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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 |
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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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\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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%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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\begin{doublespace} |
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\begin{abstract} |
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We examine potential surface reconstructions of Pt and Au (557) under various CO coverages using molecular dynamics in order to find possible mechanisms and dynamics for the restructuring. The metal-CO interactions were parameterized as part of this work so that a large scale treatment of this system could be undertaken. The relative binding strengths of the metal-CO interactions were found to play a large role with regards to step edge stability and adatom diffusion. A small correlation between coverage and the size of the diffusion constant was also determined. These results appear sufficient to explain the reconstructions observed on the Pt systems and the lack of reconstructions on the Au systems. |
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\end{abstract} |
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\newpage |
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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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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 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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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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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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This work is an attempt 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 are of particular interest, we employ classical force |
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fields that represent a compromise between chemical accuracy and the |
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computational efficiency necessary to simulate the 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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Restructuring can occur as a result of specific interactions of the |
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catalyst with adsorbates. In this work, two metal systems exposed |
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to carbon monoxide were examined. 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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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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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\cite{EAM}, 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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\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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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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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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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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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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$\phi_{ij}(r_{ij})$ is a pairwise term that is meant to represent the |
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repulsive overlap of the two positively charged cores. |
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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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% 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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The EAM, Finnis-Sinclair, MEAM, and the Quantum Sutton-Chen potentials |
