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\begin{document} |
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\title{A Random Sequential Adsorption model for the differential |
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coverage of Gold (111) surfaces by two related Silicon |
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phthalocyanines} |
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\author{Matthew A. Meineke and J. Daniel Gezelter\\ |
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Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\begin{abstract} |
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We present a simple model for the discrepancy in the coverage of a |
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Gold (111) surface by two silicon phthalocyanines. The model involves |
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Random Sequential Adsorption (RSA) simulations with two different |
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landing molecules, one of which is tilted relative to the substrate |
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surface and can (under certain conditions) allow neighboring molecules |
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to overlap. This results in a jamming limit that is near full |
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coverage of the surface. The non-overlapping molecules reproduce the |
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half-monolayer jamming limit that is common in continuum RSA models |
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with ellipsoidal landers. Additionally, the overlapping molecules |
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exhibit orientational correlation and orientational domain formation |
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evolving out of a purely random adsorption process. |
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\end{abstract} |
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\newpage |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%%%%% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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In a recent series of experiments, Li, Lieberman, and Hill found some |
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remarkable differences in the coverage of Gold (111) surfaces by a |
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related set of silicon phthalocyanines.\cite{Li2001} The molecules |
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come in two basic varieties, the ``octopus,'' which has eight thiol |
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groups distributed around the edge of the molecule, and the |
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``umbrella,'' which has a single thiol group at the end of a central |
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arm. The molecules are roughly the same size, and were expected to |
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yield similar coverage properties when the thiol groups attached to |
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the gold surface. Fig. \ref{fig:lieberman} shows the structures of |
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the two molecules. |
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\begin{figure} |
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\begin{center} |
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\epsfxsize=6in |
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\epsfbox{octo-umbrella.eps} |
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\end{center} |
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\caption{Structures of representative umbrella and octopus silicon |
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phthalocyanines.} |
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\label{fig:lieberman} |
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\end{figure} |
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Analysis of the coverage properties using ellipsometry, X-ray |
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photoelectron spectroscopy (XPS) and surface-enhanced Raman scattering |
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(SERS) showed some remarkable behavioral differences. The octopus |
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silicon phthalocyanines formed poorly-organized self-assembled |
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monolayers (SAMs), with a sub-monolayer coverage of the surface. The |
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umbrella molecule, on the other hand, formed well-ordered films |
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approaching a full monolayer of coverage. |
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This behavior is surprising for a number of reasons. First, one would |
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expect the eight thiol groups on the octopus to provide additional |
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attachment points for the molecule. Additionally, the eight arms of |
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the octopus should be able to interdigitate and allow for a relatively |
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high degree of interpenetration of the molecules on the surface if |
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only a few of the arms have attached to the surface. |
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The question that these experiments raise is: Will a simple |
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statistical model be sufficient to explain the differential coverage |
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of a gold surface by such similar molecules that permanently attach to |
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the surface? |
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We have attempted to model this behavior using a simple Random |
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Sequential Adsorption (RSA) approach. In the continuum RSA |
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simulations of disks adsorbing on a plane,\cite{Evans1993} disk-shaped |
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molecules attempt to land on the surface at random locations. If the |
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landing molecule encounters another disk blocking the chosen position, |
