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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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%Title |
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\title{Molecular Dynamics simulations of the surface reconstructions |
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of Pt(557) and Au(557) under exposure to CO} |
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\author{Joseph R. Michalka, Patrick W. McIntyre 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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\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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%Date |
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\date{Dec 15, 2012} |
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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 that exhibit a |
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high concentration of steps, kink sites, and vacancies at the edges of |
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the facets. These 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 that solid surfaces are often structurally, |
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compositionally, and chemically modified by reactants under operating |
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conditions.\cite{Tao2008,Tao:2010,Tao2011} The coupling between |
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surface oxidation states and catalytic activity for CO oxidation on |
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Pt, for instance, is widely documented.\cite{Ertl08,Hendriksen:2002} |
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Despite the well-documented role of these effects on reactivity, the |
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ability to capture or predict them in atomistic models is somewhat |
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limited. While these effects are perhaps unsurprising on the highly |
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disperse, multi-faceted nanoscale particles that characterize |
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industrial catalysts, they are manifest even on ordered, well-defined |
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surfaces. The Pt(557) surface, for example, exhibits substantial and |
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reversible restructuring under exposure to moderate pressures of |
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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 likely 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: high pressures, temperatures, etc. are able to cause reconstructions of the 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 giving a clearer picture of what processes are occurring on metal surfaces when exposed to \emph{in situ} conditions. But all of these techniques still have difficulties, especially in observing what is occurring on the surfaces 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 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 absorption energies, preferred absorption 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 will undergo 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 an effort to understand the mechanism and timescale for |
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surface restructuring using molecular simulations. Since the dynamics |
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of the process is of particular interest, we utilize classical force |
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fields that represent a compromise between chemical accuracy and the |
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computational efficiency necessary to observe the process of interest. |
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Since restructuring occurs as a result of specific interactions of the |
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catalyst with adsorbates, two metal systems exposed to carbon monoxide |
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were examined in this work. The Pt(557) surface has already been shown |
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to reconstruct under certain conditions. The Au(557) surface, because |
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of a weaker interaction with CO, is less likely to undergo this kind |
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of reconstruction. MORE HERE ON PT AND AU PREVIOUS WORK. |
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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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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 used classical molecular |
