Langmuir 1993,9, 2351-2355
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Domain Formation and System-Size Dependence in Simulations of Self -Assembled Monolayers J. Ilja Siepmann*pj Zurich Research Laboratory, IBM Research Diuision, Saumerstrasse 4, CH-8803 Riischlikon, Switzerland
Ian R. McDonald Department of Chemistry, Uniuersity of Cambridge, Lensfield Road, Cambridge CB2 1E W, United Kingdom Received February 8, 1993 Monte Carlo studies have been made of the role of sample-size effects in the simulation of monolayer systems consisting of long-chain hydrocarbons. The potential parameters were chosen to model the selfassembled monolayers formed by chemisorption of CH3(CH&,SH admolecules on a gold substrate, and results were obtained for three system sizes, N = 30, 90, and 224. Investigation of the translational, orientational, and conformational order within the monolayers reveals a marked dependence on N . In particular, the 224-molecule simulation is characterizedby the appearance of well-differentiated domains that are absent in the smaller systems. It therefore appears that the system sizes used in many monolayer simulations ( N 5 100) are too small to capture certain important structural features of the system of interest. Comparison is made with experimental results and with related molecular-dynamicscalculations.
Introduction Current experimental interest in the properties of selfassembled monolayers formed by adsorption of long-chain alkanethiols onto gold and other m e t a l ~ l is - ~fueled by the fact that systems of this type serve as convenient models of a variety of organic thin films of biological or technological i m p ~ r t a n c e . ~Features ,~ of the alkanethiol monolayers that make them particularly attractive to experimentalists include the relative ease of preparation, thermodynamic and chemical stability, and mechanical strength. The stability comes about through the formation of a chemical bond between the sulfur headgroup and the metallic substrate, an effect associated with the loss ofthe headgroup hydrogen during the process of adsorpti~n.~,' Chemisorption of the (111) face of gold gives rise to a well-defined, two-dimensional lattice structure in which the headgroups form a ( 4 3 x d3)R3O0 overlayer with a lattice constant of 4.99 8, and a surface area of approximately 21.4 A2per hai in;^^^ the ordering of the headgroups persists over regions that are at least a few hundred angstroms in length and probably much larger. On the outer surface of the monolayer, however, the indications from helium-beam diffraction,lOJl scanning tunneling + Current address: Department of Chemistry, University of Pennsylvania, Philadelphia, P A 19104-6323. (1) Nuzzo, R. G.; Allara, D. L. J . Am. Chem. SOC.1983,105, 4481. (2) Bain, C. D.; Troughton, E. B.; Tao, Y.-T.; Evall, J.; Whitesides, G. M.; Nuzzo, R. G. J . Am. Chem. SOC.1989,111, 321. (3) Laibini5,P.E.; Whitesides,G. M.;Allara, D. L.;Tao,Y.-T.;Parikh, A. N.; Nuzzo, R. G. J. Am. Chem. SOC.1991,113, 7152. (4) Swalen, J. D.;Allara, D. L.; Andrade, J. D.; Chandros,E. A,; Garoff, S., Israelachvili, J.; McCarthy, T. J.; Murray, R.; Pease, R. F.; Rabolt, J. F.; Wynne, K. J.; Yu, H. Langmuir 1987,3, 932. (5) Bain, C. D.; Whitesides, G. M. Angew. Chem., Int. Ed. Engl. 1989, 28,506. (6)Nuzzo, R. G.; Zegarski, B. R.; Dubois, L. H. J . Am. Chem. SOC. 1987,109,733. (7) Stole, S. M.; Porter, M. D. Langmuir 1990,6, 1199. (8) Strong, L.; Whitesides, G. M. Langmuir 1988,4, 546. (9) Nuzzo, R. G.; Korenic, E. M.; Dubois, L. H. J . Chem. Phys. 1990, 93,767. (10) Chidsey, C. E. D.; Liu, G.-Y.; Rowntree, P.; Scoles, G. J . Chem. P ~ Y S 1989, . si,4421. (11)Camillone, N.; Chidsey, C. E. D.; Liu, G.-Y.; Putvinski, T. M.; Scoles, G. J. Chem. Phys. 1991,94,8493.
