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%!TEX root = /Users/charles/Documents/chuckDissertation/dissertation.tex |
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\chapter{\label{chap:conclusions} CURRENT WORK AND CONCLUSIONS} |
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Computational work on metallic nanoparticles presented in Chapters \ref{chap:bulkmod} and \ref{chap:nanoglass} has presented several interesting questions about the interaction of the NP with the surrounding solvent. This includes questions about the role capping agents play in transferring heat between the nanoparticle and its environment. The implicit solvent method used in previous chapters allowed for particle dynamics to be explored in a computationally feasible manner, but did not provide any insight into the dynamics at the solvent-particle interface. Current work involves extending the implicit solvent methods to allow for an explicit solvent and capping agents. The method presented here will provide a constant pressure bath that resembles the interaction with a bulk solvent at atmospheric pressure that can then be coupled to an explicit solvent model. The explicit solvent model is allowed to interact with both the capping agent and the NP. This model without the capping agent is represented as a cartoon in Figure \ref{fig:solvated_np}. |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=3in]{images/solvated_np.pdf} |
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\caption{Implicit Constant Pressure Langevin Bath(blue) Coupled with Explict Water and Metallic Nanoparticle.} |
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\label{fig:solvated_np} |
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\end{figure} |
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\section{Constant Pressure Langevin Dynamics} |
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Current methods for conducting Molecular Dynamics simulations in the NPT ensemble use an affine transformation to scale distances within the simulation to maintain a constant pressure.\cite{melchionna93} In mixed phase (liquid-solid) simulations, this scaling may lead to unphysical consequences for the crystalline lattice in the nanoparticle (particluarly when there are large differences in compressibility). Alternative techniques for constant pressure baths have been proposed that do not use the affine transformation method.\cite{0953-8984-18-39-037,Baltazar:2006lr,calvo:125414,Kohanoff:2005,0953-8984-14-26-101} Baltazar {\it et al.} provide a good overview of currently available constant pressure techniques for finite systems.\cite{Baltazar:2006lr} As suggested by Baltazar {\it et al.}, for such systems we can write the Lagrangian (Section \ref{introSec:eom}) as |
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\begin{equation} |
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L = \sum_{i}^{N} \left[\frac{p^2_i}{2m_i}\right] - U(\vec{r}_i)-P_{\mathrm{ext}}V(\vec{r}_i) |
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\end{equation} |
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where particle $i$ of a system consisting of $N$ particles has a mass $m_i$ and momentum $p_i$. $U$ is the inter-atomic potential and $V(\vec{r}_i)$ is the volume(dependent on spatial coordinates). Equations of motion for an $N$ particle system are then given by |
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\begin{equation} |
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m_i\frac{\partial^2 \vec{r}_i}{\partial t^2} = \vec{F}_i - P_{\mathrm{ext}} \frac{\partial V}{\partial \vec{r}_i}. |
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\end{equation} |
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This equation describes a constant pressure ensemble so long as the volume can be described by a function of the system coordinates. Assuming thermal equilibrium, the external applied pressure is balanced by an average internal pressure such that |
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\begin{equation} |
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P_{\mathrm{ext}} = P_{\mathrm{int}} = \left<\frac{1}{3V}\left(\sum_i^N \frac{p^2_i}{m_i} - \sum_i^N \mathbf{r}_i \cdot \nabla_i U \right) \right>. |
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\end{equation} |
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The remaining issue then becomes how to obtain the volume of the finite sized system. Baltazar {\it et al.} review several methods that can be used to obtain the volume of such finite size systems. The convex hull method as described in Section \ref{bulkmod:sec:analysis} provides an accurate volume for the system provided the cluster does not assume a concave shape in which the volume will be overestimated. The new constant pressure method is similar to the Kohanoff {\it et al.} method in that the external pressure is applied directly to the triangles comprising the convex hull.\cite{Kohanoff:2005} The external pressure (applied at the centroid in a direction opposing the surface normal) is then mapped to the surface atoms that are defined as part of the hull facet. A Langevin thermostat is coupled to the surface atoms to maintain a constant temperature bath. The forces on the surface atoms due to the constant temperature-pressure bath are thus given by |
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\begin{equation} |
