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\chapter{\label{chap:nanoglass}GLASS FORMATION IN METALLIC NANOPARTICLES} |
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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.\cite{West:2003fk,Hu:2006lr} One |
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of the side effects of absorption of laser radiation at these |
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frequencies is the rapid (sub-picosecond) heating of the electronic |
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degrees of freedom in the metal. This hot electron gas quickly |
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transfers heat to the phonon modes of the particle, resulting in a |
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rapid heating of the lattice of the metal particles. Since metallic |
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nanoparticles have a large surface area to volume ratio, many of the |
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metal atoms are at surface locations and experience relatively weak |
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bonding. This is observable in a lowering of the melting temperatures |
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of these particles when compared with bulk metallic |
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samples.\cite{Buffat:1976yq,Dick:2002qy} One of the side effects of |
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the excitation of small metallic nanoparticles at the plasmon |
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resonance is the facile creation of liquid metal |
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droplets.\cite{Mafune01,HartlandG.V._jp0276092,Link:2000lr,Plech:2003yq,plech:195423,Plech:2007rt} |
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Much of the experimental work on this subject has been carried out in |
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the Hartland, El-Sayed and Plech |
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groups.\cite{HartlandG.V._jp0276092,Hodak:2000rb,Hartland:2003lr,Petrova:2007qy,Link:2000lr,plech:195423,Plech:2007rt} |
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These experiments mostly use the technique of time-resolved optical |
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pump-probe spectroscopy, where a pump laser pulse serves to excite |
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conduction band electrons in the nanoparticle and a following probe |
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laser pulse allows observation of the time evolution of the |
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electron-phonon coupling. Hu and Hartland have observed a direct |
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relation between the size of the nanoparticle and the observed cooling |
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rate using such pump-probe techniques.\cite{Hu:2004lr} Plech {\it et |
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al.} have use pulsed x-ray scattering as a probe to directly access |
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changes to atomic structure following pump |
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excitation.\cite{plech:195423} They further determined that heat |
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transfer in nanoparticles to the surrounding solvent is goverened by |
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interfacial dynamics and not the thermal transport properties of the |
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solvent. This is in agreement with Cahill,\cite{Wilson:2002uq} |
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but opposite to the conclusions in Reference \citen{Hu:2004lr}. |
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Since these experiments are 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 a relatively rapid process. In our |
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recent simulation study of the laser excitation of gold |
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nanoparticles,\cite{VardemanC.F._jp051575r} we observed that the |
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cooling rate for these particles (10$^{11}$-10$^{12}$ K/s) is in |
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excess of the cooling rate required for glass formation in bulk |
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metallic alloys.\cite{Greer:1995qy} Given this fact, it may be |
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possible to use laser excitation to melt, alloy and quench metallic |
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nanoparticles in order to form glassy 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{Vardeman-II:2001jn} The Hume-Rothery rules suggest that |
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alloys composed of Copper and Silver should be miscible in the solid |
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state, because their lattice constants are within 15\% of each |
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another.\cite{Kittel:1996fk} Experimentally, however Ag-Cu alloys are |
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a well-known exception to this rule and are only miscible in the |
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liquid state given equilibrium conditions.\cite{Massalski:1986rt} |
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Below the eutectic temperature of 779 $^\circ$C and composition |
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(60.1\% Ag, 39.9\% Cu), the solid alloys of Ag and Cu will phase |
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separate into Ag and Cu rich $\alpha$ and $\beta$ phases, |
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respectively.\cite{Banhart:1992sv,Ma:2005fk} This behavior is due to a |
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positive heat of mixing in both the solid and liquid phases. For the |
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one-to-one composition fcc solid solution, $\Delta H_{\rm mix}$ is on |
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the order of +6~kJ/mole.\cite{Ma:2005fk} Non-equilibrium solid |
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solutions may be formed by undercooling, and under these conditions, a |
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compositionally-disordered $\gamma$ fcc phase can be |
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formed.\cite{najafabadi:3144} |
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Metastable alloys composed of Ag-Cu were first reported by Duwez in |
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1960 and were created by using a ``splat quenching'' technique in |
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which a liquid droplet is propelled by a shock wave against a cooled |
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metallic target.\cite{duwez:1136} Because of the small positive |
