--- trunk/nanoglass/experimental.tex 2007/09/11 15:23:24 3222 +++ trunk/nanoglass/experimental.tex 2007/10/15 21:02:43 3261 @@ -1,3 +1,5 @@ +%!TEX root = /Users/charles/Desktop/nanoglass/nanoglass.tex + \section{Computational Methodology} \label{sec:details} @@ -5,12 +7,17 @@ $\mathrm{Ag}_6\mathrm{Cu}_4$. Three different sizes o Cu-core / Ag-shell and random alloy 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 of nanoparticles -corresponding to a 20 \AA radius (1961 atoms), 30 {\AA} radius (6603 -atoms) and 40 {\AA} radius (15683 atoms) were constructed. 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. +$\mathrm{Ag}_6\mathrm{Cu}_4$. Both initial geometries were considered +although experimental results suggest that the random structure is the +most likely structure to be found following +synthesis.\cite{Jiang:2005lr,gonzalo:5163} Three different sizes of +nanoparticles corresponding to a 20 \AA radius (1961 atoms), 30 {\AA} +radius (6603 atoms) and 40 {\AA} radius (15683 atoms) were +constructed. 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. All simulations were conducted using the {\sc oopse} +molecular dynamics package.\cite{Meineke:2004uq} To mimic the effects of the heating due to laser irradiation, the particles were allowed to melt by sampling velocities from the Maxwell @@ -29,43 +36,59 @@ the nanoparticle evolved under Langevin Dynamics with To approximate the effects of rapid heat transfer to the solvent following a heating at the plasmon resonance, we utilized a methodology in which atoms contained in the outer $4$ {\AA} radius of -the nanoparticle evolved under Langevin Dynamics with a solvent -friction approximating the contribution from the solvent and capping -agent. Atoms located in the interior of the nanoparticle evolved -under Newtonian dynamics. The set-up of our simulations is nearly -identical with the ``stochastic boundary molecular dynamics'' ({\sc -sbmd}) method that has seen wide use in the protein simulation +the nanoparticle evolved under Langevin Dynamics, +\begin{equation} +m \frac{\partial^2 \vec{x}}{\partial t^2} = F_\textrm{sys}(\vec{x}(t)) +- 6 \pi a \eta \vec{v}(t) + F_\textrm{ran} +\label{eq:langevin} +\end{equation} +with a solvent friction ($\eta$) approximating the contribution from +the solvent and capping agent. Atoms located in the interior of the +nanoparticle evolved under Newtonian dynamics. The set-up of our +simulations is nearly identical with the ``stochastic boundary +molecular dynamics'' ({\sc sbmd}) method that has seen wide use in the +protein simulation community.\cite{BROOKS:1985kx,BROOKS:1983uq,BRUNGER:1984fj} A sketch -of this setup can be found in Fig. \ref{fig:langevinSketch}. For a -spherical atom of radius $a$, the Langevin frictional forces can be -determined by Stokes' law -\begin{equation} -\mathbf{F}_{\mathrm{frictional}}=6\pi a \eta \mathbf{v} +of this setup can be found in Fig. \ref{fig:langevinSketch}. In +Eq. (\ref{eq:langevin}) the frictional forces of a spherical atom +of radius $a$ depend on the solvent viscosity. The random forces are +usually taken as gaussian random variables with zero mean and a +variance tied to the solvent viscosity and temperature, +\begin{equation} +\langle F_\textrm{ran}(t) \cdot F_\textrm{ran} (t') +\rangle = 2 k_B T (6 \pi \eta a) \delta(t - t') +\label{eq:stochastic} \end{equation} -where $\eta$ is the effective viscosity of the solvent in which the -particle is embedded. Due to the presence of the capping agent and -the lack of details about the atomic-scale interactions between the -metallic atoms and the solvent, the effective viscosity is a -essentially a free parameter that must be tuned to give experimentally -relevant simulations. +Due to the presence of the capping agent and the lack of details about +the atomic-scale interactions between the metallic atoms and the +solvent, the effective viscosity is a essentially a free parameter +that must be tuned to give experimentally relevant simulations. \begin{figure}[htbp] \centering -\includegraphics[width=\linewidth]{images/stochbound.pdf} -\caption{Methodology for nanoparticle cooling. Equations of motion for metal atoms contained in the outer 4 {\AA} were determined by Langevins' Equations of motion. Metal atoms outside this region were allowed to evolve under Newtonian dynamics.} +\includegraphics[width=5in]{images/stochbound.pdf} +\caption{Methodology used to mimic the experimental cooling conditions +of a hot nanoparticle surrounded by a solvent. Atoms in the core of +the particle evolved under Newtonian dynamics, while atoms that were +in the outer skin of the particle evolved under Langevin dynamics. +The radius of the spherical region operating under Newtonian dynamics, +$r_\textrm{Newton}$ was set to be 4 {\AA} smaller than the original +radius ($R$) of the liquid droplet.