matt_papers/MWTCC03/chuck.tex

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\begin{document}
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      \begin{center}
        \textbf{\Huge {Breathing Mode Dynamics and Elastic Properties of Gold Nanoparticles}}\\[0.5em]
        \textsc{\LARGE Patrick F. ~Conforti, Megan M. ~Sprague, \underline{Charles F. ~Vardeman~II}, and J. ~Daniel ~Gezelter}\\[0.3em]
        {\large Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, IN 46556, USA\\
{\tt\ Vardeman.1@nd.edu}
}
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{\large{\color{red}    \underline{ABSTRACT}  } }
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{\color{ndblue}
We present molecular dynamics based calculations of the bulk modulus, heat capacity, and the period of the breathing mode for spherical nanoparticles following excitation by ultrafast laser pulses\cite{Gezelter2003}. The bulk modulus exhibits relatively sharp transitions at both the surface-melting and bulk melting transitions, while the heat capacity shows only a relatively broad transition at the bulk melting temperature.  Equilibrium calculations of the heat capacity show that the melting transition is sharper and occurs at a lower temperature than one would observe from an ultrafast experiment\cite{Hartland2003}. We also observe an intriguing splitting in the low-frequency spectra of the nanoparticles and analyze this splitting in terms of Lamb's classical theory of elastic spheres\cite{Lamb1882}.  We conclude that the particles either 1) melt during the observation period following laser excitation, or 2) melt an outer shell while maintaining a crystalline core.  Both mechanisms for melting are commensurate with our observations.
 }
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        \begin{center}\psframebox[framearc=0.60,fillstyle=solid,fillcolor=blue]{\makebox[0.99\textwidth]{{{\LARGE 1 \hspace{0.1cm} \color{red} Introduction} }}}\end{center}
Nanoparticles exhibit behavior not displayed in bulk because of a relatively high surface area to volume volume ratio.Since the time scale for heating is faster than a single period of the breating mode for spherical nanoparticles, energy due to photons absorbed during ultrafast laser excitation is immediately transfered into thermal excitation of atomic degrees of freedom. This excitation is rapid enough to coherently excite the breathing mode of a spherical nanoparticle. We mimic this excitation using atomistic simulation methods.
   
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      \begin{kasten}
          \section*{2 \hspace{0.1cm} {\color{red} \underline{Simulation Methodology}}}
              \subsection*{2.1 {\color{blue} Model Construction}  }
\begin{wrapfigure}[13]{o}{200pt}
\includegraphics[scale=0.28]{35K_lattice.eps}
\end{wrapfigure}
Spherical Au nanoparticles were created in a standard FCC lattice at four different radii [20{\AA} (1926 atoms), 25{\AA} (3884 atoms), 30{\AA} (6602 atoms), and 35{\AA} (10606 atoms)].  To create spherical nanoparticles, a FCC lattice was built at the normal Au lattice spacing (4.08 \AA) and any atoms outside the target radius were excluded.
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         \subsection*{2.2 {\color{blue} Embedded Atom Method}  }
Potentials were calculated using the Embedded atom method due to Daw {\it et.al.} \cite{Daw84,FBD86,johnson89,Lu97} and mixing rules formulated by Johnson\cite{johnson89}.
The {\sc eam} potential has the form:
\[
V  =  \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} \phi_{ij}({\bf r}_{ij})
\]
where $\phi_{ij}$ is a primarily repulsive pairwise interaction between atoms $i$ and $j$,  $F_{i}[\rho_{i}]$ is a embedding function that determines the energy to embedded the positvely-charged core of atom $i$, in a electron density $\rho_{i}$ that is determined by the valence electrons of the surrounding $j$ atoms an has the form:
\[
\rho_{i}   =  \sum_{j \neq i} f_{j}({\bf r}_{ij})
\]
There is a cutoff distance, $r_{cut}$, which limits the summations to the few dozen atoms surrounding atom $i$.  In these simulations, a cutoff radius of 10~\AA\ was used.


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      \begin{kasten}
        \subsection*{2.3 {\color{blue} Simulation Details}  }
Excitation of Au Nanoparticles were carried out as follows:
\begin{enumerate}\itemindent=0.25cm
\item A relatively short steepest-decent minimization to relax the lattice.
\item Mimic events after ultrafast laser excitation event.
\begin{enumerate}\itemindent=0.5cm
\item Instantaneous heating by sampling from a Boltzmann Distribution at twice the simulation target temperature. By equipartition, approximately half of the initial kinetic energy disperses into the potential energy of the system.
\item A short (10 fs) evolution period with the new velocities.
\end{enumerate}
\item Evolution in a Cannonical Ensemble using Nos\'{e}-Hoover NVT dynamics at the ta
rget temperature for 40 ps.
\end{enumerate}
     \begin{itemize}
     \item Time intervals of 5 fs were used.

