Methoxy Formation Induced Defects on MoS 2 - ACS Publications

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C: Surfaces, Interfaces, Porous Materials, and Catalysis 2

Methoxy Formation Induced Defects on MoS

Prescott Evans, Hae Kyung Jeong, Zahra Hooshmand, Duy Le, Takat B Rawal, Sahar Naghibi Alvillar, Ludwig Bartels, Talat Rahman, and Peter A. Dowben J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02053 • Publication Date (Web): 10 Apr 2018 Downloaded from http://pubs.acs.org on April 11, 2018

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The Journal of Physical Chemistry

Methoxy Formation Induced Defects on MoS2 , *

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Prescott E. Evans†, Hae Kyung Jeong† ∥ , Zahra Hooshmand , Duy Le , Takat B. Rawal ⊥, Sahar Naghibi Alvillar , # §,%* * Ludwig Bartels , Talat S. Rahman , and Peter A. Dowben† † Department of Physics and Astronomy, University of Nebraska-Lincoln, 68588-0299 (USA)

∥ Department of Physics, Institute of Basic Science, Daegu University, Gyeongsan 712-714, Korea (ROK) §

Department of Physics, University of Central Florida, Orlando, FL 32816, U.S.A.

⊥ Center

for Molecular Biophysics, Oak Ridge National Laboratory, Oak Ridge, TN, 37830 U.S.A and the Department of Biochemistry and Cellular and Molecular Biology, University of Tennessee, Knoxville, TN, 37996, U.S.A.

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Pierce Hall, University of California - Riverside, Riverside, CA 92521, U.S.A.

%

Department of Physics, University of California, Berkeley, CA 94720, USA

Syngas, MoS2 catalysis, defect creation, density functional theory, methoxy ABSTRACT: We find that exposure of the MoS2 basal plane to methanol leads to the formation of adsorbed methoxy and coincides with sulfur vacancy generation. The conversion of methanol to methoxy on MoS2 is temperature dependent. Density functional theory simulations and experiment indicate that the methoxy moieties are bound to molybdenum, not sulfur, while some adsorbed methanol is readily desorbed near or slightly above room temperature. Our calculations also suggest that the dissociation of methanol via O-H bond scission occurs at the defect site (sulfur vacancy), followed subsequently by formation of a weakly bound H2S species that promptly desorbs from the surface with creation of sulfur vacancy. Photoluminescence and scanning tunneling microscopy show clear evidence of the sulfur vacancy creation on the MoS2 surface, after exposure to methanol.

Introduction The possible application of transition-metal dicalcogenides, such as MoS2, for catalytic conversion of syngas (CO, CO2, and H2) to methanol (CH3OH),1-7 rests upon a strong foundation identifying MoS2 as a catalyst for production of methanol and higher alcohols.8-16 Thus, there is an excellent motivation to study some of the simpler observed reaction steps predicted in theoretical calculations. The last key reaction step for methanol formation is the conversion of a surface methoxy species and hydrogen to methanol (CH3O* + H* → CH3OH).1,3 While this final intermediate reaction would be difficult to observe under standard ultrahigh vacuum surface science techniques, the reverse reaction using methanol to produce the more stable intermediate methoxy (CH3O) species on the MoS2 substrate is amenable to experimental investigation. The adsorption and subsequent dissociative reaction of methanol at the edge sites of MoSx clusters has certainly been predicted using density functional theory (DFT), with the methanol dissociation via O–H bond scission predicted to be the most favorable.17 On the MoS2(0001) surface, however, methanol dissociation, via O–H bond scission, should be less facile, unless accompanied by defect creation. Significant efforts have been made by the several theoretical18-21 and experimental22-26 studies in understanding

the mechanism and energetics for methanol dissociation on the metal-oxide surfaces. None of these studies, however, put forward suggestions that the mechanism for methanol dissociation includes the reaction-induced anion defect (vacancy) creation on the surface, which are critical for facile methanol dissociation. Defects not only offer a favorable site for methanol adsorption.18,27 Methanol may also migrate to a defect site, which may enhance catalytic activity. It is clear that defects remain an integral part of MoS2based catalysis,1,28-30 and this is no less true with the methanol to methoxy reaction. A previous DFT study1 indicates that there are barriers of 1.40 eV and 1.19 eV between the methoxy and methanol moieties on MoS2, at a sulfur vacancy row and patch, respectively. Another DFT study17 suggests that the most favorable pathway for methanol dissociation is via O-H bond scission with the barrier of 1.0 eV at the 50% Mo-edge, and that the dominant species, after the MoS2 surface is exposed to methanol, is methoxy (CH3O). Here, we attempt to gain a better understanding the energetics of the methanol to methoxy reaction process on MoS2. We propose a mechanism for methoxy formation and induced S vacancy formation on MoS2, based on the scanning tunneling microscopy, core level photoemission, photoluminescence measurements, and DFT simulations.