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The EAM, Finnis-Sinclair, and the Quantum Sutton-Chen (QSC) 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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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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parameterized.\cite{Foiles86,PhysRevB.37.3924,Rifkin1992,mishin99:_inter,mishin01:cu,mishin02:b2nial,zope03:tial_ap,mishin05:phase_fe_ni} |
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\subsection{CO} |
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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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\subsection{Carbon Monoxide model} |
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Previous explanations for the surface rearrangements center on |
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the large linear quadrupole moment of carbon monoxide. |
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We used a model first proposed by Karplus and Straub to study |
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the photodissociation of CO from myoglobin because it reproduces |
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the quadrupole moment well.\cite{Straub} The Straub and |
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Karplus model, treats CO as a rigid three site molecule which places a massless M |
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site at the center of mass position along the CO bond. The geometry used along |
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with the interaction parameters are reproduced in Table~\ref{tab:CO}. 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{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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\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 | c | ccc |} |
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\hline |
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\multicolumn{5}{|c|}{\textbf{Self-Interactions}}\\ |
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\hline |
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& {\it z} & $\sigma$ & $\epsilon$ & q\\ |
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\hline |
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\textbf{C} & -0.6457 & 0.0262 & 3.83 & -0.75 \\ |
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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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\label{tab:CO} |
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\end{table} |
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\subsection{Cross-Interactions} |
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\subsection{Cross-Interactions between the metals and carbon monoxide} |
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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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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). These parameters are reproduced in Table~\ref{tab:co_parameters} |
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This resulted in binding energies that are slightly higher |
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than the experimentally-reported values as shown in Table~\ref{tab:co_energies}. 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 at a larger |
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minimum distance, ($r_o$). This was chosen so that the C would be preferred |
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over O as the binder to the surface. In most cases, this parameterization 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 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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The Au-C and Au-O cross-interactions were also fit using Lennard-Jones and |
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Morse potentials, respectively, to reproduce Au-CO binding energies. |
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The limited experimental data for CO adsorption on Au lead us to refine our fits against DFT. |
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Adsorption energies were obtained from 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) |
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method,\cite{PhysRevB.50.17953,PhysRevB.59.1758} with plane waves |