microscopy,12J3and atomic force microscopy14 are that true long-range order is absent and that the hexagonal ordering of the terminal methyl groups is confined to domains having linear dimensions of the order of tens of angstroms. The results of several molecular-dynamics and Monte Carlo simulations of alkanethiol monolayer systems have recently been reported.15-18 In all the papers cited here, the calculations were made for periodic systems of 90 or 100 admolecules, corresponding to a basic simulation cell of dimensions smaller than 50 A. The experimental observations suggest that such a cell is too small to encompass more than one tailgroup domain, which in turn raises the question whether system-size effects are playing a role, since the absence of domain boundaries could well have an influence on other properties of the monolayers. To our knowledge, the only systematic study of sizedependence effects in monolayer simulations is that carried out by Bishop and Clarkelg for a model of a LangmuirBlodgett film, and even in that work the largest system considered was one consisting of 100chains. Furthermore, recently Bocker et aL20reported results of simulations of a Langmuir monolayer for two different system sizes (32 and 64 amphiphilic molecules), but at much lower density than considered in our work. Here we present results of Monte Carlo calculations for a self-assembled monolayer formed by the chemisorption of hexadecyl mercaptan, CH3(CH&SH, on the (111)face of gold. Three different system sizes (30, 90, and 224 admolecules) have been (12) Michel, B.; Rohrer, H.; Haussling, L.; Ringsdorf, H. Angew. Chem., Int. Ed. Engl. 1991,30, 569. (13) Kim, Y.-T.; Bard, A. J. Langmuir 1992,8 , 1096. (14) Durig, U.;Zuger, 0.; Michel, B.; Haussling, L.; Ringsdorf, H. Phys. Reu. B , in press. (15) Hautman, J.; Klein, M. L. J. Chem. Phys. 1989,91,4994. Ibid. 1991,93,7483. (16) Hautman, J.;Bareman, J. P.; Mar, W.; Klein, M. L. J. Chem. SOC., Faraday Trans. 1991,87,2031. Hautman, J.; Klein, M. L. Phys. Reu. Lett. 1991,67, 1763. (17) Siepmann, J. I.; McDonald, I. R. Mol. Phys. 1992, 75, 255. Siepmann, J. I.; McDonald, I. R. Phys. Reu. Lett. 1993,70, 453. (18) Siepmann, J. I.; McDonald, I. R. Mol. Phys. 1993,79,457. (19) Bishop, M.; Clarke, J. H. R. J . Chem. Phys. 1991,95,540. (20)Bocker, J.; Schlenkrich, M.; Bopp, P.; Brickmann, J. J. Phys. Chem. 1992,96,9915.
0 1993 American Chemical Society
Siepmann and McDonald
2352 Langmuir, Vol. 9, No. 9,1993 studied. As we shall see, the results for the largest system differ in important ways from those obtained for the two smaller ones.
Model All our calculations were made for the same potential model used in our earlier calculationsls (model CL*) which is a minor variation of that adopted in the moleculardynamics simulations of Hautman and Klein15(their model I). The model is based on a united-atom treatment of the admolecules: each alkanethiol molecule is represented by a chain of 17 pseudoatoms, corresponding to the sulfur headgroup, the 15 methylene units, and the methyl tailgroup. Pseudoatoms in different molecules, and those belonging to the same chain if separated by more than three bonds, interact with each other through LennardJones 12-6 potentials truncated at 12.0 A. The LennardJones parameters were mostly taken from the work of Jorgensenz1Sz2on the liquid phases of sulfur-containing organic molecules. The only departure from this scheme was the use of a headgroup-headgroup size parameter a s s equal to 4.97 A in place of the value (3.55 A) suggested by Jorgensen; the resulting enhancement of the sulfur-sulfur potential is designed to mimic the limited translational mobility characteristic of a chemisorbed molecule. The interaction between a pseudoatom and the surface is described by a 12-3 potential without truncation, with parameters taken from Hautman and Klein.15 