trunk/COonPt/firstTry.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{Molecular Dynamics simulations of the surface reconstructions
  of Pt(557) and Au(557) under exposure to CO}

\author{Joseph R. Michalka, Patrick W. McIntyre 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, because of gold's
weaker interaction with CO, is less likely to undergo such a large reconstruction.
%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 used along
with the interaction parameters are reproduced in Table 1. The effective 
dipole moment, calculated from the assigned charges, 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
&  {\it z} & $\sigma$ & $\epsilon$ & q\\
\hline
\textbf{C} & -0.6457 &  0.0262  & 3.83   &   -0.75 \\
\textbf{O} &  0.4843 &   0.1591 &   3.12 &   -0.85 \\
\textbf{M} & 0.0 & -  &  -  &    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 the potential interactions present
in the system. Since the adsorption of CO onto 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}
there is a large amount of data in the literature to fit too. 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. This 
constraint led to the binding energies being on the higher side 
of reported values. 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 with a large $r_o$ 
to contribute a weak repulsion. The resultant potential-energy
surface suitably recovers the calculated Pt-C bond length ( 1.6\AA)\cite{Beurden:2002ys} and affinity
for the atop binding position.\cite{Deshlahra:2012, Hopster:1978}

%where did you actually get the functionals for citation?
%scf calculations, so initial relaxation was of the four layers, but two layers weren't kept fixed, I don't think
%same cutoff for slab and slab + CO ? seems low, although feibelmen had values around there...
The Au-C and Au-O cross-interactions were fit using Lennard-Jones and
Morse potentials, respectively, to reproduce Au-CO binding energies.

The fits were refined against gas-surface calculations using DFT with
a periodic supercell plane-wave basis approach, as implemented in the
{\sc Quantum ESPRESSO} package.\cite{QE-2009} Electron cores are
described with the projector augmented-wave (PAW)
method,\cite{PhysRevB.50.17953,PhysRevB.59.1758} with plane waves
included to an energy cutoff of 20 Ry. Electronic energies are
computed with the PBE implementation of the generalized gradient
approximation (GGA) for gold, carbon, and oxygen that was constructed
by Rappe, Rabe, Kaxiras, and Joannopoulos.\cite{Perdew_GGA,RRKJ_PP}
Ionic relaxations were performed until the energy difference between
subsequent steps was less than 0.0001 eV.  In testing the CO-Au
interaction, Au(111) supercells were constructed of four layers of 4
Au x 2 Au surface planes and separated from vertical images by six
layers of vacuum space. The surface atoms were all allowed to relax.
Supercell calculations were performed nonspin-polarized, and energies
were converged to within 0.03 meV per Au atom with a 4 x 4 x 4
Monkhorst-Pack\cite{Monkhorst:1976,PhysRevB.13.5188} {\bf k}-point
sampling of the first Brillouin zone.  The relaxed gold slab was then
used in numerous single point calculations with CO at various heights
(and angles relative to the surface) to allow fitting of the empirical
force field. 

%Hint at future work
The fit parameter sets employed in this work are shown in Table 2 and their
reproduction of the binding energies are displayed in Table 3. Currently,
charge transfer is not being treated in this system, however, that is a goal
for future work as the effect has been seen to affect binding energies and 
binding site preferences. \cite{Deshlahra:2012}


\subsection{Construction and Equilibration of 557 Metal interfaces}

Our model systems are composed of approximately 4000 metal atoms 
cut along the 557 plane so that they are periodic in the {\it x} and {\it y} 
directions exposing the 557 plane in the {\it z} direction. Runs at various
temperatures ranging from 300~K to 1200~K were started with the intent
of viewing relative stability of the surface when CO was not present in the
system.  Owing to the different melting points (1337~K for Au and 2045~K for Pt),
the bare crystal systems were initially run in the Canonical ensemble at
800~K and 1000~K respectively for 100 ps. Various amounts of CO were
placed in the vacuum region, which upon full adsorption to the surface
corresponded to 5\%, 25\%, 33\%, and 50\% coverages. Because of the 
high temperature and the difference in binding energies, the platinum systems
very rarely had CO that was not adsorbed to the surface whereas the gold systems
often had a substantial minority of CO away from the surface.
These systems were again allowed to reach thermal equilibrium before being run in the
microcanonical ensemble. All of the systems examined in this work were
run for at least 40 ns. A subset that were undergoing interesting effects
have been allowed to continue running with one system approaching 200 ns.
All simulations were run using the open source molecular dynamics package, OpenMD. \cite{Ewald, OOPSE}