} \label{fig:langevinSketch} \end{figure} + The viscosity ($\eta$) can be tuned by comparing the cooling rate that a set of nanoparticles experience with the known cooling rates for -those particles obtained via the laser heating experiments. +similar particles obtained via the laser heating experiments. Essentially, we tune the solvent viscosity until the thermal decay profile matches a heat-transfer model using reasonable values for the interfacial conductance and the thermal conductivity of the solvent. Cooling rates for the experimentally-observed nanoparticles were calculated from the heat transfer equations for a spherical particle -embedded in a ambient medium that allows for diffusive heat -transport. The heat transfer model is a set of two coupled -differential equations in the Laplace domain, +embedded in a ambient medium that allows for diffusive heat transport. +Following Plech {\it et al.},\cite{plech:195423} we use a heat +transfer model that consists of two coupled differential equations +in the Laplace domain, \begin{eqnarray} 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\\ \left(\frac{\partial}{\partial r} T_{f}(r,s)\right)_{r=R} + @@ -90,21 +113,25 @@ Eq. \ref{eq:laplacetransform}. \end{equation} For simplicity, we have introduced the thermal diffusivity $\kappa = K/(\rho c)$, and defined $k=4\pi R^3 \rho c /(M c_p)$ and $g = G/K$ in -Eq. \ref{eq:laplacetransform}. +Eq. (\ref{eq:laplacetransform}). -Eq. \ref{eq:laplacetransform} was solved numerically for the Ag-Cu +Eq. (\ref{eq:laplacetransform}) was solved numerically for the Ag-Cu system using mole-fraction weighted values for $c_p$ and $\rho_p$ of 0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g m^{-3}})$ respectively. Since most of the laser excitation experiments have been done in aqueous solutions, 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 $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. +$K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ +$(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. Values for the interfacial conductance have been determined by a number of groups for similar nanoparticles and range from a low -$87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ to $120\times 10^{6}$ -$(\mathrm{Wm^{-2}K^{-1}})$.\cite{hartlandPrv2007} Plech {\it et al.} reported a value for the interfacial conductance of $G=105\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ and -$G=130\pm 15$ $(\mathrm{Wm^{-2}K^{-1}})$ for Pt nanoparticles.\cite{plech:195423,PhysRevB.66.224301} +$87.5\times 10^{6} (\mathrm{Wm^{-2}K^{-1}})$ to $130\times 10^{6} +(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Wilson {\it +et al.} worked with Au, Pt, and AuPd nanoparticles and obtained an +estimate for the interfacial conductance of $G=130 +(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Similarly, Plech {\it +et al.} reported a value for the interfacial conductance of $G=105\pm +15 (\mathrm{Wm^{-2}K^{-1}})$ for Au nanoparticles.\cite{plech:195423} We conducted our simulations at both ends of the range of experimentally-determined values for the interfacial conductance. @@ -115,37 +142,57 @@ experiments on Au nanospheres, we believe that the tru the fastest heat transfer, a value of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have done using raw data from the Hartland group's thermal half-time -experiments on Au nanospheres, we believe that the true G values are -closer to the faster regime: $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$. +experiments on Au nanospheres,\cite{HuM._jp020581+} the true G values +are probably in the faster regime: $117\times 10^{6}$ +$(\mathrm{Wm^{-2}K^{-1}})$. - The rate of cooling for the nanoparticles in a molecular dynamics simulation can then be tuned by changing the effective solvent viscosity ($\eta$) until the nanoparticle cooling rate matches the -cooling rate described by the heat-transfer equations -(\ref{eq:heateqn}). The effective solvent viscosity (in poise) for a G -of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.17, 0.20, and -0.22 for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The -effective solvent viscosity (again in poise) for an interfacial -conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is 0.23, -0.29, and 0.30 for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. Cooling -traces for each particle size are presented in +cooling rate described by the heat-transfer Eq. +(\ref{eq:heateqn}). The effective solvent viscosity (in Pa s) for a G +of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $4.2 \times +10^{-6}$, $5.0 \times 10^{-6}$, and +$5.5 \times 10^{-6}$ for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The +effective solvent viscosity (again in Pa s) for an interfacial +conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $5.7 +\times 10^{-6}$, $7.2 \times 10^{-6}$, and $7.5 \times 10^{-6}$ +for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. These viscosities are +essentially gas-phase values, a fact which is consistent with the +initial temperatures of the particles being well into the +super-critical region for the aqueous environment. Gas bubble +generation has also been seen experimentally around gold nanoparticles +in water.