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\input{header.tex} |
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\section{Simulation Method} \label{sec:details} Several different potential models have been developed that reasonably describe interactions in transition metals. In particular, the Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study a wide range of phenomena in both bulk materials and nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r} Both potentials are based on a model of a metal which treats the nuclei and core electrons as pseudo-atoms embedded in the electron density due to the valence electrons on all of the other atoms in the system. The {\sc sc} potential has a simple form that closely resembles that of the ubiquitous Lennard Jones potential, |
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\begin{equation} |
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\label{eq:SCP1} U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
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\end{equation} |
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where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
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\begin{equation} |
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\label{eq:SCP2} V^{pair}_{ij}(r)=\left( \frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( \frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}}. |
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\end{equation} |
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$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for interactions of the pseudo-atom cores. The $\sqrt{\rho_i}$ term in Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models the interactions between the valence electrons and the cores of the pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy scale, $c_i$ scales the attractive portion of the potential relative to the repulsive interaction and $\alpha_{ij}$ is a length parameter that assures a dimensionless form for V and $\rho$. These parameters are tuned to various experimental properties such as density, cohesive energy, elastic moduli for FCC transition metals. The quantum Sutton-Chen ({\sc q-sc}) formulation matches these properties while including zero-point quantum corrections for different transition metals.\cite{PhysRevB.59.3527} This particular parametarization has been shown to reproduce the experimentally available heat of mixing data for both FCC solid solutions of Ag-Cu and the high-temperature liquid.\cite{sheng:184203} Alternatively, the {\sc eam} potential does not reproduce the experimentally observed heat of mixing for the liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were taken to be the arithmatic average of the atomic parameters with the exception of $c_i$ since its values is only dependent on the identity of the atom where the density is evaluated. Cutoff distances are defined by this potential to be $2\alpha_{ij}$ and include up to the sixth coordination shell in FCC metals. |
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Cooling rates for the various sized nanoparticles were calculated from the heat transfer equations for a spherical particle embedded in a ambient medium that allows for diffusive heat transport. These equate to a set of coupled differential equations in the Laplace domain and are given by |
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\begin{eqnarray} |
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Mc_{P}\cdot(s\cdot T_{p}(s)-T_{0})+4\pi R^{2} G\cdot(T_{p}(s)-T_{f}(r=R,s)=0\\ |
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\left(\frac{ |
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\partial}{ |
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\partial r} T_{f}(r,s)\right)_{r=R} + \frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0 \label{eq:heateqn} |
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\end{eqnarray} |
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where s is the time-conjugated variable in Laplace space. The variables in these equations describe a nanoparticle of radius $R$, mass $M$, and specific heat $c_{p}$ at an initial temperature $T_0$. The surrounding solvent at a temperature $T_f(r,t)$ has a thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$ is the interface conductance between the nanoparticle and the surrounding solvent. The temperature of the nanoparticle as a function of time is then obtained by the inverse Laplace transformation |
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\begin{equation} |
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T_{p}(t)=\frac{2 k R^2 g^2 T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2 t/R^2)u^2}{(u^2(1+Rg)-kRg)^2+(u^3-kRgu)^2}\mathrm{d}u. |
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\end{equation} |
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For clarity, we introduce the diffusivity $\kappa = K/(pc)$, and define $k=4\pi R^3 pc /(Mc_p)$ and $g = G/K$. This equation is solved numerically for the Ag-Cu system using the mole-fraction weighted values for $c_p$ and $\rho$ of 0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g m^{-3}})$ respectively. Parameters used for the fluid are $K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ $(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. Interfacial conductance have been determined by experiment and range from a low $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ in the high end of the heat transfer regime. We have chosen to conduct simulations at both low and high values of G to present both the slowest and fastest heat transfer from the nanoparticle to the solvent consistent with experimental observation. For the slowest heat transfer, a value for G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for the fastest heat transfer, a value of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used. We believe that the that the true G values are closer to the faster regime then the slower regime. |
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To approximate laser heating-cooling experiments, we utilized a methodology in which atoms contained in the outer $4$ {\AA} radius of the nanoparticle evolve under Langevin Dynamics approximating the solvent-nanoparticle interaction and atoms outside of this region evolve under Newtonian Dynamics. For a spherical particle of radius $a$, the Langevin frictional force can be determined by Stokes' law |
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\begin{equation} |
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\mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v} |
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\end{equation} |
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where $\eta$ is the viscosity of the solvent in which the particle is embedded. The rate of cooling for the nanoparticle can be tuned by changing the solvent viscosity until the nanoparticle cooling rate matches that of the cooling rate as described by the heat-transfer equations (\ref{eq:heateqn}). The effective solvent viscosity for a G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17 for the 20 {\AA}, 0.20 for the 30 {\AA} and 0.22 for the 40 {\AA} particle. The effective solvent viscosity for a G of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23 for the 20 {\AA}, 0.29 for the 30 {\AA} and 0.30 for the 40 {\AA} particle. Cooling traces for each particle size are presented in Figure[\ref{fig:images_cooling_plot}]. It should be noted that the Langevin thermostat produces cooling that is consistent with Newtonian cooling in comparison with |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=3in]{images/cooling_plot.pdf} |
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\caption{} |
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\label{fig:images_cooling_plot} |
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\end{figure} |
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Core-shell (Cu core) and random initial structures were constructed on an underlying FCC lattice (4.09 {\AA}) at the bulk eutectic composition $\mathrm{Ag}_6\mathrm{Cu}_4$. Three different sizes corresponding to 20 \AA (1961 atoms), 30 {\AA} (6603 atoms) and 40 {\AA} (1683 atoms) in radius. Only the random initial structure was studied for the 40 {\AA} nanoparticle. These initial structures were relaxed to their equilibrium structures at 20 K for 20 ps and again at 300 K for 100 ps sampling from a Maxwell-Boltzmann distribution at each temperature. |
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