571
Langmuir 1989,5, 571-575
The Structure near Transitions in Thin Films? James M. Phillips Department of Physics, University of Missouri-Kansas City, Kansas City, Missouri 64110 Received October 27, 1988. I n Final Form: December 13, 1988 Computer simulation has been used to study the methane/graphite, methane/Au(lll), and argon/graphite systems at T = 25 and 100 K for a coverage of 4.2 ( ~ ' 3 x 4 3 )monolayers. The structure of these films is given by a vertical density profile and the two-dimensional pair distribution for each individual layer. Comparisons between the results of the three model systems are given, and a brief discussion of both solid and fluid interfacial theories is offered.
It is well-known that adsorbed crystalline films of finite thickness may coexist at low temperatures with their own vapor at pressures below saturation.' Thin physisorbed films undergo a series of changes and transitions as the temperature and/or coverage is increased. It is often assumed that the detailed microscopic structure within these films will dictate the manner of growth and the character of the transitions experienced by these evolving systems. As the temperature is raised at fixed coverage, the film will experience a number of structural transitions. Depending on the interaction strengths of the adsorbateadsorbate and the adsorbate-substrate potentials, the film could have roughening, wetting, surface melting, melting, layering, and commensurate-incommensurate transitions. Also, the structural character of the film can vary as it evolves from a thin to a thick multilayer. In an attempt to add to the information obtained from diffraction2 and thermodynamic3 experiments, this study uses computer simulation to improve our understanding of the structural behavior in these interesting systems. It has been the goal of theory to develop a formalism which realistically describes the properties of the film and gives a unique signature to each of the many different transitions and structural changes that occur throughout the phase diagram. A well-developed formalism for the realistic models of our finite-sized crystalline system, capable of the signatures to the transitions mentioned above, does not exist at this time. In the conclusions of this paper, I must rely on a simple qualitative comparison of the three model systems, which have been shown experimentally to have different phase diagrams.* A very useful criteria for multilayer coexistence has been given by Bruchs and applied to several systems by Bruch and co-workers? Also, there are significant bodies of work on two similar systems, namely, lattice gas models7 and fluid adsorbates.8 The use of lattice gas models has provided the basic topology of the phase diagrams for many adsorbed systems. Also, the classical statistical mechanics of nonuniform fluids and capillarity theory8 has given a formalism for the interpretation of vertical densities. However, the development of the fluid model formulas is based on assumptions not met by low-temperature solids. The extent of the quantitative accuracy of such computations is under study and will be reported separately. A theoretical equivalent to the signature heat capacity peaks of Zhu and Dash3 is needed for the various model calculations and simulations. A very brief outline of some of the interesting fluid theory relationships by Foiles and Ashcroftg and Tarazona Presented a t the symposium on 'Adsorption on Solid Surfaces",
62nd Colloid and Surface Science Symposium, Pennsylvania State University, State College, PA, June 19-22, 1988; W. A. Steele, Chairman.