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m_i\frac{\partial^2 \vec{r}_i}{\partial t^2} = \mathbf{F}_{\mathrm{sys}}^{(i)} + \mathbf{F}_{\mathrm{ep}}^{(i)} - 6\pi a\eta \vec{v}_i(t) + \mathbf{R}_i(t) |
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\end{equation} |
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where $\eta$ is the viscosity of the implicit solvent, $\mathbf{R}_i(t)$ is the random stochastic force defined in Section \ref{introSec:LD}, $a$ is the hydrodynamic radius, and $\vec{v}_i$ is the velocity of the surface atom. $F_{\mathrm{ep}}$ is the force do to the external pressure bath and is defined as |
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\begin{equation} |
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\mathbf{F}_{\mathrm{facet}}^{(i)} = -\hat{n}_{\mathrm{facet}}(P_{\mathrm{target}} \cdot A_{\mathrm{facet}}). |
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\end{equation} |
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Each of the facets has three vertices where a surface atom is located. The force at each vertex, $\mathbf{F}_{\mathrm{ep}}^{(i)}$, is then $1/3$ the force at the centroid of the facet ($\mathbf{F}_\mathrm{facet}$) summed over all adjacent triangles |
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\begin{equation} |
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\mathbf{F}_{\mathrm{ep}}^{(i)} = \sum_{j=1}^{N_{\mathrm{Tri}}^{(i)} }\frac{1}{3}\mathbf{F}_{\mathrm{facet}}^{(j)} |
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\end{equation} |
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as illustrated in Figure \ref{fig:images_LD_method}. |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=3in]{images/LD_method.pdf} |
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\caption{Method for Computing Constant Pressure Forces on Surface Atoms in a Nanoparticle Using the Convex Hull.} |
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\label{fig:images_LD_method} |
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\end{figure} |
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Simulations were conducted using the new method to determine the compressibility, $\kappa_T =-(\partial V/\partial P)/V_{eq}$, for gold nanoparticles similar to the method outlined in Chapter \ref{chap:bulkmod}. Figure \ref{fig:images_pv} shows $P_{int}$ versus the normalized volume ($V/V_{eq}$) where $V_{eq}$ is the zero pressure volume for a system comprised of 1926 Au atoms. From this plot, $\kappa_T$ at 301 K is calculated to be $0.00551$ $\mathrm{GPa}^{-1}$ using the {\sc QSC} model, which is in good agreement with the bulk value of $0.00577$ $\mathrm{GPa}^{-1}$ as reported by Kohanoff {\it et al.}.\cite{Kohanoff:2005} |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=3in]{images/pv.pdf} |
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\caption{Constant Pressure Langevin Dynamics $P_{int}$ vs Normalized Volume $V/V_{eq}$ for a 1926 atom Au nanoparticle using the {\sc QSC} potential (\ref{introSec:tightbind}) measured at 301 K.} |
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\label{fig:images_pv} |
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\end{figure} |
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\section{Angular Water-Metal Potential} |
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Calculation of thermal transport between the nanoparticle-solvent interface requires a potential model that accurately describes water-metal interactions. One model that has been used for simulation water-metal interactions and thermal transport from metal surfaces to water is do to Spohr.\cite{SpohrE._j100353a043,Spohr:1995lr,DouY._jp003913o} The Spohr model for water-metal interactions is given by a Morse function, |
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\begin{subeqnarray} |
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\label{con:eq:spohr} |
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U_{\mathrm{M-H_2O}} &=& U_{\mathrm{M-O}}(r_{\mathrm{M-O}}) + U_{\mathrm{M-H1}}(r_{\mathrm{M-H1}}) + U_{\mathrm{M-H2}}(r_{\mathrm{M-H2}}) \\ |
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U_{\mathrm{M-O}}(r) &=& D_0\left[\exp(-2\beta_O(r-r_{e1}))-2\exp(-\beta_O(r-r_{e1}))\right] \\ |
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U_{\mathrm{M-H}}(r) &=& \gamma D_0\exp(-2\beta_H(r-r_{e2})), |
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\end{subeqnarray} |
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where $D_0$ is the surface dissociation energy calculated from thermal desorption experiments. $U_{\mathrm{M-O}}(r)$ and $U_{\mathrm{M-H}}(r)$ are pairwise interactions between the metal and the O and H atoms in the $\mathrm{H}_2\mathrm{O}$ molecule. Unfortunately, the Spohr model does not reproduce the correct geometry of water at the metal surface. Density Functional Theory calculations have shown that the correct water orientation is given by the top panel in Figure \ref{fig:images_mnm_model_h2o} with the principle water-metal interaction provided by the Highest Occupied Molecular Orbital ($1b_1$) which is mainly $p_{z^2}$ in character as illustrated by Figure \ref{fig:images_water_1b1}.\cite{Meng:2004p151,Meng:2003p289} According to these calculations, water preferentially adopts a nearly parallel orientation to the metal surface almost directly above the atop site in the metal lattice. |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=5in]{images/mnm_model_h2o.pdf} |
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\caption{Orientation of water molecules relative to Au surface. The correct orientation as calculated by DFT(Top) and the classical Spohr model (Bottom).} |