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the landing molecule bounces back out into the solution and makes |
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another attempt at a new randomly-chosen location. RSA models have |
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been used to simulate many related chemical situations, from |
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dissociative chemisorption of water on a Fe (100) |
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surface~\cite{Dwyer1977} and the arrangement of proteins on solid |
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surfaces~\cite{Macritche1978,Feder1980,Ramsden1993} to the deposition |
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of colloidal particles on mica surfaces.\cite{Semmler1998} RSA can |
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provide a very powerful model for understanding surface phenomena when |
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the molecules become permanently bound to the surface. There are some |
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RSA models that allow for a window of movement when the molecule first |
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adsorbs.\cite{Dobson1987,Egelhoff1989} However, even in the dynamic |
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approaches to RSA, at some point the molecule becomes a fixed feature |
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of the surface. |
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There is an immense literature on the coverage statistics of RSA |
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models with a wide range of landing shapes including |
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squares,\cite{Solomon1986,Bonnier1993} ellipsoids,\cite{Viot1992a} and |
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lines.\cite{Viot1992b} In general, RSA models of surface coverage |
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approach a jamming limit, $\theta_{J}$, which depends on the shape of |
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the landing molecule and the underlying lattice of attachment |
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points.\cite{Evans1993} For disks on a continuum surface (i.e. no |
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underlying lattice), the jamming limit is $\theta_{J} \approx |
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0.547$.\cite{Evans1993} For ellipsoids, rectangles,\cite{Viot1992a} |
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and 2-dimensional spherocylinders,\cite{Ricci1994} there is a small |
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(4\%) initial rise in $\theta_{J}$ as a function of particle |
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anisotropy. However, the jamming limit {\it decreases} with |
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increasing particle anisotropy once the length-to-breadth ratio rises |
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above 2. I.e. ellipsoids landing randomly on a surface will, in |
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general, cover a smaller surface area than disks. Randomly thrown thin |
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lines cover an even smaller area.\cite{Viot1992b} |
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How, then, can one explain a near-monolayer coverage by the umbrella |
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molecules? There are really two approaches, one static and one |
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dynamic. In this paper, we present a static RSA model with {\em |
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tilted} disks that allows near-monolayer coverage and which can |
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explain the differences in coverage between the octopus and umbrella. |
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In section \ref{sec:model} we outline the model for the two adsorbing |
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molecules. The computational details of our simulations are given in |
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section \ref{sec:meth}. Section \ref{sec:results} presents the |
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results of our simulations, and section \ref{sec:conclusion} concludes. |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%% The Model |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Model} |
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\label{sec:model} |
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Two different landers were investigated in this work. The first, |
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representing the octopus phthalocyanine, was modeled as a flat disk of |
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fixed radius ($\sigma = 14 \mbox{\AA}$) with eight equally spaced |
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``legs'' around the perimeter, each of length $\ell = 5 \mbox{\AA}$. |
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The second type of lander, representing the umbrella phthalocyanine, |
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was modeled by a tilted disk (also of radius $\sigma = 14 \mbox{\AA}$) |
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which was supported by a central handle (also of length $\ell = 5 |
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\mbox{\AA}$). The surface normal for the disk of the umbrella, |
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$\hat{n}$ was tilted relative to the handle at an angle $\psi = |
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109.5^{\circ}$. This angle was chosen, as it is the normal |
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tetrahedral bond angle for $sp^{3}$ hybridized carbon atoms, and |
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therefore the likely angle the top makes with the plane. The two |
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particle types are compared in Fig. \ref{fig:landers}, and the |
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coordinates of the tilted umbrella lander are shown in Fig. |
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\ref{fig:t_umbrella}. The angle $\phi$ denotes the angle that the |
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projection of $\hat{n}$ onto the x-y plane makes with the y-axis. In |
