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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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%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex |
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|
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\section{Computational Methodology} |
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\label{sec:details} |
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|
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\subsection{Initial Geometries and Heating} |
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|
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Cu-core / Ag-shell and random alloy structures were constructed on an |
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underlying FCC lattice (4.09 {\AA}) at the bulk eutectic composition |
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$\mathrm{Ag}_6\mathrm{Cu}_4$. Both initial geometries were considered |
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although experimental results suggest that the random structure is the |
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most likely structure to be found following |
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synthesis.\cite{Jiang:2005lr,gonzalo:5163} Three different sizes of |
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nanoparticles corresponding to a 20 \AA radius (1961 atoms), 30 {\AA} |
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radius (6603 atoms) and 40 {\AA} radius (15683 atoms) were |
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constructed. These initial structures were relaxed to their |
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equilibrium structures at 20 K for 20 ps and again at 300 K for 100 ps |
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sampling from a Maxwell-Boltzmann distribution at each temperature. All simulations were conducted using the {\sc OOPSE} molecular dynamics package.\cite{Meineke:2004uq} |
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|
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To mimic the effects of the heating due to laser irradiation, the |
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particles were allowed to melt by sampling velocities from the Maxwell |
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Boltzmann distribution at a temperature of 900 K. The particles were |
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run under microcanonical simulation conditions for 1 ns of simualtion |
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time prior to studying the effects of heat transfer to the solvent. |
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In all cases, center of mass translational and rotational motion of |
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the particles were set to zero before any data collection was |
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undertaken. Structural features (pair distribution functions) were |
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used to verify that the particles were indeed liquid droplets before |
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cooling simulations took place. |
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|
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\subsection{Modeling random alloy and core shell particles in solution |
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phase environments} |
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|
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To approximate the effects of rapid heat transfer to the solvent |
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following a heating at the plasmon resonance, we utilized a |
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methodology in which atoms contained in the outer $4$ {\AA} radius of |
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the nanoparticle evolved under Langevin Dynamics, |
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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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m \frac{\partial^2 \vec{x}}{\partial t^2} = F_\textrm{sys}(\vec{x}(t)) |
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- 6 \pi a \eta \vec{v}(t) + F_\textrm{ran} |
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\label{eq:langevin} |
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\end{equation} |
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where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
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with a solvent friction ($\eta$) approximating the contribution from |
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the solvent and capping agent. Atoms located in the interior of the |
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nanoparticle evolved under Newtonian dynamics. The set-up of our |
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simulations is nearly identical with the ``stochastic boundary |
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molecular dynamics'' ({\sc sbmd}) method that has seen wide use in the |
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protein simulation |
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community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch |
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of this setup can be found in Fig. \ref{fig:langevinSketch}. In |
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equation \ref{eq:langevin} the frictional forces of a spherical atom |
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of radius $a$ depend on the solvent viscosity. The random forces are |
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usually taken as gaussian random variables with zero mean and a |
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variance tied to the solvent viscosity and temperature, |
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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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\langle F_\textrm{ran}(t) \cdot F_\textrm{ran} (t') |
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\rangle = 2 k_B T (6 \pi \eta a) \delta(t - t') |
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\label{eq:stochastic} |
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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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Due to the presence of the capping agent and the lack of details about |
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the atomic-scale interactions between the metallic atoms and the |
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solvent, the effective viscosity is a essentially a free parameter |
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that must be tuned to give experimentally relevant simulations. |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=5in]{images/stochbound.pdf} |
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\caption{Methodology used to mimic the experimental cooling conditions |
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of a hot nanoparticle surrounded by a solvent. Atoms in the core of |
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the particle evolved under Newtonian dynamics, while atoms that were |
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in the outer skin of the particle evolved under Langevin dynamics. |
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The radius of the spherical region operating under Newtonian dynamics, |
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$r_\textrm{Newton}$ was set to be 4 {\AA} smaller than the original |
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radius ($R$) of the liquid droplet.} |
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\label{fig:langevinSketch} |
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\end{figure} |
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|
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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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The viscosity ($\eta$) can be tuned by comparing the cooling rate that |
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a set of nanoparticles experience with the known cooling rates for |
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similar particles obtained via the laser heating experiments. |
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Essentially, we tune the solvent viscosity until the thermal decay |
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profile matches a heat-transfer model using reasonable values for the |
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interfacial conductance and the thermal conductivity of the solvent. |
