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%!TEX root = /Users/charles/Documents/chuckDissertation/dissertation.tex |
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\chapter{\label{chap:nanodiffusion} SIZE DEPENDENT SPONTANEOUS ALLOYING OF Au-Ag NANOPARTICLES} |
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Metallic nanoparticles ({\sc NPs}) exhibit many physical and chemical properties that differ significantly from their bulk counterparts. These unique properties may be partly due to the enormous surface area to mass (Volume) ratio of NPs which leads to excess surface free energy. If the excess surface free energy is comparable in magnitude to the lattice or binding energies of the NPs, structural instability can result. One example of this behavior is the depression of melting temperature ($T_m$) seen in metallic and semiconductor nanoparticles.\cite{Buffat:1976yq,Chen:1997p2142,GOLDSTEIN:1992p2138,Pawlow:1909p2134,SOLLIARD:1985p2137,TOLBERT:1996p2141} Size effects are also exhibited as size dependent alloy formation.\cite{MORI:1991p2144,MORI:1994p2372,YASUDA:1996p2387,yasuda:1100,PhysRevLett.69.3747} Spontaneous alloying has been observed in bimetallic NPs where Cu\cite{MORI:1991p2144,MORI:1994p2372,yasuda:1100}, Zn\cite{PhysRevLett.69.3747}, Pb\cite{YASUDA:1996p2387}, and Sb\cite{Mori1996244} have been observed to alloy into Au at ambient or sub-ambient temperatures\cite{yasuda:1100}. These experiments use transmission electron microscopy (TEM) and diffraction techniques to determine the change in lattice parameters that result from alloying. In this chapter, the mechanism for spontaneous alloying of silver deposited onto a gold core (Au-Ag) NPs will be investigated using computational techniques. |
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Previous studies for the mechanism of alloying in these small particles have been inconclusive in providing an explanation. From a purely kinetic perspective, studies have shown that for the two metals to alloy, the diffusion coefficient would have to be many orders of magnitude larger than those that have been observed in bulk materials.\cite{yasuda:1100,PhysRevLett.69.3747} It has been suggested by Shimizu {\it et al.} that surface melting plays an important role in the alloying process in small ($\approx 100$ atoms) NPs. Also, studies of Sn alloying at Cu surfaces suggest that the surface free energy allows the migration of Sn atoms at the surface.\cite{Schmid:2000ul} This is not surprising since $T_m$ in NPs is size-dependent and lower than in the bulk material due to surface melting effects. Surface melting as a possible explanation has been contradicted by the observation by Mori {\it el al.} that twinning boundaries are preserved during the alloying process in Cu-Au NPs suggesting that the particles remain solid during the process.\cite{MORI:1994p2372} A second possible explanation is that defects created at the interface result in mechanical alloying and enhance diffusion.\cite{Das:1999p2397} It is not evident which of these processes or if a combination of processes is responsible for alloying in small particles. |
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Most of the previous structural studies were carried out on NPs deposited on solid supports. We therefore chose to study the structure of NPs in the absence of a solid support in order to approach their intrinsic properties. However, the NPs in our study maintain a layer of stabilizer molecules, which is necessary to prevent their coagulation. |
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\section{Background and Methodology} |
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\label{nanodiffusion:sec:experimental} |
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Working with experimental collaborators, X-ray absorption fine structure spectroscopy ({\sc XAFS}) was chosen as a technique to gain insight into the mechanism for alloying in NPs. {\sc XAFS} can be used to identify preferred occupation of sites surrounding a given atom as well as to measure inter-atomic distances, variation in those distances, the identity and number of nearest neighbor atoms.\cite{ShibataT._ja026764r} Even though X-ray structure information obtained from {\sc XAFS} is averaged over all atoms that are X-ray excited, information about the bimetallic interface can still be obtained. Because a large fraction of the total atoms in NPs are located at the interfacial region, {\sc XAFS} will reveal structural preferences due to this region compared to the overall structure. NPs constructed from Au and Ag were chosen because of the similarity in atomic size and lattice parameters between the two atoms. Solid solutions may be obtained at any atomic composition with no change in the lattice parameter of the resultant alloy. Additionally, random bulk alloys composed of these atoms have been shown to be free from local strain.\cite{Frenkel:2000p2400} NPs of Au-Ag can be prepared in solution to eliminate the effects due to depositing NPs on a solid support. However, the Au-Ag NPs are over-coated with a layer of stabilizer molecules necessary to prevent their coagulation. These features make the Au-Ag system particularly attractive for studying the mechanism of diffusion in nanoparticles. |