and Evans'O provides an analogy for how the vertical density profiles of the first few layers in a film may cast the template for the growth of the film to bulk. The extent to which these ideas might be applied to the models in this report or what formalism would be appropriate are open but intriguing questions. A most important question is how to distinguish a wetting transition from melting. It has been shown'O that the divergence of transverse structure factor is a signature of the approach to complete wetting of a substrate by a fluid system. The method studies, at small Q , the behavior of the coverage =
Jm[N(Z)
- NB]dz
(1)
and the transverse structure factor
H(z,Q) = 1 + j m d z ' N ( z ? j d R eiQ'Rh(z,z',R) (2) 0
where NB is the bulk number density and h(z,z',R) is the total inhomogeneous two-particle correlation function from the Ornstein-Zernike equation written in cylindrical coordinates. The variable Q is the wave vector associated with the 2D Fourier transform of the correlation functions with respect to the planar R. For nonuniform fluid systems, it is observed that If(%,&) becomes very large for small values of Q , provided z is in the liquid-vapor interface. In the longwavelength limit, the generalized compressibility relationlo is written
(3) (1) Krim, J. Ph.D. Thesis, University of Washington, 1984. Dash, J. G. Phys. Today 1986,38(12),26. Bienfait, M. Surf. Sei. 1985,162,411. Pandit, R.; Schick, M.; Wortis, M. Phys. Reu. B 1982,26,5112. Ebner, C.; Rottman, C.; Wortis, M. Phys. Rev. B 1983,28,4186. Passell, L.; Satija, S. K.; Sutton, M.; Suzanne, J. In Chemistry and Physics of Solid Surfaces Vl; Vanselow, R., Ed.; Springer-Verlag: New York, 1986;p 609. (2)Larese, J. Z.; Harada, M.; Passell, L.; Krim, J.; Satija, S. Phys. Rev. B 1988,37,4735. (3)Zhu, Da-Ming; Dash, J. G. Phys. Reu. Lett. 1986,57,2969. (4)Seguin, J. L.; Suzanne, J.; Bienfait, M.; Dash, J. G.; Venables, J. A. Phys. Rev. Lett. 1983,51,122.Krim, J.; Dash, J. G.; Suzanne, J. Phys. Rev. Lett. 1984,52,640. Krim, J.; Gay, J. M.; Suzanne, J.; Lerner, E. J. Phys (Les Ulis, Fr.) 1986,47, 1757. Hamilton, J. J.; Goodstein, D. L. Phys. Rev.B 1983,28,3838.Pettersen, M. S.;Lysek, M. J.; Goodstein, D. L. Surf. Sci. 1986,175,141. (5)Bruch, L. W.; Wei, M. S. Surf. Sci. 1980,100,481. Wei, M.S.; Bruch, L. W. J. Chem. Phys. 1981,75,4130. (6)Bruch, L. W.; Ni, X.-Z. Trans. Faraday Discuss. 1986,80, 217. Bruch, L. W.; Phillips, J. M.; Ni, X.-Z. Surf. Sci. 1984,136,361 (see ref 141. --,.
(7)Saam, W. F.;Ebner, C. Phys. Reu. A. 1978,17, 1768. Ebner, C. Phys. Rev. A 1981,23,1925.Pandit, R.; Schick, M.; Wortis, M. Phys. Rev. B 1982,26,5112.Ebner, C.;Rottman, C.; Wortis, M. Phys. Rev.B 1983, 28,4186. (8)Evans, R. Adu. Phys. 1979,28,143.Rowlinson, J. S.;Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982. Schofield, P.; Henderson, J. R. Proc. R. SOC. A 1982,379,231. (9)Foiles, S. M.; Ashcroft, N. W. Phys. Rev. B 1982,25,1366. (IO) Evans, R.; Tarazona, P. Phys. Reu. A 1983,28,1864.
