matt_papers/RSA/RSA.tex

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\begin{document}

\title{A Random Sequential Adsorption model for the differential
coverage of Gold (111) surfaces by two related Silicon
phthalocyanines}

\author{Matthew A. Meineke and J. Daniel Gezelter\\
Department of Chemistry and Biochemistry\\ University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}
\maketitle

\begin{abstract}
We present a simple model for the discrepancy in the coverage of a
Gold (111) surface by two silicon phthalocyanines.  The model involves
Random Sequential Adsorption (RSA) simulations with two different
landing molecules, one of which is tilted relative to the substrate
surface and can (under certain conditions) allow neighboring molecules
to overlap.  This results in a jamming limit that is near full
coverage of the surface.  The non-overlapping molecules reproduce the
half-monolayer jamming limit that is common in continuum RSA models
with ellipsoidal landers.  Additionally, the overlapping molecules
exhibit orientational correlation and orientational domain formation
evolving out of a purely random adsorption process.
\end{abstract}

\newpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%                       BODY OF TEXT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

In a recent series of experiments, Li, Lieberman, and Hill found some
remarkable differences in the coverage of Gold (111) surfaces by a
related set of silicon phthalocyanines.\cite{Li2001} The molecules
come in two basic varieties, the ``octopus,'' which has eight thiol
groups distributed around the edge of the molecule, and the
``umbrella,'' which has a single thiol group at the end of a central
arm.  The molecules are roughly the same size, and were expected to
yield similar coverage properties when the thiol groups attached to
the gold surface.  Fig. \ref{fig:lieberman} shows the structures of
the two molecules.

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{octo-umbrella.eps}
\end{center}
\caption{Structures of representative umbrella and octopus silicon
phthalocyanines.}
\label{fig:lieberman}
\end{figure}

Analysis of the coverage properties using ellipsometry, X-ray
photoelectron spectroscopy (XPS) and surface-enhanced Raman scattering
(SERS) showed some remarkable behavioral differences.  The octopus
silicon phthalocyanines formed poorly-organized self-assembled
monolayers (SAMs), with a sub-monolayer coverage of the surface.  The
umbrella molecule, on the other hand, formed well-ordered films
approaching a full monolayer of coverage.

This behavior is surprising for a number of reasons.  First, one would
expect the eight thiol groups on the octopus to provide additional
attachment points for the molecule.  Additionally, the eight arms of
the octopus should be able to interdigitate and allow for a relatively
high degree of interpenetration of the molecules on the surface if
only a few of the arms have attached to the surface.

The question that these experiments raise is: Will a simple
statistical model be sufficient to explain the differential coverage
of a gold surface by such similar molecules that permanently attach to
the surface?

We have attempted to model this behavior using a simple Random
Sequential Adsorption (RSA) approach.  In the continuum RSA
simulations of disks adsorbing on a plane,\cite{Evans1993} disk-shaped
molecules attempt to land on the surface at random locations.  If the
landing molecule encounters another disk blocking the chosen position,
the landing molecule bounces back out into the solution and makes
another attempt at a new randomly-chosen location.  RSA models have
been used to simulate many related chemical situations, from
dissociative chemisorption of water on a Fe (100)
surface~\cite{Dwyer1977} and the arrangement of proteins on solid
surfaces~\cite{Macritche1978,Feder1980,Ramsden1993} to the deposition
of colloidal particles on mica surfaces.\cite{Semmler1998} RSA can
provide a very powerful model for understanding surface phenomena when
the molecules become permanently bound to the surface. There are some
RSA models that allow for a window of movement when the molecule first
adsorbs.\cite{Dobson1987,Egelhoff1989} However, even in the dynamic
approaches to RSA, at some point the molecule becomes a fixed feature
of the surface.

