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\begin{document} |
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%Introduction |
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% Experimental observations |
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%Summary |
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– |
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\begin{document} |
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%Title |
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\title{Investigation of the Pt and Au 557 Surface Reconstructions under a CO Atmosphere} |
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\title{Investigation of the Pt and Au 557 Surface Reconstructions |
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under a CO Atmosphere} |
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\author{Joseph R. Michalka, Patrick W. MacIntyre and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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%Date |
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\date{Dec 15, 2012} |
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%authors |
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\author{Joseph R.~Michalka, Patrick W. McIntyre, \& J.~Daniel Gezelter} |
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\doublespacing |
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\begin{doublespace} |
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\begin{abstract} |
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\end{abstract} |
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\newpage |
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\section{Introduction} |
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% Importance: catalytically active metals are important |
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% Sub: Knowledge of how their surface structure affects their ability to catalytically facilitate certain reactions is growing, but is more reactionary than predictive |
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% Sub: Also, easier to observe what is going on and provide reasons and explanations |
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% |
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Industrial catalysts usually consist of small particles exposing |
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different atomic terminations that exhibit a high concentration of |
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step, kink sites, and vacancies at the edges of the facets. These |
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sites are thought to be the locations of catalytic |
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activity.\cite{ISI:000083038000001,ISI:000083924800001} There is now |
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significant evidence to demonstrate that solid surfaces are often |
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structurally, compositionally, and chemically {\it modified} by |
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reactants under operating conditions.\cite{Tao2008,Tao:2010,Tao2011} |
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The coupling between surface oxidation state and catalytic activity |
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for CO oxidation on Pt, for instance, is widely |
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documented.\cite{Ertl08,Hendriksen:2002} Despite the well-documented |
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role of these effects on reactivity, the ability to capture or predict |
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them in atomistic models is currently somewhat limited. While these |
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effects are perhaps unsurprising on the highly disperse, multi-faceted |
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nanoscale particles that characterize industrial catalysts, they are |
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manifest even on ordered, well-defined surfaces. The Pt(557) surface, |
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for example, exhibits substantial and reversible restructuring under |
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exposure to moderate pressures of carbon monoxide.\cite{Tao:2010} |
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High-index surfaces of catalytically active metals are an important area of exploration because they are typically more reactive than an ideal surface of the same metal. The greater number of low-coordinated surface atoms is believed responsible for this increased reactivity \cite{}. Additionally, the activity and specificity of many metals towards certain chemical processes has been shown to strongly depend on the structure of the surface \cite{}. Prior work has also shown that reaction conditions, such as high pressures and high temperatures are able to cause reconstructions of the metallic surface, either through changing the displayed surface facets or by changing the number and types of high-index sites available for reactions \cite{doi:10.1126/science.1197461,doi:10.1021/nn3015322, doi:10.1021/jp302379x}. A greater understanding of these high-index surfaces and the restructuring processes they undergo is needed as a prerequisite for more intelligent catalyst design. While current experimental work has started exploring systems at \emph{in situ} conditions, for a long time such experiments were limited to ideal surfaces in high vacuum. New techniques, such as ambient pressure XPS (AP-XPS) \cite{}, high-pressure XPS (HP-XPS) \cite{}, high-pressure STM \cite{}, environmental transmission electron microscopy (E-TEM) \cite{} and many others, are providing clearer pictures of the processes that are occurring on metal surfaces under these conditions. Nevertheless, all of these techniques still have limitations, especially in observing what is occurring at an atomic level. Theoretical models and simulations in combination with experiment have proven their worth in explaining the underlying reasons for some of these reconstructions \cite{}. |
