Investigation of the vibrations of linearly growing nanostructures.
Surulere, Samuel Abayomi
Surulere, Samuel Abayomi
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Abstract
Nanostructures are structures on the nanoscale which measure approximately 10−9m. Structures on this scale are comprised of molecules and atoms. Interatomic potentials describe the interactions between atoms and molecules which make up nanostructures. Several potentials have been formulated over the years but most widely used is the Morse potential which was formulated to describe the vibrational energy levels of diatomic molecules, it is also applicable to some metals. Estimation of the parameters of Morse potential has been studied extensively by scientists over the years. Estimated Morse potential parameters are used to construct potential energy curves (PEC), which contain information about the structure of molecules. Parameters of the Classical and Generalized Morse potential would be estimated using the approach of multiple goal function (MGF), and the differential numerical method. This method is called Objective
Least Squares functions (ObLSf) method. In this work, the Classical and Generalized Morse potential (which are hybrid forms of the Morse potential) are transcendental least squares problems that are transformed to ordinary least squares problem using the quadratic goal function. The MGF approach consists of constructing two objective functions, minimizing them through the differential numerical method and obtaining estimated values using experimental data sets of gold atom. The built-in “Minimize” function in the computer algebra system (CAS), Mathematica R and MathCadR were used to minimize the ordinary least squares problem. PECs are constructed to analyze the efficiency (and applicability) of the MGF in comparison to the “Minimize” function in the CAS Mathematica R and experimental data. The built-in function fails to give parameter estimates that guarantee convergence to the optimal solution. Complex conjugate eigenvalues were obtained while estimating the parameters ˜and ˜ of the first objective function for the Generalized Morse potential. A new interatomic potential, the Modified Generalized Morse potential was then proposed. A constructed PEC comparing experimental data sets, the Classical Morse potential and the Modified Generalized Morse potential clearly shows that the proposed ObLSf method approximates the Classical Morse potential more accurately.
Vibration of linearly growing atoms in a nanostructure (which can be described by the mass-spring-damper system) will be investigated by formulating the respective system of equations using the Euler-Lagrange equation with Rayleigh’s modification, which models a vibrating atom attached to a stationary wall, the second atom attached to the first atom and vibrating with the same velocity as the first, the same applies for the ten attached atoms. The system of equations which are second order ordinary differential equations ODE, are transformed to first order ODE and their respective column vector forms will be formulated. Simulation of the systems of first order ODE would be carried out using CAS Mathematica R and MathCadR for two cases, the regular case (which is a continuous and differentiable function where the attached atom vibrates with the same boundary conditions of the previously vibrating atom that is to say having the same displacement and velocity coordinates as the previously vibrating atom) and the random case (which is a differentiable function where the attached atom vibrates with different positions from the previously vibrating atom). From the simulations, it was observed that the amplitudes of vibration increased for the discrete attachment of atoms for the regular case and the frequency of vibration decreased. For the random case, the amplitude of vibration decreased on attachment of the second atom and increased on linear attachment of subsequent atoms. The simulation results obtained from both cases
compares satisfactorily with results from existing result in the literature.
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Date
2018-03-01
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Tshwane University of Technology
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Keywords
Nanostructures, Molecules, Atoms, Morse, Potential energy curves