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Figure 1: XPS of the C 1s core level after 1 x 10 Torr·sec methanol exposure on the left and 6 x 10 Torr·sec exposure on the right, note that at low temperature, there are two contributions to C 1s core level, indicated by shading of area under XPS spectra, indicative carbon in both methanol (blue) and methoxy (red) moieties.

Experimental Details Single crystals of MoS2 (SPI-Structure Probe Inc.) of 1x1 cm2 surface area were used. The top surface layers were exfoliated before measurements in an ultrahigh vacuum (UHV) chamber. Methanol (Alfa Aesar HPLC-ultrapure grade) was deposited onto MoS2 at liquid nitrogen temperature (~ 80 K), controlled by a leak valve into the UHV chamber, in 1 x 10-8 Torr for specific times, depending on the MoS2 coverages. Monolayer of MoS2 grown by chemical vapor deposition (CVD) method was used for the photoluminescence measurement. The CVD conditions and sample preparation is described in detail elsewhere.31 The core level X-ray photoemission spectra (XPS) were taken in-situ, with a SPECS X-ray source with a Mg anode (hv = 1253.6 eV) and a Phi hemispherical electron analyzer (PHI Model 10-360) with an angular acceptance of ±10° or more, as described in detail elsewhere.33-34 Temperature dependent XPS was performed to determine the activation energy or activation barrier of MoS2 from methanol to methoxy. The temperature of the sample was controlled by a combination of cooling with liquid nitrogen and resistive Joule heating. A scanning tunnelling microscope was used to characterize the surface morphology at the atomic level, using an Omicron modular UHV system with a variable temperature scanning tunneling microscope (VT-STM).35,36 The photoluminescence of chemical vapor deposition (CVD) grown monolayers of MoS2, before and after immersion in hot methanol (388 K) for 4 hours, was mapped at room temperature using a Horiba Labram instrument with a laser operated at a wavelength of 532 nm. Computational Methods We performed density functional theory simulations employing the projector-augmented wave (PAW)37,38 and

plane wave basis set methods. We used the PerdewBurke-Ernzerhof functional (PBE)39 to describe the electronic exchange-correlation. For explicit inclusion of van der Waals interactions in the adsorption and reaction of methanol and methoxy on pristine and defected MoS2, we used the DFT-D3 correction,40 which provides reasonably accurate results while keeping computational cost low. The energy cutoff for plane-wave expansion was set at 500 eV, which is sufficient for energy and force convergences. The simulation supercell consists of a 6×6 single layer MoS2 constructed using the optimized lattice parameter (a= 3.16 Å) and a vacuum of about 15 Å to separate the resultant structure with its normal periodical image. Because of the large supercell, we sampled the Brillouin Zone with one point located at the zone center, for ascertaining the structural (ionic) relaxation of all atomic configurations included in this study (with residual force of less than 0.01 eV/Å). In calculating energy barriers for the diffusion of vacancies, we used the Nudged Elastic Band (NEB) method41 for preliminary determination of a minimum energy path. We then applied the more accurate Climbing Image Nudged Elastic Band (CI-NEB) method42 to refine the calculation of the transition pathway. Methanol Oxidation and Methoxy Formation The formation of an adsorbed methoxy species, from the exposure of MoS2 to methanol is evident from the presence of both methanol reactants and a separate methoxy product apparent on the surface after the exposure of MoS2 to methanol. Methanol is predicted to be only weakly adsorbed on MoS2(0001),1 and is expected to be a minority species and easily desorbed from the MoS2(0001) surface. In fact, present experiments confirm this. The C 1s XPS core level spectra after both short and


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long exposures of MoS2 to methanol at low temperature (roughly at a MoS2(0001) substrate temperature of 100 K),

shows two carbon species with different chemical environments on the MoS2(0001)

