J. Phys. Chem. B 2001, 105, 6515-6519
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Random Sequential Adsorption Model for the Differential Coverage of Gold (111) Surfaces by Two Related Silicon Phthalocyanines† Matthew A. Meineke and J. Daniel Gezelter* Department of Chemistry and Biochemistry, UniVersity of Notre Dame, Notre Dame, Indiana 46556 ReceiVed: March 15, 2001; In Final Form: May 7, 2001
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 nonoverlapping 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.
1. Introduction In a recent series of experiments, Li et al. found some remarkable differences in the coverage of gold (111) surfaces by a related set of silicon phthalocyanines.1 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. Figure 1 shows the structures and dimensions of the two molecules. 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 submonolayer coverage of the surface. The umbrella molecule, on the other hand, formed well-ordered films approaching a full monolayer of coverage (when compared to close-packed alkane thiols which are known to form complete well-ordered monolayers). 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 modeled this behavior using a simple random sequential adsorption (RSA) approach. In the continuum RSA simulations of disks adsorbing on a plane,2 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 †
Part of the special issue “Bruce Berne Festschrift”. * To whom correspondence should be addressed. E-mail:
[email protected].
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 an Fe (100) surface3 and the arrangement of proteins on solid surfaces4-6 to the deposition of colloidal particles on mica surfaces.7 RSA can provide a very powerful model for understanding surface phenomena when the molecules become permanently bound to the surface. There is immense literature on the coverage statistics of RSA models with a wide range of landing shapes including squares,8,9 ellipsoids,10 and lines.11 In general, RSA models of surface coverage approach a jamming limit, θJ, which depends on the shape of the landing molecule and the underlying lattice of attachment points.2 For disks on a continuum surface (i.e., no underlying lattice), the jamming limit is θJ ≈ 0.547.2 For ellipsoids, rectangles,10 and two-dimensional spherocylinders,12 there is a small (4%) initial rise in θJ as a function of particle anisotropy. However, the jamming limit decreases with increasing particle anisotropy once the length-to-breadth ratio rises above 2. That is, 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.11 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 tilted disks that allows near-monolayer coverage and which can explain the differences in coverage between the octopus and umbrella. In section 2, we outline the model for the two adsorbing molecules. The computational details of our simulations are given in section 3. Section 4 presents the results of our simulations, and section 5 concludes. 2. Model Two different landers were investigated in this work. The first, representing the octopus phthalocyanine, was modeled as a flat disk of fixed radius (σ) with eight equally spaced “legs” around the perimeter, each of length l. Because the average distance of the eight sulfur atoms from the central silicon atom is approximately 14.4 Å, disk sizes ranging from 7.5 to 14.4 Å are physically reasonable. Smaller disks correspond to a lander
10.1021/jp010985m CCC: $20.00 © 2001 American Chemical Society Published on Web 06/15/2001
6516 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Meineke and Gezelter
Figure 1. Structures of representative umbrella and octopus silicon phthalocyanines.
Figure 2. Models for the adsorbing species. Both the octopus and umbrella models have circular disks of radius σ and are supported away from the surface by arms of length l. The disk for the umbrella is tilted relative to the plane of the substrate.
Figure 4. Model thiol groups attach at the interstitial sites in the Au (111) surface. These sites are arranged in a graphitic trigonal lattice.
Figure 3. Coordinates for the umbrella lander. The vector nˆ is normal to the disks. The disks are angled at an angle of ψ to the handle, and the projection of nˆ onto the substrate surface defines the angle φ.