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$\Delta H_{\rm mix}$, supersaturated crystalline solutions are |
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typically obtained rather than an amorphous phase. Higher $\Delta |
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H_{\rm mix}$ systems, such as Ag-Ni, are immiscible even in liquid |
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states, but they tend to form metastable alloys much more readily than |
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Ag-Cu. If present, the amorphous Ag-Cu phase is usually seen as the |
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minority phase in most experiments. Because of this unique |
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crystalline-amorphous behavior, the Ag-Cu system has been widely |
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studied. Methods for creating such bulk phase structures include splat |
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quenching, vapor deposition, ion beam mixing and mechanical |
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alloying. Both structural \cite{sheng:184203} and |
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dynamic\cite{Vardeman-II:2001jn} computational studies have also been |
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performed on this system. |
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Although bulk Ag-Cu alloys have been studied widely, this alloy has |
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been mostly overlooked in nanoscale materials. The literature on |
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alloyed metallic nanoparticles has dealt with the Ag-Au system, which |
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has the useful property of being miscible on both solid and liquid |
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phases. Nanoparticles of another miscible system, Au-Cu, have been |
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successfully constructed using techniques such as laser |
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ablation,\cite{Malyavantham:2004cu} and the synthetic reduction of |
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metal ions in solution.\cite{Kim:2003lv} Laser induced alloying has |
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been used as a technique for creating Au-Ag alloy particles from |
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core-shell particles.\cite{Hartland:2003lr} To date, attempts at |
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creating Ag-Cu nanoparticles have used ion implantation to embed |
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nanoparticles in a glass matrix.\cite{De:1996ta,Magruder:1994rg} These |
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attempts have been largely unsuccessful in producing mixed alloy |
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nanoparticles, and instead produce phase segregated or core-shell |
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structures. |
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One of the more successful attempts at creating intermixed Ag-Cu |
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nanoparticles used alternate pulsed laser ablation and deposition in |
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an amorphous Al$_2$O$_3$ matrix.\cite{gonzalo:5163} Surface plasmon |
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resonance (SPR) of bimetallic core-shell structures typically show two |
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distinct resonance peaks where mixed particles show a single shifted |
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and broadened resonance.\cite{Hodak:2000rb} The SPR for pure silver |
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occurs at 400 nm and for copper at 570 nm.\cite{HengleinA._jp992950g} |
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On Al$_2$O$_3$ films, these resonances move to 424 nm and 572 nm for |
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the pure metals. For bimetallic nanoparticles with 40\% Ag an |
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absorption peak is seen between 400-550 nm. With increasing Ag |
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content, the SPR shifts towards the blue, with the peaks nearly |
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coincident at a composition of 57\% Ag. Gonzalo {\it et al.} cited the |
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existence of a single broad resonance peak as evidence of an alloyed |
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particle rather than a phase segregated system. However, spectroscopy |
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may not be able to tell the difference between alloyed particles and |
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mixtures of segregated particles. High-resolution electron microscopy |
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has so far been unable to determine whether the mixed nanoparticles |
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were an amorphous phase or a supersaturated crystalline phase. |
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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, $\langle r^2(t) \rangle$. Liquids exhibit a mean squared |
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displacement that is linear in time (at long times). Glassy materials |
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deviate significantly from this linear behavior at intermediate times, |
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entering a sub-linear regime with a return to linear behavior in the |
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infinite time limit.\cite{Kob:1999fk} However, diffusion in |
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nanoparticles differs significantly from the bulk in that atoms are |
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confined to a roughly spherical volume and cannot explore any region |
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larger than the particle radius ($R$). In these confined geometries, |
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$\langle r^2(t) \rangle$ approaches a limiting value of |
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$3R^2/40$.\cite{ShibataT._ja026764r} This limits the utility of |
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dynamical measures of glass formation when studying nanoparticles. |
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However, glassy materials exhibit strong icosahedral ordering among |
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nearest-neghbors (in contrast with crystalline and liquid-like |
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configurations). Local icosahedral structures are the |
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three-dimensional equivalent of covering a two-dimensional plane with |