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Figure 2: Comparison and visualization of critical temperature for methanol to methoxy conversion in 1 x 10 Torr·sec methanol -6 exposure on the left and 6 x 10 Torr·sec exposure on the right, note the increase in critical conversion temperature at higher methanol exposures (300 K versus 400 K)

substrate, as indicated in Figure 1. The methanol C 1s XPS core level peak, seen in Figure 1, that appears at binding energies greater than 287 eV is consistent with the methanol C 1s binding energy of 288.0±0.3 eV43 and 286.6 eV44 reported for methanol adsorbed on CuO2, 287.07 eV45 and 286.746 for methanol on Cu(111), 287.6 eV for methanol on TiO2,47 and the methanol C 1s binding energy of 286.5 eV.48 This XPS C 1s photoemission component is quickly quenched as temperature increases above room temperature. Methanol, both volatile and weakly bound to the substrate,1 is driven off with increased temperature leaving a strongly bound species with smaller binding energy of 285.0±0.2 eV, corresponding to methoxy, the dominant surface species after methanol exposure to MoS2(0001). Such a behavior has been seen for methanol decomposition to methoxy on CuO2, where again, with increasing temperature, methanol is lost from the surface and a methoxy C 1s XPS feature is observed at lower binding energies of 285.9 eV,43 286.0 eV,44 and 285.5 eV.45 Similar is the example of methanol decomposition to methoxy on TiO2, where again, with increasing temperature, methanol is lost from the surface and a methoxy C 1s XPS feature is observed at lower binding energies of 286.2 eV, as opposed to the core level binding energy of adsorbed methanol at 287.6 eV.47 This behavior is also true for methanol decomposition to methoxy on Cu(111),46 where methanol is lost from the surface well below room temperature and a methoxy C 1s XPS feature is observed at lower binding energies of 285.8 eV, as opposed to the core level binding energy of adsorbed methanol at 286.7 eV. Indeed, C 1s core level binding energies for adsorbed methoxy in the range of 285.2 to 286.243-47,49-53 are pretty typical. The much lower bonding energies for methoxy, of 285.0±0.2 eV, and is indicative of methoxy bonding to the surface at

a MoS2 defect, as discussed below, where charge donation from the substrate to the methoxy moiety is more facile. Figure 2 shows the binding energies of the various contributions to the C 1s core level spectra as a function of temperature, following 1 x 10-6 Torr•sec methanol exposure on the left and 6 x 10-6 Torr•sec exposure to MoS2(0001) at 100 K. At the low exposure of methanol two peaks from methanol and methoxy, respectively, co-exist until 300 K, as noted above, but evolve into a single C 1s feature corresponding to methoxy, as shown in Figure 2a. However, methanol remains resident on MoS2 surface until 400 K, following higher exposures to methanol, as shown in Figure 2b. We note that the higher desorption temperature of methanol, at the longer methanol exposures, as summarized in Figure 2, may be the result of a much higher sulfur vacancy defect concentration, and the greater possible coordination and thus stronger interaction with the substrate for adsorbed methanol.1 Sulfur Vacancies and Induced Defects The effects of the methanol to methoxy reaction, on the MoS2 surface, are directly observed in scanning tunneling microscopy (STM) and photoluminescence (PL). Under multiple low temperature methanol exposure and annealing cycles (Figure 3) the substrate begins to show point defects, as shown in Figure 3b, consistent with sulfur vacancy creation. Under further methanol exposure and annealing, line vacancies and larger multipoint defects appear and became abundant with larger multipoint defects nominally 2 nm in diameter as shown in Figure 3c. The scale of the line defects and larger multipoint defects, nominally 2 nm in diameter are highlighted in the inset to Figure 3.