with legs that are perpendicular to the body of the molecule, whereas the larger disks represent molecules which attach with legs that are fully extended on the surface. The second type of lander, representing the umbrella phthalocyanine, was modeled by a tilted disk (also of radius σ) which was supported by a central handle (with length l ) 5 Å). The surface normal for the disk of the umbrella, nˆ was tilted relative to the handle at an angle ψ. Two different groups of umbrella simulations were done, one with ψ ) 109.5°, the normal tetrahedral bond angle for sp3 hybridized carbon atoms, and one with ψ ) 90°, yielding umbrellas that cannot overlap. The two particle types are compared in Figure 2, and the coordinates of the tilted umbrella lander are shown in Figure 3. The angle φ denotes the angle that the projection of nˆ 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 φ at random as it is dropped onto the surface. A range of physically reasonable disk sizes (σ) was chosen for each of the landers. For the octopus landers, the smallest
radius chosen (7.5 Å) represents the situation with the eight thiol arms folded down toward the surface, whereas the largest radius (14.4 Å) represents the molecule with it’s legs splayed out to their maximum extent. For the umbrella landers, the alkane appendages are also somewhat flexible, so simulations were run with σ taking values from 7.5 (the arms folded completely underneath the disk) to 16.6 Å (the arms extended well beyond their normal lengths). Physically, the most reasonable radius for the umbrella is 11.6 Å. 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 latticebased 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,1 giving a trigonal (i.e., graphitic) underlying lattice for the RSA simulations that is illustrated in Figure 4. The hcp nearest neighbor distance was 2.3 Å, corresponding to Au (111) lattice spacing. The lattice of attachment points therefore had a nearest neighbor spacing of 1.33 Å. Figure 4 also defines the xˆ and yˆ axes. 3. Computational Methodology The simulation box was 4000 repeated hcp units in both the x and y directions. This gave a rectangular plane
Random Sequential Adsorption Model (4600 Å × 7967 Å), 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 2σ 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 intersection of two neighboring umbrella tops involved three steps. In the first step, the surface normals for the umbrella tops were used to calculate the parametric equation for the line defined by the intersection of the planes of the two disks. This parametric line was then checked for intersection with the circles which defined the spatial extent of the two umbrellas. If the line intersected these circles, then the points of intersection along the line were checked to ensure sequential intersection of the two circles. That is, the line was required to enter and exit the first umbrella’s disk before it could enter and exit the second disk. This series of tests ensured that there was no overlap of the two umbrella tops but was somewhat demanding of computational resources. Therefore, it was only attempted if the original handle-projection 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. That is, if the initial position and orientation placed one of the thiol legs within a small distance ( ) 0.1 Å) 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 × 131 grid of equally sized rectangular bins. The overlap test then cycled over all of the molecules within the bins located in a 3 × 3 grid centered on the bin in which the test molecule was attempting to land. Surface coverage calculations were handled differently for the umbrella and octopus simulations. Because the octopi were essentially flat disks, the surface coverage was computed by multiplying the number of successfully landed particles by the area of its circular top. This number was then divided by the total surface area of the plane to obtain the fractional coverage. For the umbrellas, a relatively simple scanning probe algorithm was used. Here, a 1 × 1 Å probe was scanned along the surface, and the center point of the probe area was tested for overlap with all of the molecules in the immediate vicinity. An intersection of the normal to the surface with the ellipsoidal projections of any of the disks implied that the 1 Å2 area was covered. At the end of the scan, the total covered area was
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6517
Figure 5. g(r) for both the octopus and umbrella molecules in the continuum (upper) and on-lattice (lower) simulations.
Figure 6. g(r) and the distance-dependent 〈cosφij〉 for the umbrella thiol in the off-lattice (left side) and on-lattice simulations.
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).13 4. Results 4.1. Octopi. The jamming limit coverage, θJ, of the off-lattice continuum simulation was found to range from 0.533 to 0.5384 depending on the size of the landing molecules. These values are within two percent of the jamming limit for circles on a 2-D plane.2 It is expected that they would approach the accepted jamming limit for a larger gold surface. Once the system is constrained by the underlying lattice, θJ drops only slightly, 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 is shown in the two left panels in Figure 5. 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
6518 J. Phys. Chem. B, Vol. 105, No. 28, 2001
Figure 7. The position of the first peak in 〈cosφij〉 is due to the forced alignment of two tightly packed umbrellas. The depletion zone at 2σ 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.
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 et al.14 4.2. Umbrellas. In the case of the umbrellas (ψ ) 109.5°), the jamming limits for the continuum simulations ranged from 0.920 to 0.959 and, for the simulation on the lattice, θJ ranged from 0.903 to 0.923, depending on the radius of the umbrella disks. 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 on the basis of random parking alone. This then is the primary result of this
Meineke and Gezelter 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 Figure 5. Because 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 (Figure 6). 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 ≈ σ. 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 because, once the particles are greater than 2σ 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 Figure 7. For the umbrellas with ψ ) 90°, the coverage statistics reflect nearly identical behavior to the octopi (as expected). The only differences are in the S:Au ratio, which is approximately 1/8 of
Figure 8. Bird’s-eye view of the orientational domains in a monolayer of the umbrella thiol. Similarly oriented particles are shaded the same color.