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Experimental collaborators prepared NPs using the radiolytic techniques developed by Henglein and coworkers that reduce Au(III), Au(I), or Ag(I) ions to their metallic state \cite{Hodak:2000rb,HENGLEIN:1999p2419,HengleinA._la981278w,Hodak:2000rb,MULVANEY:1993p2409,Hodak:2000ek}. The construction of these particles was done in a two-step procedure. First, collaborators synthesized small seed core particles (2.5 nm) which were subsequently enlarged to provide variably sized core particles on which Ag was deposited. This procedure is advantageous because it allows for a narrow size distribution in the resultant NPs.\cite{HengleinA._la981278w} The solid-electrolyte interface contains stabilizing molecules (citrate or poly-vinal alcohol) contained in these synthetic procedures. Later {\sc XAFS} measurements indicate that the stabilizing species has no effect at the interface. After synthesis, the result NPs were characterized by {\sc TEM} and optical spectroscopy. Variation in the size distribution was found to decrease upon enlargement of the particles. Particles of 2.5 nm $\pm$ 10 \% and 20 nm $\pm$ 5 \% were observed. Analysis of the plasmon band immediately after deposition of Ag indicated that the NPs had a core-shell structure. |
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Detailed analysis of the spectrum of 2.5 nm Au core particles coated with 3 atomic layers of Ag indicate large changes during the first few days following preparation. After three days, no further changes were observed and the spectrum remained unchanged for another week. The spectrum of the particle after three days resembles that of a random alloy.\cite{Link:1999p2468} Assuming that the spectrum is representative of the alloying process, one can obtain an estimate of the diffusion constant using the diffusion equation, |
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\begin{equation} |
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D=\frac{(d_0/2)^2}{6t}. |
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\end{equation} |
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This equation yields $D\approx 1.0\times 10^{-20}cm^2s^{-1}$. Additionally, absorption bands observed at 205 and 211 nm are indicative of some of the Au(I) cyanide complex migrating from the core to the surface of the NP. Particles with 5 atomic layers of Ag required 4 days for equilibration indicating that the rate of alloying depends on the total size of the NP and not just on the size of the core. {\sc XAFS} spectra of the 2.5 Au-core coated with 1.1, 3.8 and 6.5 atomic layers of Ag all confirm that a significant amount of Ag is present in the first nearest neighbors shell around Au. |
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\subsection{Computational Methodology} |
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To gain insight into the alloying process, molecular dynamics simulations were conducted on model spherical particles composed of Ag and Au atoms interacting under the Embedded Atom Method potential as described in \ref{introSec:eam} using Johnson’s mixing rules \cite{JOHNSON:1989p2479} for the Ag-Au interactions. Because the lattice constants for Ag and Au are nearly identical, the particles were constructed in a perfect fcc lattice using the average of the two-lattice constants (4.085 \AA). Those atoms inside the core radius ($r_{\mathrm{core}}$) were initialized with Au parameters, while those between the core and shell radius ($r_{\mathrm{core}} < r < r_{\mathrm{shell}}$) were initialized as Ag atoms ($r_{\mathrm{shell}} = R$). A 2 \AA -thick shell centered on $r_{\mathrm{core}}$ was designated as the ``interface'' region, as shown in Figure \ref{nanodiff:fig:vac_diag}. For calculations involving interfacial vacancies, 5 \% or 10 \% of these interfacial atoms were chosen at random and were removed from the initial configuration. For the simulations described here, $r_{\mathrm{core}} = 12.5$ \AA\ and $r_{\mathrm{shell}} = 19.98$ \AA. These radii match the smallest of the NPs studied by experimental collaborators. Figure \ref{fig:vac_cross} illustrates the model used for these simulations. The total number of atoms in each nanoparticle was 1926 (no vacancies), 1914 (5 \% interfacial vacancies), and 1902 (10 \% interfacial vacancies). |
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\begin{figure}[htbp] |
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\centering |
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\centerline{\includegraphics[height=5in]{images/vac_diag.pdf}} |
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\caption{Methodology for Molecular Dynamics simulations. Atoms were removed at random from the 2 \AA\ region surrounding the core-shell interface.} |
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\label{nanodiff:fig:vac_diag} |
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\end{figure} |