There are experimental indications of the mechanisms for sulfur vacancy creation. The pristine MoS2(0001) surface contains very few defects (Figure 3a). Repeated exposure of MoS2(0001) to methanol at 100 K has little to no

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effect on the defect density. Exposure to MoS2(0001) vapor at low temperatures to methanol, followed by annealing to

Figure 3: The STM images validating the creation of point defects and larger defects on MoS2(0001) with methanol exposure. a) -6 the clean MoS2(0001) surface (0.900 V gap voltage, 800 pA constant current), b) areas of point defects after 1 x 10 Torr·sec -6 methanol exposure at 350 K (0.900 V gap voltage, 800 pA constant current) and c) larger 1 nm divots on sample after 2 x 10 -6 Torr·sec methanol exposure to MoS2(0001) at 350 K, followed by 1 x 10 Torr·sec methanol exposure to MoS2(0001) on 110 K sample then annealing at 350 K (2.00 V gap voltage, 400 pA constant current). The inset gives a larger view of defects within c) with dotted lines tracing line defects and circled multipoint defects.

room temperature or above, or direct exposure at elevated temperatures results in an increased number of point defects, some of which have grown into larger, isolated defects. It is important to note that defect creation was not clearly obvious on the MoS2 surface until a multiple sequence of methanol exposure, followed by annealing of the sample was performed, thus providing a greater defect density in the STM measurement. The formation of line defects is consistent with the sulfur-vacancy row formation as discussed elsewhere.1 As discussed below, we need prior sulfur vacancies to create other sulfur vacancies on MoS2. Without a sulfur vacancy, methanol does not chemisorb on pristine MoS2. So it is probable that methoxy formation initially starts from a small concentration of isolated sulfur vacancies. The density functional theory results indicate that the process of methanol dissociation on pristine MoS2 is endothermic. There must be the enthalpy penalty for this process to be feasible. We performed complementary experiments at the micron scale using MoS2 single layer islands deposited by chemical vapor deposition.31 Pristine single-layer MoS2 has a pronounced photoluminescence response at 1.87 eV,54,55 that decays with increased defect density of the material.56 We have characterized the material from our chemical vapor deposition process in detail with regards to photoluminescence yield57 and the results are indicative of a very low native defect density prior to methanol exposure. Rather than using argon ion beam based desulfurization, as in Ref. (56), here we find that exposure to metha-

nol leads to the same bleaching of the MoS2 photoluminescence. Figure 4 shows two single-layer MoS2 islands on 300 nm SiO2/Si before and after boiling them in methanol under reflux for 4 hours. While this procedure leaves the overall structure of the islands intact, we find a significant reduction of the photoluminescence yield indicative of defect formation. Notably, the photoluminescence bleaching is not confined to nor particularly strong at the island edges, but rather is more complete on the basal face. This clearly reveals that the basal plane is directly activated by the methanol species and a methanol to methoxy reaction at the material edges is not required. Although these result are for monolayer MoS2, they are consistent with the above STM results.

Figure 4: The photoluminescence mapping of CVD-grown single-layer molybdenum disulfide (MoS2) islands (triangles), performed before (left) and after (right) exposure to methanol at 338 K, over a period of 4 hours. Decrease in photolu-


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minescence indicates defect formation and/or inhomogeneity creation in the MoS2 lattice. Defect formation is not limited to edges, but occurs throughout the basal plane.

Our DFT simulations indicate that the weak chemisorption and dissociation of methanol on basal plane of pristine MoS2 is unrealistic. We only found physisorption of methanol on MoS2, with a binding energy of -0.24 eV while that of a co-adsorption of a methoxy and an atomic H is +4.02 eV. It is worth mentioning that the presence of atomic H, on pristine MoS2 does not substantially affect the binding energy of the methanol moieties, which is found to be -0.23 or -0.31 eV for methanol adsorption on top of or adjacent to S atom on which an atomic H binds. On the other hand, as interpreted from results shown in Ref. (1), co-adsorption of a methoxy and an atomic H on sulfur vacancies structures, such as vacancy row and vacancy "patch", as is experimentally seen in Figure 3c, is energetically favorable over the weakly bound adsorption or chemisorption of methanol. This leads us to a conclusion that the adsorption and conversion to methoxy of methanol occurs only at defected sites on the basal plane of MoS2. Our calculations also show methoxy and atomic H prefer to adsorb at defect sites. Based on these results, we suggest a mechanism for the creation of vacancy on MoS2 under the exposure of methanol: Step 1: methanol adsorbs and dissociates at sulfur vacancy sites to form methoxy and H atoms occupying the vacancy sites; Step 2: Once the sites are populated, a migration of atomic H onto the pristine terrace of MoS2 occurs. Step 3: atomic H diffuses on the pristine terrace of MoS2 to meet with other H to form H2S, which, in turn, desorbs from the MoS2 surface to create S vacancy. The schematic representation of these steps is shown in Figure 5.