     \item Target temperatures spanned from 300 K to 1300 K in 100 K intervals.
     \item Five independent samples were run foreach particle and temperature.
     \item Simulations were run in parallel using Plimpton's force-decomposition method. \cite{Plimpton93}

     \end{itemize}


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        \section*{3 \hspace{0.1cm} {\color{red} \underline{Analysis }}}
           \subsection*{3.1 {\color{blue} Volume Determination}}
We are interested in the dynamics of the low-frequency breathing mode of the particles. To study this motion, we need access to accurate measures of both the volume and surface area as a function of time.
\begin{wrapfigure}[12]{o}{120pt}
\includegraphics[scale=0.25]{convex_hull.eps}
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\begin{itemize}
\item We used the convex hull to determine both volume and surface area as a function of time.
\item A convex hull is the smallest convex polyhedron which includes all of the atoms.
\item The program qhull was used to compute the convex hull as a function of time.
\end{itemize}


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       \begin{kasten}
         \subsection*{3.2 {\color{blue} Bulk Modulus}}
The bulk modulus, which is the inverse of the compressibility,
\[
K = \frac{1}{\kappa} = - V \left(\frac{\partial P}{\partial V}\right)_{T} = V \left(\frac{\partial^{2} U }{\partial V^{2}}\right)_{T}
\]
is relatively easily determined when the relationship between the total energy $U$ and volume $V$ of the system are available at a fixed temperature.
\begin{wrapfigure}[12]{o}{150pt}
\includegraphics[scale=0.30]{Sample_Ener_vs_Vol_900_35.eps}
\end{wrapfigure}

Instantaneous heating excites coherent oscillations in the breathing mode allowing us to sample a much wider range of volumes (and energies) then long time equilibrium temperature runs.
$K$ was directly calculated for the 30 and 35 \AA\  particles by multiplying the quadratic coefficient obtained from quadratic fits by the average volume of the nanoparticle for the whole run.For the very small particles (20 \AA\ and 25 \AA\ ), the breathing oscillations decohere rapidly, and we were unable to obtain convincing fits.

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        \subsection*{3.3 {\color{blue} Bulk Modulus and Heat Capacity }}
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\includegraphics[scale=0.30]{Stacked_bm_cp.eps}
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The upper panel shows the temperature dependence of the Bulk Modulus ($K$) for the two larger particles (30 \AA\ and 35 \AA\ ).  Notice, there is a dramatic drop in $K$ at temperatures below the bulk melting temperature coinciding with the surface melting transition (the temperature at which the outermost layer of Au atoms are able to move along the surface of the nanoparticle). Note, also, that the peak in $C_p$ coincides with the {\em start} of the peak in the bulk modulus.  This indicates that compression of the nanoparticles above $T_{m}$ requires an energetically expensive reformation of the lattice.

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       \subsection*{3.4 {\color{blue} Heat Capacity}}
\begin{wrapfigure}[11]{o}{150pt}
\includegraphics[scale=0.30]{Cp_vs_T.eps}
\end{wrapfigure}
Another feature of these transient (non-equilibrium) calculations is the width of the peak in the heat capacity.  Calculation of $C_{p}$ from longer equilibrium trajectories (300 ps) should indicate {\it sharper} features in $C_{p}$ for the larger particles then those observed in the 40 ps simulation.  
Since we are initiating and observing the melting process itself in these calculation, the smaller particles melt more rapidly, and thus exhibit sharper features in $C_{p}$.


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      \begin{kasten}
            \subsection*{3.5 {\color{blue} Breathing Mode Dynamics}}
\begin{wrapfigure}[12]{o}{160pt}
\includegraphics[scale=0.30]{Vol_vs_time.eps}
\end{wrapfigure}
We present representative samples of the volume vs. time traces for the 20 \AA\ and 35 \AA\ particles at a number of different temperatures. It can be seen that the period of the breathing mode is strongly dependent on temperature, and that the coherent oscillations of the particles' volume are destroyed after only a few ps in the smaller particles, while they live on for 10-20 ps in the larger particles.  The de-coherence is also strongly temperature dependent, with the high temperature samples decohering much more rapidly than lower temperatures.
      \end{kasten}

    \vspace{0.5cm}
   \begin{kasten}
       \subsection*{3.6 {\color{blue} Volume Autocorrelation Function}}
\begin{wrapfigure}[12]{o}{160pt}
\includegraphics[scale=0.30]{volcorr.eps}
\end{wrapfigure}
Although $V$ vs. $t$ traces can say a great deal, computing the autocorrelation function for volume fluctuations gives us more accurate short-time information.  Although many traces exhibit a single frequency with decaying amplitude, a number of the samples show distinct beat patterns indicating the presence of multiple frequency components in the breathing motion of the nanoparticles.  In particular, the 20 \AA\ particle shows a distinct beat in the volume fluctuations in the 800 K trace.

