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%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex |
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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 (2382 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 |
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temperature. All simulations were conducted using the {\sc oopse} |
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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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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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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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Eq. (\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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\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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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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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{\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-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 |
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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 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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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 $130\times 10^{6} |
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(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Wilson {\it |
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et al.} worked with Au, Pt, and AuPd nanoparticles and obtained an |
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estimate for the interfacial conductance of $G=130 |
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(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Similarly, Plech {\it |
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et al.} reported a value for the interfacial conductance of $G=105\pm |
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15 (\mathrm{Wm^{-2}K^{-1}})$ for Au nanoparticles.\cite{plech:195423} |
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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,\cite{HuM._jp020581+} the true G values |
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are probably in the faster regime: $117\times 10^{6}$ |
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$(\mathrm{Wm^{-2}K^{-1}})$. |
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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 Eq. |
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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. These viscosities are |
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essentially gas-phase values, a fact which is consistent with the |
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initial temperatures of the particles being well into the |
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super-critical region for the aqueous environment. Gas bubble |
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generation has also been seen experimentally around gold nanoparticles |
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in water.\cite{kotaidis:184702} Instead of a single value for the |
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effective viscosity, a time-dependent parameter might be a better |
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mimic of the cooling vapor layer that surrounds the hot particles. |
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This may also be a contributing factor to the size-dependence of the |
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effective viscosities in our simulations. |
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Cooling traces for 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[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 |
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describe interactions in transition metals. In particular, the |
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Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and |
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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 |
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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 |
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core electrons as pseudo-atoms embedded in the electron density due to |
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the valence electrons on all of the other atoms in the system. The |
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{\sc sc} potential has a simple form that closely resembles that of |
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the ubiquitous Lennard Jones potential, |
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\begin{equation} |
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\label{eq:SCP1} |
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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 |
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interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
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Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
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the interactions between the valence electrons and the cores of the |
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pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
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scale, $c_i$ scales the attractive portion of the potential relative |
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to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
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that assures a dimensionless form for $\rho$. These parameters are |
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tuned to various experimental properties such as the density, cohesive |
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energy, and elastic moduli for FCC transition metals. The quantum |
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Sutton-Chen ({\sc q-sc}) formulation matches these properties while |
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including zero-point quantum corrections for different transition |
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metals.\cite{PhysRevB.59.3527} This particular parametarization has |
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been shown to reproduce the experimentally available heat of mixing |
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data for both FCC solid solutions of Ag-Cu and the high-temperature |
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liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does |
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not reproduce the experimentally observed heat of mixing for the |
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liquid alloy.\cite{MURRAY:1984lr} In this work, we have utilized the |
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{\sc q-sc} formulation for our potential energies and forces. |
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Combination rules for the alloy were taken to be the arithmetic |
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average of the atomic parameters with the exception of $c_i$ since its |
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values is only dependent on the identity of the atom where the density |
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is evaluated. For the {\sc q-sc} potential, cutoff distances are |
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traditionally taken to be $2\alpha_{ij}$ and include up to the sixth |
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coordination shell in FCC metals. |
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|
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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. Fig. |
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\ref{fig:images_cooling_time_traces} illustrates this sampling. |
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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[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} |