There is an immense literature on the coverage statistics of RSA
models with a wide range of landing shapes including
squares,\cite{Solomon1986,Bonnier1993} ellipsoids,\cite{Viot1992a} and
lines.\cite{Viot1992b} In general, RSA models of surface coverage
approach a jamming limit, $\theta_{J}$, which depends on the shape of
the landing molecule and the underlying lattice of attachment
points.\cite{Evans1993} For disks on a continuum surface (i.e. no
underlying lattice), the jamming limit is $\theta_{J} \approx
0.547$.\cite{Evans1993} For ellipsoids, rectangles,\cite{Viot1992a}
and 2-dimensional spherocylinders,\cite{Ricci1994} there is a small
(4\%) initial rise in $\theta_{J}$ as a function of particle
anisotropy.  However, the jamming limit {\it decreases} with
increasing particle anisotropy once the length-to-breadth ratio rises
above 2. I.e. ellipsoids landing randomly on a surface will, in
general, cover a smaller surface area than disks. Randomly thrown thin
lines cover an even smaller area.\cite{Viot1992b}

How, then, can one explain a near-monolayer coverage by the umbrella
molecules?  There are really two approaches, one static and one
dynamic.  In this paper, we present a static RSA model with {\em
tilted} disks that allows near-monolayer coverage and which can
explain the differences in coverage between the octopus and umbrella.
In section \ref{sec:model} we outline the model for the two adsorbing
molecules.  The computational details of our simulations are given in
section \ref{sec:meth}.  Section \ref{sec:results} presents the
results of our simulations, and section \ref{sec:conclusion} concludes.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% The Model
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Model}
\label{sec:model}

Two different landers were investigated in this work. The first,
representing the octopus phthalocyanine, was modeled as a flat disk of
fixed radius ($\sigma = 14 \mbox{\AA}$) with eight equally spaced
``legs'' around the perimeter, each of length $\ell = 5 \mbox{\AA}$.
The second type of lander, representing the umbrella phthalocyanine,
was modeled by a tilted disk (also of radius $\sigma = 14 \mbox{\AA}$)
which was supported by a central handle (also of length $\ell = 5
\mbox{\AA}$).  The surface normal for the disk of the umbrella,
$\hat{n}$ was tilted relative to the handle at an angle $\psi =
109.5^{\circ}$.  This angle was chosen, as it is the normal
tetrahedral bond angle for $sp^{3}$ hybridized carbon atoms, and
therefore the likely angle the top makes with the plane.  The two
particle types are compared in Fig. \ref{fig:landers}, and the
coordinates of the tilted umbrella lander are shown in Fig.
\ref{fig:t_umbrella}.  The angle $\phi$ denotes the angle that the
projection of $\hat{n}$ onto the x-y plane makes with the y-axis.  In
keeping with the RSA approach, each of the umbrella landers is
assigned a value of $\phi$ at random as it is dropped onto the
surface.

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{octopus.eps}
\end{center}
\caption{Models for the adsorbing species.  Both the octopus and
umbrella models have circular disks of radius $\sigma$ and are
supported away from the surface by arms of length $\ell$.  The disk
for the umbrella is tilted relative to the plane of the substrate.}
\label{fig:landers}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{t_umbrella.eps}
\end{center}
\caption{Coordinates for the umbrella lander.  The vector $\hat{n}$ is
normal to the disks.  The disks are angled at an angle of $109.5^{\circ}$
to the handle, and the projection of $\hat{n}$ onto the substrate
surface defines the angle $\phi$.}
\label{fig:t_umbrella}
\end{figure}

For each type of lander, we investigated both the continuum
(off-lattice) RSA approach as well as a more typical RSA approach
utilizing an underlying lattice for the possible attachment points of
the thiol groups.  In the continuum case, the landers could attach
anywhere on the surface.  For the lattice-based RSA simulations, an
underlying gold hexagonal closed packed (hcp), lattice was employed.
The thiols attach at the interstitial locations between three gold
atoms on the Au (111) surface,\cite{Li2001} giving a trigonal (i.e.
graphitic) underlying lattice for the RSA simulations that is
illustrated in Fig. \ref{fig:hcp_lattice}.  The hcp nearest neighbor
distance was $2.3\mbox{\AA}$, corresponding to gold's lattice spacing.
This set the graphitic lattice to have a nearest neighbor distance of
$1.33\mbox{\AA}$.  Fig. \ref{fig:hcp_lattice} also defines the
$\hat{x}$ and $\hat{y}$ directions for the simulation.