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        \subsection*{3.7 {\color{blue} Volume Autocorrelation Function}}
\begin{wrapfigure}[12]{o}{160pt}
\includegraphics[scale=0.30]{Period_vs_T.eps}
\end{wrapfigure}
The power spectrum of the volume autocorrelation functions does indeed show multiple peaks in the power spectrum. Here we plot the period corresponding to the two lowest frequency peaks.  The spectra exhibit a split peak at lower temperatures (which is particularly noticeable in the larger particles).  This splitting disappears at intermediate temperatures, but becomes evident again at the higher temperatures.

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\vspace{0.5cm}
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         \section*{4 \hspace{0.1cm} {\color{red} \underline{Discussion}}}
           \subsection*{4.1 {\color{blue} Lamb Theory of Elastic Spheres}}
Lamb's classical theory of elastic spheres~\cite{Lamb1882} provides a number of possible explanations for the split peak in the vibrational spectrum.  The periods of the longitudinal and transverse vibrations in an elastic sphere of radius $R$ are given by:
\[
\tau_{t} = \frac{2 \pi R}{\theta c_{t}}
\hspace{0.1cm} {\text{and}} \hspace{0.1cm}
\tau_{l} = \frac{2 \pi R}{\eta c_{l}}
\]
where $\theta$ and $n$ are obtained from the solutions to the transcendental equations
\[
\tan \theta = \frac{3 \theta}{3 - \theta^{2}}
\hspace{0.1cm} {\text{and}} \hspace{0.1cm}
\tan \eta = \frac{4 \eta}{4 - \eta^{2}\frac{c_{l}^{2}}{c_{t}^{2}}}
\]
$c_{l}$ and $c_{t}$ are the longitudinal and transverse speeds of sound in the material.  In an isotropic material, these speeds are simply related to the elastic constants and the density ($\rho$),
\[
c_{l} = \sqrt{c_{11}/\rho}
\hspace{0.1cm}\text{\LARGE{;}}\hspace{0.1cm}
c_{t} = \sqrt{c_{44}/\rho}
\]
In crystalline materials, the speeds depend on the direction of propagation of the wave relative to the crystal plane.\cite{Kittel} We assume the nanoparticles are isotropic (which should be valid only above the melting transition). A more detailed analysis of the lower temperature particles would take the crystal lattice into account.
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      \subsection*{4.2 {\color{blue} Power Spectrum Splitting}}
     Using experimental values for the elastic constants for 35 \AA\ Au particles at 300K, the low-frequency longitudinal (breathing) mode should have a period of 2.56 ps while the low-frequency transverse (toroidal) mode should have a period of 2.46 ps.  Although the actual calculated frequencies in our simulations are off of these values, the difference in the periods (0.1 ps) is nearly identical to the splitting observed room-temperature simulations.  This, therefore, may be an explanation for low-temperature splitting.
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      \subsection*{4.3 {\color{blue} Melted and Partially-Melted Particles }}
 Hartland {\it et al.} have extended the Lamb analysis to include surface stress ($\gamma$)\cite{Hartland2003}. The vibrational period of the breathing mode for liquid droplets may be written
\[
\tau = \frac{2 R}{c_{l}(l)}
\]
where $c_{l}(l)$ is the longitudinal speed of sound in the liquid. Using the experimental speed of sound in liquid Au\cite{Iida1988}, a molten 35 \AA\ particle just above $T_{m}$ would have a vibrational period of 2.73 ps, and this would be markedly different from the vibrational period just below $T_{m}$ if the melting transition were sharp.
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\subsection*{4.4 {\color{blue} Observations}}
From our calculations of $C_{p}$, the complete melting of the particles is {\it not} sharp, and should take longer than the 40 ps observation time.  There are wo explanations which are commensurate with our observations.
\begin{enumerate}
\item The melting may occur at some time partway through observation
of the response to instantaneous heating.  The early part
of the simulation would then show a higher-frequency breathing mode
than would be evident during the latter parts of the simulation.
\item The melting may take place by softening the outer layers of the
particle first, followed by a melting of the core at higher
temperatures.  The liquid-like outer layer would then contribute a
lower frequency component than the interior of the particle.
\end{enumerate}
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         \section*{6 \hspace{0.1cm} {\color{red} \underline{Conclusions}}}
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\vspace{0.5cm}
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        \begin{center}  
        {\large{\color{red} \underline{Acknowledgments}}}
     \end{center}
The authors would like to thank Dr. Greg Hartland for a number of helpful discussions.  PFC was supported by an REU fellowship from the National Science Foundation. Computational time was provided on the Notre Dame Bunch-of-Boxes (B.o.B.) cluster under NSF grant DMR 00 79647.  MMS, CFV, and JDG acknowledge support under NSF grant CHE-0134881.
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Revision:	547
Committed:	Wed Jun 4 19:24:02 2003 UTC (21 years ago) by mmeineke
Content type:	application/x-tex
File size:	21988 byte(s)
Log Message:	updated the logo a little as well as some of the colors