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dynamics with potential energy surfaces that are specifically tuned |
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for transition metals. In particular, we used the EAM potential for |
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Au-Au and Pt-Pt interactions, while modeling the CO using a rigid |
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three-site model developed by Straub and Karplus for studying |
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photodissociation of CO from myoglobin.\cite{Straub} The Au-CO and |
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Pt-CO 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 modeling transition metals are based |
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on a non-pairwise additive functional of the local electron |
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density. The embedded atom method (EAM) is perhaps the best known of |
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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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\section{Simulation Methods} |
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Our model systems are composed of nearly 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 large number of low-coordination atoms along the step edges provide a suitable model for industrial catalysts which tend to have a prevalence of lower CN, i.e. more reactive, sites. Drawing from experimental conclusions, the reconstructions seen for the Pt 557 surface involve doubling of the step height and the formation of triangular motifs along the steps \cite{doi:10.1126/science.1182122}. To properly observe these changes, our system size need to be greater than the periodic phenomena we are examining. The large size and the long time scales needed precluded us from using expensive quantum approaches. Thus, a forcefield describing the Metal-Metal, CO-CO, and CO-Metal interactions was parameterized. |
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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. This description of the embedding energy allows this method to more accurately treat surfaces, alloys, and other non-bulk systems. 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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\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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The EAM, Finnis-Sinclair, 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{Carbon Monoxide model} |
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Since previous explanations for the surface rearrangements center on |
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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, Lennard-Jones parameters ($\sigma$ and |
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$\epsilon$), and charges for the CO-CO |
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interactions borrowed from Ref. \bibpunct{}{}{,}{n}{}{,} \protect\cite{Straub}. Distances are in \AA~, energies are |
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in kcal/mol, and charges are in atomic units.} |
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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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& {\it z} & $\sigma$ & $\epsilon$ & q\\ |
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\hline |
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& $\sigma$ & $\epsilon$ & q\\ |
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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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\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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\hline |
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\end{tabular} |
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\end{table} |
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%Cross |
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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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|
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\subsection{Cross-Interactions between the metals and carbon monoxide} |
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|
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Since the adsorption of CO onto a platinum surface has been the focus |
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of much experimental \cite{Yeo, Hopster:1978, Ertl:1977, Kelemen:1979} |
