| 11 |
|
rapid heat transfer to the surrounding |
| 12 |
|
medium.\cite{HuM._jp020581+,hartland02d} |
| 13 |
|
|
| 14 |
< |
This paper reports on atomic-level simulations of the transient |
| 14 |
> |
In this chapter, atomic-level simulations of the transient |
| 15 |
|
response of metallic nanoparticles to the nearly instantaneous heating |
| 16 |
|
undergone when photons are absorbed during ultrafast laser excitation |
| 17 |
< |
experiments. The time scale for heating (determined by the {\it e-ph} |
| 17 |
> |
experiments are discussed. The time scale for heating (determined by the {\it e-ph} |
| 18 |
|
coupling constant) is faster than a single period of the breathing |
| 19 |
|
mode for spherical nanoparticles.\cite{Simon2001,HartlandG.V._jp0276092} |
| 20 |
|
Hot-electron pressure and direct lattice heating contribute to the |
| 33 |
|
|
| 34 |
|
We outline our simulation techniques in section \ref{bulkmod:sec:details}. |
| 35 |
|
Results are presented in section \ref{bulkmod:sec:results}. We discuss our |
| 36 |
< |
results in terms of Lamb's classical theory of elastic spheres in |
| 36 |
> |
results in terms of Lamb's classical theory of elastic spheres \cite{Lamb1882} in |
| 37 |
|
section \ref{bulkmod:sec:discussion}. |
| 38 |
|
|
| 39 |
< |
\section{COMPUTATIONAL DETAILS} |
| 39 |
> |
\section{Computational Details} |
| 40 |
|
\label{bulkmod:sec:details} |
| 41 |
|
|
| 42 |
|
Spherical Au nanoparticles were created in a standard FCC lattice at |
| 46 |
|
spacing (4.08 \AA) and any atoms outside the target radius were |
| 47 |
|
excluded from the simulation. |
| 48 |
|
|
| 49 |
< |
\subsection{SIMULATION METHODOLOGY} |
| 49 |
> |
\subsection{Simulation Methodology} |
| 50 |
|
|
| 51 |
|
Potentials were calculated using the Voter-Chen |
| 52 |
|
parameterization~\cite{Voter:87} of the Embedded Atom Method ({\sc |
| 53 |
< |
eam}), which has been widely used for MD simulations of metallic |
| 54 |
< |
particles.%\cite{Voter:87,Daw84,FBD86,johnson89,Lu97} |
| 55 |
< |
Like other transition metal potentials,%\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} |
| 56 |
< |
{\sc eam} describes bonding in metallic systems by including an |
| 57 |
< |
attractive interaction which models the stabilization of a positively |
| 58 |
< |
charged metal core ion in a sea of surrounding valence electrons. A |
| 59 |
< |
repulsive pair potential describes the interactions of the core ions |
| 60 |
< |
with each other. The {\sc eam} potential has the form: |
| 61 |
< |
\begin{eqnarray} |
| 62 |
< |
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 63 |
< |
\phi_{ij}({\bf r}_{ij}) \\ |
| 64 |
< |
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
| 65 |
< |
\end{eqnarray} |
| 66 |
< |
where $\phi_{ij}$ is the (primarily repulsive) pairwise interaction |
| 67 |
< |
between atoms $i$ and $j$. $F_{i}[\rho_{i}]$ is an embedding function |
| 68 |
< |
that determines the energy to embed the positively-charged core, $i$, |
| 69 |
< |
in the electron density, $\rho_{i}$, due to the valence electrons of |
| 70 |
< |
the surrounding $j$ atoms, and $f_{j}(r)$ describes the radial |
| 71 |
< |
dependence of the field due to atom $j$. There is a cutoff distance, |
| 72 |
< |
$r_{cut}$, which limits the summations in the {\sc eam} equation to |
| 73 |
< |
the few dozen atoms surrounding atom $i$. In these simulations, a |
| 74 |
< |
cutoff radius of 10~\AA\ was used. |
| 75 |
< |
|
| 76 |
< |
%Mixing rules as outlined by Johnson~\cite{johnson89} were used to |
| 77 |
< |
%compute the heterogenous pair potential, |
| 78 |
< |
% |
| 79 |
< |
%\begin{eqnarray} |
| 80 |
< |
%\label{eq:johnson} |
| 81 |
< |
%\phi_{ab}(r)=\frac{1}{2}\left( |
| 82 |
< |
%\frac{f_{b}(r)}{f_{a}(r)}\phi_{aa}(r)+ |
