phase in the simulations. Many researchers have proposed the preparation methods of fl-Si based on MD simulations and their computer-generated properties were in good agreement with those obtained by experiments. However, details of the microstructures have not yet fully examined [22, 23] because the calculation domain sizes in the ah initio MD simulations were quite limited [24, 25], and the preparation methods of a-Si using various simple empirical inter-atomic potentials involved unphysical treatments. 

For ionic solids interacting with Coulomb pair potentials, similar calculations can be carried out. However, this is a rather complex matter because Coulomb, van der Waals attraction and Pauli repulsion should all be taken into account. In addition, there are uncertainties in the choice of suitable pair-potential equation (many inter-atomic potential equations, including Lennard-Iones were tried), and the calculated Gf results are highly dependent on the particular choice of pair-potential model. As an example, Gf = 212m) m 2 was calculated theoretically for the NaCl (100) crystal, which is near to the experimental value of Gf = 190 m) m 2 from extrapolation of the molten salt surface tension values, but far away from Gf = 300 mj m 2, which was found from crystal cleavage experiments. 

In spite the MD and MC methods depend on the potentials that the forces acting on atoms by differentiating inter atomic potential functions, the Ab initio techniques are accurate methods which are based on an accurate solution of the Schrodinger equation. Furthermore, the Ab initio techniques are potential-free methods wherein the atoms forces are determined by electronic stmcture calculations progressively. 

Different researchers have been doing many efforts to simulate mechanical properties of CNT, in generally the main trends of these methods employed by different researchers to predict the elastic modulus of SWCNTs results in terms of three main parameters of morphology radius, chirality, and wall thickness. The dependency of results to the diameter of CNT becomes less pronounced when non-linear inter atomic potentials are employed instead of linear ones. 

Linking between inter-atomic potential energies of lattice 2010 molecular structure and strain energies of equivalent discrete frame structure... 

Mazur et al. [103, 104] demonstrated the conformational dynamics of biomacromolecules. However, their method scaled exponentially with size and relied on an expensive expression for the inter-atomic potentials in internal coordinates. Subsequently, our group pioneered the development of internal coordinate constrained MD methods, based on ideas initially developed by the robotics community [102, 105-107], reaching 0(n) serial implementations, using the Newton-Euler Inverse Mass Operator or NEIMO [108-110] and Comodyn [111] based on a variant of the Articulated Body Inertia algorithm [112], as well as a parallel implementation of 0(log n) in 0(n) processors using the Modified Constraint Force Algorithm... 

Considering as starting point the general form of the inter-atomic potential. Figure 4.77, there is noted that, essentially, the potential function, as the electronegativity, in a qualitative meaning, contains the competition of two tendencies repulsive and attractive. 

Molecular motion can theoretically be described using molecular dynamics. In this context, molecular dynamics describes the motion of a molecule in a fluid by accurately characterizing the force between any two atoms in the system including the molecules of the surrounding fluid. All forces in the system are summed and Newton s law is applied to each atom to deduce its individual motion. The motion of the molecule is deduced from the motion of its constituent atoms. Because of its fundamental nature, a molecular dynamics approach is often considered ab initio or from first principles , meaning that there are virtually no fitting parameters required to obtain a physical solution. Nevertheless, the force between atoms, or inter-atomic potential, arises from a variety of chemical... 

In general, in addition to the inter-atomic potential (2.13), will also appear the intra-atomic forces, exerted by the nuclei on the electrons from the belonging atoms. To evaluate the intra-atomic associated potential, will be used the shape of the wave functions (2.2) for the energy of the electrons in the fundamental vibrational quantum state, here rewritten as ... 

The non-Newtonian flow of the Eyring type in arm (B) represents relaxation of inter-atomic potentials by stress-biased, thermally-activated segmental diffusion. In its previously published form, the PET model assumed the materid to be in equilibrium, with fictive temperature 7) equated to the actual temperature T. In practice, experimental data show a significant yield... 

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