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|
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Cooling rates for the experimentally-observed nanoparticles were |
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calculated from the heat transfer equations for a spherical particle |
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embedded in a ambient medium that allows for diffusive heat transport. |
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Following Plech {\it et al.},\cite{plech:195423} we use a heat |
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transfer model that consists of two coupled differential equations |
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in the Laplace domain, |
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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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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{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} + |
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\frac{G}{K}(T_{p}(s)-T_{f}(r,s) = 0 |
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\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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where $s$ is the time-conjugate variable in Laplace space. The |
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variables in these equations describe a nanoparticle of radius $R$, |
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mass $M$, and specific heat $c_{p}$ at an initial temperature |
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$T_0$. The surrounding solvent has a thermal profile $T_f(r,t)$, |
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thermal conductivity $K$, density $\rho$, and specific heat $c$. $G$ |
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is the interfacial conductance between the nanoparticle and the |
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surrounding solvent, and contains information about heat transfer to |
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the capping agent as well as the direct metal-to-solvent heat loss. |
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The temperature of the nanoparticle as a function of time can then |
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obtained by the inverse Laplace transform, |
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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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T_{p}(t)=\frac{2 k R^2 g^2 |
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T_0}{\pi}\int_{0}^{\infty}\frac{\exp(-\kappa u^2 |
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t/R^2)u^2}{(u^2(1 + R g) - k R g)^2+(u^3 - k R g u)^2}\mathrm{d}u. |
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\label{eq:laplacetransform} |
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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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For simplicity, we have introduced the thermal diffusivity $\kappa = |
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K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in |
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Eq. \ref{eq:laplacetransform}. |
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|
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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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Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu |
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system using mole-fraction weighted values for $c_p$ and $\rho_p$ of |
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0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g |
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m^{-3}})$ respectively. Since most of the laser excitation experiments |
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have been done in aqueous solutions, parameters used for the fluid are |
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$K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ |
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$(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. |
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|
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Values for the interfacial conductance have been determined by a |
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number of groups for similar nanoparticles and range from a low |
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$87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$ |
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$(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Similarly, Plech |
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{\it et al.} reported a value for the interfacial conductance of |
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$G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and $G=130\pm 15$ |
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$(\mathrm{Wm^{-2}K^{-1}})$ for Pt |
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nanoparticles.\cite{plech:195423,PhysRevB.66.224301} |
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|
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We conducted our simulations at both ends of the range of |
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experimentally-determined values for the interfacial conductance. |
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This allows us to observe both the slowest and fastest heat transfers |
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from the nanoparticle to the solvent that are consistent with |
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experimental observations. For the slowest heat transfer, a value for |
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G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for |
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the fastest heat transfer, a value of $117\times 10^{6}$ |
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$(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have |
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done using raw data from the Hartland group's thermal half-time |
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experiments on Au nanospheres, the true G values are probably in the |
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faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$. |
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|
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The rate of cooling for the nanoparticles in a molecular dynamics |
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simulation can then be tuned by changing the effective solvent |
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viscosity ($\eta$) until the nanoparticle cooling rate matches the |
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cooling rate described by the heat-transfer equations |
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(\ref{eq:heateqn}). The effective solvent viscosity (in Pa s) for a G |
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of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $4.2 \times |
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10^{-6}$, $5.0 \times 10^{-6}$, and |
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$5.5 \times 10^{-6}$ for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The |
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effective solvent viscosity (again in Pa s) for an interfacial |
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conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $5.7 |
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\times 10^{-6}$, $7.2 \times 10^{-6}$, and $7.5 \times 10^{-6}$ |
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for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling traces for |
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each particle size are presented in |
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Fig. \ref{fig:images_cooling_plot}. It should be noted that the |
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Langevin thermostat produces cooling curves that are consistent with |
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Newtonian (single-exponential) cooling, which cannot match the cooling |
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profiles from Eq. \ref{eq:laplacetransform} exactly. Fitting the |
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Langevin cooling profiles to a single-exponential produces |
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$\tau=25.576$ ps, $\tau=43.786$ ps, and $\tau=56.621$ ps for the 20, |
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30 and 40 {\AA} nanoparticles and a G of $87.5\times 10^{6}$ |