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and theoretical work |
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\cite{Beurden:2002ys,Pons:1986,Deshlahra:2009,Feibelman:2001,Mason:2004} |
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there is a significant amount of data on adsorption energies for CO on |
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clean metal surfaces. Parameters reported by Korzeniewski {\it et |
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al.}\cite{Pons:1986} were a starting point for our fits, which were |
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modified to ensure that the Pt-CO interaction favored the atop binding |
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position on Pt(111). This resulting binding energies are on the higher |
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side of the experimentally-reported values. Following Korzeniewski |
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{\it et al.},\cite{Pons:1986} the Pt-C interaction was fit to a deep |
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Lennard-Jones interaction to mimic strong, but short-ranged partial |
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binding between the Pt $d$ orbitals and the $\pi^*$ orbital on CO. The |
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Pt-O interaction was parameterized to a Morse potential with a large |
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range parameter ($r_o$). In most cases, this contributes a weak |
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repulsion which favors the atop site. The resulting potential-energy |
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surface suitably recovers the calculated Pt-C separation length |
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(1.6~\AA)\cite{Beurden:2002ys} and affinity for the atop binding |
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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 cross-interactions were fit using Lennard-Jones and |
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Morse potentials, respectively, to reproduce Au-CO binding energies. |
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|
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%\subsection{System} |
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%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. |
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The fits were refined against gas-surface DFT calculations with a |
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periodic supercell plane-wave basis approach, as implemented in the |
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{\sc Quantum ESPRESSO} package.\cite{QE-2009} Electron cores are |
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> |
described with the projector augmented-wave (PAW) |
| 255 |
> |
method,\cite{PhysRevB.50.17953,PhysRevB.59.1758} with plane waves |
| 256 |
> |
included to an energy cutoff of 20 Ry. Electronic energies are |
| 257 |
> |
computed with the PBE implementation of the generalized gradient |
| 258 |
> |
approximation (GGA) for gold, carbon, and oxygen that was constructed |
| 259 |
> |
by Rappe, Rabe, Kaxiras, and Joannopoulos.\cite{Perdew_GGA,RRKJ_PP} |
| 260 |
> |
Ionic relaxations were performed until the energy difference between |
| 261 |
> |
subsequent steps was less than $10^{-8}$ Ry. In testing the CO-Au |
| 262 |
> |
interaction, Au(111) supercells were constructed of four layers of 4 |
| 263 |
> |
Au x 2 Au surface planes and separated from vertical images by six |
| 264 |
> |
layers of vacuum space. The surface atoms were all allowed to relax. |
| 265 |
> |
Supercell calculations were performed nonspin-polarized with a 4 x 4 x |
| 266 |
> |
4 Monkhorst-Pack {\bf k}-point sampling of the first Brillouin |
| 267 |
> |
zone.\cite{Monkhorst:1976,PhysRevB.13.5188} The relaxed gold slab was |
| 268 |
> |
then used in numerous single point calculations with CO at various |
| 269 |
> |
heights (and angles relative to the surface) to allow fitting of the |
| 270 |
> |
empirical force field. |
| 271 |
|
|
| 272 |
+ |
%Hint at future work |
| 273 |
+ |
The parameters employed in this work are shown in Table 2 and the |
| 274 |
+ |
binding energies on the 111 surfaces are displayed in Table 3. To |
| 275 |
+ |
speed up the computations, charge transfer and polarization are not |
| 276 |
+ |
being treated in this model, although these effects are likely to |
| 277 |
+ |
affect binding energies and binding site |
| 278 |
+ |
preferences.\cite{Deshlahra:2012} |
| 279 |
|
|
| 280 |
|
%Table of Parameters |
| 281 |
|
%Pt Parameter Set 9 |
| 282 |
|
%Au Parameter Set 35 |
| 283 |
|