To validate the proposed mechanism, we investigated the decomposition of methanol on MoS2 with a single sulfur vacancy site within DFT. We find, as illustrated in Figure 6, that methanol physically adsorbs at the defect site with a binding energy of -0.3 eV. It then needs to overcome a barrier of 0.26 eV to become chemisorbed, via an Mo-O bond. Next, the molecule dissociates to methoxy and bound atomic H with a barrier of 0.2 eV occupying the vacancy site. The resultant H atom, in turn, can migrate toward a neighboring S atom with a barrier of 0.8 eV and start to move away with a barrier of 0.12 eV, leaving methoxy adsorbed at the vacancy site. The potential energy of each state of the above chain reaction is summarized in Figure 6. Even though the potential energy profile indicates an uphill chain reaction with a noticeably high barrier, the values are in a reasonable range that does not require extremely high temperatures to occur, suggesting that such a proposed mechanism for H atoms to diffuse from vacancy sites to pristine terrace of MoS2 is feasible.

Figure 6: Chain reaction of methanol on basal plane of MoS2 with a sulfur vacancy. Horizontal bars and values (in eV) printed on it represent each intermediate state and corresponding potential energy (zero potential corresponds to a state in which methanol and MoS2 with a S vacancy are far apart) in this chain reaction. Potential of transition states (TS) are noted. Values (in eV) along the arrows are the barrier for the corresponding reaction. Blue, yellow, black, pink, and red balls represent Mo, S, C, H, and O atoms, respectively.

Figure 5: Schematic representation of the methoxyformation-induced sulfur vacancy on MoS2 with the various reaction steps: 1 adsorption of methanol, 2 dissociation of methanol into methoxy and hydrogen, 3 diffusion of atomic hydrogen, 4 coalescence of two hydrogen atoms with surface sulfur atom, and 5 desorption of hydrogen sulfide with the formation of sulfur vacancy. Blue, yellow, black, pink, and red balls represent the Mo, S, C, H, and O, atoms respectively.



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mains a smaller binding energy than uncoordinated methoxy of 533.7 eV.48

Figure 8. Reaction of atomic H forming H2 and H2S. The nomenclature "TS" indicates transition state. Blue, yellow, and pink balls represent Mo, S, and H atoms respectively.

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Figure 7: The core level O 1s XPS peak after 6 x 10 Torr·sec exposure of methanol to MoS2(0001) at 150 K (bottom) and 350 K (top) note the shift in binding energy.

Confirmation that the methoxy is bound to molybdenum, not sulfur, is evident in Figure 7. The O 1s core level binding energy is far less at elevated temperatures, where sulfur vacancy formation is more apparent, while the higher O 1s binding energies are observed for the adsorbed methanol/methoxy species upon initial adsorption at 150 K, as seen in the O 1s XPS spectra of Figure 7. At 150 K, the core level O 1s binding energy is higher, this can be attributed to the presence of both methanol and methoxy species coexisting at low temperature as suggested by the C 1s XPS spectra Figures 1a and 2a. The shift of the O 1s feature to lower binding energies at higher temperatures may be attributed solely to methoxy bound to the MoS2 substrate, as the methanol is no longer present at 350 K (Figure 2). The higher binding energy at low temperature coincides with a more electronegative chemical environment requiring greater energy for the ejection of an electron, such as would occur with weak binding of methanol and methoxy species to the surface. The O 1s binding energy of 533.8±0.2 eV for methoxy adsorbed on MoS2 (Figure 7) is much larger than the 530.1 eV to 530.9 eV binding energies reported for methoxy bound to copper surface,49-52 but more in line with the 533.1 eV binding energy reported for methoxy on Cu(111)55 and the binding energy of 532.8 eV observed for methanol condensed on ZnO.58 Yet MoS2, even with sulfur vacancies, the molybdenum atoms are still well coordinated to sulfur and a higher binding energy is expected than for a methoxy species on a bare metal surface. We may additionally conclude support of methoxy-molybdenum bonding, through an additional electronegativity argument, as molybdenum will contribute electron density through the Mo-O bond, decreasing the O 1s binding energy. The O 1s binding energy of 532.1 eV is close to the binding energy of hydroxylated molybdenum oxide59 while the methoxy-sulfur bonding seen at lower temperatures is larger, as expected, this re-