The pseudoatoms in a given chain are assumed to be connected by rigid bonds; bond bending is modeled by a harmonic potentialz3with a force constant equal to 62 500 K rad-z, and changes in the torsional angles are controlled by the Ryckaert-Bellemans potential.z4 A more detailed description of the potential model and a complete list of the parameters can be found elsewhere.15Ja The differences with respect to the model of Hautman-Klein15 are limited to the choice of as-^ (4.97 A rather than 4.25 A) and of the cutoff in the 12-6 potentials (12.0 A rather than 9.0 A). The first of these changes was introduced in order to stabilize the headgroup spacing at its experimental value, while the second was designed to ensure that the cutoff in the potentials lay well beyond the spacing of next-nearest-neighbor chains. In practice, however, neither of these changes has more than a modest effect on the monolayer structure.18 As in our earlier work,18we have adopted a Monte Carlo, canonical-ensemble approach rather than the moleculardynamics method used in most related simulations of Because the Monte Carlo monolayer systems.15J9~z5-z9 method is not tied to the natural time evolution of the system, it is possible to adapt the calculations to the study of molecular transformations that occur only slowly on the real time scale. The computer program used in the present calculations is built around three different types of Monte Carlo “moves”: translational displacements are treated in the standard way,3orotations are handled by (21) Jorgensen, W. L. J. Phys. Chem. 1986, 90,6379. (22) Jorgensen, W. L.; Tirado-Rives, J. J . Am. Chem. SOC.1988,110,
1657. (23) van der Ploeg, P.; Berendsen, H. J. C. J . Chem. Phys. 1982, 76, 3271. (24) Ryckaert, J. P.; Bellemans, A. Faraday Discuss. Chem. SOC.1978, 66, 95. (25) Harris, J.; Rice, J. A. J. Chem. Phys. 1988, 89, 5898. (26) Moller, M. A,; Tildesley, D. J.; Kim, K. S.; Quirke, N. J. Chem. Phys. 1991,94, 8390. (27) Karaborni, S.; Toxvaerd, S. J. Chem. Phys. 1992, 96, 5505; 97, 5876. (28) Karaborni, S.;Toxvaerd, S.;Olsen, 0.H. J.Phys. Chem. 1992,96, 4965. Karaborni, S. Langmuir 1993, 9, 1334. (29) Collazo, N.; Shin, S.; Rice, S. A. J. Chem. Phys. 1992, 96, 4735.
Table I. Details of the Monte Carlo Runs run
N
Nm NW
30 90 224
N224 a
cell dimensionsiA
Cadi
Cord
(t1C)Vs
24.85 X 25.82 44.73 X 43.04 69.58 X 68.87
30000 30000 100000
20000 20000 50000
3.1 14.4 64.8
cpu time per Monte Carlo cycle on a n I B M RS60001550.
Table 11. Selected Results (with Statistical Uncertainties) of the Monte Carlo Calculations N30
NW
(U,JN)/(104 K) -2.42 f 0.02
-2.410 f 0.004 2.410 f 0.001 2.407 f 0.001 19.97 f 0.06 20.05 f 0.08 29.2 f 0.5 29.0 f 0.5 28.3 f 0.5 28.8 f 0.5
(Zhead)/A
(ztd)/A (0, )/deg (Bdldeg
N224
-2.347 f 0.002 2.4076 f 0.0004 20.15 f 0.03 27.8 f 0.2 25.5 f 0.3
Table 111. Results (with Statistical Uncertainties) for Conformational Defects NW
N224
1.18 f 0.03 57f 1 88.7 f 0.3 6.8 4.1 0.3 0.1
3.27 f 0.09 59 f 2 76.4 f 0.5 10.2 8.2 2.8 1.7
N30
fT@)/
%
f&l% fM@O)/ fM@l)/ fM@2)/
fM@3)/
% %I % %
fM@4)/%
1.06 f 0.09 45 9 89.2 f 0.4 7.6 2.7 0.6
0.0
the method of Barker and Watts,3l and the conformations of the chain molecules are sampled by the configurationalbias Monte Carlo (CBMC) scheme.32 Both the molecule to be moved and the type of move are chosen at random with equal probabilities; a detailed description is given in our previous paper.18 Changes in the molecular conformations belong to the class of slow processes mentioned above, and the efficiencywith which the program explores the accessible configurations of the monolayer as a whole is determined very largely by the way in which the conformations are sampled. The CBMC method has proved to be a very efficient tool in this r e ~ p e c t , ~ ~ , ~ ~ - ~ ~ since its use permits large-scale conformational changes to take place in a single move.