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{hcp_lattice.eps}
\end{center}
\caption{The model thiol groups attach at the interstitial sites in
the Au (111) surface.  These sites are arranged in a graphitic
trigonal lattice.}
\label{fig:hcp_lattice}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%    Computational Methods
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Computational Methodology}
\label{sec:meth}

The simulation box was 4,000 repeated hcp units in both the x and y
directions. This gave a rectangular plane ($4600 \mbox{\AA} \times
7967 \mbox{\AA}$), to which periodic boundary conditions were
applied. Each molecule's attempted landing spot was then chosen
randomly.  In the continuum simulations, the landing molecule was then
checked for overlap with all previously adsorbed molecules.  For the
octopus molecules, which lie parallel to the surface, the check was a
simple distance test.  If the center of the landing molecule was at
least $2\sigma$ away from the centers of all other molecules, the new
molecule was allowed to stay.

For the umbrella molecule, the test for overlap was slightly more
complex.  To speed computation, several sequential tests were made.
The first test was the simplest, i.e. a check to make sure that the
new umbrella's attachment point, or ``handle'', did not lie within the
elliptical projection of a previously attached umbrella's top onto the
xy-plane.  If the lander passed this first test, the disk was tested
for intersection with any of the other nearby umbrellas.  

The test for the interection of two neighboring umbrella tops involved
three steps. In the first step, the surface normals for the umbrella
tops were used to caclulate the parametric line equation that was
defined by the intersection of the two planes.  This parametric line
was then checked for intersection with both of the umbrella tops.  If
the line did indeed intersect the tops, then the points of
intersection along the line were checked to insure sequential
intersection of the two tops. ie. The line most enter then leave the
first top before it can enter and leave the second top.  These series
of tests were demanding of computational resources, and were therefore
only attempted if the original handle - projection overlap test had
been passed.

Once all of these tests had been passed, the random location and
orientation for the molecule were accepted, and the molecule was added
to the pool of particles that were permanently attached to the
surface.

For the on-lattice simulations, the initially chosen location on the
plane was used to pick an attachment point from the underlying
lattice.  I.e. if the initial position and orientation placed one of
the thiol legs within a small distance ($\epsilon = 0.1 \mbox{\AA}$)
of one of the interstitial attachment points, the lander was moved so
that the thiol leg was directly over the lattice point before checking
for overlap with other landers.  If all of the molecule's legs were
too far from the attachment points, the molecule bounced back into
solution for another attempt. 

To speed up the overlap tests, a modified 2-D neighbor list method was
employed. The plane was divided into a $131 \times 131$ grid of
equally sized rectangular bins. The overlap test then cycled over all
of the molecules within the bins located in a $3 \times 3$ grid
centered on the bin in which the test molecule was attempting to land.

Surface coverage calculations were handled differently between the
umbrella molecule simulation, and the octopus model simulation.  In
the case of the umbrella molecule, the surface coverage was tracked by
multiplying the number of succesfully landed particles by the area of
its circular top.  This number was then divided by the total surfacew
area of the plane, to obtain the fractional coverage.  In the case of
the umbrella molecule, a scanning probe algorithm was used.  Here, a
$1\mbox{\AA} \times 1\mbox{\AA}$ probe was scanned along the surface,
and each point was tested for overlap with the neighboring molecules.
At the end of the scan, the total covered area was divided by the
total surface area of the plane to determine the fractional coverage.

Radial and angular correlation functions were computed using standard
methods from liquid theory (modified for use on a planar
surface).\cite{Hansen86}

\section{Results}
\label{sec:results}

\subsection{Octopi}

The jamming limit coverage, $\theta_{J}$, of the off-lattice continuum
simulation was found to be 0.5384. This value is within one percent of
the jamming limit for circles on a 2D plane.\cite{Evans1993} It is
expected that we would approach the accepted jamming limit for a
larger gold surface.

Once the system is constrained by the underlying lattice, $\theta_{J}$
drops to 0.5378, showing that the lattice has an almost
inconsequential effect on the jamming limit.  If the spacing between
the interstitial sites were closer to the radius of the landing
particles, we would expect a larger effect, but in this case, the
jamming limit is nearly unchanged from the continuum simulation.