0743~7463/89/2405-0571$01.50/0 0 1989 American Chemical Society
572 Langmuir, Vol. 5, No. 3, 1989
Phillips
where p is the chemical potential of the system and N ( z ) is the vertical density profile. The behavior of H(z,O) above and below the transition temperature could be a possible signature for the nature of the film growth. Henderson and van Swol'l have applied the method to a square-well fluid and a hard-wall system in a study of the approach to complete wetting by a gas at a liquid-wall interface. Bruno et a1.12have applied the method to the wetting transitions for the Ar/C02 interface by using the modified hypernetted-chain theory. Evans and Tarazona13 have also studied the Ar/C02 system. They used the free energy functional method to examine wetting and thickthin film transitions. Our future work should evaluate the applicability of the nonuniform fluid theory to the thin solidlike systems. The physical model used in the simulations of this report was developed and applied to quantum mechanical cell model calculation^^^ and to earlier
simulation^.'^ The methane-methane pair interaction is assumed to be a Lennard-Jones LJl2v6potential with t/k = 137 K and u = 3.6814 A. As before, it is assumed that thermal effects have the molecule in a state of rotational diffusion and a spherical model is reasonable. The methane to graphite potential is the Steele potential for a molecule at a height ( z ) above the substrate u(z) =
z
+jd
where tln/k = 1468.5 K, ues= 3.297 A, and d = 3.37 A. The comparative studies for argon/graphite used the values Elk = 143.2 K, u = 3.35 A, t,/k = 995.4 K, and ags= 3.1243 A. The methane/Au(lll) system used tls/k = 2253.6 K. The number of particles in the system is 672. Topological defects and long-range correlations whose lengths exceed the size of the simulation cell are, by definition, constrained out of the system. It is important to realize that two effects in the quantum mechanical calculations are not present in the classical simulation. Zero-point effects are, of course, not present. Substrate-mediated forces are also omitted. The simulation cell has a rectangular base scaled to fit a triangular lattice and extended with periodic boundary conditions active along the sides. The bottom of the cell is a hard smooth substrate. The molecules are attracted to the substrate with an interaction potential given by u(z), eq 4. The top of the cell is a hard reflective wall at a height of 20 molecular cores (a). The initial condition was three layers of the (111)faces of a face-centered cubic lattice (ABC stacking). Each simulation was equilibrated by a minimum of 20000 Monte Carlo steps (MCS) per particle. In most cases, the equilibration was several times longer. Computer simulations of the physiorption process using grand ensembles have been published.16 We, however, monitor the microstructure within an existing solid multilayer film after the example of Vernov and Steelel' rather than attempt to observe the condensation of the threedimensional vapor into islands of adsorbate. To monitor the approach of the simulation to computational equili(11)Henderson, J. R.;van Swol, F. Mol. Phys. 1984,51,991. (12)Bruno, E.; Caccamo, C.; Tarazona, P. Phys. Rev. A 1987,35,1210. (13)Tarazona, P.; Evans, R. Mol. Phys. 1982,47,1033. (14)Phillips, J. M.Phys. Reu. B 1986,34,2823(paper 111); 1984,29, 4821; 1984,29,5865 (paper 11). Phillips, J. M.;Hammerbacher, M. D. Phys. Reu. B 1984,29,5859 (paper I). (15)Hruska, C. D.;Phillips, J. M. Phys. Rev. B 1988,37,3801. (16)Nicholson, D.;Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic: London, 1982. Lane, J. E.;Spurling, T. Aust. J . Chem. 1976,29,2103. (17)Vernov, A. V.; Steele, W. A. Surf. Sci. 1986,171,83.