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\\ |
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By examining two different metal-CO systems the effect that the metal and the metal-CO interaction plays can be elucidated. Our first system is composed of platinum and CO and has been the subject of many experimental and theoretical studies primarily because of platinum's strong reactivity toward CO oxidation. The focus has primarily been on adsorption energies, preferred adsorption sites, and catalytic activities. The second system we examined is composed of gold and CO. The gold-CO interaction is much weaker than the platinum-CO interaction and it seems likely that this difference in attraction would lead to differences in any potential surface reconstructions. |
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%It has also been a good test for new quantum methods because of the difficulty with modeling the preference CO has for the atop binding site \cite{doi:10.1021/jp002302t}. |
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%Now that dynamic surface events are known to play a role in many catalytic systems, additional research is being done to more closely examine many systems. Recent work by Tao et al. \cite{doi:10.1126/science.1182122} shows that a high-index platinum surface undergoes surface reconstructions when exposed to a small amount of CO, $\sim$~1 torr. Unexpectedly, the reconstruction was metastable and reverted upon removal of the CO. Work by McCarthy et al. \cite{doi:10.1021/jp302379x} examined temperature programmed desorption's of CO from various platinum samples and saw that species which had large amounts of low-coordinated surface atoms, highly sputtered surfaces or small nano particles, developed a higher temperature desorption peak, suggesting that binding of CO to the platinum surface is strongly dependent on local geometry. |
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This work is part of an ongoing effort to understand the causes, |
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mechanisms and timescales for surface restructuring using molecular |
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simulation methods. Since the dynamics of the process is of |
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particular interest, we utilize classical molecular dynamic methods |
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with force fields that represent a compromise between chemical |
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accuracy and the computational efficiency necessary to observe the |
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process of interest. |
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Since restructuring occurs as a result of specific interactions of the catalyst |
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with adsorbates, two metals systems exposed to the same adsorbate, CO, |
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were examined in this work. The Pt(557) surface has already been shown to |
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reconstruct under certain conditions. The Au(557) surface, because of gold's |
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weaker interaction with CO, is less likely to undergo such a large reconstruction. |
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%Platinum molecular dynamics |
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%gold molecular dynamics |
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\section{Simulation Methods} |
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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. |
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%Metal |
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\subsection{Metal} |
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Recent metallic forcefields, inspired by density-functional theory, including EAM\cite{doi:10.1103/PhysRevB.29.6443, doi:10.1103/PhysRevB.33.7983} and QSC\cite{} have become very popular for modeling novel metallic systems. What makes these forcefields more suitable for metals than their pair-wise predecessors is that they work with the total electron density of the system in a manner akin to DFT. The energy contributed by a single atom is a function of the total background electron density at the point where the atom is to be embedded. The density at any given point is well-approximated by a linear superposition of the electron density as contributed by all the other atoms in the system. The function describing the energy as related to the density is parameterized for each element, rather than by solving the Kohn-Sham equations which is what allows this method to be used for large systems. The embedding energy is completely enclosed within the functional $F_i[\rho_{h,i}]$ which is dependent on the host density $\rho_{h}$ at atom $i$. The density at $i$ is the sum of the density as generated by the rest of the metal. The $\phi_{ij}$ term is a purely repulsive pair-pair interaction parameterized from effective charge repulsions. |
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%Can I increase the \sum size, not sure how... |
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\begin{equation} |
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E_{tot} = \sum_i F_i[\rho_{h,i}] + \frac{1}{2}\sum_i\sum_{j(\ne i)} \phi_{ij}(R_{ij}) |