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included to an energy cutoff of 20 Ry. Electronic energies are |
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computed with the PBE implementation of the generalized gradient |
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approximation (GGA) for gold, carbon, and oxygen that was constructed |
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by Rappe, Rabe, Kaxiras, and Joannopoulos.\cite{Perdew_GGA,RRKJ_PP} |
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In testing the Au-CO interaction, Au(111) supercells were constructed of four layers of 4 |
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Au x 2 Au surface planes and separated from vertical images by six |
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layers of vacuum space. The surface atoms were all allowed to relax |
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before CO was added to the system. Electronic relaxations were |
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performed until the energy difference between subsequent steps |
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was less than $10^{-8}$ Ry. Nonspin-polarized supercell calculations |
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were performed with a 4~x~4~x~4 Monkhorst-Pack {\bf k}-point sampling of the first Brillouin |
| 270 |
> |
zone.\cite{Monkhorst:1976,PhysRevB.13.5188} The relaxed gold slab was |
| 271 |
> |
then used in numerous single point calculations with CO at various |
| 272 |
> |
heights (and angles relative to the surface) to allow fitting of the |
| 273 |
> |
empirical force field. |
| 274 |
|
|
| 275 |
|
%Hint at future work |
| 276 |
< |
The fit parameter sets employed in this work are shown in Table 2 and their |
| 277 |
< |
reproduction of the binding energies are displayed in Table 3. Currently, |
| 278 |
< |
charge transfer is not being treated in this system, however, that is a goal |
| 279 |
< |
for future work as the effect has been seen to affect binding energies and |
| 280 |
< |
binding site preferences. \cite{Deshlahra:2012} |
| 276 |
> |
The parameters employed for the metal-CO cross-interactions in this work |
| 277 |
> |
are shown in Table~\ref{co_parameters} and the binding energies on the |
| 278 |
> |
(111) surfaces are displayed in Table~\ref{co_energies}. Charge transfer |
| 279 |
> |
and polarization are neglected in this model, although these effects are likely to |
| 280 |
> |
affect binding energies and binding site preferences, and will be added in |
| 281 |
> |
a future work.\cite{Deshlahra:2012,StreitzMintmire} |
| 282 |
|
|
| 268 |
– |
|
| 269 |
– |
|
| 270 |
– |
|
| 271 |
– |
\subsection{Construction and Equilibration of 557 Metal interfaces} |
| 272 |
– |
|
| 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 |
– |
|
| 297 |
– |
|
| 283 |
|
%Table of Parameters |
| 284 |
|
%Pt Parameter Set 9 |
| 285 |
|
%Au Parameter Set 35 |
| 286 |
|
\begin{table}[H] |
| 287 |
< |
\caption{Best fit parameters for metal-adsorbate interactions. Distances are in \AA~and energies are in kcal/mol} |
| 287 |
> |
\caption{Best fit parameters for metal-CO cross-interactions. Metal-C |
| 288 |
> |
interactions are modeled with Lennard-Jones potential, while the |
| 289 |
> |
(mostly-repulsive) metal-O interactions were fit to Morse |
| 290 |
> |
potentials. Distances are given in \AA~and energies in kcal/mol. } |
| 291 |
|
\centering |
| 292 |
|
\begin{tabular}{| c | cc | c | ccc |} |
| 293 |
|
\hline |
| 294 |
< |
\multicolumn{7}{| c |}{\textbf{Cross-Interactions} }\\ |
| 294 |
> |
& $\sigma$ & $\epsilon$ & & $r$ & $D$ & $\gamma$ (\AA$^{-1}$) \\ |
| 295 |
|
\hline |
| 308 |
– |
& $\sigma$ & $\epsilon$ & & $r$ & $D$ & $\gamma$ \\ |
| 309 |
– |
\hline |
| 296 |
|
\textbf{Pt-C} & 1.3 & 15 & \textbf{Pt-O} & 3.8 & 3.0 & 1 \\ |
| 297 |
|
\textbf{Au-C} & 1.9 & 6.5 & \textbf{Au-O} & 3.8 & 0.37 & 0.9\\ |
| 298 |
|
|
| 299 |
|
\hline |
| 300 |
|
\end{tabular} |
| 301 |
+ |
\label{tab:co_parameters} |
| 302 |
|
\end{table} |
| 303 |
|
|
| 304 |
|
%Table of energies |
| 305 |
|
\begin{table}[H] |
| 306 |
< |
\caption{Adsorption energies in eV} |
| 306 |
> |
\caption{Adsorption energies for CO on M(111) using the potentials |
| 307 |
> |
described in this work. All values are in eV} |
| 308 |
|
\centering |
| 309 |
|
\begin{tabular}{| c | cc |} |