\begin{table}[H] |
| 284 |
< |
\caption{Best fit parameters for metal-adsorbate interactions. Distances are in \AA~and energies are in kcal/mol} |
| 284 |
> |
\caption{Best fit parameters for metal-CO cross-interactions. Metal-C |
| 285 |
> |
interactions are modeled with Lennard-Jones potential, while the |
| 286 |
> |
(mostly-repulsive) metal-O interactions were fit to Morse |
| 287 |
> |
potentials. Distances are given in \AA~and energies in kcal/mol. } |
| 288 |
|
\centering |
| 289 |
|
\begin{tabular}{| c | cc | c | ccc |} |
| 290 |
|
\hline |
| 291 |
< |
\multicolumn{7}{| c |}{\textbf{Cross-Interactions} }\\ |
| 291 |
> |
& $\sigma$ & $\epsilon$ & & $r$ & $D$ & $\gamma$ (\AA$^{-1}$) \\ |
| 292 |
|
\hline |
| 115 |
– |
& $\sigma$ & $\epsilon$ & & $r$ & $D$ & $\gamma$ \\ |
| 116 |
– |
\hline |
| 293 |
|
\textbf{Pt-C} & 1.3 & 15 & \textbf{Pt-O} & 3.8 & 3.0 & 1 \\ |
| 294 |
|
\textbf{Au-C} & 1.9 & 6.5 & \textbf{Au-O} & 3.8 & 0.37 & 0.9\\ |
| 295 |
|
|
| 299 |
|
|
| 300 |
|
%Table of energies |
| 301 |
|
\begin{table}[H] |
| 302 |
< |
\caption{Absorption energies in eV} |
| 302 |
> |
\caption{Adsorption energies for CO on M(111) using the potentials |
| 303 |
> |
described in this work. All values are in eV} |
| 304 |
|
\centering |
| 305 |
|
\begin{tabular}{| c | cc |} |
| 306 |
< |
\hline |
| 307 |
< |
& Calc. & Exp. \\ |
| 308 |
< |
\hline |
| 309 |
< |
\textbf{Pt-CO} & -1.9 & -1.4~\cite{Kelemen}-- -1.9~\cite{Yeo} \\ |
| 310 |
< |
\textbf{Au-CO} & -0.39 & -0.44~\cite{TPD_Gold_CO} \\ |
| 311 |
< |
\hline |
| 306 |
> |
\hline |
| 307 |
> |
& Calculated & Experimental \\ |
| 308 |
> |
\hline |
| 309 |
> |
\multirow{2}{*}{\textbf{Pt-CO}} & \multirow{2}{*}{-1.9} & -1.4 \bibpunct{}{}{,}{n}{}{,} |
| 310 |
> |
(Ref. \protect\cite{Kelemen:1979}) \\ |
| 311 |
> |
& & -1.9 \bibpunct{}{}{,}{n}{}{,} (Ref. \protect\cite{Yeo}) \\ \hline |
| 312 |
> |
\textbf{Au-CO} & -0.39 & -0.40 \bibpunct{}{}{,}{n}{}{,} (Ref. \protect\cite{TPD_Gold}) \\ |
| 313 |
> |
\hline |
| 314 |
|
\end{tabular} |
| 315 |
|
\end{table} |
| 316 |
|
|
| 317 |
< |
|
| 139 |
< |
|
| 317 |
> |
\subsection{Pt(557) and Au(557) metal interfaces} |
| 318 |
|
|
| 319 |
+ |
Our model systems are composed of 3888 Pt atoms and 3384 Au atoms in a |
| 320 |
+ |
FCC crystal that have been cut along the 557 plane so that they are |
| 321 |
+ |
periodic in the {\it x} and {\it y} directions, and have been rotated |
| 322 |
+ |
to expose two parallel 557 cuts along the positive and negative {\it |
| 323 |
+ |
z}-axis. Simulations of the bare metal interfaces at temperatures |
| 324 |
+ |
ranging from 300~K to 1200~K were done to observe the relative |
| 325 |
+ |
stability of the surfaces without a CO overlayer. |
| 326 |
|
|
| 327 |
+ |
The different bulk (and surface) melting temperatures (1337~K for Au |
| 328 |
+ |
and 2045~K for Pt) suggest that the reconstruction may happen at |
| 329 |
+ |
different temperatures for the two metals. To copy experimental |
| 330 |
+ |
conditions for the CO-exposed surfaces, the bare surfaces were |
| 331 |
+ |
initially run in the canonical (NVT) ensemble at 800~K and 1000~K |
| 332 |
+ |
respectively for 100 ps. Each surface was exposed to a range of CO |
| 333 |
+ |
that was initially placed in the vacuum region. Upon full adsorption, |
| 334 |
+ |
these amounts correspond to 0\%, 5\%, 25\%, 33\%, and 50\% surface |
| 335 |
+ |
coverage. Because of the difference in binding energies, the platinum |
| 336 |
+ |
systems very rarely had CO that was not bound to the surface, while |
| 337 |
+ |
the gold surfaces often had a significant CO population in the gas |
| 338 |
+ |
phase. These systems were allowed to reach thermal equilibrium (over |
| 339 |
+ |
5 ns) before being shifted to the microcanonical (NVE) ensemble for |
| 340 |
+ |
data collection. All of the systems examined had at least 40 ns in the |
| 341 |
+ |
data collection stage, although simulation times for some of the |
| 342 |
+ |
systems exceeded 200ns. All simulations were run using the open |
| 343 |
+ |
source molecular dynamics package, OpenMD.\cite{Ewald,OOPSE,OpenMD} |
| 344 |
|
|
| 345 |
|
% Just results, leave discussion for discussion section |
| 346 |
+ |
% structure |
| 347 |
+ |
% Pt: step wandering, double layers, no triangular motifs |
| 348 |
+ |
% Au: step wandering, no double layers |
| 349 |
+ |
% dynamics |
| 350 |
+ |
% diffusion |
| 351 |
+ |
% time scale, formation, breakage |
| 352 |
|
\section{Results} |
| 353 |
< |
\subsection{Diffusion} |
| 354 |
< |
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. |
| 353 |
> |
\subsection{Structural remodeling} |
| 354 |
> |
Tao {\it et al.} showed experimentally that the Pt(557) surface undergoes |
| 355 |
> |
two separate reconstructions upon CO adsorption.\cite{Tao:2010} The first |