From our DFT simulations, we also find that on the basal plane of MoS2, without defects, the barrier for H to rotate about one S atom is 0.19 eV and to hop from one S atom to the other is 0.29 eV. These indicate the high mobility of bound atomic H on basal plane of MoS2, suggesting that eventually two H atoms may land on two neighboring S atoms from which either H2 or H2S is formed. The barrier for the formation of the former product is found to be 0.42 eV (red reaction in Figure 8) and that of the latter product is found to be 0.38 eV, suggesting a slight preference for the formation of H2S complex. Once the H2S complex formed, it can desorb, with a barrier of 0.53 eV, creating S vacancy on the MoS2 surface. As summarized in Figure 8, the formation and desorption of H2S are downhill reactions (exothermic). This indicates that should H atoms exist on the basal plane of MoS2, a sulfur vacancy will be formed. Clearly, the conversion of the adsorbed methanol to methoxy consumes sulfur atoms of MoS2, leaving the sulfur vacancies in the lattice, especially at the surface. This is evident in experiment (Figures 3 and 4). In the conversion process, sulfur is removed from the surface into a mobile state, likely producing H2S. As both methanol and methoxy moieties are present on the MoS2 surface at low temperature (in the vicinity of 100 to 200 K), and the conversion process is sulfur eliminating, at temperatures where H2S desorbs, additional methanol exposure and conversion to methoxy is required to further increase the sulfur vacancy density. The process, for methanol to methoxy conversion, must be accompanied by an activation barrier, as is evident from the energetics just discussed. From theory,17 the activation barrier has been estimated to be in the vicinity of 0.45 eV on the bald Moedge and about 1.0 eV on the 50% Mo-edge and 50% Sedge of MoS2, where methanol adsorption is more facile, and must be significant on the basal face, as just discussed.


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The Journal of Physical Chemistry *Talat Rahman: [email protected]

Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Funding Sources U.S. Department of Energy National Science Foundation; National Research Foundation of Korea; National Energy Research Scientific Computing Center.

ACKNOWLEDGMENT

Figure 9: The oxygen to molybdenum XPS intensity ratios, -6 for exposure of 1 x 10 Torr·sec exposure of methanol, plotted as a temperature dependent rate (R), for given temperature. The line indicates the best fit to the data and the shaded region, the range of fits based on a statistical analysis of the data (using a jackknife indicator).

An activation barrier of the reaction is apparent from ratio of the O 1s to Mo 2p core level intensities as a function of time and temperature. The rate of increase in the O 1s to Mo 2p core level intensity ratio, as a function of temperature has been plotted as an Arrhenius plot in Figure 9. The measured activation energy is small, 12±6 meV/molecule, but experimentally, subsumes a number of different process, as indicated. Nonetheless, an activation barrier for methoxy formation is both anticipated and observed. Conclusion The coverage and temperature dependent XPS measurements confirm the existence of the methoxy on MoS2 after methanol exposure. The STM and PL measurements indicate that sulfur vacancies are generated during methoxy formation via O-H scission of methanol, resulting in defects on the MoS2(0001) surface. While methoxy binds to vacancy sites, on the MoS2 surface, the detached hydrogen atoms spill over onto the pristine terrace of MoS2. The coalescence of these hydrogen atoms with surface S atom results in a volatile sulfur product in the form of H2S, as suggested by density functional theory. The continual of such process leads to the higher concentration of the S vacancies, which ultimately give rise to larger defect patches or line defects. Thus, the methanol to methoxy conversion is most facile on MoS2 surface in the presence of defects, and defect creation sequentially takes place.

AUTHOR INFORMATION Corresponding Author *Peter Dowben: [email protected] *Hae-Kyung Jeong: [email protected]

This work was supported by the U.S. Department of Energy through grant #DE-FG02-07ER15842 while HKJ was partly supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF2016R1D1A3B04931018). The DFT calculations were performed using resource from the National Energy Research Scientific Computing Center (NERSC, project 1996) and the Advanced Research Computing Center at UCF. The authors would like to thank Sumit Beniwal, for technical support. TSR would also like to thank the Miller Institute for Basic Research in Science, University of California Berkeley, for facilitating this work through the award of a Miller visiting professorship.

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