| 310 |
< |
\hline |
| 311 |
< |
& Calc. & Exp. \\ |
| 312 |
< |
\hline |
| 313 |
< |
\textbf{Pt-CO} & -1.9 & -1.4~\cite{Kelemen:1979}-- -1.9~\cite{Yeo} \\ |
| 314 |
< |
\textbf{Au-CO} & -0.39 & -0.44~\cite{TPD_Gold_CO} \\ |
| 315 |
< |
\hline |
| 310 |
> |
\hline |
| 311 |
> |
& Calculated & Experimental \\ |
| 312 |
> |
\hline |
| 313 |
> |
\multirow{2}{*}{\textbf{Pt-CO}} & \multirow{2}{*}{-1.9} & -1.4 \bibpunct{}{}{,}{n}{}{,} |
| 314 |
> |
(Ref. \protect\cite{Kelemen:1979}) \\ |
| 315 |
> |
& & -1.9 \bibpunct{}{}{,}{n}{}{,} (Ref. \protect\cite{Yeo}) \\ \hline |
| 316 |
> |
\textbf{Au-CO} & -0.39 & -0.40 \bibpunct{}{}{,}{n}{}{,} (Ref. \protect\cite{TPD_Gold}) \\ |
| 317 |
> |
\hline |
| 318 |
|
\end{tabular} |
| 319 |
+ |
\label{tab:co_energies} |
| 320 |
|
\end{table} |
| 321 |
|
|
| 322 |
+ |
\subsection{Pt(557) and Au(557) metal interfaces} |
| 323 |
|
|
| 324 |
+ |
Our model systems are composed of 3888 Pt atoms and 3384 Au atoms in a |
| 325 |
+ |
FCC crystal that have been cut along the 557 plane so that they are |
| 326 |
+ |
periodic in the {\it x} and {\it y} directions, and have been rotated |
| 327 |
+ |
to expose two parallel 557 cuts along the positive and negative {\it |
| 328 |
+ |
z}-axis. Simulations of the bare metal interfaces at temperatures |
| 329 |
+ |
ranging from 300~K to 1200~K were done to observe the relative |
| 330 |
+ |
stability of the surfaces without a CO overlayer. |
| 331 |
|
|
| 332 |
+ |
The different bulk (and surface) melting temperatures (1337~K for Au |
| 333 |
+ |
and 2045~K for Pt) suggest that the reconstruction may happen at |
| 334 |
+ |
different temperatures for the two metals. To copy experimental |
| 335 |
+ |
conditions for the CO-exposed surfaces, the bare surfaces were |
| 336 |
+ |
initially run in the canonical (NVT) ensemble at 800~K and 1000~K |
| 337 |
+ |
respectively for 100 ps. Each surface was exposed to a range of CO |
| 338 |
+ |
that was initially placed in the vacuum region. Upon full adsorption, |
| 339 |
+ |
these amounts correspond to 0\%, 5\%, 25\%, 33\%, and 50\% surface |
| 340 |
+ |
coverage. Because of the difference in binding energies, the platinum |
| 341 |
+ |
systems very rarely had CO that was not bound to the surface, while |
| 342 |
+ |
the gold surfaces often had a significant CO population in the gas |
| 343 |
+ |
phase. These systems were allowed to reach thermal equilibrium (over |
| 344 |
+ |
5 ns) before being shifted to the microcanonical (NVE) ensemble for |
| 345 |
+ |
data collection. All of the systems examined had at least 40 ns in the |
| 346 |
+ |
data collection stage, although simulation times for some of the |
| 347 |
+ |
systems exceeded 200ns. All simulations were run using the open |
| 348 |
+ |
source molecular dynamics package, OpenMD.\cite{Ewald,OOPSE,OpenMD} |
| 349 |
|
|
| 334 |
– |
|
| 335 |
– |
|
| 350 |
|
% Just results, leave discussion for discussion section |
| 351 |
+ |
% structure |
| 352 |
+ |
% Pt: step wandering, double layers, no triangular motifs |
| 353 |
+ |
% Au: step wandering, no double layers |
| 354 |
+ |
% dynamics |
| 355 |
+ |
% diffusion |
| 356 |
+ |
% time scale, formation, breakage |
| 357 |
|
\section{Results} |
| 358 |
< |
\subsection{Diffusion} |
| 359 |
< |
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. |
| 358 |
> |
\subsection{Structural remodeling} |
| 359 |
> |
Tao {\it et al.} showed experimentally that the Pt(557) surface undergoes |
| 360 |
> |
two separate reconstructions upon CO adsorption.\cite{Tao:2010} The first |
| 361 |
> |
reconstruction involves a doubling of the step height and plateau length. Similar |
| 362 |
> |
behavior has been seen to occur on numerous surfaces at varying conditions.\cite{Williams:1994,Williams:1991,Pearl} |
| 363 |
> |
Of the two systems we examined, the Platinum system showed the most surface |
| 364 |
> |
reconstruction. Additionally, the amount of reconstruction appears to be |
| 365 |
> |
dependent on the amount of CO adsorbed upon the surface. This result is likely |
| 366 |
> |
related to the effect that coverage has on surface diffusion. While both systems |
| 367 |
> |
displayed step edge wandering, only the Pt surface underwent doubling within |
| 368 |
> |
the time scales we were modeling. Specifically only the 50 \% coverage Pt system |
| 369 |
> |