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\end{equation} |
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\begin{equation} |
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\rho_{h,i} = \sum_{j (\ne i)} \rho_j^a(R_{ij}) |
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\end{equation} |
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The EAM functional forms are used to model the Au and Pt self-interactions in all of our simulations. |
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%CO |
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The challenge in modeling any solid/gas interface problem is the |
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development of a sufficiently general yet computationally tractable |
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model of the chemical interactions between the surface atoms and |
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adsorbates. Since the interfaces involved are quite large (10$^3$ - |
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10$^6$ atoms) and respond slowly to perturbations, {\it ab initio} |
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molecular dynamics |
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(AIMD),\cite{KRESSE:1993ve,KRESSE:1993qf,KRESSE:1994ul} Car-Parrinello |
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methods,\cite{CAR:1985bh,Izvekov:2000fv,Guidelli:2000fy} and quantum |
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mechanical potential energy surfaces remain out of reach. |
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Additionally, the ``bonds'' between metal atoms at a surface are |
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typically not well represented in terms of classical pairwise |
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interactions in the same way that bonds in a molecular material are, |
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nor are they captured by simple non-directional interactions like the |
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Coulomb potential. For this work, we have been using classical |
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molecular dynamics with potential energy surfaces that are |
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specifically tuned for transition metals. In particular, we use the |
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EAM potential for Au-Au and Pt-Pt interactions, while modeling the CO |
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using a model developed by Straub and Karplus for studying |
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photodissociation of CO from myoglobin.\cite{Straub} The Au-CO and Pt-CO |
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cross interactions were parameterized as part of this work. |
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|
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\subsection{Metal-metal interactions} |
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Many of the potentials used for classical simulation of transition |
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metals are based on a non-pairwise additive functional of the local |
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electron density. The embedded atom method (EAM) is perhaps the best |
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known of these |
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methods,\cite{Daw84,Foiles86,Johnson89,Daw89,Plimpton93,Voter95a,Lu97,Alemany98} |
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but other models like the Finnis-Sinclair\cite{Finnis84,Chen90} and |
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the quantum-corrected Sutton-Chen method\cite{QSC,Qi99} have simpler |
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parameter sets. The glue model of Ercolessi {\it et al.} is among the |
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fastest of these density functional approaches.\cite{Ercolessi88} In |
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all of these models, atoms are conceptualized as a positively charged |
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core with a radially-decaying valence electron distribution. To |
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calculate the energy for embedding the core at a particular location, |
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the electron density due to the valence electrons at all of the other |
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atomic sites is computed at atom $i$'s location, |
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\begin{equation*} |
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\bar{\rho}_i = \sum_{j\neq i} \rho_j(r_{ij}) |
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\end{equation*} |
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Here, $\rho_j(r_{ij})$ is the function that describes the distance |
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dependence of the valence electron distribution of atom $j$. The |
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contribution to the potential that comes from placing atom $i$ at that |
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location is then |
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\begin{equation*} |
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V_i = F[ \bar{\rho}_i ] + \sum_{j \neq i} \phi_{ij}(r_{ij}) |
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\end{equation*} |
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where $F[ \bar{\rho}_i ]$ is an energy embedding functional, and |