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$(\mathrm{Wm^{-2}K^{-1}})$. For comparison's sake, similar |
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single-exponential fits with an interfacial conductance of G of |
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$117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ produced a $\tau=13.391$ |
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ps, $\tau=30.426$ ps, $\tau=43.857$ ps for the 20, 30 and 40 {\AA} |
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nanoparticles. |
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|
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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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\centering |
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\includegraphics[width=5in]{images/cooling_plot.pdf} |
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\caption{Thermal cooling curves obtained from the inverse Laplace |
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transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as |
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well as from molecular dynamics simulations (circles). Effective |
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solvent viscosities of 4.2-7.5 $\times 10^{-6}$ Pa s (depending on the |
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radius of the particle) give the best fit to the experimental cooling |
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curves. This viscosity suggests that the nanoparticles in these |
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experiments are surrounded by a vapor layer (which is a reasonable |
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assumptions given the initial temperatures of the particles). } |
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\label{fig:images_cooling_plot} |
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\end{figure} |
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|
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\subsection{Potentials for classical simulations of bimetallic |
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nanoparticles} |
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|
|
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Several different potential models have been developed that reasonably |
| 189 |
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describe interactions in transition metals. In particular, the |
| 190 |
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Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and |
| 191 |
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Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study |
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a wide range of phenomena in both bulk materials and |
| 193 |
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nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} Both |
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potentials are based on a model of a metal which treats the nuclei and |
| 195 |
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core electrons as pseudo-atoms embedded in the electron density due to |
| 196 |
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the valence electrons on all of the other atoms in the system. The |
| 197 |
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{\sc sc} potential has a simple form that closely resembles that of |
| 198 |
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the ubiquitous Lennard Jones potential, |
| 199 |
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\begin{equation} |
| 200 |
+ |
\label{eq:SCP1} |
| 201 |
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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} |
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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 |
| 209 |
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interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 210 |
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Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 211 |
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the interactions between the valence electrons and the cores of the |
| 212 |
+ |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 213 |
+ |
scale, $c_i$ scales the attractive portion of the potential relative |
| 214 |
+ |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 215 |
+ |
that assures a dimensionless form for $\rho$. These parameters are |
| 216 |
+ |
tuned to various experimental properties such as the density, cohesive |
| 217 |
+ |
energy, and elastic moduli for FCC transition metals. The quantum |
| 218 |
+ |
Sutton-Chen ({\sc q-sc}) formulation matches these properties while |
| 219 |
+ |
including zero-point quantum corrections for different transition |
| 220 |
+ |
metals.\cite{PhysRevB.59.3527} This particular parametarization has |
| 221 |
+ |
been shown to reproduce the experimentally available heat of mixing |
| 222 |
+ |
data for both FCC solid solutions of Ag-Cu and the high-temperature |
| 223 |
+ |
liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does |
| 224 |
+ |
not reproduce the experimentally observed heat of mixing for the |
| 225 |
+ |
liquid alloy.\cite{MURRAY:1984lr} Combination rules for the alloy were |
| 226 |
+ |
taken to be the arithmatic average of the atomic parameters with the |
| 227 |
+ |
exception of $c_i$ since its values is only dependent on the identity |
| 228 |
+ |
of the atom where the density is evaluated. For the {\sc q-sc} |
| 229 |
+ |
potential, cutoff distances are traditionally taken to be |
| 230 |
+ |
$2\alpha_{ij}$ and include up to the sixth coordination shell in FCC |
| 231 |
+ |
metals. |
| 232 |
|
|
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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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%\subsection{Sampling single-temperature configurations from a cooling |
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%trajectory} |
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|
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To better understand the structural changes occurring in the |
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nanoparticles throughout the cooling trajectory, configurations were |
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sampled at regular intervals during the cooling trajectory. These |
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configurations were then allowed to evolve under NVE dynamics to |
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sample from the proper distribution in phase space. Figure |
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\ref{fig:images_cooling_time_traces} illustrates this sampling. |
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< |
\input{footer.tex} |
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|
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=3in]{images/cooling_time_traces.pdf} |
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\caption{Illustrative cooling profile for the 40 {\AA} |
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nanoparticle evolving under stochastic boundary conditions |
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corresponding to $G=$$87.5\times 10^{6}$ |
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$(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling |
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trajectory, configurations were sampled and allowed to evolve in the |
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NVE ensemble. These subsequent trajectories were analyzed for |
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structural features associated with bulk glass formation.} |
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\label{fig:images_cooling_time_traces} |
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\end{figure} |
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|
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|
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[width=5in]{images/cross_section_array.jpg} |
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\caption{Cutaway views of 30 \AA\ Ag-Cu nanoparticle structures for |
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random alloy (top) and Cu (core) / Ag (shell) initial conditions |
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(bottom). Shown from left to right are the crystalline, liquid |
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droplet, and final glassy bead configurations.} |
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\label{fig:cross_sections} |
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\end{figure} |