Results Three simulations were carried out, each for a temperature of 300 K. The systems studied in the different runs consisted of N = 30 (run N3o), 90 (run Ngo) or 224 (run N224) alkanethiol molecules confined to a rectangular cell at a surface area of 21.4 A2 per chain and with periodic boundary conditions imposed in the plane ( x , y ) of the surface. The cell dimensions in the three runs are listed in Table I, and Tables I1 and I11 summarize the most important results. The choices of N were dictated by a variety of considerations: a 30-molecule system is the smallest one for which the cell length is at least twice the required cutoff distance in the Lennard-Jones potentials, N = 90 is the system size used in several earlier simulations, and the 224-molecule system was considered to have an area sufficient to accommodate more than one tailgroup domain while being small enough to study with the available computing resources. Each of the simulations was initiated from an ordered configuration in which the sulfur headgroups occupy the sites of a perfect triangular (30) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (31) Barker, J. A.; Watts, R. 0. Chem. Phys. Lett. 1969, 3, 144. (32) Siepmann, J. I.; Frenkel, D. Mol. Phys. 1992, 75, 59. (33) Siepmann, J. I.; Sprik, M. Chem. Phys. Lett. 1992, 199, 220. (34) Frenkel, D.; Mooij, G.C. A. M.; Smit, B. J . Phys.; Cond. Mutt. 1992, 4 , 3053.
Simulations of Monolayers
lattice and the molecules are in all-trans conformations, with the planes containing the carbon backbones arranged in a two-sublattice, herringbone structure. The lengths of the simulations, expressed as numbers of Monte Carlo cycles (1 cycle = N trial moves), are shown in Table I; the columns labeled Cequil and Cprod give the numbers of cycles in the equilibration and production stages, respectively, and the column labeled t / C gives the cpu time per cycle on an IBM RS6000/550 workstation. Note that run N 2 2 4 required a total of 2700 cpu hours. Molecular-graphicsprograms have been used to generate computer images of configurations produced in the course of the simulations. The final configurations from the three runs are pictured in Figure 1. It is clear from the figure that whatever the system size the alkanethiol chains order in hexagonal fashion and are tilted with respect to the surface normal in what are locally well-defined directions. However, there is a striking difference in the degree of inhomogeneity seen for the largest system compared with the two smaller ones. In Figure IC,taken from run N224, a boundary between domains is clearlyvisible, across which the preferred direction of tilt changes; the boundary itself is marked by molecules that have larger than average numbers of conformational defects (see below). Both the shape and the location of the domain boundary were seen to change in the course of the production stage of the calculation. By contrast, the views of the smaller systems pictured in Figure la,b reveal no significant inhomogeneities. In each case, the system behaves as a single domain, throughout which the tilt direction is virtually uniform. Note also (Table 11) that the mean potential energiesper chain ( U,,t), in runs N30 and Ngo are identical, within statistical uncertainties, while that in run N 2 2 4 is significantly higher, reflecting the greater disorder in the largest system. The radial distribution functions for headgroups and tailgroups are shown in Figure 2. There is no measurable shift in peak positions as the sample size increases, but the peaks become broader. Once again, however, the differences between runs N30 and N90 are small. Figure 2a shows that the headgroup positions are highly ordered and that the ordering persists up to the largest distances studied. This picture is consistent with the results of diffraction experiments, which suggest a solidlike packing with a lattice constant of 4.99 A and domain sizes of hundreds of angstrom^.*^^ The tailgroup-tailgroup distributions are much less structured. The nearest-neighbor spacing is approximately 5.0 A, independent of the system size, which is to be compared with a value of 5.01 f 0.02 A deduced from helium-scattering datalOJ1 and with a surface X-ray diffraction estimate35 of 4.32 A for the interrow spacing, measured relative to the underlying gold surface. The tailgroup-tailgroup ordering in run N 2 2 4 is already weak a t half the cell length (note the difference in scale of the vertical axes in the two parts of Figure 2). There is also a significant size dependence: the peak heights for N = 224 (with values for N = 90 in parentheses) are 2.30 (2.44), 1.50 (1.62), 1.35 (1.58), 1.27 (1.56), 1.18, 1.14, and 1.11. A rough extrapolation of the results for run " 2 2 4 suggests that the structure in the tailgroup distribution would be undetectable for distances greater than about 50 A,and at somewhat smaller separations for still larger systems. Again, therefore, the Monte Carlo data are consistent with the available experimental data, in this case with measurements of the likely maximum (35) Samant, M. G.; Brown, C. A.; Gordon, J. G., 11. Langmuir 1991, 7, 437.
h
Figure I. Final configurationsof runs N30 (a),N ~(b), o and N224 ( c )viewed from the direction of the surfacenormal. For runs N30 and NW parts of the periodic surroundings are also shown, while for the largest system only the basic simulation cell is pictured. Molecules are color-coded according to their number of gauche defects, ranging from red for all-transchains to purple for chains incorporating eight or more gauche defects.