The radial distribution function, $g(r)$, for the continuum and
lattice simulations are shown in the two left panels in
Fig. \ref{fig:octgofr}.  It is clear that the lattice has no
significant contribution to the distribution other than slightly
raising the peak heights.  $g(r)$ for the octopus molecule is not
affected strongly by the underlying lattice because each molecule can
attach with any of it's eight legs.  Additionally, the molecule can be
randomly oriented around each attachment point.  The effect of the
lattice on the distribution of molecular centers is therefore
inconsequential.

The features of both radial distribution functions are quite
simple. An initial peak at twice the radius of the octopi
corresponding to the first shell being the closest two circles can
approach without overlapping each other. The second peak at four times
the radius is simply a second ``packing'' shell.  These features agree
almost perfectly with the Percus-Yevick-like expressions for $g(r)$
for a two dimensional RSA model that were derived by Boyer {\em et
al.}\cite{Boyer1995}

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{gofr.eps}
\end{center}
\caption{$g(r)$ for both the octopus and umbrella molecules in the
continuum (upper) and on-lattice (lower) simulations.}
\label{fig:octgofr}
\end{figure}

\subsection{Umbrellas}

In the case of the umbrellas, the jamming limit for the continuum
simulation was $0.920$ and for the simulation on the lattice,
$\theta_{J} = 0.915$ .  Once again, the lattice has an almost
inconsequential effect on the jamming limit.  The overlap allowed by
the umbrellas allows for almost total surface coverage based on random
parking alone.  This then is the primary result of this work: the
observation of a jamming limit or coverage near unity for molecules
that can (under certain conditions) allow neighboring molecules to
overlap.

The underlying lattice has a strong effect on $g(r)$ for the
umbrellas.  The umbrellas do not have the eight legs and orientational
freedom around each leg available to the octopi.  The effect of the
lattice on the distribution of molecular centers is therefore quite
pronounced, as can be seen in Fig. \ref{fig:octgofr}.  Since the total
number of particles is similar to the continuum simulation, the
apparent noise in $g(r)$ for the on-lattice umbrellas is actually an
artifact of the underlying lattice.

Because a molecule's success in sticking is closely linked to its
orientation, the radial distribution function and the angular
distribution function show some very interesting features
(Fig. \ref{fig:tugofr}). The initial peak is located at approximately
one radius of the umbrella. This corresponds to the closest distance
that a perfectly aligned landing molecule may approach without
overlapping.  The angular distribution confirms this, showing a
maximum angular correlation at $r = \sigma$. The location of the
second peak in the radial distribution corresponds to twice the radius
of the umbrella. This peak is accompanied by a dip in the angular
distribution.  The angular depletion can be explained easily since
once the particles are greater than $2 \sigma$ apart, the landing
molecule can take on any orientation and land successfully.  The
recovery of the angular correlation at slightly larger distances is
due to second-order correlations with intermediate particles.  The
alignments associated with all three regions are illustrated in
Fig. \ref{fig:peaks}.

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{angular.eps}
\end{center}
\caption{$g(r)$ and the distance-dependent $\langle cos \phi_{ij}
\rangle$ for the umbrella  thiol in the off-lattice (left side) and
on-lattice simulations.}
\label{fig:tugofr}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{peaks.eps}
\end{center}
\caption{The position of the first peak in $\langle cos \phi_{ij}
\rangle$ is due to the forced alignment of two tightly-packed
umbrellas.  The depletion zone at 2$\sigma$ is due to the availability
of all alignments at this separation.  Recovery of the angular
correlation at longer distances is due to second-order correlations.}
\label{fig:peaks}
\end{figure}

\subsection{Comparison with Experiment}

Considering the lack of atomistic detail in this model, the coverage
statistics are in relatively good agreement with those observed by Li
{\it et al.}\cite{Li2001} Their experiments directly measure the ratio
of Sulfur atoms to Gold surface atoms.  In this way, they are able to
estimate the average area taken up by each adsorbed molecule.  Rather
than relying on area estimates, we have computed the S:Au ratio for
both types of molecule from our simulations.  The ratios are given in
Table \ref{tab:coverage}.