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5-sided tiles; they cannot be used to tile space in a periodic |
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fashion, and are therefore an indicator of non-periodic packing in |
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amorphous solids. Steinhart {\it et al.} defined an orientational bond |
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order parameter that is sensitive to icosahedral |
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ordering.\cite{Steinhardt:1983mo} This bond order parameter can |
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therefore be used to characterize glass formation in liquid and solid |
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solutions.\cite{wolde:9932} |
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Theoretical molecular dynamics studies have been performed on the |
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formation of amorphous single component nanoclusters of either |
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gold,\cite{Chen:2004ec,Cleveland:1997jb,Cleveland:1997gu} or |
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nickel,\cite{Gafner:2004bg,Qi:2001nn} by rapid cooling($\thicksim |
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10^{12}-10^{13}$ K/s) from a liquid state. All of these studies found |
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icosahedral ordering in the resulting structures produced by this |
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rapid cooling which can be evidence of the formation of an amorphous |
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structure.\cite{Strandburg:1992qy} The nearest neighbor information |
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was obtained from pair correlation functions, common neighbor analysis |
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and bond order parameters.\cite{Steinhardt:1983mo} It should be noted |
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that these studies used single component systems with cooling rates |
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that are only obtainable in computer simulations and particle sizes |
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less than 20\AA. Single component systems are known to form amorphous |
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states in small clusters,\cite{Breaux:rz} but do not generally form |
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amorphous structures in bulk materials. |
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Since the nanoscale Ag-Cu alloy has been largely unexplored, many |
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interesting questions remain about the formation and properties of |
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such a system. Does the large surface area to volume ratio aid Ag-Cu |
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nanoparticles in rapid cooling and formation of an amorphous state? |
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Nanoparticles have been shown to have a size dependent melting |
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transition ($T_m$),\cite{Buffat:1976yq,Dick:2002qy} so we might expect |
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a similar trend to follow for the glass transition temperature |
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($T_g$). By analogy, bulk metallic glasses exhibit a correlation |
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between $T_m$ and $T_g$ although such dependence is difficult to |
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establish because of the dependence of $T_g$ on cooling rate and the |
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process by which the glass is formed.\cite{Wang:2003fk} It has also |
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been demonstrated that there is a finite size effect depressing $T_g$ |
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in polymer glasses in confined geometries.\cite{Alcoutlabi:2005kx} |
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In the sections below, we describe our modeling of the laser |
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excitation and subsequent cooling of the particles {\it in silico} to |
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mimic real experimental conditions. The simulation parameters have |
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been tuned to the degree possible to match experimental conditions, |
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and we discusss both the icosahedral ordering in the system, as well |
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as the clustering of icosahedral centers that we observed. |
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\section{COMPUTATIONAL METHODOLOGY} |
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\label{nanoglass:sec:details} |
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\subsection{INITIAL GEOMETRIES AND HEATING} |
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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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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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\subsection{MODELING RANDOM ALLOY AND CORE SHELL PARTICLES IN SOLUTION PHASE ENVIRONMENTS} |
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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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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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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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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 |
311 |
|
|
0.295 $(\mathrm{J g^{-1} K^{-1}})$ and $9.826\times 10^6$ $(\mathrm{g |
312 |
|
|
m^{-3}})$ respectively. Since most of the laser excitation experiments |
313 |
|
|
have been done in aqueous solutions, parameters used for the fluid are |
314 |
|
|
$K$ of $0.6$ $(\mathrm{Wm^{-1}K^{-1}})$, $\rho$ of $1.0\times10^6$ |
315 |
|
|
$(\mathrm{g m^{-3}})$ and $c$ of $4.184$ $(\mathrm{J g^{-1} K^{-1}})$. |
316 |
|
|
|
317 |
|
|
Values for the interfacial conductance have been determined by a |
318 |
|
|
number of groups for similar nanoparticles and range from a low |
319 |
|
|
$87.5\times 10^{6} (\mathrm{Wm^{-2}K^{-1}})$ to $130\times 10^{6} |
320 |
|
|
(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Wilson {\it |
321 |
|
|
et al.} worked with Au, Pt, and AuPd nanoparticles and obtained an |
322 |
|
|
estimate for the interfacial conductance of $G=130 |
323 |
|
|
(\mathrm{Wm^{-2}K^{-1}})$.\cite{Wilson:2002uq} Similarly, Plech {\it |
324 |
|
|
et al.} reported a value for the interfacial conductance of $G=105\pm |
325 |
|
|
15 (\mathrm{Wm^{-2}K^{-1}})$ for Au nanoparticles.\cite{plech:195423} |
326 |
|
|
|
327 |
|
|
We conducted our simulations at both ends of the range of |
328 |
|
|
experimentally-determined values for the interfacial conductance. |