widths of tailgroup domains, namely, 50 A (from helium scatteringll) or 70 A (from X-ray d i f f r a ~ t i o n ~ ~ ) . Results for the mean distances between the sulfur atom and the surface, (Zhead), and between the terminal methyl group and the surface, (zhil), are given in Table 11. The
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r /A Figure 2. Radial distribution functions for (a) the sulfur headgroups and (b) the methyl tailgroups from runs N ~(dotted o lines), Nw (dashed lines), and N224 (solid lines).
differences between runs are small, but the thickness of the monolayer, as measured by the quantity (Ztail), is apparently a weakly increasing function of the system size. Monolayer thicknesses can be determined experimentally by optical ellipsometry.2 The values so far reported are similar to those listed in Table 11, but the system-size effects (-0.1 A) seen in the calculations are at least an order of magnitude smaller than the experimental uncertainties, and in any event a knowledge of (Ztail) alone provides little insight into the nature of the monolayer disorder. More detailed information is provided by pictures such as Figure 3, which illustrates the topology of the 224-molecule monolayer a t its outer surface. The figure was constructed by taking a configuration of the chains and allowing a probe sphere of diameter equal to the Lennard-Jones parameter for the methyl group (a = 3.905 A) to move over the surface. The thickness, t, recorded in the figure as a function of position in the (x, y) plane is the maximum height above the gold surface at which an overlap ( R < a) occurs between the probe sphere and any pseudoatom of the monolayer. In the center of the figure is a large domain (t == 23 A) consisting predominantly of molecules tilted toward nearest-neighbor chains, in the upper left part is a region (t = 22 A)in which the moleculeshave tilts directed between nearest-neighbor and next-nearest-neighbor chains, and in the lower left part, separating the other two areas, there is a depression of approximate dimensions 15A X 20 A in the (x,y) plane and a depth between 2 and 4 A (18 A < t < 20 A). The corresponding maps (not shown) for the 30- and 90molecule simulations reveal no significant variations in height; this is to be expected, given the more homogeneous character of the smaller systems. Though a direct comparison between simulation and experiment is not yet possible, there is no doubt that the results for the largest system are more in line with the existing experimental evideqce. Until recently it was generally believed that the monolayers produced either by the Langmuir-Blodgett technique or by a self-assembly process are homogeneous systems with smooth outer s u r f a c e ~ . ~ ?Since ~ J ~ 1991, however, experimental results based on atomic imaging
thickness / 18.0
20.5
A 2b.u
Figure 3. Topology of the final configuration of run N224. The picture is a guide for the images expected in STM and AFM experiments: see text for details. The sulfur headgroups are highlighted by the white circles.
techniques12-14(scanning tunneling microscopy, or STM, and atomic force microscopy, or AFM) have become available. The new results indicate that the surfaces are more heterogeneous than was previously supposed and are extensively pitted rather than smooth. For example, holes with lateral dimensions around 30 A and a depth between 2 and 3 A have been observed by combined STM/ AFM for a self-assembled monolayer consisting of CH2OH(CH2)1sSH molecules chemisorbed on Au(lll),14although other systems, such as mixtures of CH3(CH2)11SH and CH&F2)7NHCO(CH2)2SH, apparently have much smoother surfaces.36 The results from the 224-molecule simulation suggest that where holes do occur, they are associated with the development of domains in which the molecules have different tilt directions. The density profiles, p ( z ) , for the direction normal to the surface are plotted in Figure 4. System-size effects show themselves in two ways: as N increases, the amplitudes of the peaks in p ( ~ )decrease, but the peak positions shift to larger values of z. The profiles for the terminal methyl group are shown in the inset to the figure. In the latter, the marked outward shift in position for run N 2 2 4 implies that within the ordered regions of the monolayer the chains are more erect than in the smaller systems; the long tail at smaller z represents the contribution from molecules in the domain wall. Since (Ztail) is an average over the density profile of the terminal pseudoatom, it follows that the insensitivity of this quantity to the system size (Table 11) is to some extent fortuitous. The mean values of the molecular (0,) and system (0,) tilt angles in the different runs are given in Table 11. The angle 8, is the angle between the surface normal and the (36) Mizutani, W.; Anselmetti, D.; Michel, B. In Computationsfor the Nano-Scale; Bloechl, P. E., Joachim, C., Fisher, A. J., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1993; p 43.
