trunk/COonPt/COonPtAu.tex

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%%
%Introduction
%       Experimental observations
%       Previous work on Pt, CO, etc.
%
%Simulation Methodology
%       FF (fits and parameters)
%       MD (setup, equilibration, collection)
%
%       Analysis of trajectories!!!
%Discussion
%       CO preferences for specific locales
%       CO-CO interactions
%       Differences between Au & Pt
%       Causes of 2_layer reordering in Pt
%Summary
%%

%Title
\title{Investigation of the Pt and Au 557 Surface Reconstructions
  under a CO Atmosphere}
\author{Joseph R. Michalka, Patrick W. MacIntyre and J. Daniel
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\
Department of Chemistry and Biochemistry,\\
University of Notre Dame\\
Notre Dame, Indiana 46556}
%Date
\date{Dec 15,  2012}
%authors

% make the title
\maketitle

\begin{doublespace}

\begin{abstract}

\end{abstract}

\newpage


\section{Introduction}
% Importance: catalytically active metals are important
%       Sub: Knowledge of how their surface structure affects their ability to catalytically facilitate certain reactions is growing, but is more reactionary than predictive
%       Sub: Designing catalysis is the future, and will play an important role in numerous processes (ones that are currently seen to be impractical, or at least inefficient)
% Theory can explore temperatures and pressures which are difficult to work with in experiments
%       Sub: Also, easier to observe what is going on and provide reasons and explanations
%

Industrial catalysts usually consist of small particles exposing
different atomic terminations that exhibit a high concentration of
step, kink sites, and vacancies at the edges of the facets.  These
sites are thought to be the locations of catalytic
activity.\cite{ISI:000083038000001,ISI:000083924800001} There is now
significant evidence to demonstrate that solid surfaces are often
structurally, compositionally, and chemically {\it modified} by
reactants under operating conditions.\cite{Tao2008,Tao:2010,Tao2011}
The coupling between surface oxidation state and catalytic activity
for CO oxidation on Pt, for instance, is widely
documented.\cite{Ertl08,Hendriksen:2002} Despite the well-documented
role of these effects on reactivity, the ability to capture or predict
them in atomistic models is currently somewhat limited.  While these
effects are perhaps unsurprising on the highly disperse, multi-faceted
nanoscale particles that characterize industrial catalysts, they are
manifest even on ordered, well-defined surfaces. The Pt(557) surface,
for example, exhibits substantial and reversible restructuring under
exposure to moderate pressures of carbon monoxide.\cite{Tao:2010}

This work is part of an ongoing effort to understand the causes,
mechanisms and timescales for surface restructuring using molecular
simulation methods.  Since the dynamics of the process is of
particular interest, we utilize classical molecular dynamic methods
with force fields that represent a compromise between chemical
accuracy and the computational efficiency necessary to observe the
process of interest.

Since restructuring occurs as a result of specific interactions of the catalyst
with adsorbates, two metals systems exposed to the same adsorbate, CO,
were examined in this work. The Pt(557) surface has already been shown to 
reconstruct under certain conditions. The Au(557) surface will provide a
useful counterpoint

%Platinum molecular dynamics
%gold molecular dynamics


\section{Simulation Methods}
The challenge in modeling any solid/gas interface problem is the
development of a sufficiently general yet computationally tractable
model of the chemical interactions between the surface atoms and
adsorbates.  Since the interfaces involved are quite large (10$^3$ -
10$^6$ atoms) and respond slowly to perturbations, {\it ab initio}
molecular dynamics
(AIMD),\cite{KRESSE:1993ve,KRESSE:1993qf,KRESSE:1994ul} Car-Parrinello
methods,\cite{CAR:1985bh,Izvekov:2000fv,Guidelli:2000fy} and quantum
mechanical potential energy surfaces remain out of reach.
Additionally, the ``bonds'' between metal atoms at a surface are
typically not well represented in terms of classical pairwise
interactions in the same way that bonds in a molecular material are,
nor are they captured by simple non-directional interactions like the
Coulomb potential.  For this work, we have been using classical
molecular dynamics with potential energy surfaces that are
specifically tuned for transition metals.  In particular, we use the
EAM potential for Au-Au and Pt-Pt interactions, while modeling the CO
using a model developed by Straub and Karplus for studying
photodissociation of CO from myoglobin.\cite{Straub} The Au-CO and Pt-CO
cross interactions were parameterized as part of this work.
  