| 356 |
> |
reconstruction involves a doubling of the step height and plateau length. Similar |
| 357 |
> |
behavior has been seen to occur on numerous surfaces at varying conditions.\cite{Williams:1994,Williams:1991,Pearl} |
| 358 |
> |
Of the two systems we examined, the Platinum system showed the most surface |
| 359 |
> |
reconstruction. Additionally, the amount of reconstruction appears to be |
| 360 |
> |
dependent on the amount of CO adsorbed upon the surface. This result is likely |
| 361 |
> |
related to the effect that coverage has on surface diffusion. While both systems |
| 362 |
> |
displayed step edge wandering, only the Pt surface underwent doubling within |
| 363 |
> |
the time scales we were modeling. Specifically only the 50 \% coverage Pt system |
| 364 |
> |
was observed to undergo a complete doubling in the time scales we were able to monitor. |
| 365 |
> |
This event encouraged us to allow that specific system to run continuously during which two |
| 366 |
> |
more double layers were created. The other systems, not displaying any large scale changes |
| 367 |
> |
of interest, were all stopped after 40 ns of simulation. Neverthless, the other Platinum systems tended to show |
| 368 |
> |
more cumulative lateral movement of the step edges when compared to the Gold systems. |
| 369 |
> |
The 50 \% Pt system is highlighted in figure \ref{fig:reconstruct} at various times along the |
| 370 |
> |
simulation showing the evolution of the system. |
| 371 |
|
|
| 372 |
< |
%Table of Diffusion Constants |
| 373 |
< |
%Add gold?M |
| 374 |
< |
\begin{table}[H] |
| 375 |
< |
\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} |
| 376 |
< |
\centering |
| 153 |
< |
\begin{tabular}{| c | ccc | ccc | c |} |
| 154 |
< |
\hline |
| 155 |
< |
\textbf{System Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & \textbf{Atoms} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & \textbf{Atoms} & \textbf{Time (ns)}\\ |
| 156 |
< |
\hline |
| 157 |
< |
&\multicolumn{3}{c|}{\textbf{Mobile}}&\multicolumn{3}{c|}{\textbf{Surface Atoms}} & \\ |
| 158 |
< |
\hline |
| 159 |
< |
50\% & 3.74 & 0.89 & 497 & 2.05 & 0.49 & 912 & 116 \\ |
| 160 |
< |
50\% & 5.81 & 1.59 & 365 & 2.41 & 0.68 & 912 & 46 \\ |
| 161 |
< |
33\% & 6.73 & 2.47 & 332 & 2.51 & 0.93 & 912 & 46 \\ |
| 162 |
< |
25\% & 5.38 & 2.04 & 361 & 2.18 & 0.84 & 912 & 46 \\ |
| 163 |
< |
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\\ |
| 168 |
< |
\hline |
| 169 |
< |
\end{tabular} |
| 170 |
< |
\end{table} |
| 372 |
> |
The second reconstruction on the Pt(557) surface observed by Tao involved the |
| 373 |
> |
formation of triangular clusters that stretched across the plateau between two step edges. |
| 374 |
> |
Neither system, within our simulated time scales, experiences this reconstruction. A constructed |
| 375 |
> |
system in which the triangular motifs were constructed on the surface will be explored in future |
| 376 |
> |
work and is shown in the supporting information. |
| 377 |
|
|
| 378 |
+ |
\subsection{Dynamics} |
| 379 |
+ |
While atomistic-like simulations of stepped surfaces have been performed before \cite{}, they tend to be |
| 380 |
+ |
performed using Monte Carlo techniques\cite{Williams:1991,Williams:1994}. This allows them to efficiently sample the thermodynamic |
| 381 |
+ |
landscape but at the expense of ignoring the dynamics of the system. Previous work, using STM \cite{Pearl}, |
| 382 |
+ |
has been able to visualize the coalescing of steps of (system). The time scale of the image acquisition, ~ 70 s/image |
| 383 |
+ |
provides an upper bounds for the time required for the doubling to actually occur. While statistical treatments |
| 384 |
+ |
of step edges are adept at analyzing such systems, it is important to remember that the edges are made |
| 385 |
+ |
up of individual atoms and thus can be examined in numerous ways. |
| 386 |
|
|
| 387 |
+ |
\subsubsection{Transport of surface metal atoms} |
| 388 |
+ |
%forcedSystems/stepSeparation |
| 389 |
+ |
The movement of a step edge is a cooperative effect arising from the individual movements of the atoms |
| 390 |
+ |
making up the step. An ideal metal surface displaying a low index facet (111, 100, 110) is unlikely to |
| 391 |
+ |
experience much surface diffusion because of the large energetic barrier to lift an atom out of the surface. |
| 392 |
+ |
For our surfaces however, the presence of step edges provide a source for mobile metal atoms. Breaking away |
| 393 |
+ |
from the step edge still imposes an energetic penalty around 40 kcal/mole, but is much less than lifting the same metal |
| 394 |
+ |
atom out from the surface, > 60 kcal/mole, and the penalty lowers even further when CO is present in sufficient quantities |
| 395 |
+ |