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$\phi_{ij}(r_{ij})$ is an pairwise term that is meant to represent the |
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overlap of the two positively charged cores. |
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|
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The {\it modified} embedded atom method (MEAM) adds angular terms to |
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the electron density functions and an angular screening factor to the |
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pairwise interaction between two |
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atoms.\cite{BASKES:1994fk,Lee:2000vn,Thijsse:2002ly,Timonova:2011ve} |
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MEAM has become widely used to simulate systems in which angular |
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interactions are important (e.g. silicon,\cite{Timonova:2011ve} bcc |
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metals,\cite{Lee:2001qf} and also interfaces.\cite{Beurden:2002ys}) |
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MEAM presents significant additional computational costs, however. |
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|
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The EAM, Finnis-Sinclair, MEAM, and the Quantum Sutton-Chen potentials |
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have all been widely used by the materials simulation community for |
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simulations of bulk and nanoparticle |
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properties,\cite{Chui:2003fk,Wang:2005qy,Medasani:2007uq} |
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melting,\cite{Belonoshko00,sankaranarayanan:155441,Sankaranarayanan:2005lr} |
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fracture,\cite{Shastry:1996qg,Shastry:1998dx} crack |
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propagation,\cite{BECQUART:1993rg} and alloying |
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dynamics.\cite{Shibata:2002hh} All of these potentials have their |
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strengths and weaknesses. One of the strengths common to all of the |
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methods is the relatively large library of metals for which these |
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potentials have been |
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parameterized.\cite{Foiles86,PhysRevB.37.3924,Rifkin1992,mishin99:_inter,mishin01:cu,mishin02:b2nial,zope03:tial_ap,mishin05:phase_fe_ni} |
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|
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\subsection{CO} |
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Our CO model was obtained from work done by Karplus and Straub\cite{}. In their description of the biological importance of CO they developed an accurate quadrupolar model of CO which we make use of in this work. It has been suggested that the strong electrostatic repulsion that arises from this linear quadrupole may play an important role in the restructuring of metal surfaces to which CO is bound\cite{}. |
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Since one explanation for the strong surface CO repulsion on metals is |
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the large linear quadrupole moment of carbon monoxide, the model |
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chosen for this molecule exhibits this property in an efficient |
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manner. We used a model first proposed by Karplus and Straub to study |
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the photodissociation of CO from myoglobin.\cite{Straub} The Straub and |
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Karplus model is a rigid three site model which places a massless M |
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site at the center of mass along the CO bond. The geometry used along |
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with the interaction parameters are reproduced in Table 1. The effective |
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dipole moment, calculated from the assigned charges, is still |
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small (0.35 D) while the linear quadrupole (-2.40 D~\AA) is close |
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to the experimental (-2.63 D~\AA)\cite{QuadrupoleCO} and quantum |
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mechanical predictions (-2.46 D~\AA)\cite{QuadrupoleCOCalc}. |
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%CO Table |
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\begin{table}[H] |
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\caption{$\sigma$, $\epsilon$ and charges for CO self-interactions\cite{}. Distances are in \AA~, energies are in kcal/mol, and charges are in $e$.} |
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\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 $e$.} |
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\centering |
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\begin{tabular}{| c | ccc |} |
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\begin{tabular}{| c | c | ccc |} |
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\hline |
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\multicolumn{4}{|c|}{\textbf{Self-Interactions}}\\ |
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\multicolumn{5}{|c|}{\textbf{Self-Interactions}}\\ |
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\hline |
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& $\sigma$ & $\epsilon$ & q\\ |