\subsection{Metal-metal interactions}
Many of the potentials used for classical simulation of transition
metals are based on a non-pairwise additive functional of the local
electron density. The embedded atom method (EAM) is perhaps the best
known of these
methods,\cite{Daw84,Foiles86,Johnson89,Daw89,Plimpton93,Voter95a,Lu97,Alemany98}
but other models like the Finnis-Sinclair\cite{Finnis84,Chen90} and
the quantum-corrected Sutton-Chen method\cite{QSC,Qi99} have simpler
parameter sets. The glue model of Ercolessi {\it et al.} is among the
fastest of these density functional approaches.\cite{Ercolessi88} In
all of these models, atoms are conceptualized as a positively charged
core with a radially-decaying valence electron distribution. To
calculate the energy for embedding the core at a particular location,
the electron density due to the valence electrons at all of the other
atomic sites is computed at atom $i$'s location, 
\begin{equation*}
\bar{\rho}_i = \sum_{j\neq i} \rho_j(r_{ij})
\end{equation*}
Here, $\rho_j(r_{ij})$ is the function that describes the distance
dependence of the valence electron distribution of atom $j$. The
contribution to the potential that comes from placing atom $i$ at that
location is then
\begin{equation*}
V_i =  F[ \bar{\rho}_i ]  + \sum_{j \neq i} \phi_{ij}(r_{ij})
\end{equation*}
where $F[ \bar{\rho}_i ]$ is an energy embedding functional, and
$\phi_{ij}(r_{ij})$ is an pairwise term that is meant to represent the
overlap of the two positively charged cores.  

The {\it modified} embedded atom method (MEAM) adds angular terms to
the electron density functions and an angular screening factor to the
pairwise interaction between two
atoms.\cite{BASKES:1994fk,Lee:2000vn,Thijsse:2002ly,Timonova:2011ve}
MEAM has become widely used to simulate systems in which angular
interactions are important (e.g. silicon,\cite{Timonova:2011ve} bcc
metals,\cite{Lee:2001qf} and also interfaces.\cite{Beurden:2002ys})
MEAM presents significant additional computational costs, however.

The EAM, Finnis-Sinclair, MEAM, and the Quantum Sutton-Chen potentials
have all been widely used by the materials simulation community for
simulations of bulk and nanoparticle
properties,\cite{Chui:2003fk,Wang:2005qy,Medasani:2007uq}
melting,\cite{Belonoshko00,sankaranarayanan:155441,Sankaranarayanan:2005lr}
fracture,\cite{Shastry:1996qg,Shastry:1998dx} crack
propagation,\cite{BECQUART:1993rg} and alloying
dynamics.\cite{Shibata:2002hh} All of these potentials have their
strengths and weaknesses.  One of the strengths common to all of the
methods is the relatively large library of metals for which these
potentials have been
parameterized.\cite{Foiles86,PhysRevB.37.3924,Rifkin1992,mishin99:_inter,mishin01:cu,mishin02:b2nial,zope03:tial_ap,mishin05:phase_fe_ni} 

\subsection{CO}
Since one explanation for the strong surface CO repulsion on metals is
the large linear quadrupole moment of carbon monoxide, the model
chosen for this molecule exhibits this property in an efficient
manner.  We used a model first proposed by Karplus and Straub to study
the photodissociation of CO from myoglobin.\cite{Straub} The Straub and
Karplus model is a rigid three site model which places a massless M
site at the center of mass along the CO bond.  The geometry along with the interaction
parameters are reproduced in Table 1. The effective dipole moment is still
small (0.35 D) while the linear quadrupole (-2.40 D~\AA) is close
to the experimental (-2.63 D~\AA)\cite{QuadrupoleCO} and quantum mechanical predictions (-2.46 D~\AA)\cite{QuadrupoleCOCalc}.
%CO Table
\begin{table}[H]
\caption{Positions, $\sigma$, $\epsilon$ and charges for CO geometry and self-interactions\cite{Straub}. Distances are in \AA~, energies are in kcal/mol, and charges are in $e$.}
\centering
\begin{tabular}{| c | c | ccc |}
\hline
\multicolumn{5}{|c|}{\textbf{Self-Interactions}}\\
\hline
&  r & $\sigma$ & $\epsilon$ & q\\
\hline
\textbf{C} & 0.0 &  0.0262  & 3.83   &   -0.75 \\
\textbf{O} &  1.13 &   0.1591 &   3.12 &   -0.85 \\
\textbf{M} & 0.6457 & -  &  -  &    1.6 \\
\hline
\end{tabular}
\end{table}

\subsection{Cross-Interactions}

One hurdle that must be overcome in classical molecular simulations 
is the proper parameterization of all of the potential interactions present
in the system. CO adsorbed on a platinum surface has been the focus of
many experimental \cite{Yeo, Hopster:1978, Ertl:1977, Kelemen:1979} and theoretical studies. 
\cite{Beurden:2002ys,Pons:1986,Deshlahra:2009,Feibelman:2001,Mason:2004} 
We started with parameters reported by Korzeniewski et al. \cite{Pons:1986} and then
modified them to ensure that the Pt-CO interaction favored 
an atop binding position for the CO upon the Pt surface. Following the method 
laid out by Korzeniewski, the Pt-C interaction was fit to a strong 
Lennard-Jones 12-6 interaction to mimic binding, while the Pt-O interaction
was parameterized to a Morse potential. The resultant potential-energy
surface suitably recovers the calculated Pt-CO bond length (1.1 \AA)\cite{Deshlahra:2012} and affinity
for the atop binding position.\cite{Deshlahra:2012, Hopster:1978}

The Au-C and Au-O interaction parameters were fit to a Lennard-Jones and Morse potential respectively. The binding energies were obtained from quantum calculations carried out using <functional> for gold. 