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keeping with the RSA approach, each of the umbrella landers is |
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assigned a value of $\phi$ at random as it is dropped onto the |
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surface. |
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\begin{figure} |
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\begin{center} |
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\epsfxsize=6in |
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\epsfbox{octopus.eps} |
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\end{center} |
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\caption{Models for the adsorbing species. Both the octopus and |
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umbrella models have circular disks of radius $\sigma$ and are |
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supported away from the surface by arms of length $\ell$. The disk |
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for the umbrella is tilted relative to the plane of the substrate.} |
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\label{fig:landers} |
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\end{figure} |
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\begin{figure} |
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\begin{center} |
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\epsfxsize=6in |
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\epsfbox{t_umbrella.eps} |
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\end{center} |
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\caption{Coordinates for the umbrella lander. The vector $\hat{n}$ is |
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normal to the disks. The disks are angled at an angle of $109.5^{\circ}$ |
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to the handle, and the projection of $\hat{n}$ onto the substrate |
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surface defines the angle $\phi$.} |
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\label{fig:t_umbrella} |
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\end{figure} |
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For each type of lander, we investigated both the continuum |
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(off-lattice) RSA approach as well as a more typical RSA approach |
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utilizing an underlying lattice for the possible attachment points of |
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the thiol groups. In the continuum case, the landers could attach |
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anywhere on the surface. For the lattice-based RSA simulations, an |
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underlying gold hexagonal closed packed (hcp), lattice was employed. |
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The thiols attach at the interstitial locations between three gold |
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atoms on the Au (111) surface,\cite{Li2001} giving a trigonal (i.e. |
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graphitic) underlying lattice for the RSA simulations that is |
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illustrated in Fig. \ref{fig:hcp_lattice}. The hcp nearest neighbor |
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distance was $2.3\mbox{\AA}$, corresponding to gold's lattice spacing. |
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This set the graphitic lattice to have a nearest neighbor distance of |
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$1.33\mbox{\AA}$. Fig. \ref{fig:hcp_lattice} also defines the |
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$\hat{x}$ and $\hat{y}$ directions for the simulation. |
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\begin{figure} |
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\begin{center} |
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\epsfxsize=6in |
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\epsfbox{hcp_lattice.eps} |
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\end{center} |
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\caption{The model thiol groups attach at the interstitial sites in |
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the Au (111) surface. These sites are arranged in a graphitic |
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trigonal lattice.} |
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\label{fig:hcp_lattice} |
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\end{figure} |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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%%%% Computational Methods |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Computational Methodology} |
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\label{sec:meth} |
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The simulation box was 4,000 repeated hcp units in both the x and y |
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directions. This gave a rectangular plane ($4600 \mbox{\AA} \times |
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7967 \mbox{\AA}$), to which periodic boundary conditions were |
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applied. Each molecule's attempted landing spot was then chosen |
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randomly. In the continuum simulations, the landing molecule was then |
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checked for overlap with all previously adsorbed molecules. For the |
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octopus molecules, which lie parallel to the surface, the check was a |
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simple distance test. If the center of the landing molecule was at |
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least $2\sigma$ away from the centers of all other molecules, the new |
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molecule was allowed to stay. |
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For the umbrella molecule, the test for overlap was slightly more |
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complex. To speed computation, several sequential tests were made. |
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The first test was the simplest, i.e. a check to make sure that the |