bration, we calculate the ensemble averages for the internal energy and the integrated planar component of the pressure tensor. The evolution of these thermal properties and their statistical fluctuations are effective indicators of equilibration.18 The steady-state populations (densities) of the individual layers are monitored throughout the simulation and are found to approach constant values well within the length of the simulation. In previous work, we have studied 21 different coverages, with each simulation reproducible from different starting configurations. The simulation agree well with the predictions of our earlier analytic calculation^.'^ The strongest case for our simulations is the fact that our predictions compare correctly with the neutron diffraction experiments of Larese et al.2 The low-temperature structure for methane on graphite is mutually incommensurate structure for the bilayer and mutually commensurate for the argon/graphite system. The experimental confirmation of these very sensitive properties is most encouraging. In this system, the uncompressed monolayer has a density of 0.0663 molecule/A2. A similar simulation test for argon on graphite gives a nearly identical density. The structure of the multilayers is characterized by sampling the phase space of the simulation for two types of distribution functions. The vertical density distribution N ( z )is the relative probability of finding a molecule in the range z to z + dz above the substrate. The averaged structural distribution within a single layer is given by the 2D pair distribution function G(R). These intralayer distributions are accumulated for all layers in all films, when the individuality of the layers remains approximately 2D. The pair distribution is defined to give the number of molecules dn in a circular ring of radius R and thickness dR to be dn = (2rR)pG(R)dR (5) where p is the 2D number density of the individual layer. For computational efficiency, the number of molecules in each layer after each Monte Carlo move (MCM) is not counted. As a result, the normalization of the G(R) results is only approximate. The essential character of the functions is retained. Some of the results of the simulations are given in the tables and figures. The structural distributions are the ensemble averages of the vertical density N(z)and the 2D pair distribution functions for each individual layer. These measurements were taken for two representative temperatures (25 and 100 K) and for the three model systems (CH,/graphite, Ar/graphite, and CH4/Au). Currently, it is generally believed that the three systems we have studied exhibit different multilayer growth modes at low temperatures. Although some controversy exists,lg argon on graphite at 25 K is thought to completely wet (Frank-van der Merwe).20 Methane on goldz1is thought to triple point wet; Le., it partially wets (Stranski-Krastanov) for T < T,, the bulk triple point temperature, and completely wets above Tt.Methane on graphite partially wetsn below 40 K and completely wetsa above 64 K. The (18)Gould, H.; Tobochnik, J. An Introduction to Computer Simulations Methods; Addison-Wesley: New York, 1987. (19)Bruschi, L.; Torzo, G.; Chan, M. H. W., private communication. (20)Seguin, J. L.; Suzanne, J.; Bienfait, M.; Dash, J. G.; Venables, J. A. Phys. Reu. Lett. 1983,51, 122. (21)Krim, J.; Dash, J. G.; Suzanne, J. Phys. Reu. Lett. 1984,52,640. (22)Krim, J.; Gay, J. M.; Suzanne, J.; Lerner, E. J.Phys. (Les Ulis, Fr.) 1986,47,1757. (23)Hamilton, J. J.; Goodstein, D. L. Phys. Rev. B 1983,28, 3838. Pettersen, M.S.;Lysek, M. J.; Goodstein, D. L. Surf. Sci. 1986,175, 141. Lysek, M.J.; Pettersen, M. S.;Goodstein, D. L. Phys. Lett. A 1986,115, 340. Goodstein, D. L.; Hamilton, J. J.;Lysek, M. J.; Vidali, G. Surf. Sci. 1984,148,187.
Langrnuir, Vol. 5, No. 3, 1989 573
Structure near Transitions in Thin Films
Table I. Position ( z / u ) of the Maxima, Value of the Maxima, and the Width at Half-Maximum (WHM) for the T = 25 K Vertical Density Distribution N ( z ) CH,/graphite Ar/graphite CHhOAu(111) z,/. N(z-1 WHM z,/n Nkrnax) WHM h x /0 N(z,) WHM 0.88 1.77 2.67 3.57 4.41
0.0613 0.0333 0.0270 0.0189 0.0014
0.0389 0.0688 0.0846 0.1118 0.1124
0.88 1.79 2.69 3.61 4.49
0.0522 0.0341 0.0285 0.0183 0.0063
0.0466 0.0690 0.0821 0.1147 0.1222
0.88 1.17 2.66 3.59 4.45
0.0762 0.0303 0.0225 0.0142 0.0005
0.0339 0.0770 0.0944 0.1141
Table 11. Position ( z / u ) of the Maxima, Value of the Maxima, and the Width at Half-Maximum (WHM) for the T = 100 K Vertical Density Distribution N ( z ) CH,/graphite Arlgraphite CH,/Au( 111) N(z-1 WHM N(%,,) NkrnaJ Zlnaxl0 ax/^ WHM WHM z,/a ~
~~~
0.0285 0.0111 0.0065 0.0039 0.0017
0.88 1.80 2.72 3.68 4.55
0.88 1.81 2.75 3.67 4.58
0.0812 0.1825 0.2780 0.4273 0.6802
0.0231 0.0110 0.0066 0.0039 0.0017
0.081
2
0.03.
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0.05
0.87 1.77 2.70 3.62 4.55
0.0985 0.1899 0.2697 0.4064 0.6820
1
!
