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\section{Introduction} |
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Excitation of the plasmon resonance in metallic nanoparticles has |
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attracted enormous interest in the past several years. This is partly |
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due to the location of the plasmon band in the near IR for particles |
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in a wide range of sizes and geometries. (Living tissue is nearly |
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transparent in the near IR, and for this reason, there is an |
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unrealized potential for metallic nanoparticles to be used in both |
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diagnostic and therapeutic settings.) One of the side effects of |
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absorption of laser radiation at these frequencies is the rapid |
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(sub-picosecond) heating of the electronic degrees of freedom in the |
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metal. This hot electron gas quickly transfers heat to the phonon |
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modes of the lattice, resulting in a rapid heating of the metal |
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particles. |
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Since metallic nanoparticles have a large surface area to volume |
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ratio, many of the metal atoms are at surface locations and experience |
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relatively weak bonding. This is observable in a lowering of the |
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melting temperatures and a substantial softening of the bulk modulus |
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of these particles when compared with bulk metallic samples. One of |
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the side effects of the excitation of small metallic nanoparticles at |
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the plasmon resonance is the facile creation of liquid metal |
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droplets. |
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Much of the experimental work on this subject has been carried out in |
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the Hartland and von~Plessen |
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groups.\cite{HartlandG.V._jp0276092,Hodak:2000rb,Hartland:2003yf,HuM._jp020581+,plech:195423} |
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They have [BRIEF SURVEY OF THE EXPERIMENTAL WORK] |
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Since these experiments are often carried out in condensed phase |
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surroundings, the large surface area to volume ratio makes the heat |
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transfer to the surrounding solvent also a relatively rapid process. |
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In our recent simulation study of the laser excitation of gold |
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nanoparticles,\cite{} we observed that the cooling rate for these |
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particles (10$^11$-10$^12$ K/s) is in excess of the cooling rate |
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required for glass formation in bulk metallic alloys. Given this |
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fact, it may be possible to use laser excitation to melt, alloy and |
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quench metallic nanoparticles in order to form metallic glass |
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nanobeads. |
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To study whether or not glass nanobead formation is feasible, we have |
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chosen the bimetallic alloy of Silver (60\%) and Copper (40\%) as a |
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model system because it is an experimentally known glass former and |
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has been used previously as a theoretical model for glassy |
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dynamics.\cite{Vardeman2001} |
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\section{Background} |
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Hume-Rothery rules suggest that alloys composed of Copper and Silver noble fcc metals should be miscible in the solid state, because their lattice constants are within 15\% of each another. |
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\begin{figure}[htbp] |
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\begin{center} |
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\includegraphics[]{agcu_phase_diagram.pdf} |
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\caption{Equilibrium Phase Diagram for Ag-Cu binary system from reference \cite{Banhart:1992sv}. The dashed line indicates the lowest temperature required to obtain the metastable crystalline phase of the same composition.} |
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\label{fig:phasedia} |
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\end{center} |
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\end{figure} |