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new umbrella's attachment point, or ``handle'', did not lie within the |
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elliptical projection of a previously attached umbrella's top onto the |
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xy-plane. If the lander passed this first test, the disk was tested |
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for intersection with any of the other nearby umbrellas. |
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The test for the interection of two neighboring umbrella tops involved |
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three steps. In the first step, the surface normals for the umbrella |
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tops were used to caclulate the parametric line equation that was |
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defined by the intersection of the two planes. This parametric line |
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was then checked for intersection with both of the umbrella tops. If |
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the line did indeed intersect the tops, then the points of |
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intersection along the line were checked to insure sequential |
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intersection of the two tops. ie. The line most enter then leave the |
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first top before it can enter and leave the second top. These series |
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of tests were demanding of computational resources, and were therefore |
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only attempted if the original handle - projection overlap test had |
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been passed. |
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Once all of these tests had been passed, the random location and |
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orientation for the molecule were accepted, and the molecule was added |
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to the pool of particles that were permanently attached to the |
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surface. |
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For the on-lattice simulations, the initially chosen location on the |
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plane was used to pick an attachment point from the underlying |
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lattice. I.e. if the initial position and orientation placed one of |
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the thiol legs within a small distance ($\epsilon = 0.1 \mbox{\AA}$) |
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of one of the interstitial attachment points, the lander was moved so |
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that the thiol leg was directly over the lattice point before checking |
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for overlap with other landers. If all of the molecule's legs were |
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too far from the attachment points, the molecule bounced back into |
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solution for another attempt. |
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To speed up the overlap tests, a modified 2-D neighbor list method was |
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employed. The plane was divided into a $131 \times 131$ grid of |
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equally sized rectangular bins. The overlap test then cycled over all |
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of the molecules within the bins located in a $3 \times 3$ grid |
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centered on the bin in which the test molecule was attempting to land. |
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Surface coverage calculations were handled differently between the |
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umbrella molecule simulation, and the octopus model simulation. In |
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the case of the umbrella molecule, the surface coverage was tracked by |
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multiplying the number of succesfully landed particles by the area of |
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its circular top. This number was then divided by the total surfacew |
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area of the plane, to obtain the fractional coverage. In the case of |
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the umbrella molecule, a scanning probe algorithm was used. Here, a |
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$1\mbox{\AA} \times 1\mbox{\AA}$ probe was scanned along the surface, |
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and each point was tested for overlap with the neighboring molecules. |
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At the end of the scan, the total covered area was divided by the |
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total surface area of the plane to determine the fractional coverage. |
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Radial and angular correlation functions were computed using standard |
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methods from liquid theory (modified for use on a planar |
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surface).\cite{Hansen86} |
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\section{Results} |
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\label{sec:results} |
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\subsection{Octopi} |
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The jamming limit coverage, $\theta_{J}$, of the off-lattice continuum |
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simulation was found to be 0.5384. This value is within one percent of |
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the jamming limit for circles on a 2D plane.\cite{Evans1993} It is |
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expected that we would approach the accepted jamming limit for a |
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larger gold surface. |
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Once the system is constrained by the underlying lattice, $\theta_{J}$ |
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drops to 0.5378, showing that the lattice has an almost |