%\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 & -0.40~\cite{TPD_Gold} \\
\hline
\end{tabular}
\end{table}


% Just results, leave discussion for discussion section
\section{Results}
\subsection{Diffusion}
An ideal metal surface displaying a low-energy facet, a (111) face for 
instance, is unlikely to experience much surface diffusion because of 
the large energy barrier associated with atoms 'lifting' from the top 
layer to then be able to explore the surface. Rougher surfaces, those 
that already contain numerous adatoms, step edges, and kinks, should 
have concomitantly higher surface diffusion rates. Tao et al. showed 
that the platinum 557 surface undergoes two separate reconstructions 
upon CO adsorption. \cite{Tao:2010} The first reconstruction involves a 
doubling of the step edge height which is accomplished by a doubling 
of the plateau length. The second reconstruction led to the formation of 
triangular motifs stretching across the lengthened plateaus. 

As shown in Figure 2, over a period of approximately 100 ns, the surface
has reconstructed from a 557 surface by doubling the step height and 
step length. Focusing on only the platinum, or gold, atoms that were 
deemed mobile on the surface, an analysis of the surface diffusion was 
performed. A particle was considered mobile once it had traveled more 
than 2~\AA between snapshots. This immediately eliminates all of the 
bulk metal and greatly limits the number of surface atoms examined. 
Since diffusion on a surface is strongly affected by overcoming energy 
barriers, the diffusion parallel to the step edge axis was determined 
separately from the diffusion perpendicular to the step edge. The results 
at various coverages on both platinum and gold are shown in Table 4.

%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 has been running continuously. The lowered diffusion constant at longer run times will be examined in-depth in the discussion section.

\begin{figure}[H]
\includegraphics[scale=0.6]{DiffusionComparison_error.png}
\caption{Diffusion parallel to the step edge will always be higher than that perpendicular to the edge because of the lower energy barrier associated with going from approximately 7 nearest neighbors to 5, as compared to the 3 of an adatom. Additionally, the observed maximum and subsequent decrease for the Pt system suggests that the CO self-interactions are playing a significant role with regards to movement of the platinum atoms around and more importantly across the surface. }
\end{figure}

%Table of Diffusion Constants
%Add gold?M
\begin{table}[H]
\caption{Platinum and gold 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. Units are \AA\textsuperscript{2}/ns}
\centering
\begin{tabular}{| c | cc | cc | c |}
\hline
\textbf{System Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$  & \textbf{Time (ns)}\\
\hline
&\multicolumn{2}{c|}{\textbf{Platinum}}&\multicolumn{2}{c|}{\textbf{Gold}} & \\
\hline
50\% & 4.32 $\pm$ 0.02 & 1.185 $\pm$ 0.008 & 1.72 $\pm$ 0.02 & 0.455 $\pm$ 0.006 & 40 \\
33\% & 5.18 $\pm$ 0.03 & 1.999 $\pm$ 0.005 & 1.95 $\pm$ 0.02 & 0.337 $\pm$ 0.004 & 40   \\
25\% & 5.01 $\pm$ 0.02 & 1.574 $\pm$ 0.004 & 1.26 $\pm$ 0.03 & 0.377 $\pm$ 0.006 & 40  \\
5\%   & 3.61 $\pm$ 0.02 & 0.355 $\pm$ 0.002 & 1.84 $\pm$ 0.03 & 0.169 $\pm$ 0.004 & 40  \\
0\%   & 3.27 $\pm$ 0.02 & 0.147 $\pm$ 0.004 & 1.50 $\pm$ 0.02 & 0.194 $\pm$ 0.002  & 40  \\
\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 of the $\frac{1}{2}$ monolayer system at various times a) 258 ps b) 19 ns c) 31.2 ns and 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!
\begin{figure}[H]
\includegraphics[scale=0.25]{doublePeaks_noCO.png}
\caption{}
\end{figure}
\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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