was observed to undergo a complete doubling in the time scales we were able to monitor. |
| 370 |
> |
This event encouraged us to allow that specific system to run continuously during which two |
| 371 |
> |
more double layers were created. The other systems, not displaying any large scale changes |
| 372 |
> |
of interest, were all stopped after 40 ns of simulation. Neverthless, the other Platinum systems tended to show |
| 373 |
> |
more cumulative lateral movement of the step edges when compared to the Gold systems. |
| 374 |
> |
The 50 \% Pt system is highlighted in figure \ref{fig:reconstruct} at various times along the |
| 375 |
> |
simulation showing the evolution of the system. |
| 376 |
|
|
| 377 |
< |
%Table of Diffusion Constants |
| 378 |
< |
%Add gold?M |
| 379 |
< |
\begin{table}[H] |
| 380 |
< |
\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} |
| 381 |
< |
\centering |
| 346 |
< |
\begin{tabular}{| c | cc | cc | c |} |
| 347 |
< |
\hline |
| 348 |
< |
\textbf{System Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & \textbf{Time (ns)}\\ |
| 349 |
< |
\hline |
| 350 |
< |
&\multicolumn{2}{c|}{\textbf{Platinum}}&\multicolumn{2}{c|}{\textbf{Gold}} & \\ |
| 351 |
< |
\hline |
| 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} |
| 377 |
> |
The second reconstruction on the Pt(557) surface observed by Tao involved the |
| 378 |
> |
formation of triangular clusters that stretched across the plateau between two step edges. |
| 379 |
> |
Neither system, within our simulated time scales, experiences this reconstruction. A constructed |
| 380 |
> |
system in which the triangular motifs were constructed on the surface will be explored in future |
| 381 |
> |
work and is shown in the supporting information. |
| 382 |
|
|
| 383 |
+ |
\subsection{Dynamics} |
| 384 |
+ |
While atomistic-like simulations of stepped surfaces have been performed before \cite{}, they tend to be |
| 385 |
+ |
performed using Monte Carlo techniques\cite{Williams:1991,Williams:1994}. This allows them to efficiently sample the thermodynamic |
| 386 |
+ |
landscape but at the expense of ignoring the dynamics of the system. Previous work, using STM \cite{Pearl}, |
| 387 |
+ |
has been able to visualize the coalescing of steps of (system). The time scale of the image acquisition, ~ 70 s/image |
| 388 |
+ |
provides an upper bounds for the time required for the doubling to actually occur. While statistical treatments |
| 389 |
+ |
of step edges are adept at analyzing such systems, it is important to remember that the edges are made |
| 390 |
+ |
up of individual atoms and thus can be examined in numerous ways. |
| 391 |
|
|
| 392 |
+ |
\subsubsection{Transport of surface metal atoms} |
| 393 |
+ |
%forcedSystems/stepSeparation |
| 394 |
+ |
The movement of a step edge is a cooperative effect arising from the individual movements of the atoms |
| 395 |
+ |
making up the step. An ideal metal surface displaying a low index facet (111, 100, 110) is unlikely to |
| 396 |
+ |
experience much surface diffusion because of the large energetic barrier to lift an atom out of the surface. |
| 397 |
+ |
For our surfaces however, the presence of step edges provide a source for mobile metal atoms. Breaking away |
| 398 |
+ |
from the step edge still imposes an energetic penalty around 40 kcal/mole, but is much less than lifting the same metal |
| 399 |
+ |
atom out from the surface, > 60 kcal/mole, and the penalty lowers even further when CO is present in sufficient quantities |
| 400 |
+ |
on the surface, ~20 kcal/mole. Once an adatom exists on the surface, its barrier for diffusion is negligible ( < 4 kcal/mole) |
| 401 |
+ |
and is well able to explore its terrace. Atoms traversing terraces is more difficult, but can be overcome through a joining and lifting stage. |
| 402 |
+ |
By tracking the mobility of individual metal atoms on the Platinum and Gold surfaces we were able to determine |
| 403 |
+ |
the relative diffusion rates and how varying coverages of CO affected the rates. Close |
| 404 |
+ |
observation of the mobile metal atoms showed that they were typically in equilibrium with the |
| 405 |
+ |
step edges, constantly breaking apart and rejoining. Additionally, at times their motion was concerted and |
| 406 |
+ |
two or more atoms would be observed moving together across the surfaces. The primary challenge in quantifying |
| 407 |