Langmuir, Vol. 9, No. 9, 1993 2366
Simulations of Monolayers
2 / A Figure 4. Density profiles normal to the surface plane from rune Nm (dotted lines),NSO(dashed lines), and NZU(solid lines). z is the height above the surface. Resulta are shown for the s u m of a l l pseudoatoms and in the inset for the terminal methyl poup alone.
cos e,
Figure 5. Normalized distributions for the molecular tilt angles, e,, from runs Nm (dottedline), NSO(dashedline), and NZU(solid line).
vector parallel to the long axis of a given molecule; 8, describes the collective tilt of the system and is defined in terms of the instantaneous mean headgroup and tailgroup positions.18 The difference between 8, and 8, is a measure of the inhomogeneity of the system, it is therefore not surprising to find that the differenceincreases with the system size. The 224-molecule system is one in which regions of different tilt directions coexist, while in the smaller systems the tilt is almost uniform and the difference between 8, and 8, is statistically insignificant. The distributions, g(c0s em), of the cosine of the molecular tilt angle are plotted in Figure 5. The results match those obtained for the density profile of the terminal methyl group (Figure 4): the curve obtained for N = 224 is broader than for N = 30 and N = 90,the peak is displaced toward smaller angles, and there is a long tail extending to larger angles. Results obtained for properties descriptive of conformational order along the chains are summarized in Table 111. The quantity f T @ ) is the mean percentage of gauche defects, calculated as an average over all molecules and their 14 dihedral angles, f,(k) is the percentages of defecta that are part of a kink defect, i.e., a pattern of defects of the type g + t g or g t g + , and f M ( g i ) , i = 1-4, are the percentages of molecules that instantaneously contain precisely i defects. The most striking result to emerge is that the number of conformational defects per molecule is roughly 3 times larger for run N224 than for either of the
other two runs: this difference is behavior is related to the large numbers of defecta formed at domain boundaries, which appear only in the largest system. Roughly a quarter of all molecules in the 224-chain system contain at least one such defect, and a significant number (-2 % ) contain four or more. By contrast, runs Nm and NW lead to structures in which about 90% of the chains are in alltrans conformations. In each of the runs, roughly one in two defects is part of a kink, but double-gauche defects (Le., two successive gauche bonds) are found only for N = 224, since their formation is sterically unfavored in situations where all chains are closely packed and nearly erect.
Discussion It is clear from our results that the degree of translational, orientational, and conformationaldisorder in the simulated monolayers is strongly system-size dependent. When N = 30 or 90, the monolayer has a homogeneous structure, with the molecules uniformly tilted. When N is increased to 224, domains appear, characterized by different tilt directions and separated by walls consisting of molecules that contain more conformational defects and have smaller tailgroup-substrate separations than elsewhere. An increase in N also leads to some reduction in the mean molecular and system tilt angles. The greater homogeneity of the smaller systems presumably has its origin in a stabilization of the ordered triangular structure by the periodic boundary conditions. Note that, even for N = 90, a system size typical of those used in other published calculations, the length of a molecular chain ( 20 A) is only little less than half the length of the periodic cell (-44 A, Table I). Our conclusions are in conflict with those reached by Bishop and Clarkelg on the basis of molecular-dynamics calculationsfor models of a LangmukBlodgett fii. Those researchers find that both the molecular tilt angle and the fraction of torsional angles in the trans state increase with the system size ( N = 16,64, and 100) at rates which in turn are dependent on the surface area per chain. They then make a linear extrapolation in N-' in order to obtain estimates of the tilt angle in the limit N =. The qualitative changes observed in our own work occur only for N > 100, i.e., for system sizes larger than those considered by Bishop and Clarke, and the qualitative nature of the differences between runs NW and N ~ makes u it inappropriate to attempt an extrapolation tothe infiitesystem limit. It is worth adding that the minimum size of the system capable of supporting a heterogeneous structure is almost certainly dependent on the surface density. Indeed, Karaborni and ToxvaerdZ7 have recently reported the observation of heterogeneous structures in simulations of Langmuir films for systems consisting of only 64chains, but at headgroup areas per molecule greater than 27 A.
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Acknowledgment. We are grateful to Urs Diirig, Bruno Michel, Michiel Sprik, Mike Klein, and Sami Karabomi for many useful discussions, and to John Shelley and Francois Gygi for supplying us with copies of their molecular-graphics programs.