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Experimentally, Ag-Cu alloys are an exception to this well-known rule and are only miscible in the liquid state given equilibrium conditions. Below the eutectic temperature of \unit{779}{\celsius} and composition (60.1 wt. \% Ag,39.9 wt .\% Cu), the solid alloy Ag and Cu phase separate into a Ag and Cu rich $\alpha$ and $\beta$ phase respectively. This behavior is due to a positive heat of mixing in both solid and liquid phases. For the equatomic composition fcc solid solution, $\Delta H$ is on the order of \unit{+6}{\kilo\joule\per\mole}\cite{Ma:2005zt}. Figure \ref{fig:phasedia} shows the equilibrium phase diagram for the Ag-Cu binary system\cite{Massalski:1986kl}\cite{Banhart:1992sv}, where the dashed line indicates the temperature at which a non-equilibrium solid solution may be formed. Under these non-equilibrium conditions, a crystalline-disordered fcc $\gamma$ phase can be created. |
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Metastable alloys composed of Ag-Cu were first reported by Dewez\cite{duwez:1136} in 1960 and were created by using a "splat quenching" technique where a liquid droplet is propelled by a shock wave against a super-cooled metallic target. Because of the small positive $\Delta H$, supersaturated crystalline solutions are typically obtained rather than an amorphous phase. Higher $\Delta H$ systems, such as Ag-Ni, are immiscible even in liquid states, but they tend to form metastable alloys much more readily than Ag-Cu. If present, the amorphous Ag-Cu phase is usually seen as the minority phase in most experiments. Because of this unique crystalline-amorphous behavior, the Ag-Cu system has been widely studied. Methods for creating such bulk phase structures include splat quenching, vapor deposition, ion beam mixing and mechanical alloying. Both structural \cite{sheng:184203} and dynamic\cite{Vardeman-II:2001jn} molecular dynamics computational studies have also been performed on this system providing an excellent model for the behavior of glass forming systems. |
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Although, Ag-Cu alloys have been studied widely studied in bulk phase, this alloy has been |
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scarcely studied in nano scale materials. Most of the literature on alloyed metallic nanoparticles has dealt with the Ag-Au system, which has the useful property of being miscible on both solid and liquid phases. Nanoparticles of another miscible system, Au-Cu, have been successfully constructed using techniques such as laser ablation\cite{Malyavantham:2004cu} and the synthetic reduction of metal ions in solution\cite{Kim:2003lv}. Laser induced alloying has been used as a technique for creating Au-Ag alloy particles from core-shell particles\cite{Hartland:2003yf}. To date, attempts at creating Ag-Cu nanoparticles have used ion implantation to embed nanoparticles in a glass matrix\cite{De:1996ta}\cite{Magruder:1994rg}. These attempts have been largely unsuccessful in producing mixed alloy nanoparticles, and instead produce a phase segregated or a core-shell structure. |
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\begin{figure}[htbp] |
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\begin{center} |
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\includegraphics[width=3in]{SPR_Ag_Cu.pdf} |
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\caption{Absorption spectra, from reference \cite{gonzalo:5163}, of films containing nanoparticles of different atomic \% Ag (0 \% being pure Cu). Inset compares nanoparticles with 57 at. \% with the simulated normalized spectrum calculated by the weighted average of spectra for pure Ag and Cu particles. } |
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\label{fig:spr} |
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\end{center} |
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\end{figure} |
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One of the more successful attempts at creating Ag-Cu mixed nanoparticles used alternate pulsed laser ablation and deposition in an amorphous $\ch Al_2O_3$ matrix\cite{gonzalo:5163}. Surface plasmon resonance (SPR) of bimetallic core-shell structures typically show two distinct resonance peaks where mixed particles show a single shifted and broadened resonance peak\cite{Hodak:2000rb}. The SPR for pure silver occurs at \unit{400}{\nano\meter} and for copper at \unit{570}{\nano\meter}. Figure \ref{fig:spr} shows the absorption spectra for pure Cu and Ag $\ch Al_2O_3$ films with SPR peaks at \unit{572}{\nano\meter} and \unit{424}{\nano\meter} respectively as a reference for the pure states. For bimetallic nanoparticles with 40 at.\% Ag an absorption peak is seen between \unit{400\mbox{-}550}{\nano\meter}. With increasing Ag content, the SPR shifts towards the blue, with the peaks nearly coincident at a composition of 57 at.\% Ag. The authors cited the existence of a single broad resonance peak as evidence of a mixed alloy particle rather than a phase segregated system. Unfortunately, it was not determined whether the mixed nanoparticles were an amorphous phase or a supersaturated crystalline phase. One consequence of embedding the Ag-Cu nanoparticles in a glass matrix is that the SPR can be shifted because of the nanoparticle-glass matrix interaction\cite{De:1996ta}\cite{Roy:2003dy}. It would be useful to create free Ag-Cu nanoparticles that could be studied independent of the surrounding environment. |