\cite{kotaidis:184702} Instead of a single value for the +effective viscosity, a time-dependent parameter might be a better +mimic of the cooling vapor layer that surrounds the hot particles. +This may also be a contributing factor to the size-dependence of the +effective viscosities in our simulations. + +Cooling traces for each particle size are presented in Fig. \ref{fig:images_cooling_plot}. It should be noted that the Langevin thermostat produces cooling curves that are consistent with Newtonian (single-exponential) cooling, which cannot match the cooling -profiles from Eq. \ref{eq:laplacetransform} exactly. +profiles from Eq. (\ref{eq:laplacetransform}) exactly. Fitting the +Langevin cooling profiles to a single-exponential produces +$\tau=25.576$ ps, $\tau=43.786$ ps, and $\tau=56.621$ ps for the 20, +30 and 40 {\AA} nanoparticles and a G of $87.5\times 10^{6}$ +$(\mathrm{Wm^{-2}K^{-1}})$. For comparison's sake, similar +single-exponential fits with an interfacial conductance of G of +$117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ produced a $\tau=13.391$ +ps, $\tau=30.426$ ps, $\tau=43.857$ ps for the 20, 30 and 40 {\AA} +nanoparticles. \begin{figure}[htbp] \centering -\includegraphics[width=\linewidth]{images/cooling_plot.pdf} +\includegraphics[width=5in]{images/cooling_plot.pdf} \caption{Thermal cooling curves obtained from the inverse Laplace -transform heat model in Eq. \ref{eq:laplacetransform} (solid line) as +transform heat model in Eq. (\ref{eq:laplacetransform}) (solid line) as well as from molecular dynamics simulations (circles). Effective -solvent viscosities of 0.23-0.30 poise (depending on the radius of the -particle) give the best fit to the experimental cooling curves. Since -this viscosity is substantially in excess of the viscosity of liquid -water, much of the thermal transfer to the surroundings is probably -due to the capping agent.} +solvent viscosities of 4.2-7.5 $\times 10^{-6}$ Pa s (depending on the +radius of the particle) give the best fit to the experimental cooling +curves. This viscosity suggests that the nanoparticles in these +experiments are surrounded by a vapor layer (which is a reasonable +assumptions given the initial temperatures of the particles). } \label{fig:images_cooling_plot} \end{figure} @@ -157,7 +204,7 @@ nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026 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 +nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} 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 @@ -189,15 +236,46 @@ liquid alloy.\cite{MURRAY:1984lr} Combination rules fo data for both FCC solid solutions of Ag-Cu and the high-temperature liquid.\cite{sheng:184203} In contrast, 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. For the {\sc q-sc} -potential, cutoff distances are traditionally taken to be -$2\alpha_{ij}$ and include up to the sixth coordination shell in FCC -metals. +liquid alloy.\cite{MURRAY:1984lr} In this work, we have utilized the +{\sc q-sc} formulation for our potential energies and forces. +Combination rules for the alloy were taken to be the arithmetic +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. For the {\sc q-sc} potential, cutoff distances are +traditionally taken to be $2\alpha_{ij}$ and include up to the sixth +coordination shell in FCC metals. -\subsection{Sampling single-temperature configurations from a cooling -trajectory} +%\subsection{Sampling single-temperature configurations from a cooling +%trajectory} -ffdsafjdksalfdsa +To better understand the structural changes occurring in the +nanoparticles throughout the cooling trajectory, configurations were +sampled at regular intervals during the cooling trajectory. These +configurations were then allowed to evolve under NVE dynamics to +sample from the proper distribution in phase space. Fig. +\ref{fig:images_cooling_time_traces} illustrates this sampling. + + +\begin{figure}[htbp] + \centering + \includegraphics[height=3in]{images/cooling_time_traces.pdf} + \caption{Illustrative cooling profile for the 40 {\AA} +nanoparticle evolving under stochastic boundary conditions +corresponding to $G=$$87.5\times 10^{6}$ +$(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling +trajectory, configurations were sampled and allowed to evolve in the +NVE ensemble. These subsequent trajectories were analyzed for +structural features associated with bulk glass formation.} + \label{fig:images_cooling_time_traces} +\end{figure} + + +\begin{figure}[htbp] +\centering +\includegraphics[width=5in]{images/cross_section_array.jpg} +\caption{Cutaway views of 30 \AA\ Ag-Cu nanoparticle structures for +random alloy (top) and Cu (core) / Ag (shell) initial conditions +(bottom). Shown from left to right are the crystalline, liquid +droplet, and final glassy bead configurations.} +\label{fig:cross_sections} +\end{figure}