Numerous single point calculations were performed at various distances of the CO


\subsection{Construction and Equilibration of 557 Metal interfaces}

Our model systems are composed of approximately 4000 metal atoms cut along the 557 plane. The bare crystals were initially run in the Canonical ensemble at 1000K and 800K respectively for Pt and Au. The difference in temperature is necessary because of the two metals different melting points. Various amounts of CO were added to the simulation box and allowed to absorb to the metal surfaces over a short period of 100 ps. After further thermal relaxation the simulations were all run for at least 40 ns. A subset of the runs that showed interesting effects were allowed to run longer. The system


Our model systems are composed of approximately 4000 metal atoms cut along the 557 plane. This cut creates a stepped surface of 6x(111) surface plateaus separated by a single (100) atomic step height. The abundance of low-coordination atoms along the step edges acts as a suitable model for industrial catalysts which tend to have a high concentration of high-index sites. Experimental work has shown that such surfaces are notable for reconstructing upon adsorption\cite{}. Reconstructions have been seen for the Pt 557 surface that involve doubling of the step height and further formation of nano clusters with a triangular motif \cite{doi:10.1126/science.1182122}. To shed insight on whether this reconstruction is limited to the platinum surface, simulations of gold under similar conditions will also be examined. To properly observe these changes, our system size needs to be greater than the periodic phenomena we are examining. The large size and the long time scales needed precluded us from using quantum approaches. Thus, a forcefield describing the Metal-Metal, CO-CO, and CO-Metal interactions was parameterized and the simulations were run using OpenMD\cite{} an open-source molecular dynamics package.


%\subsection{System}
%Once equilibration was reached, the systems were exposed to various sub-monolayer coverage of CO: $0, \frac{1}{10}, \frac{1}{4}, \frac{1}{3},\frac{1}{2}$. The CO was started many \AA~above the surface with random velocity and rotational velocity vectors sampling from a Gaussian distribution centered on the temperature of the equilibrated metal block.  Full adsorption occurred over the period of approximately 10 ps for Pt, while the binding energy between Au and CO is smaller and led to an incomplete adsorption. The metal-metal interactions were treated using the Embedded Atom Method while the Pt-CO and Au-CO interactions were fit to experimental data and quantum calculations. The raised temperature helped shorten the length of the simulations by allowing the activation barrier of reconstruction to be more easily overcome. A few runs at lower temperatures showed the very beginnings of reconstructions, but their simulation lengths limited their usefulness.


%Table  of Parameters
%Pt Parameter Set 9
%Au Parameter Set 35
\begin{table}[H]
\caption{Best fit parameters for metal-adsorbate interactions. Distances are in \AA~and energies are in kcal/mol}
\centering
\begin{tabular}{| c | cc | c | ccc |}
\hline
\multicolumn{7}{| c |}{\textbf{Cross-Interactions} }\\
\hline
 &  $\sigma$ & $\epsilon$ & & $r$ & $D$ & $\gamma$ \\
\hline
\textbf{Pt-C} & 1.3 & 15  & \textbf{Pt-O} & 3.8 & 3.0 & 1 \\
\textbf{Au-C} & 1.9 & 6.5  & \textbf{Au-O} & 3.8 & 0.37 & 0.9\\

\hline
\end{tabular}
\end{table}

%Table of energies
\begin{table}[H]
\caption{Adsorption energies in eV}
\centering
\begin{tabular}{| c | cc |}
\hline
 & Calc. & Exp. \\
\hline
\textbf{Pt-CO} & -1.9 & -1.4~\cite{Kelemen:1979}-- -1.9~\cite{Yeo} \\
\textbf{Au-CO} & -0.39 & -0.44~\cite{TPD_Gold_CO} \\
\hline
\end{tabular}
\end{table}


% Just results, leave discussion for discussion section
\section{Results}
\subsection{Diffusion}
While an ideal metallic surface is unlikely to experience much surface diffusion, high-index surfaces have large numbers of low-coordinated atoms which have a much easier time overcoming the energetic barriers limiting diffusion, leading to easier surface reconstructions. Surface movement was divided between the parallel ($\parallel$) and perpendicular ($\perp$) directions relative to the step edge. We were then able to calculate diffusion constants as a function of CO coverage. As can be seen in Table 4, the presence and amount of CO directly affects the diffusion constants of surface platinum atoms. The presence of two 50\% coverage systems is to show how the diffusion process is affected by time. The majority of the systems were run for approximately 50 ns while the half monolayer system been running continuously. The lowered diffusion constant at longer run times will be examined in-depth in the discussion section.