329 |
|
|
This allows us to observe both the slowest and fastest heat transfers |
330 |
|
|
from the nanoparticle to the solvent that are consistent with |
331 |
|
|
experimental observations. For the slowest heat transfer, a value for |
332 |
|
|
G of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ was used and for |
333 |
|
|
the fastest heat transfer, a value of $117\times 10^{6}$ |
334 |
|
|
$(\mathrm{Wm^{-2}K^{-1}})$ was used. Based on calculations we have |
335 |
|
|
done using raw data from the Hartland group's thermal half-time |
336 |
|
|
experiments on Au nanospheres,\cite{HuM._jp020581+} the true G values |
337 |
|
|
are probably in the faster regime: $117\times 10^{6}$ |
338 |
|
|
$(\mathrm{Wm^{-2}K^{-1}})$. |
339 |
|
|
|
340 |
|
|
The rate of cooling for the nanoparticles in a molecular dynamics |
341 |
|
|
simulation can then be tuned by changing the effective solvent |
342 |
|
|
viscosity ($\eta$) until the nanoparticle cooling rate matches the |
343 |
|
|
cooling rate described by the heat-transfer Eq. |
344 |
|
|
(\ref{eq:heateqn}). The effective solvent viscosity (in Pa s) for a G |
345 |
|
|
of $87.5\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $4.2 \times |
346 |
|
|
10^{-6}$, $5.0 \times 10^{-6}$, and |
347 |
|
|
$5.5 \times 10^{-6}$ for 20 {\AA}, 30 {\AA}, and 40 {\AA} particles, respectively. The |
348 |
|
|
effective solvent viscosity (again in Pa s) for an interfacial |
349 |
|
|
conductance of $117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ is $5.7 |
350 |
|
|
\times 10^{-6}$, $7.2 \times 10^{-6}$, and $7.5 \times 10^{-6}$ |
351 |
|
|
for 20 {\AA}, 30 {\AA} and 40 {\AA} particles. These viscosities are |
352 |
|
|
essentially gas-phase values, a fact which is consistent with the |
353 |
|
|
initial temperatures of the particles being well into the |
354 |
|
|
super-critical region for the aqueous environment. Gas bubble |
355 |
|
|
generation has also been seen experimentally around gold nanoparticles |
356 |
|
|
in water.\cite{kotaidis:184702} Instead of a single value for the |
357 |
|
|
effective viscosity, a time-dependent parameter might be a better |
358 |
|
|
mimic of the cooling vapor layer that surrounds the hot particles. |
359 |
|
|
This may also be a contributing factor to the size-dependence of the |
360 |
|
|
effective viscosities in our simulations. |
361 |
|
|
|
362 |
|
|
Cooling traces for each particle size are presented in |
363 |
|
|
Fig. \ref{fig:images_cooling_plot}. It should be noted that the |
364 |
|
|
Langevin thermostat produces cooling curves that are consistent with |
365 |
|
|
Newtonian (single-exponential) cooling, which cannot match the cooling |
366 |
|
|
profiles from Eq. (\ref{eq:laplacetransform}) exactly. Fitting the |
367 |
|
|
Langevin cooling profiles to a single-exponential produces |
368 |
|
|
$\tau=25.576$ ps, $\tau=43.786$ ps, and $\tau=56.621$ ps for the 20, |
369 |
|
|
30 and 40 {\AA} nanoparticles and a G of $87.5\times 10^{6}$ |
370 |
|
|
$(\mathrm{Wm^{-2}K^{-1}})$. For comparison's sake, similar |
371 |
|
|
single-exponential fits with an interfacial conductance of G of |
372 |
|
|
$117\times 10^{6}$ $(\mathrm{Wm^{-2}K^{-1}})$ produced a $\tau=13.391$ |
373 |
|
|
ps, $\tau=30.426$ ps, $\tau=43.857$ ps for the 20, 30 and 40 {\AA} |
374 |
|
|
nanoparticles. |
375 |
|
|
|
376 |
|
|
\begin{figure}[htbp] |
377 |
|
|
\centering |
378 |
|
|
\includegraphics[width=5in]{images/cooling_plot.pdf} |
379 |
|
|
\caption{Thermal cooling curves obtained from the inverse Laplace |
380 |
|
|
transform heat model in Eq. (\ref{eq:laplacetransform}) (solid line) as |
381 |
|
|
well as from molecular dynamics simulations (circles). Effective |
382 |
|
|
solvent viscosities of 4.2-7.5 $\times 10^{-6}$ Pa s (depending on the |
383 |
|
|
radius of the particle) give the best fit to the experimental cooling |
384 |
|
|
curves. This viscosity suggests that the nanoparticles in these |
385 |
|
|
experiments are surrounded by a vapor layer (which is a reasonable |
386 |
|
|
assumptions given the initial temperatures of the particles). } |
387 |
|
|
\label{fig:images_cooling_plot} |
388 |
|
|
\end{figure} |
389 |
|
|
|
390 |
|
|
\subsection{POTENIALS FOR CLASSICAL SIMULATIONS OF BIMETALLIC NANOPARTICLES} |
391 |
|
|
|
392 |
|
|
Several different potential models have been developed that reasonably |
393 |
|
|
describe interactions in transition metals. In particular, the |
394 |
|
|
Embedded Atom Model ({\sc eam})~\cite{PhysRevB.33.7983} and |
395 |
|
|
Sutton-Chen ({\sc sc})~\cite{Chen90} potential have been used to study |
396 |
|
|
a wide range of phenomena in both bulk materials and |
397 |
|
|
nanoparticles.\cite{Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} Both |
398 |
|
|
potentials are based on a model of a metal which treats the nuclei and |
399 |
|
|
core electrons as pseudo-atoms embedded in the electron density due to |
400 |
|
|
the valence electrons on all of the other atoms in the system. The |
401 |
|
|
{\sc sc} potential has a simple form that closely resembles that of |
402 |
|
|
the ubiquitous Lennard Jones potential, |
403 |
|
|
\begin{equation} |
404 |
|
|
\label{eq:SCP1} |
405 |
|
|
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] , |
406 |
|
|
\end{equation} |
407 |
|
|
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
408 |
|
|
\begin{equation} |
409 |
|
|
\label{eq:SCP2} |
410 |
|
|
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}}. |
411 |
|
|
\end{equation} |
412 |
|
|
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
413 |
|
|
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
414 |
|
|
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
415 |
|
|
the interactions between the valence electrons and the cores of the |
416 |
|
|
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
417 |
|
|
scale, $c_i$ scales the attractive portion of the potential relative |
418 |
|
|
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
419 |
|
|
that assures a dimensionless form for $\rho$. These parameters are |
420 |
|
|
tuned to various experimental properties such as the density, cohesive |
421 |
|
|
energy, and elastic moduli for FCC transition metals. The quantum |
422 |
|
|
Sutton-Chen ({\sc q-sc}) formulation matches these properties while |
423 |
|
|
including zero-point quantum corrections for different transition |
424 |
|
|
metals.\cite{PhysRevB.59.3527} This particular parametarization has |
425 |
|
|
been shown to reproduce the experimentally available heat of mixing |
426 |
|
|
data for both FCC solid solutions of Ag-Cu and the high-temperature |
427 |
|
|
liquid.\cite{sheng:184203} In contrast, the {\sc eam} potential does |
428 |
|
|
not reproduce the experimentally observed heat of mixing for the |
429 |
|
|
liquid alloy.\cite{MURRAY:1984lr} In this work, we have utilized the |
430 |
|
|
{\sc q-sc} formulation for our potential energies and forces. |
431 |
|
|
Combination rules for the alloy were taken to be the arithmetic |
432 |
|
|
average of the atomic parameters with the exception of $c_i$ since its |
433 |
|
|
values is only dependent on the identity of the atom where the density |
434 |
|
|
is evaluated. For the {\sc q-sc} potential, cutoff distances are |