+ |
the overall surface mobility was in defining ``mobile" vs. ``static" atoms. |
| 408 |
|
|
| 409 |
+ |
A particle was considered mobile once it had traveled more than 2~\AA~ between saved configurations |
| 410 |
+ |
of the system (10-100 ps). An atom that was truly mobile would typically travel much greater than this, but |
| 411 |
+ |
the 2~\AA~ cutoff was to prevent the in-place vibrational movement of atoms from being included in the analysis. |
| 412 |
+ |
Since diffusion on a surface is strongly affected by local structures, in this case the presence of single and double |
| 413 |
+ |
layer step edges, the diffusion parallel to the step edges was determined separately from the diffusion perpendicular |
| 414 |
+ |
to these edges. The parallel and perpendicular diffusion constants are shown in figure \ref{fig:diff}. |
| 415 |
+ |
|
| 416 |
+ |
\subsubsection{Double layer formation} |
| 417 |
+ |
The increased amounts of diffusion on Pt at the higher CO coverages appears to play a role in the |
| 418 |
+ |
formation of double layers, seeing as how that was the only system within our observed simulation time |
| 419 |
+ |
that showed the formation. Despite this being the only system where this reconstruction occurs, three separate layers |
| 420 |
+ |
were formed over the extended run time of this system. As mentioned earlier, previous experimental work has given some insight into |
| 421 |
+ |
the upper bounds of the time required for enough atoms to move around to allow two steps to coalesce\cite{Williams:1991,Pearl}. |
| 422 |
+ |
As seen in figure \ref{fig:reconstruct}, the first appearance of a double layer, a nodal site, appears at 19 ns into |
| 423 |
+ |
the simulation. Within 12 ns, nearly half of the step has formed the double layer and by 86 ns, a smooth complete |
| 424 |
+ |
layer has formed. The double layer is complete by 37 ns but is a bit rough. |
| 425 |
+ |
From the appearance of the first node to the initial doubling of the layers ignoring their roughness took ~20 ns. |
| 426 |
+ |
Another ~40 ns was necessary for the layer to completely straighten. The other two layers in this simulation form |
| 427 |
+ |
over a period of 22 ns and 42 ns respectively. |
| 428 |
+ |
|
| 429 |
+ |
%Evolution of surface |
| 430 |
+ |
\begin{figure}[H] |
| 431 |
+ |
\includegraphics[width=\linewidth]{ProgressionOfDoubleLayerFormation_yellowCircle.png} |
| 432 |
+ |
\caption{The Pt(557) / 50\% CO system at a sequence of times after |
| 433 |
+ |
initial exposure to the CO: (a) 258 ps, (b) 19 ns, (c) 31.2 ns, and |
| 434 |
+ |
(d) 86.1 ns. Disruption of the 557 step edges occurs quickly. The |
| 435 |
+ |
doubling of the layers appears only after two adjacent step edges |
| 436 |
+ |
touch. The circled spot in (b) nucleated the growth of the double |
| 437 |
+ |
step observed in the later configurations.} |
| 438 |
+ |
\label{fig:reconstruct} |
| 439 |
+ |
\end{figure} |
| 440 |
+ |
|
| 441 |
+ |
\begin{figure}[H] |
| 442 |
+ |
\includegraphics[width=\linewidth]{DiffusionComparison_errorXY_remade.pdf} |
| 443 |
+ |
\caption{Diffusion constants for mobile surface atoms along directions |
| 444 |
+ |
parallel ($\mathbf{D}_{\parallel}$) and perpendicular |
| 445 |
+ |
($\mathbf{D}_{\perp}$) to the 557 step edges as a function of CO |
| 446 |
+ |
surface coverage. Diffusion parallel to the step edge is higher |
| 447 |
+ |
than that perpendicular to the edge because of the lower energy |
| 448 |
+ |
barrier associated with going from approximately 7 nearest neighbors |
| 449 |
+ |
to 5, as compared to the 3 of an adatom. Additionally, the observed |
| 450 |
+ |
maximum and subsequent decrease for the Pt system suggests that the |
| 451 |
+ |
CO self-interactions are playing a significant role with regards to |
| 452 |
+ |
movement of the platinum atoms around and more importantly across |
| 453 |
+ |
the surface. } |
| 454 |
+ |
\label{fig:diff} |
| 455 |
+ |
\end{figure} |
| 456 |
+ |
|
| 457 |
+ |
|
| 458 |
+ |
|
| 459 |
+ |
|
| 460 |
|
%Discussion |
| 461 |
|
\section{Discussion} |
| 462 |
< |
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. |
| 462 |
> |
In this paper we have shown that we were able to accurately model the initial reconstruction of the |
| 463 |
> |
Pt (557) surface upon CO adsorption as shown by Tao et al. \cite{Tao:2010}. More importantly, we |