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& {\it z} & $\sigma$ & $\epsilon$ & q\\ |
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\hline |
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\textbf{C} & 0.0262 & 3.83 & -0.75 \\ |
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\textbf{O} & 0.1591 & 3.12 & -0.85 \\ |
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\textbf{M} & - & - & 1.6 \\ |
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\textbf{C} & -0.6457 & 0.0262 & 3.83 & -0.75 \\ |
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\textbf{O} & 0.4843 & 0.1591 & 3.12 & -0.85 \\ |
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\textbf{M} & 0.0 & - & - & 1.6 \\ |
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\hline |
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\end{tabular} |
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\end{table} |
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%Cross |
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|
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\subsection{Cross-Interactions} |
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To finish the forcefield, the cross-interactions between the metals and the CO needed to be parameterized. Previous attempts at parameterization have used two different functional forms to model these interactions\cite{}. A LJ model was fit for the Metal-Carbon interaction and a Morse potential was parameterized for the Metal-Oxygen interaction. The parameter sets chosen, as shown in Table 2, did a suitable job at reproducing experimental adsorption energies as shown in Table 3, but more importantly, they were able to capture the binding site preference. The Pt-CO parameters show a slight preference for the atop binding site which matches the experimental observations. |
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One hurdle that must be overcome in classical molecular simulations |
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is the proper parameterization of the potential interactions present |
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in the system. Since the adsorption of CO onto a platinum surface has been |
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the focus of many experimental \cite{Yeo, Hopster:1978, Ertl:1977, Kelemen:1979} |
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and theoretical studies \cite{Beurden:2002ys,Pons:1986,Deshlahra:2009,Feibelman:2001,Mason:2004} |
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there is a large amount of data in the literature to fit too. We started with parameters |
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reported by Korzeniewski et al. \cite{Pons:1986} and then |
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modified them to ensure that the Pt-CO interaction favored |
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an atop binding position for the CO upon the Pt surface. This |
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constraint led to the binding energies being on the higher side |
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of reported values. Following the method laid out by Korzeniewski, |
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the Pt-C interaction was fit to a strong Lennard-Jones 12-6 |
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interaction to mimic binding, while the Pt-O interaction |
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was parameterized to a Morse potential with a large $r_o$ |
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to contribute a weak repulsion. The resultant potential-energy |
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surface suitably recovers the calculated Pt-C bond length ( 1.6\AA)\cite{Beurden:2002ys} and affinity |
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for the atop binding position.\cite{Deshlahra:2012, Hopster:1978} |
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|
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%where did you actually get the functionals for citation? |
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%scf calculations, so initial relaxation was of the four layers, but two layers weren't kept fixed, I don't think |
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%same cutoff for slab and slab + CO ? seems low, although feibelmen had values around there... |
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The Au-C and Au-O interaction parameters were also fit to a Lennard-Jones |
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and Morse potential respectively, to reproduce Au-CO binding energies. |
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These energies were obtained from quantum calculations carried out using |
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the PBE GGA exchange-correlation functionals\cite{Perdew_GGA} for gold, carbon, and oxygen |
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constructed by Rappe, Rabe, Kaxiras, and Joannopoulos. \cite{RRKJ_PP}. |
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All calculations were run using the {\sc Quantum ESPRESSO} package. \cite{QE-2009} |
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First, a four layer slab of gold comprised of 32 atoms displaying a (111) surface was |
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converged using a 4X4X4 grid of Monkhorst-Pack \emph{k}-points.\cite{Monkhorst:1976} |
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The kinetic energy of the wavefunctions were truncated at 20 Ry while the |
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cutoff for the charge density and potential was set at 80 Ry. This relaxed |
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gold slab was then used in numerous single point calculations with CO at various heights |