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inconsequential effect on the jamming limit. If the spacing between |
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the interstitial sites were closer to the radius of the landing |
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particles, we would expect a larger effect, but in this case, the |
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jamming limit is nearly unchanged from the continuum simulation. |
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The radial distribution function, $g(r)$, for the continuum and |
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lattice simulations are shown in the two left panels in |
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Fig. \ref{fig:octgofr}. It is clear that the lattice has no |
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significant contribution to the distribution other than slightly |
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raising the peak heights. $g(r)$ for the octopus molecule is not |
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affected strongly by the underlying lattice because each molecule can |
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attach with any of it's eight legs. Additionally, the molecule can be |
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randomly oriented around each attachment point. The effect of the |
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lattice on the distribution of molecular centers is therefore |
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inconsequential. |
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The features of both radial distribution functions are quite |
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simple. An initial peak at twice the radius of the octopi |
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corresponding to the first shell being the closest two circles can |
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approach without overlapping each other. The second peak at four times |
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the radius is simply a second ``packing'' shell. These features agree |
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almost perfectly with the Percus-Yevick-like expressions for $g(r)$ |
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for a two dimensional RSA model that were derived by Boyer {\em et |
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al.}\cite{Boyer1995} |
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\begin{figure} |
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\begin{center} |
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\epsfxsize=6in |
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\epsfbox{gofr.eps} |
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\end{center} |
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\caption{$g(r)$ for both the octopus and umbrella molecules in the |
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continuum (upper) and on-lattice (lower) simulations.} |
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\label{fig:octgofr} |
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\end{figure} |
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\subsection{Umbrellas} |
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In the case of the umbrellas, the jamming limit for the continuum |
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simulation was $0.920$ and for the simulation on the lattice, |
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$\theta_{J} = 0.915$ . Once again, the lattice has an almost |
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inconsequential effect on the jamming limit. The overlap allowed by |
347 |
|
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the umbrellas allows for almost total surface coverage based on random |
348 |
|
|
parking alone. This then is the primary result of this work: the |
349 |
|
|
observation of a jamming limit or coverage near unity for molecules |
350 |
|
|
that can (under certain conditions) allow neighboring molecules to |
351 |
|
|
overlap. |
352 |
|
|
|
353 |
|
|
The underlying lattice has a strong effect on $g(r)$ for the |
354 |
|
|
umbrellas. The umbrellas do not have the eight legs and orientational |
355 |
|
|
freedom around each leg available to the octopi. The effect of the |
356 |
|
|
lattice on the distribution of molecular centers is therefore quite |
357 |
|
|
pronounced, as can be seen in Fig. \ref{fig:octgofr}. Since the total |
358 |
|
|
number of particles is similar to the continuum simulation, the |
359 |
|
|
apparent noise in $g(r)$ for the on-lattice umbrellas is actually an |
360 |
|
|
artifact of the underlying lattice. |
361 |
|
|
|
362 |
|
|
Because a molecule's success in sticking is closely linked to its |
363 |
|
|
orientation, the radial distribution function and the angular |
364 |
|
|
distribution function show some very interesting features |
365 |
|
|
(Fig. \ref{fig:tugofr}). The initial peak is located at approximately |
366 |
|
|
one radius of the umbrella. This corresponds to the closest distance |
367 |
|
|
that a perfectly aligned landing molecule may approach without |
368 |
|
|
overlapping. The angular distribution confirms this, showing a |
369 |
|
|
maximum angular correlation at $r = \sigma$. The location of the |
370 |
|
|
second peak in the radial distribution corresponds to twice the radius |
371 |
|
|
of the umbrella. This peak is accompanied by a dip in the angular |
372 |
|
|
distribution. The angular depletion can be explained easily since |
373 |
|
|
once the particles are greater than $2 \sigma$ apart, the landing |
374 |
|
|
molecule can take on any orientation and land successfully. The |
375 |
|
|
recovery of the angular correlation at slightly larger distances is |
376 |
|
|
due to second-order correlations with intermediate particles. The |
377 |
|
|
alignments associated with all three regions are illustrated in |
378 |
|
|
Fig. \ref{fig:peaks}. |
379 |
|
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|
380 |
|
|
\begin{figure} |
381 |
|
|
\begin{center} |
382 |
|
|
\epsfxsize=6in |
383 |
|
|
\epsfbox{angular.eps} |
384 |
|
|
\end{center} |
385 |
|
|
\caption{$g(r)$ and the distance-dependent $\langle cos \phi_{ij} |
386 |
|
|
\rangle$ for the umbrella thiol in the off-lattice (left side) and |
387 |
|
|
on-lattice simulations.} |
388 |
|
|
\label{fig:tugofr} |
389 |
|
|
\end{figure} |
390 |
|
|
|
391 |
|
|
\begin{figure} |
392 |
|
|
\begin{center} |
393 |
|