%Table of Diffusion Constants
%Add gold?M
\begin{table}[H]
\caption{Platinum diffusion constants parallel and perpendicular to the 557 step edge. As the coverage increases, the diffusion constants parallel and  perpendicular to the step edge both initially increase and then decrease slightly. There were two approaches of analysis. One looking at the surface atoms that had moved more than a prescribed amount over the run time and the other looking at all surface atoms. Units are \AA\textsuperscript{2}/ns}
\centering
\begin{tabular}{| c | ccc | ccc | c |}
\hline
\textbf{System Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & \textbf{Atoms} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & \textbf{Atoms} & \textbf{Time (ns)}\\
\hline
&\multicolumn{3}{c|}{\textbf{Mobile}}&\multicolumn{3}{c|}{\textbf{Surface Atoms}} & \\
\hline
50\% & 3.74 & 0.89 & 497 & 2.05 & 0.49 & 912 & 116 \\
50\% & 5.81 & 1.59 & 365 & 2.41 & 0.68 & 912 & 46   \\
33\% & 6.73 & 2.47 & 332 & 2.51 & 0.93 & 912 & 46   \\
25\% & 5.38 & 2.04 & 361 & 2.18 & 0.84 & 912 & 46  \\
5\% & 5.54 & 0.63 & 230 & 1.44 & 0.19 & 912 & 46  \\
0\% & 3.53 & 0.61 & 282 & 1.11 & 0.22 & 912 & 56  \\
\hline
50\%-r & 6.91 & 2.00 & 198 & 2.23 & 0.68  & 925 & 25\\
0\%-r  & 4.73 & 0.27 & 128 & 0.72 & 0.05 & 925 & 43\\
\hline 
\end{tabular}
\end{table}


%Discussion
\section{Discussion}
Comparing the results from simulation to those reported previously by Tao et al. the similarities in the platinum and CO system are quite strong. As shown in figure 1, the simulated platinum system under a CO atmosphere will restructure slightly by doubling the terrace heights. The restructuring appears to occur slowly, one to two platinum atoms at a time. Looking at individual snapshots, these adatoms tend to either rise on top of the plateau or break away from the step edge and then diffuse perpendicularly to the step direction until reaching another step edge. This combination of growth and decay of the step edges appears to be in somewhat of a state of dynamic equilibrium. However, once two previously separated edges meet as shown in figure 1.B, this point tends to act as a focus or growth point for the rest of the edge to meet up, akin to that of a zipper. From the handful of cases where a double layer was formed during the simulation. Measuring from the initial appearance of a growth point, the double layer tends to be fully formed within $\sim$~35 ns.

\subsection{Diffusion}
As shown in the results section, the diffusion parallel to the step edge tends to be much faster than that perpendicular to the step edge. Additionally, the coverage of CO appears to play a slight role in relative rates of diffusion, as shown in Table 4. Thus, the bottleneck of the double layer formation appears to be the initial formation of this growth point, which seems to be somewhat of a stochastic event. Once it appears, parallel diffusion, along the now slightly angled step edge, will allow for a faster formation of the double layer than if the entire process were dependent on only perpendicular diffusion across the plateaus. Thus, the larger $D_{\perp}$, the more likely a growth point is to be formed. One driving force behind this reconstruction appears to be the lowering of surface energy that occurs by doubling the terrace widths. (I'm not really proving this... I have the surface flatness to show it, but surface energy?)
\\
\\
%Evolution of surface
\begin{figure}[H]
\includegraphics[scale=0.5]{ProgressionOfDoubleLayerFormation_yellowCircle.png}
\caption{Four snapshots at various times a) 258 ps b) 19 ns c) 31.2 ns d) 86.1 ns. Slight disruption of the surface occurs fairly quickly. However, the doubling of the layers seems to be very dependent on the initial linking of two separate step edges. The focal point in b, appears to be a growth spot for the rest of the double layer.}
\end{figure}


%Peaks!
\includegraphics[scale=0.25]{doublePeaks_noCO.png}
\section{Conclusion}


\section{Acknowledgments}
Support for this project was provided by the National Science
Foundation under grant CHE-0848243 and by the Center for Sustainable
Energy at Notre Dame (cSEND). Computational time was provided by the
Center for Research Computing (CRC) at the University of Notre Dame.

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