on the surface, ~20 kcal/mole. Once an adatom exists on the surface, its barrier for diffusion is negligible ( < 4 kcal/mole) |
| 396 |
+ |
and is well able to explore its terrace. Atoms traversing terraces is more difficult, but can be overcome through a joining and lifting stage. |
| 397 |
+ |
By tracking the mobility of individual metal atoms on the Platinum and Gold surfaces we were able to determine |
| 398 |
+ |
the relative diffusion rates and how varying coverages of CO affected the rates. Close |
| 399 |
+ |
observation of the mobile metal atoms showed that they were typically in equilibrium with the |
| 400 |
+ |
step edges, constantly breaking apart and rejoining. Additionally, at times their motion was concerted and |
| 401 |
+ |
two or more atoms would be observed moving together across the surfaces. The primary challenge in quantifying |
| 402 |
+ |
the overall surface mobility was in defining ``mobile" vs. ``static" atoms. |
| 403 |
|
|
| 404 |
+ |
A particle was considered mobile once it had traveled more than 2~\AA~ between saved configurations |
| 405 |
+ |
of the system (10-100 ps). An atom that was truly mobile would typically travel much greater than this, but |
| 406 |
+ |
the 2~\AA~ cutoff was to prevent the in-place vibrational movement of atoms from being included in the analysis. |
| 407 |
+ |
Since diffusion on a surface is strongly affected by local structures, in this case the presence of single and double |
| 408 |
+ |
layer step edges, the diffusion parallel to the step edges was determined separately from the diffusion perpendicular |
| 409 |
+ |
to these edges. The parallel and perpendicular diffusion constants are shown in figure \ref{fig:diff}. |
| 410 |
+ |
|
| 411 |
+ |
\subsubsection{Double layer formation} |
| 412 |
+ |
The increased amounts of diffusion on Pt at the higher CO coverages appears to play a role in the |
| 413 |
+ |
formation of double layers, seeing as how that was the only system within our observed simulation time |
| 414 |
+ |
that showed the formation. Despite this being the only system where this reconstruction occurs, three separate layers |
| 415 |
+ |
were formed over the extended run time of this system. As mentioned earlier, previous experimental work has given some insight into |
| 416 |
+ |
the upper bounds of the time required for enough atoms to move around to allow two steps to coalesce\cite{Williams:1991,Pearl}. |
| 417 |
+ |
As seen in figure \ref{fig:reconstruct}, the first appearance of a double layer, a nodal site, appears at 19 ns into |
| 418 |
+ |
the simulation. Within 12 ns, nearly half of the step has formed the double layer and by 86 ns, a smooth complete |
| 419 |
+ |
layer has formed. The double layer is complete by 37 ns but is a bit rough. |
| 420 |
+ |
From the appearance of the first node to the initial doubling of the layers ignoring their roughness took ~20 ns. |
| 421 |
+ |
Another ~40 ns was necessary for the layer to completely straighten. The other two layers in this simulation form |
| 422 |
+ |
over a period of 22 ns and 42 ns respectively. |
| 423 |
+ |
|
| 424 |
+ |
%Evolution of surface |
| 425 |
+ |
\begin{figure}[H] |
| 426 |
+ |
\includegraphics[width=\linewidth]{ProgressionOfDoubleLayerFormation_yellowCircle.png} |
| 427 |
+ |
\caption{The Pt(557) / 50\% CO system at a sequence of times after |
| 428 |
+ |
initial exposure to the CO: (a) 258 ps, (b) 19 ns, (c) 31.2 ns, and |
| 429 |
+ |
(d) 86.1 ns. Disruption of the 557 step edges occurs quickly. The |
| 430 |
+ |
doubling of the layers appears only after two adjacent step edges |
| 431 |
+ |
touch. The circled spot in (b) nucleated the growth of the double |
| 432 |
+ |
step observed in the later configurations.} |
| 433 |
+ |
\label{fig:reconstruct} |
| 434 |
+ |
\end{figure} |
| 435 |
+ |
|
| 436 |
+ |
\begin{figure}[H] |
| 437 |
+ |
\includegraphics[width=\linewidth]{DiffusionComparison_errorXY_remade.pdf} |
| 438 |
+ |
\caption{Diffusion constants for mobile surface atoms along directions |
| 439 |
+ |
parallel ($\mathbf{D}_{\parallel}$) and perpendicular |
| 440 |
+ |
($\mathbf{D}_{\perp}$) to the 557 step edges as a function of CO |
| 441 |
+ |
surface coverage. Diffusion parallel to the step edge is higher |
| 442 |
+ |
than that perpendicular to the edge because of the lower energy |
| 443 |
+ |
barrier associated with going from approximately 7 nearest neighbors |
| 444 |
+ |
to 5, as compared to the 3 of an adatom. Additionally, the observed |
| 445 |
+ |
maximum and subsequent decrease for the Pt system suggests that the |
| 446 |
+ |
CO self-interactions are playing a significant role with regards to |
| 447 |
+ |
movement of the platinum atoms around and more importantly across |
| 448 |
+ |
the surface. } |
| 449 |
+ |
\label{fig:diff} |