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\label{fig:images_mnm_model_h2o} |
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\end{figure} |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=5in]{images/water_1b1.pdf} |
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\caption{The $1b_1$ HOMO in water as calculated by Gaussian 03 using the B3LYP/6-311+G(d,p) level of theory.} |
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\label{fig:images_water_1b1} |
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\end{figure} |
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A computationally efficient water model has been developed by Ichiye |
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\emph{et al.}\cite{liu96:new_model} is the Soft Sticky Dipole (SSD) model and is a modified form of the hard-sphere model proposed by Bratko, Blum, and |
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Luzar.\cite{Bratko85,Bratko95} This model has been re-parametrized by Fennell \emph{et al.} as the extended Soft Sticky Dipole (SSD/E) model to give more a more accurate description of the liquid state properties of water at 300 K.\cite{fennell04} It consists of a single point dipole with a Lennard-Jones core and a sticky potential that directs the particles to assume the proper hydrogen bond orientation in the first solvation shell. One could imagine further modifying the SSD/E model to include orientational bonding to metal surfaces. This model is centered in the body frame of the water molecule relative to the metal center as is illustrated in Figure \ref{fig:images_mnm_diagram}. |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=4in]{images/mnm_diagram.pdf} |
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\caption{Coordinate reference frame for the orientational water-metal model. } |
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\label{fig:images_mnm_diagram} |
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\end{figure} |
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The general functional form for this potential is a Morse potential, $U_{\mathrm{Morse}}$ as in the Spohr model that is modulated by an orientational function, $f(\theta,\phi)$ consisting of mostly $p_{z^2}$ character with a small amount of mixing of $1a_1$ and $s$ orbital character. This potential is then described by |
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\begin{subeqnarray} |
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\label{con:eq:mnm} |
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U(r,\theta,\phi) &=& U_{\mathrm{Morse}} \cdot f(\theta,\phi)\\ |
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U_{\mathrm{Morse}} &=& D_0\left[\exp(-2\beta_0(r-r_{0}))-2\exp(-\beta_0(r-r_{0}))\right] \\ |
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f(\theta,\phi) &=& c_{a1}(1-\cos\theta)^2 + c_{b1}\sin^2\theta\sin^2\phi +c_{s}. |
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\end{subeqnarray} |
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This water model is being fit to single water-metal DFT calculations performed in Jaguar using the PWP91 DFT model with the LACV3P**++ basis set including a relativistic effective core potential for the heavy atoms. This model includes a Slater local functional,\cite{Slater} and the Perdew-Wang 1991 gradient correction for |
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exchange,\cite{Perdew1991,PERDEW:1992xi} and the Perdew-Wang 1991 |
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(GGA-II) local and nonlocal functionals for correlation.\cite{Perdew1991,PERDEW:1992xi} The basis set we are using is (LACV3P**++)\cite{HAY:1985xt,LACV3P,MCLEAN:1980xi,KRISHNAN:1980aw,CLARK:1983sb,FRISCH:1984dp} |
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Counterpoise corrections were made using standard methods for the basis set superposition error. |
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Many-body surface effects are included in the functional fit by single water and metal surface calculations performed in VASP using using DFT with the periodic supercell plane-wave basis approach.\cite{Kresse:1996zm} In this DFT method, electron cores are described with the projector augmented-wave (PAW) |
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method,\cite{PhysRevB.50.17953,PhysRevB.59.1758} with plane waves |
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included to an energy cutoff of 400 eV. Electronic energies are |
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computed with the PW91 implementation of the generalized gradient |
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approximation (GGA).\cite{PhysRevB.45.13244,PhysRevB.46.6671} Ionic |
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relaxations are performed until the energy difference between |
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subsequent steps was less than 0.0001 eV. For these cluster calculations, gold (111) supercells were constructed consisting of 4 Au x 4 Au surface planes and separated from their vertical images by six layers of vacuum. Atoms that reside in the bottom-most layer of the supercell are fixed according to bulk conditions. The remaining layers are allow to relax under the DFT potential. These supercell calculations were performed nonspin-polarized, and energies were converged to within 0.03 meV per Au atom with a 6 x 6 x 1 Monkhorst-Pack\cite{PhysRevB.13.5188} {\bf k}-point sampling of the |