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to create a potential energy surface for the Au-CO interaction. |
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|
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%Hint at future work |
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The fit parameter sets employed in this work are shown in Table 2 and their |
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reproduction of the binding energies are displayed in Table 3. Currently, |
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charge transfer is not being treated in this system, however, that is a goal |
| 265 |
+ |
for future work as the effect has been seen to affect binding energies and |
| 266 |
+ |
binding site preferences. \cite{Deshlahra:2012} |
| 267 |
|
|
| 268 |
+ |
|
| 269 |
+ |
|
| 270 |
+ |
|
| 271 |
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\subsection{Construction and Equilibration of 557 Metal interfaces} |
| 272 |
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|
| 273 |
+ |
Our model systems are composed of approximately 4000 metal atoms |
| 274 |
+ |
cut along the 557 plane so that they are periodic in the {\it x} and {\it y} |
| 275 |
+ |
directions exposing the 557 plane in the {\it z} direction. Runs at various |
| 276 |
+ |
temperatures ranging from 300~K to 1200~K were started with the intent |
| 277 |
+ |
of viewing relative stability of the surface when CO was not present in the |
| 278 |
+ |
system. Owing to the different melting points (1337~K for Au and 2045~K for Pt), |
| 279 |
+ |
the bare crystal systems were initially run in the Canonical ensemble at |
| 280 |
+ |
800~K and 1000~K respectively for 100 ps. Various amounts of CO were |
| 281 |
+ |
placed in the vacuum region, which upon full adsorption to the surface |
| 282 |
+ |
corresponded to 5\%, 25\%, 33\%, and 50\% coverages. These systems |
| 283 |
+ |
were again allowed to reach thermal equilibrium before being run in the |
| 284 |
+ |
microcanonical ensemble. All of the systems examined in this work were |
| 285 |
+ |
run for at least 40 ns. A subset that were undergoing interesting effects |
| 286 |
+ |
have been allowed to continue running with one system approaching 200 ns. |
| 287 |
+ |
All simulations were run using the open source molecular dynamics package, OpenMD. \cite{Ewald, OOPSE} |
| 288 |
+ |
|
| 289 |
+ |
|
| 290 |
+ |
|
| 291 |
+ |
|
| 292 |
+ |
|
| 293 |
+ |
|
| 294 |
|
%\subsection{System} |
| 295 |
|
%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. |
| 296 |
|
|
| 322 |
|
\hline |
| 323 |
|
& Calc. & Exp. \\ |
| 324 |
|
\hline |
| 325 |
< |
\textbf{Pt-CO} & -1.9 & -1.4~\cite{Kelemen}-- -1.9~\cite{Yeo} \\ |
| 325 |
> |
\textbf{Pt-CO} & -1.9 & -1.4~\cite{Kelemen:1979}-- -1.9~\cite{Yeo} \\ |
| 326 |
|
\textbf{Au-CO} & -0.39 & -0.44~\cite{TPD_Gold_CO} \\ |
| 327 |
|
\hline |
| 328 |
|
\end{tabular} |
| 341 |
|
%Table of Diffusion Constants |
| 342 |
|
%Add gold?M |
| 343 |
|
\begin{table}[H] |
| 344 |
< |
\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} |
| 344 |
> |
\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} |
| 345 |
|
\centering |
| 346 |
< |
\begin{tabular}{| c | ccc | ccc | c |} |
| 346 |
> |
\begin{tabular}{| c | cc | cc | c |} |
| 347 |
|
\hline |
| 348 |
< |
\textbf{System Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & \textbf{Atoms} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & \textbf{Atoms} & \textbf{Time (ns)}\\ |
| 348 |
> |
\textbf{System Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & \textbf{Time (ns)}\\ |
| 349 |
|
\hline |
| 350 |
< |
&\multicolumn{3}{c|}{\textbf{Mobile}}&\multicolumn{3}{c|}{\textbf{Surface Atoms}} & \\ |
| 350 |
> |
&\multicolumn{2}{c|}{\textbf{Platinum}}&\multicolumn{2}{c|}{\textbf{Gold}} & \\ |
| 351 |
|
\hline |
| 352 |
< |
50\% & 3.74 & 0.89 & 497 & 2.05 & 0.49 & 912 & 116 \\ |
| 353 |
< |
50\% & 5.81 & 1.59 & 365 & 2.41 & 0.68 & 912 & 46 \\ |
| 354 |
< |
33\% & 6.73 & 2.47 & 332 & 2.51 & 0.93 & 912 & 46 \\ |
| 355 |
< |
25\% & 5.38 & 2.04 & 361 & 2.18 & 0.84 & 912 & 46 \\ |
| 356 |
< |
5\% & 5.54 & 0.63 & 230 & 1.44 & 0.19 & 912 & 46 \\ |
| 164 |
< |
0\% & 3.53 & 0.61 & 282 & 1.11 & 0.22 & 912 & 56 \\ |
| 165 |
< |
\hline |
| 166 |
< |
50\%-r & 6.91 & 2.00 & 198 & 2.23 & 0.68 & 925 & 25\\ |
| 167 |
< |
0\%-r & 4.73 & 0.27 & 128 & 0.72 & 0.05 & 925 & 43\\ |
| 352 |
> |
50\% & 4.32 $\pm$ 0.02 & 1.185 $\pm$ 0.008 & 1.72 $\pm$ 0.02 & 0.455 $\pm$ 0.006 & 40 \\ |
| 353 |
> |
33\% & 5.18 $\pm$ 0.03 & 1.999 $\pm$ 0.005 & 1.95 $\pm$ 0.02 & 0.337 $\pm$ 0.004 & 40 \\ |
| 354 |
> |
25\% & 5.01 $\pm$ 0.02 & 1.574 $\pm$ 0.004 & 1.26 $\pm$ 0.03 & 0.377 $\pm$ 0.006 & 40 \\ |
| 355 |
> |
5\% & 3.61 $\pm$ 0.02 & 0.355 $\pm$ 0.002 & 1.84 $\pm$ 0.03 & 0.169 $\pm$ 0.004 & 40 \\ |
| 356 |
> |
0\% & 3.27 $\pm$ 0.02 & 0.147 $\pm$ 0.004 & 1.50 $\pm$ 0.02 & 0.194 $\pm$ 0.002 & 40 \\ |
| 357 |
|
\hline |
| 358 |
|
\end{tabular} |
| 359 |
|
\end{table} |
| 382 |
|
\section{Conclusion} |
| 383 |
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|
| 384 |
|
|
| 385 |
+ |
\section{Acknowledgments} |
| 386 |
+ |
Support for this project was provided by the National Science |
| 387 |
+ |
Foundation under grant CHE-0848243 and by the Center for Sustainable |
| 388 |
+ |
Energy at Notre Dame (cSEND). Computational time was provided by the |
| 389 |
+ |
Center for Research Computing (CRC) at the University of Notre Dame. |
| 390 |
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< |
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< |
\end{document} |
| 391 |
> |
\newpage |
| 392 |
> |
\bibliography{firstTryBibliography} |
| 393 |
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\end{doublespace} |
| 394 |
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\end{document} |