435 |
|
|
traditionally taken to be $2\alpha_{ij}$ and include up to the sixth |
436 |
|
|
coordination shell in FCC metals. |
437 |
|
|
|
438 |
|
|
%\subsection{Sampling single-temperature configurations from a cooling |
439 |
|
|
%trajectory} |
440 |
|
|
|
441 |
|
|
To better understand the structural changes occurring in the |
442 |
|
|
nanoparticles throughout the cooling trajectory, configurations were |
443 |
|
|
sampled at regular intervals during the cooling trajectory. These |
444 |
|
|
configurations were then allowed to evolve under NVE dynamics to |
445 |
|
|
sample from the proper distribution in phase space. Fig. |
446 |
|
|
\ref{fig:images_cooling_time_traces} illustrates this sampling. |
447 |
|
|
|
448 |
|
|
|
449 |
|
|
\begin{figure}[htbp] |
450 |
|
|
\centering |
451 |
|
|
\includegraphics[height=3in]{images/cooling_time_traces.pdf} |
452 |
|
|
\caption{Illustrative cooling profile for the 40 {\AA} |
453 |
|
|
nanoparticle evolving under stochastic boundary conditions |
454 |
|
|
corresponding to $G=$$87.5\times 10^{6}$ |
455 |
|
|
$(\mathrm{Wm^{-2}K^{-1}})$. At temperatures along the cooling |
456 |
|
|
trajectory, configurations were sampled and allowed to evolve in the |
457 |
|
|
NVE ensemble. These subsequent trajectories were analyzed for |
458 |
|
|
structural features associated with bulk glass formation.} |
459 |
|
|
\label{fig:images_cooling_time_traces} |
460 |
|
|
\end{figure} |
461 |
|
|
|
462 |
|
|
|
463 |
|
|
\begin{figure}[htbp] |
464 |
|
|
\centering |
465 |
|
|
\includegraphics[width=5in]{images/cross_section_array.jpg} |
466 |
|
|
\caption{Cutaway views of 30 \AA\ Ag-Cu nanoparticle structures for |
467 |
|
|
random alloy (top) and Cu (core) / Ag (shell) initial conditions |
468 |
|
|
(bottom). Shown from left to right are the crystalline, liquid |
469 |
|
|
droplet, and final glassy bead configurations.} |
470 |
|
|
\label{fig:cross_sections} |
471 |
|
|
\end{figure} |
472 |
|
|
|
473 |
|
|
\section{ANALYSIS} |
474 |
|
|
|
475 |
|
|
Frank first proposed local icosahedral ordering of atoms as an |
476 |
|
|
explanation for supercooled atomic (specifically metallic) liquids, |
477 |
|
|
and further showed that a 13-atom icosahedral cluster has a 8.4\% |
478 |
|
|
higher binding energy the either a face centered cubic ({\sc fcc}) or |
479 |
|
|
hexagonal close-packed ({\sc hcp}) crystal structures.\cite{19521106} |
480 |
|
|
Icosahedra also have six five-fold symmetry axes that cannot be |
481 |
|
|
extended indefinitely in three dimensions; long-range translational |
482 |
|
|
order is therefore incommensurate with local icosahedral ordering. |
483 |
|
|
This does not preclude icosahedral clusters from possessing long-range |
484 |
|
|
{\it orientational} order. The ``frustrated'' packing of these |
485 |
|
|
icosahedral structures into dense clusters has been proposed as a |
486 |
|
|
model for glass formation.\cite{19871127} The size of the icosahedral |
487 |
|
|
clusters is thought to increase until frustration prevents any further |
488 |
|
|
growth.\cite{HOARE:1976fk} Molecular dynamics simulations of a |
489 |
|
|
two-component Lennard-Jones glass showed that clusters of face-sharing |
490 |
|
|
icosahedra are distributed throughout the |
491 |
|
|
material.\cite{PhysRevLett.60.2295} Simulations of freezing of single |
492 |
|
|
component metalic nanoclusters have shown a tendency for icosohedral |
493 |
|
|
structure formation particularly at the surfaces of these |
494 |
|
|
clusters.\cite{Gafner:2004bg,PhysRevLett.89.275502,Chen:2004ec} |
495 |
|
|
Experimentally, the splitting (or shoulder) on the second peak of the |
496 |
|
|
X-ray structure factor in binary metallic glasses has been attributed |
497 |
|
|
to the formation of tetrahedra that share faces of adjoining |
498 |
|
|
icosahedra.\cite{Waal:1995lr} |
499 |
|
|
|
500 |
|
|
Various structural probes have been used to characterize structural |
501 |
|
|
order in molecular systems including: common neighbor analysis, |
502 |
|
|
Voronoi tesselations, and orientational bond-order |
503 |
|
|
parameters.\cite{HoneycuttJ.Dana_j100303a014,Iwamatsu:2007lr,hsu:4974,nose:1803} |
504 |
|
|
The method that has been used most extensively for determining local |
505 |
|
|
and extended orientational symmetry in condensed phases is the |
506 |
|
|
bond-orientational analysis formulated by Steinhart {\it et |
507 |
|
|
al.}\cite{Steinhardt:1983mo} In this model, a set of spherical |
508 |
|
|
harmonics is associated with each of the near neighbors of a central |
509 |
|
|
atom. Neighbors (or ``bonds'') are defined as having a distance from |
510 |
|
|
the central atom that is within the first peak in the radial |
511 |
|
|
distribution function. The spherical harmonic between a central atom |
512 |
|
|
$i$ and a neighboring atom $j$ is |
513 |
|
|
\begin{equation} |
514 |
|
|
Y_{lm}\left(\vec{r}_{ij}\right)=Y_{lm}\left(\theta(\vec{r}_{ij}),\phi(\vec{r}_{ij})\right) |
515 |
|
|
\label{eq:spharm} |
516 |
|
|
\end{equation} |
517 |
|
|
where, $\{ Y_{lm}(\theta,\phi)\}$ are spherical harmonics, and |
518 |
|
|
$\theta(\vec{r})$ and $\phi(\vec{r})$ are the polar and azimuthal |
519 |
|
|
angles made by the bond vector $\vec{r}$ with respect to a reference |
520 |
|
|
coordinate system. We chose for simplicity the origin as defined by |
521 |
|
|
the coordinates for our nanoparticle. (Only even-$l$ spherical |
522 |
|
|
harmonics are considered since permutation of a pair of identical |
523 |
|
|
particles should not affect the bond-order parameter.) The local |
524 |
|
|
environment surrounding atom $i$ can be defined by |
525 |
|
|
the average over all neighbors, $N_b(i)$, surrounding that atom, |
526 |
|
|
\begin{equation} |
527 |
|
|
\bar{q}_{lm}(i) = \frac{1}{N_{b}(i)}\sum_{j=1}^{N_b(i)} Y_{lm}(\vec{r}_{ij}). |
528 |
|
|
\label{eq:local_avg_bo} |
529 |
|
|
\end{equation} |
530 |
|
|
We can further define a global average orientational-bond order over |
531 |
|
|
all $\bar{q}_{lm}$ by calculating the average of $\bar{q}_{lm}(i)$ |
532 |
|
|
over all $N$ particles |
533 |
|
|
\begin{equation} |
534 |
|
|
\bar{Q}_{lm} = \frac{\sum^{N}_{i=1}N_{b}(i)\bar{q}_{lm}(i)}{\sum^{N}_{i=1}N_{b}(i)} |
535 |
|
|
\label{eq:sys_avg_bo} |
536 |
|
|
\end{equation} |
537 |
|
|
The $\bar{Q}_{lm}$ contained in Eq. (\ref{eq:sys_avg_bo}) is not |
538 |
|
|
necessarily invariant under rotations of the arbitrary reference |
539 |
|
|
coordinate system. Second- and third-order rotationally invariant |
540 |
|
|
combinations, $Q_l$ and $W_l$, can be taken by summing over $m$ values |
541 |
|
|
of $\bar{Q}_{lm}$, |
542 |
|
|
\begin{equation} |
543 |
|
|
Q_{l} = \left( \frac{4\pi}{2 l+1} \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{1/2} |
544 |
|
|
\label{eq:sec_ord_inv} |
545 |
|
|
\end{equation} |
546 |
|
|
and |
547 |
|
|
\begin{equation} |
548 |
|
|
\hat{W}_{l} = \frac{W_{l}}{\left( \sum^{l}_{m=-l} \left| \bar{Q}_{lm} \right|^2\right)^{3/2}} |
549 |
|
|
\label{eq:third_ord_inv} |
550 |
|
|
\end{equation} |
551 |
|
|
where |
552 |
|
|
\begin{equation} |
553 |
|
|
W_{l} = \sum_{\substack{m_1,m_2,m_3 \\m_1 + m_2 + m_3 = 0}} \left( \begin{array}{ccc} l & l & l \\ m_1 & m_2 & m_3 \end{array}\right) \\ \times \bar{Q}_{lm_1}\bar{Q}_{lm_2}\bar{Q}_{lm_3}. |