| 464 |
> |
were able to capture the dynamic processes inherent within this reconstruction. |
| 465 |
|
|
| 466 |
+ |
\subsection{Mechanism for restructuring} |
| 467 |
+ |
The increased computational cost to examine this system using molecular dynamics rather than |
| 468 |
+ |
a Monte Carlo based approach was necessary so that our predictions on possible mechanisms |
| 469 |
+ |
and driving forces would have support not only from thermodynamic arguments but also from the |
| 470 |
+ |
actual dynamics of the system. |
| 471 |
+ |
|
| 472 |
+ |
Comparing the results from simulation to those reported previously by |
| 473 |
+ |
Tao et al. the similarities in the platinum and CO system are quite |
| 474 |
+ |
strong. As shown in figure \ref{fig:reconstruct}, the simulated platinum system under a CO |
| 475 |
+ |
atmosphere will restructure slightly by doubling the terrace |
| 476 |
+ |
heights. The restructuring appears to occur slowly, one to two |
| 477 |
+ |
platinum atoms at a time. Looking at individual snapshots, these |
| 478 |
+ |
adatoms tend to either rise on top of the plateau or break away from |
| 479 |
+ |
the step edge and then diffuse perpendicularly to the step direction |
| 480 |
+ |
until reaching another step edge. This combination of growth and decay |
| 481 |
+ |
of the step edges appears to be in somewhat of a state of dynamic |
| 482 |
+ |
equilibrium. However, once two previously separated edges meet as |
| 483 |
+ |
shown in figure 1.B, this point tends to act as a focus or growth |
| 484 |
+ |
point for the rest of the edge to meet up, akin to that of a |
| 485 |
+ |
zipper. From the handful of cases where a double layer was formed |
| 486 |
+ |
during the simulation, measuring from the initial appearance of a |
| 487 |
+ |
growth point, the double layer tends to be fully formed within |
| 488 |
+ |
$\sim$~35 ns. |
| 489 |
+ |
|
| 490 |
+ |
There are a number of possible mechanisms to explain the role of |
| 491 |
+ |
adsorbed CO in restructuring the Pt surface. Quadrupolar repulsion |
| 492 |
+ |
between adjacent CO molecules adsorbed on the surface is one |
| 493 |
+ |
possibility. However, the quadrupole-quadrupole interaction is |
| 494 |
+ |
short-ranged and is attractive for some orientations. If the CO |
| 495 |
+ |
molecules are ``locked'' in a specific orientation relative to each other however, |
| 496 |
+ |
this explanation gains some weight. The energetic repulsion between two CO |
| 497 |
+ |
located a distance of 2.77~\AA~apart (nearest-neighbor distance of Pt) with both in a |
| 498 |
+ |
vertical orientation is 8.62 kcal/mole. Moving the CO apart to the second nearest-neighbor |
| 499 |
+ |
distance of 4.8~\AA~or 5.54~\AA~drops the repulsion to nearly 0 kcal/mole. SHOW A NUMBER FOR ROTATION. |
| 500 |
+ |
As mentioned above, the energy barrier for surface diffusion of a platinum adatom is only 4 kcal/mole. So this |
| 501 |
+ |
repulsion between CO can help increase the surface diffusion. However, the residence time of CO was examined |
| 502 |
+ |
and while the majority of the CO is on or near the surface throughout the run, it is extremely mobile. This mobility |
| 503 |
+ |
suggests that the CO are more likely to shift their positions without necessarily dragging the platinum along |
| 504 |
+ |
with them. |
| 505 |
+ |
|
| 506 |
+ |
Another possible and more likely mechanism for the restructuring is in the |
| 507 |
+ |
destabilization of strong Pt-Pt interactions by CO adsorbed on surface |
| 508 |
+ |
Pt atoms. This could have the effect of increasing surface mobility |
| 509 |
+ |
of these atoms. To test this hypothesis, numerous configurations of |
| 510 |
+ |
CO in varying quantities were arranged on the higher and lower plateaus |
| 511 |
+ |
around a step on a otherwise clean Pt (557) surface. One representative |
| 512 |
+ |
configuration is displayed in figure \ref{fig:lambda}. Single or concerted movement |
| 513 |
+ |
of platinum atoms was then examined to determine possible barriers. Because |
| 514 |
+ |
of the forced movement along a pre-defined reaction coordinate that may differ |
| 515 |
+ |
from the true minimum of this path, only the beginning and ending energies |
| 516 |
+ |
are displayed in table \ref{tab:energies}. The presence of CO at suitable |
| 517 |
+ |
sites can lead to lowered barriers for platinum breaking apart from the step edge. |
