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 that exhibit a
high concentration of steps, 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 that solid surfaces are often structurally,
compositionally, and chemically modified by reactants under operating
conditions.\cite{Tao2008,Tao:2010,Tao2011} The coupling between
surface oxidation states 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 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 an effort to understand the mechanism and timescale for
surface restructuring using molecular simulations.  Since the dynamics
of the process is of particular interest, we utilize classical 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 metal systems exposed to carbon monoxide
were examined in this work. The Pt(557) surface has already been shown
to reconstruct under certain conditions. The Au(557) surface, because
of a weaker interaction with CO, is less likely to undergo this kind
of reconstruction.  MORE HERE ON PT AND AU PREVIOUS WORK.

%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 used classical molecular
dynamics with potential energy surfaces that are specifically tuned
for transition metals.  In particular, we used the EAM potential for
Au-Au and Pt-Pt interactions, while modeling the CO using a rigid
three-site 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 modeling 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, 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{Carbon Monoxide model}
Since previous explanations for the surface rearrangements center on
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, Lennard-Jones parameters ($\sigma$ and
    $\epsilon$), and charges for the CO-CO 
    interactions borrowed from Ref. \bibpunct{}{}{,}{n}{}{,} \protect\cite{Straub}. Distances are in \AA~, energies are 
    in kcal/mol, and charges are in atomic units.}
\centering
\begin{tabular}{| c | c | ccc |}
\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 between the metals and carbon monoxide}

Since the adsorption of CO onto a platinum surface has been the focus
of much experimental \cite{Yeo, Hopster:1978, Ertl:1977, Kelemen:1979}
and theoretical work
\cite{Beurden:2002ys,Pons:1986,Deshlahra:2009,Feibelman:2001,Mason:2004}
there is a significant amount of data on adsorption energies for CO on
clean metal surfaces. Parameters reported by Korzeniewski {\it et
  al.}\cite{Pons:1986} were a starting point for our fits, which were
modified to ensure that the Pt-CO interaction favored the atop binding
position on Pt(111). This resulting binding energies are on the higher
side of the experimentally-reported values. Following Korzeniewski
{\it et al.},\cite{Pons:1986} the Pt-C interaction was fit to a deep
Lennard-Jones interaction to mimic strong, but short-ranged partial
binding between the Pt $d$ orbitals and the $\pi^*$ orbital on CO. The
Pt-O interaction was parameterized to a Morse potential with a large
range parameter ($r_o$).  In most cases, this contributes a weak
repulsion which favors the atop site.  The resulting potential-energy
surface suitably recovers the calculated Pt-C separation 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 DFT calculations 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 $10^{-8}$ Ry.  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 with a 4 x 4 x
4 Monkhorst-Pack {\bf k}-point sampling of the first Brillouin
zone.\cite{Monkhorst:1976,PhysRevB.13.5188} 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 parameters employed in this work are shown in Table 2 and the
binding energies on the 111 surfaces are displayed in Table 3.  To
speed up the computations, charge transfer and polarization are not
being treated in this model, although these effects are likely to
affect binding energies and binding site
preferences.\cite{Deshlahra:2012}

%Table  of Parameters
%Pt Parameter Set 9
%Au Parameter Set 35
\begin{table}[H]
  \caption{Best fit parameters for metal-CO cross-interactions.   Metal-C
    interactions are modeled with Lennard-Jones potential, while the
    (mostly-repulsive) metal-O interactions were fit to Morse
    potentials.  Distances are given in \AA~and energies in kcal/mol. }
\centering
\begin{tabular}{| c | cc | c | ccc |}
\hline
 &  $\sigma$ & $\epsilon$ & & $r$ & $D$ & $\gamma$ (\AA$^{-1}$) \\
\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 for CO on M(111) using the potentials
    described in this work.  All values are in eV}
\centering
\begin{tabular}{| c | cc |}
  \hline
  & Calculated & Experimental \\
  \hline
  \multirow{2}{*}{\textbf{Pt-CO}} & \multirow{2}{*}{-1.9} & -1.4 \bibpunct{}{}{,}{n}{}{,}
  (Ref. \protect\cite{Kelemen:1979}) \\
 & &  -1.9 \bibpunct{}{}{,}{n}{}{,} (Ref. \protect\cite{Yeo}) \\ \hline
  \textbf{Au-CO} & -0.39 & -0.40 \bibpunct{}{}{,}{n}{}{,}  (Ref. \protect\cite{TPD_Gold}) \\
  \hline
\end{tabular}
\end{table}

\subsection{Pt(557) and Au(557) metal interfaces}

Our model systems are composed of 3888 Pt atoms and 3384 Au atoms in a
FCC crystal that have been cut along the 557 plane so that they are
periodic in the {\it x} and {\it y} directions, and have been rotated
to expose two parallel 557 cuts along the positive and negative {\it
  z}-axis.  Simulations of the bare metal interfaces at temperatures
ranging from 300~K to 1200~K were done to observe the relative
stability of the surfaces without a CO overlayer.  