554 |
|
|
\label{eq:third_inv} |
555 |
|
|
\end{equation} |
556 |
|
|
The factor in parentheses in Eq. (\ref{eq:third_inv}) is the Wigner-3$j$ |
557 |
|
|
symbol, and the sum is over all valid ($|m| \leq l$) values of $m_1$, |
558 |
|
|
$m_2$, and $m_3$ which sum to zero. |
559 |
|
|
|
560 |
|
|
\begin{table} |
561 |
|
|
\caption{Values of bond orientational order parameters for |
562 |
|
|
simple structures cooresponding to $l=4$ and $l=6$ spherical harmonic |
563 |
|
|
functions.\cite{wolde:9932} $Q_4$ and $\hat{W}_4$ have values for {\it |
564 |
|
|
individual} icosahedral clusters, but these values are not invariant |
565 |
|
|
under rotations of the reference coordinate systems. Similar behavior |
566 |
|
|
is observed in the bond-orientational order parameters for individual |
567 |
|
|
liquid-like structures.} |
568 |
|
|
\begin{center} |
569 |
|
|
\begin{tabular}{ccccc} |
570 |
|
|
\hline |
571 |
|
|
\hline |
572 |
|
|
& $Q_4$ & $Q_6$ & $\hat{W}_4$ & $\hat{W}_6$\\ |
573 |
|
|
|
574 |
|
|
fcc & 0.191 & 0.575 & -0.159 & -0.013\\ |
575 |
|
|
|
576 |
|
|
hcp & 0.097 & 0.485 & 0.134 & -0.012\\ |
577 |
|
|
|
578 |
|
|
bcc & 0.036 & 0.511 & 0.159 & 0.013\\ |
579 |
|
|
|
580 |
|
|
sc & 0.764 & 0.354 & 0.159 & 0.013\\ |
581 |
|
|
|
582 |
|
|
Icosahedral & - & 0.663 & - & -0.170\\ |
583 |
|
|
|
584 |
|
|
(liquid) & - & - & - & -\\ |
585 |
|
|
\hline |
586 |
|
|
\end{tabular} |
587 |
|
|
\end{center} |
588 |
|
|
\label{table:bopval} |
589 |
|
|
\end{table} |
590 |
|
|
|
591 |
|
|
For ideal face-centered cubic ({\sc fcc}), body-centered cubic ({\sc |
592 |
|
|
bcc}) and simple cubic ({\sc sc}) as well as hexagonally close-packed |
593 |
|
|
({\sc hcp}) structures, these rotationally invariant bond order |
594 |
|
|
parameters have fixed values independent of the choice of coordinate |
595 |
|
|
reference frames. For ideal icosahedral structures, the $l=6$ |
596 |
|
|
invariants, $Q_6$ and $\hat{W}_6$ are also independent of the |
597 |
|
|
coordinate system. $Q_4$ and $\hat{W}_4$ will have non-vanishing |
598 |
|
|
values for {\it individual} icosahedral clusters, but these values are |
599 |
|
|
not invariant under rotations of the reference coordinate systems. |
600 |
|
|
Similar behavior is observed in the bond-orientational order |
601 |
|
|
parameters for individual liquid-like structures. Additionally, both |
602 |
|
|
$Q_6$ and $\hat{W}_6$ are thought to have extreme values for the |
603 |
|
|
icosahedral clusters.\cite{Steinhardt:1983mo} This makes the $l=6$ |
604 |
|
|
bond-orientational order parameters particularly useful in identifying |
605 |
|
|
the extent of local icosahedral ordering in condensed phases. For |
606 |
|
|
example, a local structure which exhibits $\hat{W}_6$ values near |
607 |
|
|
-0.17 is easily identified as an icosahedral cluster and cannot be |
608 |
|
|
mistaken for distorted cubic or liquid-like structures. |
609 |
|
|
|
610 |
|
|
One may use these bond orientational order parameters as an averaged |
611 |
|
|
property to obtain the extent of icosahedral ordering in a supercooled |
612 |
|
|
liquid or cluster. It is also possible to accumulate information |
613 |
|
|
about the {\it distributions} of local bond orientational order |
614 |
|
|
parameters, $p(\hat{W}_6)$ and $p(Q_6)$, which provide information |
615 |
|
|
about individual atomic sites that are central to local icosahedral |
616 |
|
|
structures. |
617 |
|
|
|
618 |
|
|
The distributions of atomic $Q_6$ and $\hat{W}_6$ values are plotted |
619 |
|
|
as a function of temperature for our nanoparticles in Fig. |
620 |
|
|
\ref{fig:q6} and \ref{fig:w6}. At high temperatures, the |
621 |
|
|
distributions are unstructured and are broadly distributed across the |
622 |
|
|
entire range of values. As the particles are cooled, however, there |
623 |
|
|
is a dramatic increase in the fraction of atomic sites which have |
624 |
|
|
local icosahedral ordering around them. (This corresponds to the |
625 |
|
|
sharp peak appearing in Fig. \ref{fig:w6} at $\hat{W}_6=-0.17$ and |
626 |
|
|
to the broad shoulder appearing in Fig. \ref{fig:q6} at $Q_6 = |
627 |
|
|
0.663$.) |
628 |
|
|
|
629 |
|
|
\begin{figure}[htbp] |
630 |
|
|
\centering |
631 |
|
|
\includegraphics[width=5in]{images/w6_stacked_plot.pdf} |
632 |
|
|
\caption{Distributions of the bond orientational order parameter |
633 |
|
|
($\hat{W}_6$) at different temperatures. The upper, middle, and lower |
634 |
|
|
panels are for 20, 30, and 40 \AA\ particles, respectively. The |
635 |
|
|
left-hand column used cooling rates commensurate with a low |
636 |
|
|
interfacial conductance ($87.5 \times 10^{6}$ |
637 |
|
|
$\mathrm{Wm^{-2}K^{-1}}$), while the right-hand column used a more |
638 |
|
|
physically reasonable value of $117 \times 10^{6}$ |
639 |
|
|
$\mathrm{Wm^{-2}K^{-1}}$. The peak at $\hat{W}_6 \approx -0.17$ is |
640 |
|
|
due to local icosahedral structures. The different curves in each of |
641 |
|
|
the panels indicate the distribution of $\hat{W}_6$ values for samples |
642 |
|
|
taken at different times along the cooling trajectory. The initial |
643 |
|
|
and final temperatures (in K) are indicated on the plots adjacent to |
644 |
|
|
their respective distributions.} |
645 |
|
|
\label{fig:w6} |
646 |
|
|
\end{figure} |
647 |
|
|
|
648 |
|
|
\begin{figure}[htbp] |
649 |
|
|
\centering |
650 |
|
|
\includegraphics[width=5in]{images/q6_stacked_plot.pdf} |
651 |
|
|
\caption{Distributions of the bond orientational order parameter |
652 |
|
|
($Q_6$) at different temperatures. The curves in the six panels in |
653 |
|
|
this figure were computed at identical conditions to the same panels in |
654 |
|
|
figure \ref{fig:w6}.} |
655 |
|
|
\label{fig:q6} |
656 |
|
|
\end{figure} |
657 |
|
|
|
658 |
|
|
The probability distributions of local order can be used to generate |
659 |
|
|
free energy surfaces using the local orientational ordering as a |
660 |
|
|
reaction coordinate. By making the simple statistical equivalence |
661 |
|
|
between the free energy and the probabilities of occupying certain |
662 |
|
|
states, |
663 |
|
|
\begin{equation} |
664 |
|
|
g(\hat{W}_6) = - k_B T \ln p(\hat{W}_6), |
665 |
|
|
\end{equation} |
666 |
|
|
we can obtain a sequence of free energy surfaces (as a function of |
667 |
|
|
temperature) for the local ordering around central atoms within our |
668 |
|
|
particles. Free energy surfaces for the 40 \AA\ particle at a range |
669 |
|
|
of temperatures are shown in Fig. \ref{fig:freeEnergy}. Note that |
670 |
|
|
at all temperatures, the liquid-like structures are global minima on |
671 |
|
|
the free energy surface, while the local icosahedra appear as local |
672 |
|
|
minima once the temperature has fallen below 528 K. As the |
673 |
|
|
temperature falls, it is possible for substructures to become trapped |
674 |
|
|
in the local icosahedral well, and if the cooling is rapid enough, |
675 |
|
|
this trapping leads to vitrification. A similar analysis of the free |
676 |
|
|
energy surface for orientational order in bulk glass formers can be |
677 |
|
|
found in the work of van~Duijneveldt and |
678 |
|
|
Frenkel.\cite{duijneveldt:4655} |
679 |
|
|
|
680 |
|
|
|
681 |
|
|
\begin{figure}[htbp] |
682 |