| 518 |
+ |
Additionally, as highlighted in figure \ref{fig:lambda}, the presence of CO makes the |
| 519 |
+ |
burrowing and lifting nature favorable, whereas without CO, the process is neutral |
| 520 |
+ |
in terms of energetics. |
| 521 |
+ |
|
| 522 |
+ |
%lambda progression of Pt -> shoving its way into the step |
| 523 |
+ |
\begin{figure}[H] |
| 524 |
+ |
\includegraphics[width=\linewidth]{lambdaProgression_atopCO.png} |
| 525 |
+ |
\caption{A model system of the Pt 557 surface was used as the framework for a reaction coordinate. |
| 526 |
+ |
Various numbers, placements, and rotations of CO were examined. The one displayed was a |
| 527 |
+ |
representative sample. As shown in Table , relative to the energy at 0\% there is a slight decrease |
| 528 |
+ |
upon insertion of the platinum atom into the step edge along with the resultant lifting of the other |
| 529 |
+ |
platinum atom.} |
| 530 |
+ |
\label{fig:lambda} |
| 531 |
+ |
\end{figure} |
| 532 |
+ |
|
| 533 |
+ |
|
| 534 |
+ |
|
| 535 |
|
\subsection{Diffusion} |
| 536 |
< |
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?) |
| 536 |
> |
As shown in the results section, the diffusion parallel to the step edge tends to be |
| 537 |
> |
much faster than that perpendicular to the step edge. Additionally, the coverage |
| 538 |
> |
of CO appears to play a slight role in relative rates of diffusion, as shown in figure \ref{fig:diff} |
| 539 |
> |
Thus, the bottleneck of the double layer formation appears to be the initial formation |
| 540 |
> |
of this growth point, which seems to be somewhat of a stochastic event. Once it |
| 541 |
> |
appears, parallel diffusion, along the now slightly angled step edge, will allow for |
| 542 |
> |
a faster formation of the double layer than if the entire process were dependent on |
| 543 |
> |
only perpendicular diffusion across the plateaus. Thus, the larger $D_{\perp}$, the |
| 544 |
> |
more likely a growth point is to be formed. |
| 545 |
|
\\ |
| 546 |
< |
\\ |
| 547 |
< |
%Evolution of surface |
| 546 |
> |
|
| 547 |
> |
|
| 548 |
> |
%breaking of the double layer upon removal of CO |
| 549 |
|
\begin{figure}[H] |
| 550 |
< |
\includegraphics[scale=0.5]{ProgressionOfDoubleLayerFormation_yellowCircle.png} |
| 551 |
< |
\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.} |
| 550 |
> |
\includegraphics[width=\linewidth]{doubleLayerBreaking_greenBlue_whiteLetters.png} |
| 551 |
> |
\caption{Hi} |
| 552 |
> |
\label{fig:breaking} |
| 553 |
|
\end{figure} |
| 554 |
|
|
| 555 |
|
|
| 556 |
|
|
| 557 |
|
|
| 558 |
|
%Peaks! |
| 559 |
< |
\includegraphics[scale=0.25]{doublePeaks_noCO.png} |
| 559 |
> |
\begin{figure}[H] |
| 560 |
> |
\includegraphics[width=\linewidth]{doublePeaks_noCO.png} |
| 561 |
> |
\caption{} |
| 562 |
> |
\label{fig:peaks} |
| 563 |
> |
\end{figure} |
| 564 |
> |
|
| 565 |
> |
%clean surface... |
| 566 |
> |
\begin{figure}[H] |
| 567 |
> |
\includegraphics[width=\linewidth]{557_300K_cleanPDF.pdf} |
| 568 |
> |
\caption{} |
| 569 |
> |
|
| 570 |
> |
\end{figure} |
| 571 |
> |
\label{fig:clean} |
| 572 |
|
\section{Conclusion} |
| 573 |
|
|
| 574 |
|
|
| 575 |
+ |
%Things I am not ready to remove yet |
| 576 |
+ |
|
| 577 |
+ |
%Table of Diffusion Constants |
| 578 |
+ |
%Add gold?M |
| 579 |
+ |
% \begin{table}[H] |
| 580 |
+ |
% \caption{} |
| 581 |
+ |
% \centering |
| 582 |
+ |
% \begin{tabular}{| c | cc | cc | } |
| 583 |
+ |
% \hline |
| 584 |
+ |
% &\multicolumn{2}{c|}{\textbf{Platinum}}&\multicolumn{2}{c|}{\textbf{Gold}} \\ |
| 585 |
+ |
% \hline |
| 586 |
+ |
% \textbf{Surface Coverage} & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ & $\mathbf{D}_{\parallel}$ & $\mathbf{D}_{\perp}$ \\ |
| 587 |
+ |
% \hline |
| 588 |
+ |
% 50\% & 4.32(2) & 1.185(8) & 1.72(2) & 0.455(6) \\ |
| 589 |
+ |
% 33\% & 5.18(3) & 1.999(5) & 1.95(2) & 0.337(4) \\ |
| 590 |
+ |
% 25\% & 5.01(2) & 1.574(4) & 1.26(3) & 0.377(6) \\ |
| 591 |
+ |
% 5\% & 3.61(2) & 0.355(2) & 1.84(3) & 0.169(4) \\ |
| 592 |
+ |
% 0\% & 3.27(2) & 0.147(4) & 1.50(2) & 0.194(2) \\ |
| 593 |
+ |
% \hline |
| 594 |
+ |
% \end{tabular} |
| 595 |
+ |
% \end{table} |
| 596 |
+ |
|
| 597 |
|
\section{Acknowledgments} |
| 598 |
|
Support for this project was provided by the National Science |
| 599 |
|
Foundation under grant CHE-0848243 and by the Center for Sustainable |