| 83 |
< |
%\frac{f_{a}(r)}{f_{b}(r)}\phi_{bb}(r) |
| 84 |
< |
%\right). |
| 85 |
< |
%\end{eqnarray} |
| 86 |
< |
|
| 53 |
> |
eam}) as described in Section \ref{introSec:eam} of this dissertation. |
| 54 |
|
Before starting the molecular dynamics runs, a relatively short |
| 55 |
|
steepest-descent minimization was performed to relax the lattice in |
| 56 |
|
the initial configuration. To facilitate the study of the larger |
| 77 |
|
The results presented below are averaged properties for each of the |
| 78 |
|
five independent samples. |
| 79 |
|
|
| 80 |
< |
\subsection{ANALYSIS} |
| 80 |
> |
\subsection{\label{bulkmod:sec:analysis}Analysis} |
| 81 |
|
|
| 82 |
|
Of primary interest when comparing our simulations to experiments is |
| 83 |
|
the dynamics of the low-frequency breathing mode of the particles. To |
| 84 |
< |
study this motion, we need access to accurate measures of both the |
| 84 |
> |
study this motion, access is needed to accurate measures of both the |
| 85 |
|
volume and surface area as a function of time. Throughout the |
| 86 |
< |
simulations, we monitored both quantities using Barber {\it et al.}'s |
| 86 |
> |
simulations, both quantities were monitored using Barber {\it et al.}'s |
| 87 |
|
very fast quickhull algorithm to obtain the convex hull for the |
| 88 |
|
collection of 3-d coordinates of all of atoms at each point in |
| 89 |
|
time.~\cite{barber96quickhull,qhull} The convex hull is the smallest convex |
| 102 |
|
\label{eq:Kdef} |
| 103 |
|
\end{equation} |
| 104 |
|
can be related to quantities that are relatively easily accessible |
| 105 |
< |
from our molecular dynamics simulations. We present here four |
| 106 |
< |
different approaches for estimating the bulk modulus directly from |
| 105 |
> |
from our molecular dynamics simulations. Four |
| 106 |
> |
different approaches are presented here for estimating the bulk modulus directly from |
| 107 |
|
basic or derived quantities: |
| 108 |
|
|
| 109 |
|
\begin{itemize} |
| 110 |
|
\item The traditional ``Energetic'' approach |
| 111 |
|
|
| 112 |
< |
Using basic thermodynamics and one of the Maxwell relations, we write |
| 112 |
> |
Using basic thermodynamics and one of the Maxwell relations, it can be written that |
| 113 |
|
\begin{equation} |
| 114 |
|
P = T\left(\frac{\partial S}{\partial V}\right)_{T} - |
| 115 |
|
\left(\frac{\partial U}{\partial V}\right)_{T}. |
| 167 |
|
|
| 168 |
|
\item The Extended System Approach |
| 169 |
|
|
| 170 |
< |
Since we are performing these simulations in the NVT ensemble using |
| 170 |
> |
Since these simulations are being performed in the NVT ensemble using |
| 171 |
|
Nos\'e-Hoover thermostatting, the quantity conserved by our integrator |
| 172 |
|
($H_{NVT}$) can be expressed as: |
| 173 |
|
\begin{equation} |
| 191 |
|
\end{equation} |
| 192 |
|
However, $H_{NVT}$ is essentially conserved during these simulations, |
| 193 |
|
so fitting fluctuations of this quantity to obtain meaningful physical |
| 194 |
< |
quantities is somewhat suspect. We also note that this method would |
| 194 |
> |
quantities is somewhat suspect. It should also noted that this method would |
| 195 |
|
fail in periodic systems because the volume itself is fixed in |
| 196 |
|
periodic NVT simulations. |
| 197 |
|
|
| 218 |
|
\begin{equation} |
| 219 |
|
\overleftrightarrow{\mathsf{W}} = \sum_{i} \sum_{j>i} \vec{r}_{ij} \otimes \vec{f}_{ij} |
| 220 |
|
\end{equation} |
| 221 |
< |
During the simulation, we record the internal pressure, $P$, as well as |
| 221 |
> |
During the simulation, the internal pressure, $P$, is recorded as well as |
| 222 |
|
the total energy, $U$, the extended system's hamiltonian, $H_{NVT}$, |
| 223 |
< |