The different bulk (and surface) melting temperatures (1337~K for Au
and 2045~K for Pt) suggest that the reconstruction may happen at
different temperatures for the two metals.  To copy experimental
conditions for the CO-exposed surfaces, the bare surfaces were
initially run in the canonical (NVT) ensemble at 800~K and 1000~K
respectively for 100 ps.  Each surface was exposed to a range of CO
that was initially placed in the vacuum region.  Upon full adsorption,
these amounts correspond to 0\%, 5\%, 25\%, 33\%, and 50\% surface
coverage.  Because of the difference in binding energies, the platinum
systems very rarely had CO that was not bound to the surface, while
the gold surfaces often had a significant CO population in the gas
phase.  These systems were allowed to reach thermal equilibrium (over
5 ns) before being shifted to the microcanonical (NVE) ensemble for
data collection. All of the systems examined had at least 40 ns in the
data collection stage, although simulation times for some of the
systems exceeded 200ns.  All simulations were run using the open
source molecular dynamics package, OpenMD.\cite{Ewald,OOPSE,OpenMD}

% Just results, leave discussion for discussion section
% structure
%       Pt: step wandering, double layers, no triangular motifs
%       Au: step wandering, no double layers
% dynamics
%       diffusion
%       time scale, formation, breakage
\section{Results}
\subsection{Structural remodeling}
Tao {\it et al.} showed experimentally that the Pt(557) surface undergoes
two separate reconstructions upon CO adsorption.\cite{Tao:2010} The first 
reconstruction involves a doubling of the step height and plateau length. Similar
behavior has been seen to occur on numerous surfaces at varying conditions.\cite{}
Of the two systems we examined, the Platinum system showed the most surface 
reconstruction. Additionally, the amount of reconstruction appears to be 
dependent on the amount of CO adsorbed upon the surface. This result is likely
related to the effect that coverage has on surface diffusion. While both systems 
displayed step edge wandering, only the Pt surface underwent doubling within 
the time scales we were modeling. Specifically only the 50 \% coverage Pt system
was observed to undergo doubling in the time scales we were able to monitor. 
Although, the other Platinum systems tended to show more cumulative lateral movement of
the step edges when compared to the Gold systems. The 50 \% Pt system is highlighted 
in figure \ref{fig:reconstruct} at various times along the simulation showing
the evolution of the system.

The second reconstruction on the Pt(557) surface observed by Tao involved the 
formation of triangular clusters that stretched across the plateau between two step edges.
Neither system, within our simulated time scales, experiences this reconstruction. A constructed
system in which the triangular motifs were constructed on the surface will be explored in future 
work and is shown in the supporting information.

\subsection{Dynamics}
While atomistic simulations of stepped surfaces have been performed before \cite{}, they tend to be
performed using Monte Carlo techniques\cite{}. This allows them to efficiently sample the thermodynamic
landscape but at the expense of ignoring the dynamics of the system. Previous work, using STM (?)\cite{},
has been able to visualize the coalescing of steps of (system). The time scale of the image acquisition 
provides an upper bounds for the time required for the doubling to actually occur. While statistical treatments
of step edges are adept at analyzing such systems, it is important to remember that the edges are made
up of individual atoms and thus can be examined in numerous ways.

\subsubsection{Transport of surface metal atoms}
The movement of a step edge is a cooperative effect arising from the individual movements of the atoms
making up the step. An ideal metal surface displaying a low index facet (111, 100, 110) is unlikely to
experience much surface diffusion because of the large energetic barrier to lift an atom out of the surface.
For our surfaces, the presence of step edges provide a source for mobile metal atoms. Breaking away
from the step edge is still an energetic penalty around (value) but is much less than lifting the same metal
atom out from the surface and the penalty lowers even further when CO is present in sufficient quantities
on the surface. Once an adatom exists on the surface, its barrier for diffusion is negligible ( < 4 kcal/mole)
and is well able to explore its terrace because both steps act as barriers constraining the area in which 
diffusion is allowed. By tracking the mobility of individual metal atoms on the surface we were able to determine
the relative diffusion rates and how varying coverages of CO affected the diffusion constants. Close
observation of the mobile metal atoms showed that they were typically in equilibrium with the
step edges, constantly breaking apart and rejoining. Additionally, at times their motion was concerted and
two or more atoms would be observed moving together across the surfaces. The primary challenge in quantifying
the overall surface mobility is in defining ``mobile" vs. ``static" atoms.