|
|
\centering |
683 |
|
|
\includegraphics[width=5in]{images/freeEnergyVsW6.pdf} |
684 |
|
|
\caption{Free energy as a function of the orientational order |
685 |
|
|
parameter ($\hat{W}_6$) for 40 {\AA} bimetallic nanoparticles as they |
686 |
|
|
are cooled from 902 K to 310 K. As the particles cool below 528 K, a |
687 |
|
|
local minimum in the free energy surface appears near the perfect |
688 |
|
|
icosahedral ordering ($\hat{W}_6 = -0.17$). At all temperatures, |
689 |
|
|
liquid-like structures are a global minimum on the free energy |
690 |
|
|
surface, but if the cooling rate is fast enough, substructures |
691 |
|
|
may become trapped with local icosahedral order, leading to the |
692 |
|
|
formation of a glass.} |
693 |
|
|
\label{fig:freeEnergy} |
694 |
|
|
\end{figure} |
695 |
|
|
|
696 |
|
|
We have also calculated the fraction of atomic centers which have |
697 |
|
|
strong local icosahedral order: |
698 |
|
|
\begin{equation} |
699 |
|
|
f_\textrm{icos} = \int_{-\infty}^{w_i} p(\hat{W}_6) d \hat{W}_6 |
700 |
|
|
\label{eq:ficos} |
701 |
|
|
\end{equation} |
702 |
|
|
where $w_i$ is a cutoff value in $\hat{W}_6$ for atomic centers that |
703 |
|
|
are displaying icosahedral environments. We have chosen a (somewhat |
704 |
|
|
arbitrary) value of $w_i= -0.15$ for the purposes of this work. A |
705 |
|
|
plot of $f_\textrm{icos}(T)$ as a function of temperature of the |
706 |
|
|
particles is given in Fig. \ref{fig:ficos}. As the particles cool, |
707 |
|
|
the fraction of local icosahedral ordering rises smoothly to a plateau |
708 |
|
|
value. The smaller particles (particularly the ones that were cooled |
709 |
|
|
in a higher viscosity solvent) show a slightly larger tendency towards |
710 |
|
|
icosahedral ordering. |
711 |
|
|
|
712 |
|
|
\begin{figure}[htbp] |
713 |
|
|
\centering |
714 |
|
|
\includegraphics[width=5in]{images/fraction_icos.pdf} |
715 |
|
|
\caption{Temperautre dependence of the fraction of atoms with local |
716 |
|
|
icosahedral ordering, $f_\textrm{icos}(T)$ for 20, 30, and 40 \AA\ |
717 |
|
|
particles cooled at two different values of the interfacial |
718 |
|
|
conductance.} |
719 |
|
|
\label{fig:ficos} |
720 |
|
|
\end{figure} |
721 |
|
|
|
722 |
|
|
Since we have atomic-level resolution of the local bond-orientational |
723 |
|
|
ordering information, we can also look at the local ordering as a |
724 |
|
|
function of the identities of the central atoms. In figure |
725 |
|
|
\ref{fig:AgVsCu} we display the distributions of $\hat{W}_6$ values |
726 |
|
|
for both the silver and copper atoms, and we note a strong |
727 |
|
|
predilection for the copper atoms to be central to icosahedra. This |
728 |
|
|
is probably due to local packing competition of the larger silver |
729 |
|
|
atoms around the copper, which would tend to favor icosahedral |
730 |
|
|
structures over the more densely packed cubic structures. |
731 |
|
|
|
732 |
|
|
\begin{figure}[htbp] |
733 |
|
|
\centering |
734 |
|
|
\includegraphics[width=5in]{images/w6_stacked_bytype_plot.pdf} |
735 |
|
|
\caption{Distributions of the bond orientational order parameter |
736 |
|
|
($\hat{W}_6$) for the two different elements present in the |
737 |
|
|
nanoparticles. This distribution was taken from the fully-cooled 40 |
738 |
|
|
\AA\ nanoparticle. Local icosahedral ordering around copper atoms is |
739 |
|
|
much more prevalent than around silver atoms.} |
740 |
|
|
\label{fig:AgVsCu} |
741 |
|
|
\end{figure} |
742 |
|
|
|
743 |
|
|
The locations of these icosahedral centers are not uniformly |
744 |
|
|
distrubted throughout the particles. In Fig. \ref{fig:icoscluster} |
745 |
|
|
we show snapshots of the centers of the local icosahedra (i.e. any |
746 |
|
|
atom which exhibits a local bond orientational order parameter |
747 |
|
|
$\hat{W}_6 < -0.15$). At high temperatures, the icosahedral centers |
748 |
|
|
are transitory, existing only for a few fs before being reabsorbed |
749 |
|
|
into the liquid droplet. As the particle cools, these centers become |
750 |
|
|
fixed at certain locations, and additional icosahedra develop |
751 |
|
|
throughout the particle, clustering around the sites where the |
752 |
|
|
structures originated. There is a strong preference for icosahedral |
753 |
|
|
ordering near the surface of the particles. Identification of these |
754 |
|
|
structures by the type of atom shows that the silver-centered |
755 |
|
|
icosahedra are evident only at the surface of the particles. |
756 |
|
|
|
757 |
|
|
\begin{figure}[htbp] |
758 |
|
|
\centering |
759 |
|
|
\begin{tabular}{c c c} |
760 |
|
|
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_liq_icosonly.pdf} |
761 |
|
|
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A__0007_icosonly.pdf} |
762 |
|
|
\includegraphics[width=2.1in]{images/Cu_Ag_random_30A_glass_icosonly.pdf} |
763 |
|
|
\end{tabular} |
764 |
|
|
\caption{Centers of local icosahedral order ($\hat{W}_6<0.15$) at 900 |
765 |
|
|
K, 471 K and 315 K for the 30 \AA\ nanoparticle cooled with an |
766 |
|
|
interfacial conductance $G = 87.5 \times 10^{6}$ |
767 |
|
|
$\mathrm{Wm^{-2}K^{-1}}$. Silver atoms (blue) exhibit icosahedral |
768 |
|
|
order at the surface of the nanoparticle while copper icosahedral |
769 |
|
|
centers (green) are distributed throughout the nanoparticle. The |
770 |
|
|
icosahedral centers appear to cluster together and these clusters |
771 |
|
|
increase in size with decreasing temperature.} |
772 |
|
|
\label{fig:icoscluster} |
773 |
|
|
\end{figure} |
774 |
|
|
|
775 |
|
|
In contrast with the silver ordering behavior, the copper atoms which |
776 |
|
|
have local icosahedral ordering are distributed more evenly throughout |
777 |
|
|
the nanoparticles. Fig. \ref{fig:Surface} shows this tendency as a |
778 |
|
|
function of distance from the center of the nanoparticle. Silver, |
779 |
|
|
since it has a lower surface free energy than copper, tends to coat |
780 |
|
|
the skins of the mixed particles.\cite{Zhu:1997lr} This is true even |
781 |
|
|
for bimetallic particles that have been prepared in the Ag (core) / Cu |
782 |
|
|
(shell) configuration. Upon forming a liquid droplet, approximately 1 |
783 |
|
|
monolayer of Ag atoms will rise to the surface of the particles. This |
784 |
|
|
can be seen visually in Fig. \ref{fig:cross_sections} as well as in |
785 |
|
|
the density plots in the bottom panel of Fig. \ref{fig:Surface}. |
786 |
|
|
This observation is consistent with previous experimental and |
787 |
|
|
theoretical studies on bimetallic alloys composed of noble |
788 |
|
|
metals.\cite{MainardiD.S._la0014306,HuangS.-P._jp0204206,Ramirez-Caballero:2006lr} |
789 |
|
|
Bond order parameters for surface atoms are averaged only over the |
790 |
|
|
neighboring atoms, so packing constraints that may prevent icosahedral |
791 |
|
|
ordering around silver in the bulk are removed near the surface. It |
792 |
|
|
would certainly be interesting to see if the relative tendency of |
793 |
|
|
silver and copper to form local icosahedral structures in a bulk glass |
794 |
|
|
differs from our observations on nanoparticles. |
795 |
|
|
|
796 |
|
|
\begin{figure}[htbp] |
797 |
|
|
\centering |
798 |
|
|
\includegraphics[width=5in]{images/dens_fracr_stacked_plot.pdf} |