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first Brillouin zone. Fits of parameters and validation for this model are on-going at the time of this dissertation. |
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\section{Conclusions} |
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This dissertation has explored many unique aspects related to the dynamics in metallic systems that no longer behave as bulk materials. Chapter \ref{chap:metalglass} explored the relaxtion and dynamics in a known glass forming system composed of copper and silver alloy. The cage correlation function was used to obtain hopping times that can then be explored using the {\sc ctrw} model for dispersive transport. It was showed that the {\sc ctrw} approach to dispersive transport gives substantially better agreement than Zwanzig's model which is based on interrupted harmonic motion in the inherent structures. The limiting factor in the {\sc ctrw} model was a need to calculate the average hopping distance $\sigma_{0}$ and the uncertainty associated with that distance do to poor statistics. In the low-temperature supercooled liquid, there are substantial deviations from the $\gamma=1$ limit of the theory. |
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Below 500 K for the copper-silver system, the mean square displacement cannot be fit well to a linear function in time and the cage correlation function no longer has a simple exponential decay. Instead, the long-time decay of the cage correlation function has been fit with the more familiar Kohlrausch-Williams-Watts law. There appears to be a relatively simple relationship between the stretching parameter in the fit to the cage correlation function and the value of $\gamma$ from the {\sc ctrw} model. One hopes that there is some more fundamental relationship between {\sc kww}-like decay and a {\sc ctrw}-analysis of the propagator for dispersive transport that can be derived. |
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In Chapter \ref{chap:nanodiffusion} it was shown presence of vacancies at the bimetallic interface provides a reasonable explanation for the spontaneous alloying observed by optical spectroscopy and {\sc XAFS}. Molecular Dynamics calculations show that only a few, (5-6\%), vacancies at the the interface are necessary to produce the alloying in the time scale observed experimentally. The presence of vacancies at the interface also provides a rationalization for why the alloying process terminates after a short period. Mechanistically, the alloying process can be viewed as a competition between percolation of the defects to the outer surface and diffusion of the metal atoms into the vacancies. Once a vacancy reaches the NP surface it is essentially lost since migration back into the lattice is prohibitively slow. It is expected that the preferred directionally of vacancy migration is toward the surface since there is a larger volume fraction available for the vacancy to explore and there is a smaller curvature in the outer layers. Additionally, due to a smaller number of nearest neighbors at the surface, there is a smaller energy barrier to generate a vacancy at the surface relative to the interior of the NP. |
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The molecular Dynamics simulations presented in Chapter \ref{chap:bulkmod} mimic laser-excitation and cooling experiments and explore relatively fast dynamics in gold nanoparticles. From these simulations, we find that there is both a size and temperature-dependent softening of these nanoparticles even at temperatures below the melting temperature. In plots of the Bulk Modulus ($\kappa$) for the various size nanoparticles studied as a function of final temperature after excitation, we can see a dramatic (and size-dependent) drop in $\kappa$ at temperature well below the melting temperature of bulk polycrystalline gold. (Polycrystalline gold has a bulk modulus of 220 GPa at a temperature of 300 K.) Surface melting occurs at even lower temperatures as indicated by our calculations of the radial-density profile which shows a merging of the crystalline peaks in the outer layer of the nanoparticle much like that seen in the Ag-Au nanoparticle with vacancies. The time-dependent estimate for the bulk modulus indicates a time scale for softening of about 10 ps independent of particle size. Correspondingly, the smaller particles exhibit coherent breathing vibrations for a few periods before melting and larger particles melt within the first vibrational period. |
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Lastly, it was shows that it is feasible to create glassy nanobeads using laser-excitation experiments similar to those presented in Chapter \ref{chap:bulkmod}. Using estimates of the interfacial heat transfer coefficient, $G$, a model for the cooling of a nanoparticle embedded in a solvent has been constructed. Cooling rates have been obtained for laser-heated nanoparticles that are in excess of 10$^{13}$ K / s which are more that sufficient to form glassy structures. From the local icosahedral ordering around the atoms in the nanoparticles (particularly Copper atoms), we deduce that it is likely that glassy nanobeads are created via laser heating of bimetallic nanoparticles, particularly when the initial temperature of the particles approaches the melting temperature of the bulk metal alloy. |
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