A particle was considered mobile once it had traveled more than 2~\AA~ between saved configurations
of the system (10-100 ps). An atom that was truly mobile would typically travel much greater than this, but
the 2~\AA~ cutoff was to prevent the in place vibrational movement of atoms from being included in the analysis.
Since diffusion on  a surface is strongly affected by local structures, in this case the presence of single and double
layer step edges, the diffusion parallel to the step edges was determined separately from the diffusion perpendicular
to these edges. The parallel and perpendicular diffusion constants are shown in figure \ref{fig:diff}.

\subsubsection{Double layer formation}
The increased amounts of diffusion on Pt at the higher CO coverages appears to play a role in the
formation of double layers. Seeing as how that was the only system within our observed simulation time
that showed the formation. As mentioned earlier, previous experimental work has given some insight into
the upper bounds of the time required for enough atoms to move around to allow two steps to coalesce\cite{}.
As seen in figure \ref{fig:reconstruct}, the first appearance of a double layer, a nodal site, appears at 19 ns into
the simulation. Within 10 ns, nearly half of the step has formed the double layer and by 86 ns, the complete
layer has formed. From the appearance of the first node to the complete doubling of the layers, only ~65 ns
have elapsed. The other two layers in this simulation form over a period of ---- and ---- ns respectively.

\begin{figure}[H]
\includegraphics[width=\linewidth]{DiffusionComparison_errorXY_remade.pdf}
\caption{Diffusion constants for mobile surface atoms along directions
  parallel ($\mathbf{D}_{\parallel}$) and perpendicular
  ($\mathbf{D}_{\perp}$) to the 557 step edges as a function of CO
  surface coverage.  Diffusion parallel to the step edge is 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. }
\label{fig:diff}
\end{figure}

%Table of Diffusion Constants
%Add gold?M
% \begin{table}[H]
%   \caption{}
%   \centering
% \begin{tabular}{| c | cc | cc | }
%   \hline
%   &\multicolumn{2}{c|}{\textbf{Platinum}}&\multicolumn{2}{c|}{\textbf{Gold}} \\
%   \hline
%   \textbf{Surface Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$  \\
%   \hline
%   50\% & 4.32(2) & 1.185(8)  & 1.72(2) & 0.455(6) \\
%   33\% & 5.18(3)  & 1.999(5)  & 1.95(2) & 0.337(4)   \\
%   25\% & 5.01(2)  & 1.574(4)  & 1.26(3) & 0.377(6) \\
%   5\%   & 3.61(2)  & 0.355(2)  & 1.84(3)  & 0.169(4)  \\
%   0\%   & 3.27(2)  & 0.147(4)  & 1.50(2)  & 0.194(2)   \\
%   \hline 
% \end{tabular}
% \end{table}

%Discussion
\section{Discussion}

Mechanism for restructuring

There are a number of possible mechanisms to explain the role of
adsorbed CO in restructuring the Pt surface. Quadrupolar repulsion
between adjacent CO molecules adsorbed on the surface is one
possibility.  However, the quadrupole-quadrupole interaction is
short-ranged and is attractive for some orientations.  If the CO
molecules are locked in a specific orientation relative to each other,
this explanation gains some weight.  

Another possible mechanism for the restructuring is in the
destabilization of strong Pt-Pt interactions by CO adsorbed on surface
Pt atoms.  This could have the effect of increasing surface mobility
of these atoms.  

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, 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[width=\linewidth]{ProgressionOfDoubleLayerFormation_yellowCircle.png}
\caption{The Pt(557) / 50\% CO system at a sequence of times after
  initial exposure to the CO: (a) 258 ps, (b) 19 ns, (c) 31.2 ns, and
  (d) 86.1 ns. Disruption of the 557 step edges occurs quickly.  The
  doubling of the layers appears only after two adjacent step edges
  touch.  The circled spot in (b) nucleated the growth of the double
  step observed in the later configurations.}
  \label{fig:reconstruct}
\end{figure}


%Peaks!
\begin{figure}[H]
\includegraphics[width=\linewidth]{doublePeaks_noCO.png}
\caption{}
\end{figure}
\begin{figure}[H]
\includegraphics[width=\linewidth]{557_300K_cleanPDF.pdf}
\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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