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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 a variety of experimental techniques 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 $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ for the value of G to present the slowest heat transfer from the nanoparticle to the solvent consistent with experimental observation. |
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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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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}). |
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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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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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\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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