| 450 |
+ |
\end{figure} |
| 451 |
+ |
|
| 452 |
+ |
|
| 453 |
+ |
|
| 454 |
+ |
|
| 455 |
|
%Discussion |
| 456 |
|
\section{Discussion} |
| 457 |
< |
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. |
| 457 |
> |
In this paper we have shown that we were able to accurately model the initial reconstruction of the |
| 458 |
> |
Pt (557) surface upon CO adsorption as shown by Tao et al. \cite{Tao:2010}. More importantly, we |
| 459 |
> |
were able to capture the dynamic processes inherent within this reconstruction. |
| 460 |
|
|
| 461 |
< |
\subsection{Diffusion} |
| 462 |
< |
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?) |
| 463 |
< |
\\ |
| 464 |
< |
\\ |
| 465 |
< |
%Evolution of surface |
| 461 |
> |
\subsection{Mechanism for restructuring} |
| 462 |
> |
The increased computational cost to examine this system using molecular dynamics rather than |
| 463 |
> |
a Monte Carlo based approach was necessary so that our predictions on possible mechanisms |
| 464 |
> |
and driving forces would have support not only from thermodynamic arguments but also from the |
| 465 |
> |
actual dynamics of the system. |
| 466 |
> |
|
| 467 |
> |
Comparing the results from simulation to those reported previously by |
| 468 |
> |
Tao et al. the similarities in the platinum and CO system are quite |
| 469 |
> |
strong. As shown in figure \ref{fig:reconstruct}, the simulated platinum system under a CO |
| 470 |
> |
atmosphere will restructure slightly by doubling the terrace |
| 471 |
> |
heights. The restructuring appears to occur slowly, one to two |
| 472 |
> |
platinum atoms at a time. Looking at individual snapshots, these |
| 473 |
> |
adatoms tend to either rise on top of the plateau or break away from |
| 474 |
> |
the step edge and then diffuse perpendicularly to the step direction |
| 475 |
> |
until reaching another step edge. This combination of growth and decay |
| 476 |
> |
of the step edges appears to be in somewhat of a state of dynamic |
| 477 |
> |
equilibrium. However, once two previously separated edges meet as |
| 478 |
> |
shown in figure 1.B, this point tends to act as a focus or growth |
| 479 |
> |
point for the rest of the edge to meet up, akin to that of a |
| 480 |
> |
zipper. From the handful of cases where a double layer was formed |
| 481 |
> |
during the simulation, measuring from the initial appearance of a |
| 482 |
> |
growth point, the double layer tends to be fully formed within |
| 483 |
> |
$\sim$~35 ns. |
| 484 |
> |
|
| 485 |
> |
There are a number of possible mechanisms to explain the role of |
| 486 |
> |
adsorbed CO in restructuring the Pt surface. Quadrupolar repulsion |
| 487 |
> |
between adjacent CO molecules adsorbed on the surface is one |
| 488 |
> |
possibility. However, the quadrupole-quadrupole interaction is |
| 489 |
> |
short-ranged and is attractive for some orientations. If the CO |
| 490 |
> |
molecules are ``locked'' in a specific orientation relative to each other however, |
| 491 |
> |
this explanation gains some weight. The energetic repulsion between two CO |
| 492 |
> |
located a distance of 2.77~\AA~apart (nearest-neighbor distance of Pt) with both in a |
| 493 |
> |
vertical orientation is 8.62 kcal/mole. Moving the CO apart to the second nearest-neighbor |
| 494 |
> |
distance of 4.8~\AA~or 5.54~\AA~drops the repulsion to nearly 0 kcal/mole. SHOW A NUMBER FOR ROTATION. |
| 495 |
> |
As mentioned above, the energy barrier for surface diffusion of a platinum adatom is only 4 kcal/mole. So this |
| 496 |
> |
repulsion between CO can help increase the surface diffusion. However, the residence time of CO was examined |
| 497 |
> |
and while the majority of the CO is on or near the surface throughout the run, it is extremely mobile. This mobility |
| 498 |
> |
suggests that the CO are more likely to shift their positions without necessarily dragging the platinum along |
| 499 |
> |
with them. |
| 500 |
> |
|
| 501 |
> |
Another possible and more likely mechanism for the restructuring is in the |
| 502 |
> |
destabilization of strong Pt-Pt interactions by CO adsorbed on surface |
| 503 |
> |
Pt atoms. This could have the effect of increasing surface mobility |
| 504 |
> |
of these atoms. To test this hypothesis, numerous configurations of |
| 505 |
> |
CO in varying quantities were arranged on the higher and lower plateaus |
| 506 |
> |
around a step on a otherwise clean Pt (557) surface. One representative |
| 507 |
> |
configuration is displayed in figure \ref{fig:lambda}. Single or concerted movement |
| 508 |
> |
of platinum atoms was then examined to determine possible barriers. Because |
| 509 |
> |
of the forced movement along a pre-defined reaction coordinate that may differ |
| 510 |
> |
