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%!TEX root = /Users/charles/Documents/chuckDissertation/dissertation.tex |
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\chapter{\label{chap:nanodiffusion} SIZE DEPENDENT SPONTANEOUS ALLOYING OF AU-AG NANOPARTICLES} |
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To gain insight into the alloying process, molecular dynamics simulations were conducted on model spherical particles composed of Ag and Au atoms interacting under the Embedded Atom Method (EAM) potential(42,43) using Johnson’s mixing rules (44) for the Ag-Au interactions. Because the lattice constants for Ag and Au are nearly identical, the particles were constructed in a perfect fcc lattice using the average of the two-lattice constants (4.085 Å). Those atoms inside the core radius (rcore) were initialized with Au parameters, while those between the core and shell radius (rcore < r < rshell) were initialized as Ag atoms (rshell =R). A 2Å-thick shell centered on rcore was designated as the “interface” region, as shown in Figure 1. For calculations involving interfacial vacancies, 5\% or 10\% of these interfacial atoms were chosen at random and were removed from the initial configuration. For the simulations described here, rcore = 12.5 Å and rshell = 19.98Å. These radii match the smallest of the NPs described in the experimental sections above (B in Table 1). The total number of atoms in each nanoparticle was 1926 (no vacancies), 1914 (5\% interfacial vacancies), and 1902 (10\% interfacial vacancies). |
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Before starting the molecular dynamics runs, a relatively short minimization was performed to relax the lattice in the initial configuration. During the initial 30 ps of each trajectory, velocities were repeatedly sampled from a Maxwell-Boltzmann distribution matching the target temperature for the run. Following this initialization procedure, trajectories were run in the constant-NVE (number, volume, and total energy) ensemble with zero initial total angular momentum. In order to compare the structural features obtained from the NVE-ensemble molecular dynamics, additional trajectories were run using a modified Nosé-Hoover thermostat to maintain constant temperature and zero total angular momentum. Data collection for all of the simulations started after the 30 ps equilibration period had been completed. We simulated particles with the above-mentioned interfacial vacancy density at 100K intervals from 500 to 1200 oK. Given the masses of the constituent atoms, we were able to use time steps of 5 fs while maintaining excellent energy conservation. Typical run times for our simulations were 24 ns for nanoparticles simulated at 500-700K and 12 ns for particles at 800-1200K. |
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Analysis of the molecular dynamics simulations centers on transport properties of atoms within the NPs. Most of the atomic motion is due to surface atoms migrating around the outer spherical shell. Since this motion does not effectively mix the two constituents of the core-shell structure, we have developed a method of estimating the relaxation time for the complete alloying process from our simulations. |
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The mean square displacement of atoms confined to a spherical volume of radius R approaches $6R^2/5$ in the infinite-time limit. Similarly, the mean square displacement in the radial coordinate only (relative to the NP center of mass) approaches 3R2/40. At shorter times, one might want to connect the observable displacements of atoms in a simulation to the solutions of the diffusion equation. For a spherically symmetric volume with a reflecting boundary at R, the solutions to the diffusion equation are given by Eq. 2, |