Random Sequential Adsorption Model
J. Phys. Chem. B, Vol. 105, No. 28, 2001 6519
TABLE 1: Coverage Statistics for the Different Types of Landing Moleculesa σ
type of simulation
θJ
S:Au ratio
7.5 Å 7.5 Å 11.6 Å 11.6 Å 14 Å 14 Å 14.4 Å 14.4 Å
lattice continuum lattice continuum lattice continuum lattice continuum
0.532 0.533 0.537 0.537 0.5378 0.5384 0.539 0.538
0.11 0.11 0.0465 0.0465 0.032 0.032 0.0303 0.0303
octopi
0.021
umbrellas (ψ ) 109.5°)
0.0065 11.6 Å 11.6 Å 14 Å 14 Å 16.6 Å 16.6 Å
lattice continuum lattice continuum lattice continuum
0.903 0.959 0.915 0.920 0.923 0.920
0.019 0.015 0.0105 0.0107 0.00771 0.00899
umbrellas (ψ ) 90°)
0.0065 7.5 Å 7.5 Å 11.6 Å 11.6 Å 14 Å 14 Å 14.4 Å 14.4 Å
a
expt ratio
lattice continuum lattice continuum lattice continuum lattice continuum
0.532 0.533 0.537 0.537 0.5378 0.5384 0.539 0.538
0.0137 0.0138 0.0058 0.0058 0.0040 0.0040 0.0038 0.0038
The experimental S:Au ratios are taken from ref 1.
the S:Au ratio for an octopus with the same radius. Because the “flat umbrellas” have their normal vectors parallel to the handles, there is no way to define a φ angle and therefore no angular correlations. 4.3. 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 et al.1 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 1. Our simulations suggest that umbrellas with a radius of 11.6 Å are probably lying somewhat flatter than they do when ψ ) 109.5°. The experimental S:Au ratio lies between the ratios obtained from the ψ ) 109.5 and 90° simulations, so an angle between these two values would be required for full agreement with the experiments. For the octopi, our lowest estimate of the S:Au ratio is for a disk of radius 14.4 Å corresponding to a phthalocyanine with it’s arms fully extended on the surface. This model therefore predicts a S:Au ratio that is still 50% higher than the experimental S:Au ratio, which means experimentally, the disks are even farther apart than their molecular geometries would require because of random packing alone. The model has left out some important physical details, namely the thickness of the molecular bodies and the finite space taken up by the legs. We are in remarkable agreement with the coverage statistics given the simplicity of the model, but these calculations suggest that the octopi are interfering with each other at larger distances than they could if they were permanently fixed to the surface. 5. Conclusions The primary result of this work is the observation of nearmonolayer 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 et al. have observed that the octopus molecules are not parallel to the substrate and that they are attached to the surface with only four legs on average.1 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 exclude the Volume underneath the disks will revert to a standard RSA model with ellipsoidal landers. Viot et al. have shown that, for ellipsoids, the maximal jamming limit is only θJ ) 0.58.10 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 Figure 8, using color to denote the orientation of each molecule. Indeed, the monolayer does show orientational domains that are surprisingly large. 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 methodology12,15 to investigate what effect this has on the properties of the monolayer. Acknowledgment. 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. The code used to generate the results in this paper may be found at www.openscience.org/∼mmeineke/RSA. References and Notes (1) Li, Z.; Lieberman, M.; Hill, W. Langmuir 2001, in press. (2) Evans, J. W. ReV. Mod. Phys. 1993, 65, 1281-1329. (3) Dwyer, D. J.; Simmons, G. W.; Wei, R. P. Surf. Sci. 1977, 64, 617. (4) MacRitche, F. AdV. Protein Chem. 1978, 32, 283. (5) Feder, J. J. Theor. Biol. 1980, 87, 237. (6) Ramsden, J. J. Phys. ReV. Lett. 1993, 71, 295. (7) Semmler, M.; Mann, E. K.; Ricka, J.; Borkovec, M. Langmuir 1998, 14, 5127-5132. (8) Solomon, H.; Weiner, H. Comm. Stat. A 1986, 15, 2571-2607. (9) Bonnier, B.; Hontebeyrie, M.; Meyers, C. Physica A 1993, 198, 1-10. (10) Viot, P.; Tarjus, G.; Ricci, S. M.; Talbot, J. J. Chem. Phys. 1992, 97, 5212-5218. (11) Viot, P.; Tarjus, G.; Ricci, S. M.; Talbot, J. Physica A 1992, 191, 248-252. (12) Ricci, S. M.; Talbot, J.; Tarjus, G.; Viot, P. J. Chem. Phys. 1994, 101, 9164. (13) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (14) Boyer, D.; Viot, P.; Tarjus, G.; Talbot, J. J. Chem. Phys. 1995, 103, 1607. (15) Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic Press: New York, 1996.