and the particle coordinates. Once we have calculated the time |
| 224 |
< |
dependent volume of the nanoparticle using the convex hull, we can use |
| 225 |
< |
any of these four methods to estimate the bulk modulus. |
| 223 |
> |
and the particle coordinates. Once the time |
| 224 |
> |
dependent volume of the nanoparticle has been calculated using the convex hull, |
| 225 |
> |
any of these four methods can be used to estimate the bulk modulus. |
| 226 |
|
\end{itemize} |
| 227 |
|
|
| 228 |
< |
We find, however, that only the fourth method (the direct pressure |
| 228 |
> |
It is found, however, that only the fourth method (the direct pressure |
| 229 |
|
approach) gives meaningful results. Bulk moduli for the 35 \AA\ |
| 230 |
|
particle were computed with the traditional (Energy vs. Volume) |
| 231 |
|
approach as well as the direct pressure approach. A comparison of the |
| 245 |
|
The Bulk moduli reported in the rest of this paper were computed using |
| 246 |
|
the direct pressure method. |
| 247 |
|
|
| 248 |
< |
To study the frequency of the breathing mode, we have calculated the |
| 249 |
< |
power spectrum for volume ($V$) fluctuations, |
| 248 |
> |
To study the frequency of the breathing mode, the |
| 249 |
> |
power spectrum for volume ($V$) fluctuations have been calculated according to, |
| 250 |
|
\begin{equation} |
| 251 |
|
\rho_{\Delta V}(\omega) = \int_{-\infty}^{\infty} \langle \Delta V(t) |
| 252 |
|
\Delta V(0) \rangle e^{-i \omega t} dt |
| 262 |
|
analysis of the breathing dynamics follows in section |
| 263 |
|
\ref{bulkmod:sec:discussion}. |
| 264 |
|
|
| 265 |
< |
We have also computed the heat capacity for our simulations to verify |
| 265 |
> |
The heat capacity for our simulations has also been computed to verify |
| 266 |
|
the location of the melting transition. Calculations of the heat |
| 267 |
|
capacity were performed on the non-equilibrium, instantaneous heating |
| 268 |
|
simulations, as well as on simulations of nanoparticles that were at |
| 269 |
|
equilibrium at the target temperature. |
| 270 |
|
|
| 271 |
< |
\section{RESULTS} |
| 271 |
> |
\section{Results} |
| 272 |
|
\label{bulkmod:sec:results} |
| 273 |
|
|
| 274 |
< |
\subsection{THE BULK MODULUS AND HEAT CAPACITY} |
| 274 |
> |
\subsection{The Bulk Modulus and Heat Capacity} |
| 275 |
|
|
| 276 |
|
The upper panel in Fig. \ref{fig:BmCp} shows the temperature |
| 277 |
|
dependence of the Bulk Modulus ($K$). In all samples, there is a |
| 298 |
|
Another feature of these transient (non-equilibrium) calculations is |
| 299 |
|
the width of the peak in the heat capacity. Calculation of $C_{p}$ |
| 300 |
|
from longer equilibrium trajectories should indicate {\it sharper} |
| 301 |
< |
features in $C_{p}$ for the larger particles. Since we are initiating |
| 335 |
< |
and observing the melting process itself in these calculation, the |
| 301 |
> |
features in $C_{p}$ for the larger particles. Since the melting process itself is being initiated and observed in these calculations, the |
| 302 |
|
smaller particles melt more rapidly, and thus exhibit sharper features |
| 303 |
|
in $C_{p}$. Indeed, longer trajectories do show that $T_{m}$ occurs |
| 304 |
|
at lower temperatures and with sharper transitions in larger particles |
| 320 |
|
|
| 321 |
|
|
| 322 |
|
|
| 323 |
< |
\subsection{BREATHING MODE DYNAMICS} |
| 323 |
> |
\subsection{Breathing Mode Dynamics} |
| 324 |
|
|
| 325 |
|
Fig.\ref{fig:VolTime} shows representative samples of the volume |
| 326 |
|
vs. time traces for the 20 \AA\ and 35 \AA\ particles at a number of |
| 327 |
< |
different temperatures. We can clearly see that the period of the |
| 327 |
> |
different temperatures. It can clearly be seen that the period of the |