\begin{table}
\caption{Ratio of Monolayer Sulfur atoms to Gold surface atoms}
\label{tab:coverage}
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
                             & umbrella & octopus \\ \hline
Li {\it et al.}\cite{Li2001} & 0.021    & 0.0065  \\ \hline
continuum                    & 0.0320   & 0.0107  \\ \hline
on-lattice                   & 0.0320   & 0.0105  \\ \hline
\end{tabular}
\end{center}
\end{table}

Our simulations give S:Au ratios that are 52\% higher than the
experiments for the umbrella and 63\% higher than the experiments for
the octopi.  There are a number of explanations for this discrepancy.
The simplest explanation is that the disks we are using to model these
molecules are too small.  Another factor leading to the discrepancy is
the lack of thickness for both the disks and the supporting legs.
Thicker disks would force the umbrellas to be farther apart, and
thicker supporting legs would effectively increase the radius of the
octopus molecules.  

However, this model does effectively capture the discrepancy in
coverage surface between the two related landing molecules.  We are in
remarkable agreement with the coverage statistics given the simplicity
of the model.

\section{Conclusions}
\label{sec:conclusion}

The primary result of this work is the observation of near-monolayer
coverage in a simple RSA model with molecules that can partially
overlap.  This is sufficient to explain the experimentally-observed
coverage differences between the octopus and umbrella molecules.
Using ellipsometry, Li {\it et al.} have observed that the octopus
molecules are {\it not} parallel to the substrate, and that they are
attached to the surface with only four legs on average.\cite{Li2001}
As long as the remaining thiol arms that are not bound to the surface
can provide steric hindrance to molecules that attempt to slide
underneath the disk, the results will be largely unchanged.  The
projection of a tilted disk onto the surface is a simple ellipsoid, so
a RSA model using tilted disks that {\em exclude the volume underneath
the disks} will revert to a standard RSA model with ellipsoidal
landers.  Viot {\it et al.}  have shown that for ellipsoids, the
maximal jamming limit is only $\theta_{J} = 0.58$.\cite{Viot1992a}
Therefore, the important feature that leads to near-monolayer coverage
is the ability of the landers to overlap.

The other important result of this work is the observation of an
angular correlation between the molecules that extends to fairly large
distances.  Although not unexpected, the correlation extends well past
the first ``shell'' of molecules.  Farther than the first shell, there
is no direct interaction between an adsorbed molecule and a molecule
that is landing, although once the surface has started to approach the
jamming limit, the only available landing spots will require landing
molecules to adopt an orientation similar to one of the adsorbed
molecules.  Therefore, given an entirely random adsorption process, we
would still expect to observe orientational ``domains'' developing in
the monolayer.  We have shown a relatively small piece of the
monolayer in Fig. \ref{fig:bent_u}, using color to denote the
orientation of each molecule.  Indeed, the monolayer does show
orientational domains that are surprisingly large.

\begin{figure}
\begin{center}
\epsfxsize=6in
\epsfbox{bent_u.eps}
\end{center}
\caption{A bird's-eye view of the orientational domains in a monolayer
of the umbrella thiol.  Similarly oriented particles are shaded the
same color.}
\label{fig:bent_u}
\end{figure}

The important physics that has been left out of this simple RSA model
is the relaxation and dynamics of the monolayer.  We would expect that
allowing the adsorbed molecules to rotate on the surface would result
in a monolayer with much longer range orientational order and a nearly
complete coverage of the underlying surface.  It should be relatively
simple to add orientational relaxation using standard Monte Carlo
methodology~\cite{Ricci1994,Frenkel1996} to investigate what effect
this has on the properties of the monolayer.

\section{Acknowledgments}
The authors would like to thank Marya Lieberman for helpful
discussions.  This work has been supported in part by a New Faculty
Award from the Camille and Henry Dreyfus Foundation.

\pagebreak

\bibliographystyle{achemso}
\bibliography{RSA}

\end{document}
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