|
\epsfxsize=6in |
394 |
|
|
\epsfbox{peaks.eps} |
395 |
|
|
\end{center} |
396 |
|
|
\caption{The position of the first peak in $\langle cos \phi_{ij} |
397 |
|
|
\rangle$ is due to the forced alignment of two tightly-packed |
398 |
|
|
umbrellas. The depletion zone at 2$\sigma$ is due to the availability |
399 |
|
|
of all alignments at this separation. Recovery of the angular |
400 |
|
|
correlation at longer distances is due to second-order correlations.} |
401 |
|
|
\label{fig:peaks} |
402 |
|
|
\end{figure} |
403 |
|
|
|
404 |
|
|
\subsection{Comparison with Experiment} |
405 |
|
|
|
406 |
|
|
Considering the lack of atomistic detail in this model, the coverage |
407 |
|
|
statistics are in relatively good agreement with those observed by Li |
408 |
|
|
{\it et al.}\cite{Li2001} Their experiments directly measure the ratio |
409 |
|
|
of Sulfur atoms to Gold surface atoms. In this way, they are able to |
410 |
|
|
estimate the average area taken up by each adsorbed molecule. Rather |
411 |
|
|
than relying on area estimates, we have computed the S:Au ratio for |
412 |
|
|
both types of molecule from our simulations. The ratios are given in |
413 |
|
|
Table \ref{tab:coverage}. |
414 |
|
|
|
415 |
|
|
\begin{table} |
416 |
|
|
\caption{Ratio of Monolayer Sulfur atoms to Gold surface atoms} |
417 |
|
|
\label{tab:coverage} |
418 |
|
|
\begin{center} |
419 |
|
|
\begin{tabular}{|l|l|l|} |
420 |
|
|
\hline |
421 |
|
|
& umbrella & octopus \\ \hline |
422 |
|
|
Li {\it et al.}\cite{Li2001} & 0.021 & 0.0065 \\ \hline |
423 |
|
|
continuum & 0.0320 & 0.0107 \\ \hline |
424 |
|
|
on-lattice & 0.0320 & 0.0105 \\ \hline |
425 |
|
|
\end{tabular} |
426 |
|
|
\end{center} |
427 |
|
|
\end{table} |
428 |
|
|
|
429 |
|
|
Our simulations give S:Au ratios that are 52\% higher than the |
430 |
|
|
experiments for the umbrella and 63\% higher than the experiments for |
431 |
|
|
the octopi. There are a number of explanations for this discrepancy. |
432 |
|
|
The simplest explanation is that the disks we are using to model these |
433 |
|
|
molecules are too small. Another factor leading to the discrepancy is |
434 |
|
|
the lack of thickness for both the disks and the supporting legs. |
435 |
|
|
Thicker disks would force the umbrellas to be farther apart, and |
436 |
|
|
thicker supporting legs would effectively increase the radius of the |
437 |
|
|
octopus molecules. |
438 |
|
|
|
439 |
|
|
However, this model does effectively capture the discrepancy in |
440 |
|
|
coverage surface between the two related landing molecules. We are in |
441 |
|
|
remarkable agreement with the coverage statistics given the simplicity |
442 |
|
|
of the model. |
443 |
|
|
|
444 |
|
|
\section{Conclusions} |
445 |
|
|
\label{sec:conclusion} |
446 |
|
|
|
447 |
|
|
The primary result of this work is the observation of near-monolayer |
448 |
|
|
coverage in a simple RSA model with molecules that can partially |
449 |
|
|
overlap. This is sufficient to explain the experimentally-observed |
450 |
|
|
coverage differences between the octopus and umbrella molecules. |
451 |
|
|
Using ellipsometry, Li {\it et al.} have observed that the octopus |
452 |
|
|
molecules are {\it not} parallel to the substrate, and that they are |
453 |
|
|
attached to the surface with only four legs on average.\cite{Li2001} |
454 |
|
|
As long as the remaining thiol arms that are not bound to the surface |
455 |
|
|
can provide steric hindrance to molecules that attempt to slide |
456 |
|
|
underneath the disk, the results will be largely unchanged. The |
457 |
|
|
projection of a tilted disk onto the surface is a simple ellipsoid, so |
458 |
|
|
a RSA model using tilted disks that {\em exclude the volume underneath |
459 |
|
|
the disks} will revert to a standard RSA model with ellipsoidal |
460 |
|
|
landers. Viot {\it et al.} have shown that for ellipsoids, the |
461 |
|
|
maximal jamming limit is only $\theta_{J} = 0.58$.\cite{Viot1992a} |
462 |
|
|
Therefore, the important feature that leads to near-monolayer coverage |
463 |
|
|
is the ability of the landers to overlap. |
464 |
|
|
|
465 |
|
|
The other important result of this work is the observation of an |
466 |
|
|
angular correlation between the molecules that extends to fairly large |
467 |
|
|
distances. Although not unexpected, the correlation extends well past |
468 |
|
|
the first ``shell'' of molecules. Farther than the first shell, there |
469 |
|
|
is no direct interaction between an adsorbed molecule and a molecule |
470 |
|
|
that is landing, although once the surface has started to approach the |
471 |
|
|
jamming limit, the only available landing spots will require landing |
472 |
|
|
molecules to adopt an orientation similar to one of the adsorbed |
473 |
|
|
molecules. Therefore, given an entirely random adsorption process, we |
474 |
|
|
would still expect to observe orientational ``domains'' developing in |
475 |
|
|
the monolayer. We have shown a relatively small piece of the |
476 |
|
|
monolayer in Fig. \ref{fig:bent_u}, using color to denote the |
477 |
|
|
orientation of each molecule. Indeed, the monolayer does show |
478 |
|
|
orientational domains that are surprisingly large. |
479 |
|
|
|
480 |
|
|
\begin{figure} |
481 |
|
|
\begin{center} |
482 |
|
|
\epsfxsize=6in |
483 |
|
|
\epsfbox{bent_u.eps} |
484 |
|
|
\end{center} |
485 |
|
|
\caption{A bird's-eye view of the orientational domains in a monolayer |
486 |
|
|
of the umbrella thiol. Similarly oriented particles are shaded the |
487 |
|
|
same color.} |
488 |
|
|
\label{fig:bent_u} |
489 |
|
|
\end{figure} |
490 |
|
|
|
491 |
|
|
The important physics that has been left out of this simple RSA model |
492 |
|
|
is the relaxation and dynamics of the monolayer. We would expect that |
493 |
|
|
allowing the adsorbed molecules to rotate on the surface would result |
494 |
|
|
in a monolayer with much longer range orientational order and a nearly |
495 |
|
|
complete coverage of the underlying surface. It should be relatively |
496 |
|
|
simple to add orientational relaxation using standard Monte Carlo |
497 |
|
|
methodology~\cite{Ricci1994,Frenkel1996} to investigate what effect |
498 |
|
|
this has on the properties of the monolayer. |
499 |
|
|
|
500 |
|
|
\section{Acknowledgments} |
501 |
|
|
The authors would like to thank Marya Lieberman for helpful |
502 |
|
|
discussions. This work has been supported in part by a New Faculty |
503 |
|
|
Award from the Camille and Henry Dreyfus Foundation. |
504 |
|
|
|
505 |
|
|
\pagebreak |
506 |
|
|
|
507 |
|
|
\bibliographystyle{achemso} |
508 |
|
|
\bibliography{RSA} |
509 |
|
|
|
510 |
|
|
\end{document} |