799 |
|
|
\caption{Appearance of icosahedral clusters around central silver atoms |
800 |
|
|
is largely due to the presence of these silver atoms at or near the |
801 |
|
|
surface of the nanoparticle. The upper panel shows the fraction of |
802 |
|
|
icosahedral atoms ($f_\textrm{icos}(r)$ for each of the two metallic |
803 |
|
|
atoms as a function of distance from the center of the nanoparticle |
804 |
|
|
($r$). The lower panel shows the radial density of the two |
805 |
|
|
constituent metals (relative to the overall density of the |
806 |
|
|
nanoparticle). Icosahedral clustering around copper atoms are more |
807 |
|
|
evenly distributed throughout the particle, while icosahedral |
808 |
|
|
clustering around silver is largely confined to the silver atoms at |
809 |
|
|
the surface.} |
810 |
|
|
\label{fig:Surface} |
811 |
|
|
\end{figure} |
812 |
|
|
|
813 |
|
|
The methods used by Sheng, He, and Ma to estimate the glass transition |
814 |
|
|
temperature, $T_g$, in bulk Ag-Cu alloys involve finding |
815 |
|
|
discontinuities in the slope of the average atomic volume, $\langle V |
816 |
|
|
\rangle / N$, or enthalpy when plotted against the temperature of the |
817 |
|
|
alloy. They obtained a bulk glass transition temperature, $T_g$ = 510 |
818 |
|
|
K for a quenching rate of $2.5 \times 10^{13}$ K/s. |
819 |
|
|
|
820 |
|
|
For simulations of nanoparticles, there is no periodic box, and |
821 |
|
|
therefore, no easy way to compute the volume exactly. Instead, we |
822 |
|
|
estimate the volume of our nanoparticles using Barber {\it et al.}'s |
823 |
|
|
very fast quickhull algorithm to obtain the convex hull for the |
824 |
|
|
collection of 3-d coordinates of all of atoms at each point in |
825 |
|
|
time.~\cite{barber96quickhull,qhull} The convex hull is the smallest convex |
826 |
|
|
polyhedron which includes all of the atoms, so the volume of this |
827 |
|
|
polyhedron is an excellent estimate of the volume of the nanoparticle. |
828 |
|
|
This method of estimating the volume will be problematic if the |
829 |
|
|
nanoparticle breaks into pieces (i.e. if the bounding surface becomes |
830 |
|
|
concave), but for the relatively short trajectories used in this |
831 |
|
|
study, it provides an excellent measure of particle volume as a |
832 |
|
|
function of time (and temperature). |
833 |
|
|
|
834 |
|
|
Using the discontinuity in the slope of the average atomic volume |
835 |
|
|
vs. temperature, we arrive at an estimate of $T_g$ that is |
836 |
|
|
approximately 488 K. We note that this temperature is somewhat below |
837 |
|
|
the onset of icosahedral ordering exhibited in the bond orientational |
838 |
|
|
order parameters. It appears that icosahedral ordering sets in while |
839 |
|
|
the system is still somewhat fluid, and is locked in place once the |
840 |
|
|
temperature falls below $T_g$. We did not observe any dependence of |
841 |
|
|
our estimates for $T_g$ on either the nanoparticle size or the value |
842 |
|
|
of the interfacial conductance. However, the cooling rates and size |
843 |
|
|
ranges we utilized are all sampled from a relatively narrow range, and |
844 |
|
|
it is possible that much larger particles would have substantially |
845 |
|
|
different values for $T_g$. Our estimates for the glass transition |
846 |
|
|
temperatures for all three particle sizes and both interfacial |
847 |
|
|
conductance values are shown in table \ref{table:Tg}. |
848 |
|
|
|
849 |
|
|
\begin{table} |
850 |
|
|
\caption{Estimates of the glass transition temperatures $T_g$ for |
851 |
|
|
three different sizes of bimetallic Ag$_6$Cu$_4$ nanoparticles cooled |
852 |
|
|
under two different values of the interfacial conductance, $G$.} |
853 |
|
|
\begin{center} |
854 |
|
|
\begin{tabular}{ccccc} |
855 |
|
|
\hline |
856 |
|
|
\hline |
857 |
|
|
Radius (\AA\ ) & Interfacial conductance & Effective cooling rate |
858 |
|
|
(K/s $\times 10^{13}$) & & $T_g$ (K) \\ |
859 |
|
|
20 & 87.5 & 2.4 & 477 \\ |
860 |
|
|
20 & 117 & 4.5 & 502 \\ |
861 |
|
|
30 & 87.5 & 1.3 & 491 \\ |
862 |
|
|
30 & 117 & 1.9 & 493 \\ |
863 |
|
|
40 & 87.5 & 1.0 & 476 \\ |
864 |
|
|
40 & 117 & 1.3 & 487 \\ |
865 |
|
|
\hline |
866 |
|
|
\end{tabular} |
867 |
|
|
\end{center} |
868 |
|
|
\label{table:Tg} |
869 |
|
|
\end{table} |
870 |
|
|
|
871 |
|
|
\section{CONCLUSIONS} |
872 |
|
|
\label{metglass:sec:conclusion} |
873 |
|
|
|
874 |
|
|
Our heat-transfer calculations have utilized the best current |
875 |
|
|
estimates of the interfacial heat transfer coefficient (G) from recent |
876 |
|
|
experiments. Using reasonable values for thermal conductivity in both |
877 |
|
|
the metallic particle and the surrounding solvent, we have obtained |
878 |
|
|
cooling rates for laser-heated nanoparticles that are in excess of |
879 |
|
|
10$^{13}$ K / s. To test whether or not this cooling rate can form |
880 |
|
|
glassy nanoparticles, we have performed a mixed molecular dynamics |
881 |
|
|
simulation in which the atoms in contact with the solvent or capping |
882 |
|
|
agent are evolved under Langevin dynamics while the remaining atoms |
883 |
|
|
are evolved under Newtonian dynamics. The effective solvent viscosity |
884 |
|
|
($\eta$) is a free parameter which we have tuned so that the particles |
885 |
|
|
in the simulation follow the same cooling curve as their experimental |
886 |
|
|
counterparts. From the local icosahedral ordering around the atoms in |
887 |
|
|
the nanoparticles (particularly Copper atoms), we deduce that it is |
888 |
|
|
likely that glassy nanobeads are created via laser heating of |
889 |
|
|
bimetallic nanoparticles, particularly when the initial temperature of |
890 |
|
|
the particles approaches the melting temperature of the bulk metal |
891 |
|
|
alloy. |
892 |
|
|
|
893 |
|
|
Improvements to our calculations would require: 1) explicit treatment |
894 |
|
|
of the capping agent and solvent, 2) another radial region to handle |
895 |
|
|
the heat transfer to the solvent vapor layer that is likely to form |
896 |
|
|
immediately surrounding the hot |
897 |
|
|
particle,\cite{Hu:2004lr,kotaidis:184702} and 3) larger particles in |
898 |
|
|
the size range most easily studied via laser heating experiments. |
899 |
|
|
|
900 |
|
|
The local icosahedral ordering we observed in these bimetallic |
901 |
|
|
particles is centered almost completely around the copper atoms, and |
902 |
|
|
this is likely due to the size mismatch leading to a more efficient |
903 |
|
|
packing of 5-membered rings of silver around a central copper atom. |
904 |
|
|
This size mismatch should be reflected in bulk calculations, and work |
905 |
|
|
is ongoing in our lab to confirm this observation in bulk |
906 |
|
|
glass-formers. |
907 |
|
|
|
908 |
|
|
The physical properties (bulk modulus, frequency of the breathing |
909 |
|
|
mode, and density) of glassy nanobeads should be somewhat different |
910 |
|
|
from their crystalline counterparts. However, observation of these |
911 |
|
|
differences may require single-particle resolution of the ultrafast |
912 |
|
|
vibrational spectrum of one particle both before and after the |
913 |
|
|
crystallite has been converted into a glassy bead. |