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Before starting the molecular dynamics runs, a relatively short minimization was performed to relax the lattice in the initial configuration. During the initial 30 ps of each trajectory, velocities were repeatedly sampled from a Maxwell-Boltzmann distribution matching the target temperature for the run. Following this initialization procedure, trajectories were run in the constant-NVE (number, volume, and total energy) ensemble with zero initial total angular momentum. In order to compare the structural features obtained from the NVE-ensemble molecular dynamics, additional trajectories were run using a modified Nos\'{e}-Hoover thermostat to maintain constant temperature and zero total angular momentum.\cite{hoover85} Data collection for all of the simulations started after the 30 ps equilibration period had been completed. We simulated particles with the above-mentioned interfacial vacancy density at 100 K intervals from 500 to 1200 K. Given the masses of the constituent atoms, we were able to use time steps of 5 fs while maintaining excellent energy conservation. Typical run times for our simulations were 24 ns for nanoparticles simulated at 500-700K and 12 ns for particles at 800-1200K. |
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\begin{figure}[htbp] |
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\centering |
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\includegraphics[height=3in]{images/vac_cross.jpg} |
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\caption{Cross Section of Model Used for Simulations. Vacancies at the Interface are Represented by the Red Translucent Spheres.} |
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\label{fig:vac_cross} |
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\end{figure} |
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\section{Computational Results} |
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Analysis of the molecular dynamics simulations centers on transport properties of atoms within the NPs. Most of the atomic motion is due to surface atoms migrating around the outer spherical shell. Since this motion does not effectively mix the two constituents of the core-shell structure, we have developed a method of estimating the relaxation time for the complete alloying process from our simulations. |
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The mean square displacement of atoms confined to a spherical volume of radius $R$ approaches $6R^2/5$ in the infinite-time limit. Similarly, the mean square displacement in the radial coordinate only (relative to the NP center of mass) approaches $3R^2/40$. At shorter times, one might want to connect the observable displacements of atoms in a simulation to the solutions of the diffusion equation. For a spherically symmetric volume with a reflecting boundary at $R$, the solutions to the diffusion equation are given by Equation \ref{nanodiff:eq:diff1}, |
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\begin{equation} |
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\rho(r,t) = \sum_{n=1}^{\infty} a_n \frac{n\pi r/R}{r}e^{-n^2\pi^2Dt/R} |
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\label{nanodiff:eq:diff1} |
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\end{equation} |
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where the coefficients are determined from the initial concentration profile. Proper mapping of the displacements of the individual atoms onto the diffusion coefficient, $D$, in this confined geometry would involve projecting the initial and final positions of each atom onto each of the diffusional modes expressed in the summation of Equation \ref{nanodiff:eq:diff1}. A simpler approach, and one that is more relevant to the situation at hand, is to compute the relaxation time of the mean square displacement using the known infinite-time limit and the famous Kohlraush-William-Watts ({\sc KWW}) relaxation law\cite{Kohlrausch:1863zv,Williams:1970fk} that can arise from a sum of exponential decays: |
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\begin{equation} |
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\left<\left| \mathbf{r}(t)-\mathbf{r}(0)\right|^2\right> \approx \frac{6R^2}{5}\left(1-e^{-(t/\tau)^\beta}\right) |
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\label{nanodiff:eq:kww1} |
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\end{equation} |
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For the alloying process, only displacements in the radial direction lead to mixing. If we use only the radial component of the particle positions to compute the mean square displacements, the pre-factor that gives the infinite-time limit must change to $3R^2/40$: |
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\begin{equation} |
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\left<\left| \mathbf{r}_{\mathrm{radial}}(t)-\mathbf{r}_{\mathrm{radial}}(0)\right|^2\right> \approx \frac{3R^2}{40}\left(1-e^{-(t/\tau)^\beta}\right) |
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\label{nanodiff:eq:kww2} |
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\end{equation} |