| 328 |
|
breathing mode is dependent on temperature, and that the coherent |
| 329 |
|
oscillations of the particles' volume are destroyed after only a few |
| 330 |
|
ps in the smaller particles, while they live on for 10-20 ps in the |
| 360 |
|
\begin{figure}[htbp] |
| 361 |
|
\centering |
| 362 |
|
\includegraphics[height=3in]{images/volcorr.pdf} |
| 363 |
< |
\caption{Volume fluctuation autocorrelation functions for the 20 \AA\ |
| 364 |
< |
and 35 \AA\ particles at a range of temperatures. Successive |
| 363 |
> |
\caption{Volume fluctuation autocorrelation functions for the 20 \AA\ (lower panel) |
| 364 |
> |
and 35 \AA\ (upper panel) particles at a range of temperatures. Successive |
| 365 |
|
temperatures have been translated upwards by one unit. Note the beat |
| 366 |
|
pattern in the 20 \AA\ particle at 800K.} |
| 367 |
|
\label{fig:volcorr} |
| 370 |
|
|
| 371 |
|
When the power spectrum of the volume autocorrelation functions are |
| 372 |
|
analyzed (Eq. (\ref{eq:volspect})), the samples which exhibit beat |
| 373 |
< |
patterns do indeed show multiple peaks in the power spectrum. We plot |
| 374 |
< |
the period corresponding to the two lowest frequency peaks in |
| 409 |
< |
Fig. \ref{fig:Period}. The smaller particles have the most evident |
| 373 |
> |
patterns do indeed show multiple peaks in the power spectrum. The period corresponding to the two lowest frequency peaks is plotted in |
| 374 |
> |
Fig. \ref{fig:Period}. Smaller particles have the most evident |
| 375 |
|
splitting, particularly as the temperature rises above the melting |
| 376 |
|
points for these particles. |
| 377 |
|
|
| 385 |
|
\end{figure} |
| 386 |
|
|
| 387 |
|
|
| 388 |
< |
\section{DISCUSSION} |
| 388 |
> |
\section{Discussion} |
| 389 |
|
\label{bulkmod:sec:discussion} |
| 390 |
|
|
| 391 |
|
Lamb's classical theory of elastic spheres~\cite{Lamb1882} provides |
| 420 |
|
|
| 421 |
|
In crystalline materials, the speeds depend on the direction of |
| 422 |
|
propagation of the wave relative to the crystal plane.\cite{Kittel:1996fk} |
| 423 |
< |
For the remainder of our analysis, we assume the nanoparticles are |
| 423 |
> |
For the remainder of our analysis, it is assumed that the nanoparticles are |
| 424 |
|
isotropic (which should be valid only above the melting transition). |
| 425 |
|
A more detailed analysis of the lower temperature particles would take |
| 426 |
|
the crystal lattice into account. |
| 427 |
|
|
| 428 |
< |
If we use the experimental values for the elastic constants for 30 |
| 428 |
> |
Using the experimental values for the elastic constants for 30 |
| 429 |
|
\AA\ Au particles at 300K, the low-frequency longitudinal (breathing) |
| 430 |
|
mode should have a period of 2.19 ps while the low-frequency |
| 431 |
|
transverse (toroidal) mode should have a period of 2.11 ps. Although |
| 432 |
< |
the actual calculated frequencies in our simulations are off of these |
| 432 |
> |
the actual calculated frequencies in the simulations are off of these |
| 433 |
|
values, the difference in the periods (0.08 ps) is approximately half |
| 434 |
|
of the splitting observed room-temperature simulations. This, |
| 435 |
|
therefore, may be an explanation for the low-temperature splitting in |
| 442 |
|
Lamb mode using the classical Lamb theory results for isotropic |
| 443 |
|
elastic spheres.\cite{Simon2001} |
| 444 |
|
|
| 445 |
< |
\subsection{MELTED AND PARTIALLY-MELTED PARTICLES} |
| 445 |
> |
\subsection{Melted and Partially-Melted Particles} |
| 446 |
|
|
| 447 |
|
Hartland {\it et al.} have extended the Lamb analysis to include |
| 448 |
|
surface stress ($\gamma$).\cite{HartlandG.V._jp0276092} In this case, the |