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Theoretical molecular dynamics computational studies have been performed on the formation of amorphous single component nanoclusters of either gold\cite{Chen:2004ec}\cite{Cleveland:1997jb}\cite{Cleveland:1997gu} or nickel\cite{Gafner:2004bg}\cite{Qi:2001nn} by rapid cooling($\thicksim \unit{10^{12}-10^{13}}{\kelvin\per\second}$) from a liquid state. All of these studies found icosahedral ordering in the resulting structures produced by this rapid cooling which can be evidence of the formation of a amorphous structure\cite{Sachdev:1992mo}. The nearest neighbor information was obtained from pair correlation functions, common neighbor analysis and bond order parameters\cite{Steinhardt:1983mo}. It should be noted that these studies used single component systems with cooling rates that are only obtainable in computer simulations and particle sizes less than 20\AA. Single component systems are known to form amorphous states in small clusters\cite{Breaux:rz} but do not generally form amorphous structures in bulk materials. Icosahedral structures have also been reported in nanoparticles, particularly multiply twinned particles\cite{Ascencio:2000qy}. |
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Since the nanoscale Ag-Cu alloy has been largely unexplored, many interesting questions remain about the formation and properties of such a system. Does the large surface to volume ratio aid Ag-Cu nanoparticles in rapid cooling and formation of an amorphous state? Would a predisposition to isosahedral ordering in nanoparticles also allow for easier formation of an amorphous state and what is the preferred ordering in a amorphous nanoparticle? Nanoparticles have been shown to have size dependent melting transition\cite{Buffat:1976yq}, one would expect a similar trend with the glass transition temperature, because cooling rate is dependent on particle size and the glass transition temperature is dependent on cooling rate. |
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XXX stuff from ORP |
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In the sections below, we describe our |
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modeling of the laser excitation and subsequent cooling of the |
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particles in silico to mimic real experimental conditions. |
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constructing and relaxing the eutectic composition (Ag$_6$Cu$_4$) on a |
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FCC lattice with a lattice constant of 4.09 \AA\ for 20, 30 and 40 |
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\AA\ radius nanoparticles. The nanoparticles are melted at 900 K and |
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allowed to mix for 1 ns. Resulting structures are then quenched using |
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a implicit solvent model where Langevin dynamics is applied to the |
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outer 4 \AA\ radius of the nanoparticle and normal Newtonian dynamics |
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are applied to the rest of the atoms. By fitting to |
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experimentally-determined cooling rates, we find that collision |
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frequencies of 3.58 fs$^-1$ for Ag and 5.00 fs$^-1$ for Cu lead to |
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nearly exact agreement with the Temperature vs. time data. The cooling |
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rates are therefore 2.37 x 10$^13$ K/s, 1.37 x 10$^13$ K/s and 1.06 x |
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10$^13$ K/s for the 20, 30 and 40 \AA\ radius nanoparticles |
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respectively. |
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Structural Measures for Glass Formation |
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Characterization of glassy behavior by molecular dynamics simulations |
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is typically done using dynamic measurements such as the mean squared |
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displacement, <r2(t)>. Liquids exhibit a mean squared displacement |
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that is linear in time. Glassy materials deviate significantly from |
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this linear behavior at intermediate times, entering a sub-linear |
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regime with a return to linear behavior in the infinite time |
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limit. Diffusion in nanoparticles differs significantly from the bulk |
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in that atoms are confined to a roughly spherical volume and cannot |
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explore any region larger than the particle radius. In these confined |
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geometries, <r2(t)> in the radial direction approaches a limiting |
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value of 6R2/40. |
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However, glassy materials exhibit strong icosahedral ordering among nearest-neghbors in contrast to crystalline or liquid structures. Steinhart, et al., defined an orientational bond order parameter that is sensitive to the nearest-neighbor environment by using invariant combinations of spherical harmonics Yl,m(?,?).[10] Spherical harmonics involving the Y6,m(?,?) are particularly sensitive to icosohedral order among nearest neighbors as can be seen in the cartoon to the left. The second and third-order invariants, Q6 and W6 are used to determine the level of icosahedral order present in a quenched nanoparticle. Perfect icosahedral structures have a maximal value of 0.663 for Q6 and -0.170 for W6. A plot of the distributions of Q6 and W6 with cooling temperature indicates increasing icosahedral order with decreasing temperature. This is a clear indication that glassy structures are forming as the nanoparticles are quenched. |