from the true minimum of this path, only the beginning and ending energies |
| 511 |
> |
are displayed in table \ref{tab:energies}. The presence of CO at suitable |
| 512 |
> |
sites can lead to lowered barriers for platinum breaking apart from the step edge. |
| 513 |
> |
Additionally, as highlighted in figure \ref{fig:lambda}, the presence of CO makes the |
| 514 |
> |
burrowing and lifting nature favorable, whereas without CO, the process is neutral |
| 515 |
> |
in terms of energetics. |
| 516 |
> |
|
| 517 |
> |
%lambda progression of Pt -> shoving its way into the step |
| 518 |
|
\begin{figure}[H] |
| 519 |
< |
\includegraphics[scale=0.5]{ProgressionOfDoubleLayerFormation_yellowCircle.png} |
| 520 |
< |
\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.} |
| 519 |
> |
\includegraphics[width=\linewidth]{lambdaProgression_atopCO.png} |
| 520 |
> |
\caption{A model system of the Pt 557 surface was used as the framework for a reaction coordinate. |
| 521 |
> |
Various numbers, placements, and rotations of CO were examined. The one displayed was a |
| 522 |
> |
representative sample. As shown in Table , relative to the energy at 0\% there is a slight decrease |
| 523 |
> |
upon insertion of the platinum atom into the step edge along with the resultant lifting of the other |
| 524 |
> |
platinum atom.} |
| 525 |
> |
\label{fig:lambda} |
| 526 |
|
\end{figure} |
| 527 |
|
|
| 528 |
|
|
| 529 |
|
|
| 530 |
+ |
\subsection{Diffusion} |
| 531 |
+ |
As shown in the results section, the diffusion parallel to the step edge tends to be |
| 532 |
+ |
much faster than that perpendicular to the step edge. Additionally, the coverage |
| 533 |
+ |
of CO appears to play a slight role in relative rates of diffusion, as shown in figure \ref{fig:diff} |
| 534 |
+ |
Thus, the bottleneck of the double layer formation appears to be the initial formation |
| 535 |
+ |
of this growth point, which seems to be somewhat of a stochastic event. Once it |
| 536 |
+ |
appears, parallel diffusion, along the now slightly angled step edge, will allow for |
| 537 |
+ |
a faster formation of the double layer than if the entire process were dependent on |
| 538 |
+ |
only perpendicular diffusion across the plateaus. Thus, the larger $D_{\perp}$, the |
| 539 |
+ |
more likely a growth point is to be formed. |
| 540 |
+ |
\\ |
| 541 |
|
|
| 191 |
– |
%Peaks! |
| 192 |
– |
\includegraphics[scale=0.25]{doublePeaks_noCO.png} |
| 193 |
– |
\section{Conclusion} |
| 542 |
|
|
| 543 |
+ |
%breaking of the double layer upon removal of CO |
| 544 |
+ |
\begin{figure}[H] |
| 545 |
+ |
\includegraphics[width=\linewidth]{doubleLayerBreaking_greenBlue_whiteLetters.png} |
| 546 |
+ |
\caption{Hi} |
| 547 |
+ |
\label{fig:breaking} |
| 548 |
+ |
\end{figure} |
| 549 |
|
|
| 550 |
|
|
| 551 |
|
|
| 552 |
|
|
| 553 |
+ |
%Peaks! |
| 554 |
+ |
\begin{figure}[H] |
| 555 |
+ |
\includegraphics[width=\linewidth]{doublePeaks_noCO.png} |
| 556 |
+ |
\caption{} |
| 557 |
+ |
\label{fig:peaks} |
| 558 |
+ |
\end{figure} |
| 559 |
|
|
| 560 |
+ |
%clean surface... |
| 561 |
+ |
\begin{figure}[H] |
| 562 |
+ |
\includegraphics[width=\linewidth]{557_300K_cleanPDF.pdf} |
| 563 |
+ |
\caption{} |
| 564 |
|
|
| 565 |
+ |
\end{figure} |
| 566 |
+ |
\label{fig:clean} |
| 567 |
+ |
\section{Conclusion} |
| 568 |
|
|
| 569 |
|
|
| 570 |
+ |
%Things I am not ready to remove yet |
| 571 |
|
|
| 572 |
+ |
%Table of Diffusion Constants |
| 573 |
+ |
%Add gold?M |
| 574 |
+ |
% \begin{table}[H] |
| 575 |
+ |
% \caption{} |
| 576 |
+ |
% \centering |
| 577 |
+ |
% \begin{tabular}{| c | cc | cc | } |
| 578 |
+ |
% \hline |
| 579 |
+ |
% &\multicolumn{2}{c|}{\textbf{Platinum}}&\multicolumn{2}{c|}{\textbf{Gold}} \\ |
| 580 |
+ |
% \hline |
| 581 |
+ |
% \textbf{Surface Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ \\ |
| 582 |
+ |
% \hline |
| 583 |
+ |
% 50\% & 4.32(2) & 1.185(8) & 1.72(2) & 0.455(6) \\ |
| 584 |
+ |
% 33\% & 5.18(3) & 1.999(5) & 1.95(2) & 0.337(4) \\ |
| 585 |
+ |
% 25\% & 5.01(2) & 1.574(4) & 1.26(3) & 0.377(6) \\ |
| 586 |
+ |
% 5\% & 3.61(2) & 0.355(2) & 1.84(3) & 0.169(4) \\ |
| 587 |
+ |
% 0\% & 3.27(2) & 0.147(4) & 1.50(2) & 0.194(2) \\ |
| 588 |
+ |
% \hline |
| 589 |
+ |
% \end{tabular} |
| 590 |
+ |
% \end{table} |
| 591 |
|
|
| 592 |
< |
\end{document} |
| 592 |
> |
\section{Acknowledgments} |
| 593 |
> |
Support for this project was provided by the National Science |
| 594 |
> |
Foundation under grant CHE-0848243 and by the Center for Sustainable |
| 595 |
> |
Energy at Notre Dame (cSEND). Computational time was provided by the |
| 596 |
> |
Center for Research Computing (CRC) at the University of Notre Dame. |
| 597 |
> |
|
| 598 |
> |
\newpage |
| 599 |
> |
\bibliography{firstTryBibliography} |
| 600 |
> |
\end{doublespace} |
| 601 |
> |
\end{document} |