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Note that the radial displacements involve different contributions from the diffusional modes in Equation \ref{nanodiff:eq:diff1}, so one should expect to observe different values of $\tau$ and $\beta$ in Equations (\ref{nanodiff:eq:kww1}) and (\ref{nanodiff:eq:kww2}). Our fits to the mean square radial displacement functions show stretching-parameters $\beta \approx 0.8$ in the liquid state, and $\beta \approx 0.4$ below the melting transition. These stretching parameters are merely indicators of multi-exponential decay due to the boundary conditions on the diffusion equation. They should not be taken to indicate a system with inhomogeneities similar to those one would see in glasses\cite{Vardeman-II:2001jn} or defective crystals.\cite{Rabani99} |
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\begin{figure}[htbp] |
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\centering |
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\centerline{\includegraphics[height=5in]{images/mixing_rate.pdf}} |
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\caption{Arrhenius plot of relaxation times for the particles of Figure \ref{nanodiff:fig:vacancy_dens} in the absence (triangles) and presence of 10\% (circles) vacancies, taken from fits of the mean square displacements of the atoms vs. time.} |
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\label{nanodiff:fig:mixing_rate} |
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\end{figure} |
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The mixing relaxation times are displayed in a standard Arrhenius plot in Figure \ref{nanodiff:fig:mixing_rate}. At the high end of the temperature scale shown in Figure \ref{nanodiff:fig:mixing_rate} the particles melt during the calculation (24 or 12 ns). Because diffusion in the melt is via a different mechanism from that in the solid, these temperatures were excluded from the linear Arrhenius extrapolation. At temperatures lower than $\sim 500$ K the displacement is too small to provide reliable estimate of the relaxation times. For the core-shell NPs without any vacancy we obtain from these plots activation energy of 1.85 eV, in close agreement with the experimentally determined value, $\Delta \mathrm{H}=1.76 \mathrm{eV}$\cite{Tu:1992uq} For 5\% and 10\% vacancy levels we obtain 0.92 and 0.11 eV, respectively. Projecting the Arrhenius plots of the relaxation times down to 300 K, we predict room temperature relaxation times of $>10^7$ years (no vacancies), a few days (5\% interfacial vacancies) and minutes to seconds (10\% vacancies). Thus, a few percent of vacancies are probably not far from the present experimental conditions. |
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Figure \ref{nanodiff:fig:vacancy_dens} shows the radial density profile, $\rho(r)/\rho$, of the two constituents as a function of distance from the center of the NP. This density profile was obtained from the last 2 ns of the 24 ns NVE runs at 800K. Remarkably, the interfacial vacancies result in a substantial smoothing of the peaks in the density profile, indicating that the particles with interfacial vacancies are closer to the melting transition than those without them. It is evident from the region near $r=12.5$ \AA\ that the interfacial vacancies result in significantly enhanced radial diffusion of Ag into the core region of the NPs. Most of the displacement, though, occurs along the bimetallic interface (i.e., at constant $r$) as is clear from the pronounced broadening of the density peaks. |
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\begin{figure}[htbp] |
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\centering |
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\centerline{\includegraphics[width=6in]{images/vacancy_dens.jpg}} |
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\caption{Density profiles of Au and Ag atoms within the NP during the last 2 ns of a 24 ns NVE simulation. Solid profile: no interfacial vacancies. Dotted profile: 10\% initial interfacial vacancies.} |
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\label{nanodiff:fig:vacancy_dens} |
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\end{figure} |
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The presence of vacancies at the bimetallic interface provides a reasonable explanation for the spontaneous alloying observed by optical spectroscopy and {\sc XAFS}. Molecular Dynamics calculations show that only a few, (5-6\%), vacancies at the the interface are necessary to produce the alloying in the time scale observed experimentally. The presence of vacancies at the interface also provides a rationalization for why the alloying process terminates after a short period. Mechanistically, the alloying process can be viewed as a competition between percolation of the defects to the outer surface and diffusion of the metal atoms into the vacancies. Once a vacancy reaches the NP surface it is essentially lost since migration back into the lattice is prohibitively slow. It is expected that the preferred directionally of vacancy migration is toward the surface since there is a larger volume fraction available for the vacancy to explore and there is a smaller curvature in the outer layers. Additionally, due to a smaller number of nearest neighbors at the surface, there is a smaller energy barrier to generate a vacancy at the surface relative to the interior of the NP. |
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