3
Figure 1. Comparative view of the relative vertical density distribution for the systems CH4/Au(lll)and Arlgraphite at 25 K and a coverage of 4.2 ( d 3 X d 3 ) monolayers.
gap in the data leaves the precise determination of the wetting transition temperature to the future. This system is unique in that it appears to have a wetting transition at a temperature well below the bulk triple point (Tt = 90.7 K). Additional features include dewetting and wetting transitions reported to be very near the bulk triple point.23 Our simulations involve only a few layers. We are taking detailed information only on the microscopic structure of layers one to four. Direct observation of wetting phenomena, in particular wetting transitions, is not plausible with such meager information. It is, however, interesting to compare the vertical densities N ( z ) for each of these systems. Assuming the laboratory systems are reasonably well approximated by the models in our simulations, the differences in the vertical structure of the three systems in question may reflect how the stresses within the film effect additional multilayer growth. The comparison of the methanelgraphite system to the argonlgraphite and methane/Au(111)films shows some interesting differences. Figure 1 shows the vertical density distribution for the Ar/graphite and CH4/Au(lll)system at T = 25 K. These two systems represent the two extreme cases; i.e., the ratios of the coefficients of the adsorption potential to the adsorbate interaction potential are quite different. The strong and sharp adsorption potential of the gold substrate compresses the first layer of the methane much more than the weaker graphite to argon potential. Table I and Figure 2 illustrate how the different peaks compare for the dif-
0.02
h U
0.01
4
HEIGHT(z/a)
0
1
2
3
4
5
HEIGHTtz/a) Figure 2. Comparative plot of the peak maxima in the relative vertical density distributions of the three model systems at 25
K. Table 111. Fraction of the Particles Found in Each Individual Layer for the Three Systemsa layer CH4/graphite Ar /graphite CH4/Au(111) 1
2 3 4 5 6
0.252 0.252 0.250 0.229 0.018 0.000
(0.250) (0.228) (0.190) (0.141) (0.119) (0.071)
0.260 0.250 0.250 0.232 0.007 0.000
(0.250) (0.240) (0.189) (0.143) (0.108) (0.070)
0.281 0.262 0.250 0.196 0.010 0.000
(0.281) (0.238) (0.190) (0.118) (0.076) (0.097)
The values in parentheses are for T = 100 K, and those without are for T = 25 K. ferent systems at the lower temperature. The most striking difference is in the maxima for the first layer. For the upper layers the CH,/graphite and Ar/graphite systems are very close. The structure within the layers for these two systems is shown in Figure 3. These films appear to be quite solidlike and are virtually commensurate internally. A t lower coverages, the first and second layers of the CH4/graphite system are incommensurate. This is true at all coverages studied for the CH,/Au(lll) system. The other difference at low temperatures is the somewhat lower peak values in the upper layers for the CH,/Au(lll) system (see Figure 2). Material is not being promoted to the upper layers quite so readily as in the other two system. This is in keeping with the difference between the growth characteristics of the two systems. The supprising feature is the nearly identical features of the Ar/graphite and
Phillips
574 Langmuir, Vol. 5,No. 3, 1989 0.030 0.028 0.026 0.024 0.022 0.020
YARGON/GRAPHITE COVERAGE. 4 2 T=2!5K
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5
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---
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--
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-
0.012 0.010 0.008 0.006 0.004 0.002 0.000 0
3 2 1
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1
2
3
4
5
6
7
DISTANCE (R/LT) Figure 3. Graph of the 2D pair distribution function for the
second, third, and fourth layers in the Ar/graphite system. Due to the commensurability of the layers in the film, the plots are superimposed.
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Figure 4. Comparative plot of the width at half-maximum (WHM) for all three systems at 25 K.
2
3
4
vertical density distributions of the three model systems at 100 K. 0.7 '
0.6 0.035 0.030. 0.025 n
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