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STRUCTURE AND DYNAMICS OF ADSORBED
FILMS. EFFECTS OF DIMENSIONALITY AND
SUBSTRATE POTENTIAL
F. Hanson, M. Mandell, J. Mctague
To cite this version:
JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Tome 38, octobre 1977, page C4-76
STRUCTURE AND DYNAMICS OF ADSORBED FILMS.
EFFECTS OF DIMENSIONALITY AND SUBSTRATE POTENTIAL
(*)F. E. HANSON, M. J. MANDELL and J. P. McTAGUE University of California, Los Angeles, California, 90024, U.S.A.
Rhsumh.
-
On reporte ici des simulations de dynamique molkculaire de couches minces d'argon sur la surface (001) du graphite et on Ies compare avec des donnCes expkrimentales. Trois diffCrents types d'approximations ont kt6 utilisCes : un modile tridimensionnel interagissant avec une surface lisse et deux modbles bidimensionnels avec ou sans potentiel de substrat pkriodique. On trouve, en particulier, une fusion continue aux alentours de 45 K. Les effets dus 5la formation de la deuxiime couche sont kgalement analysks. L'accord convenable entre ces trois types de calcul entre eux et avec les exphriences faites sur couches ayant un degr6 de recouvrement infkrieur B I'unitk, suggire que les phCnomines observks dans ce rCgime sont
caractkristiques de systimes bidimensionnels.
Abstract.
-
Molecular dynamics simulations of argon films on the (001) surface of graphite are reported and compared with experimental data. Three different levels of approximation have been investigated : three-dimensional systems interacting with a smooth substrate, and two- dimensional ones both with and without lateral periodic substrate potentials. In particular, a broad melting transition at about T = 45 K is identified and characterized. Effects of second layer promotion are also investigated. The substantial agreement of all three calculations with each other and with experiments on submonolayer films suggests that the observed phenomena in this regime are characteristic of two-dimensional systems.1 . Introduction.
-
The availability of forms of graphite with large amounts of smooth, relatively well oriented surface planes has made possible detailed thermodynamic, structural, and even dynamic studies of such simple systems a s adsorbed rare gas films, thereby greatly increasing our knowledge of physical adsorption and of the condensed phases of ideal thin films. The purpose of the present investigation is twofold. First, we wish t o determine the extent t o which the experimentally observed phenomena can be reproduced by classical systems of particles interacting via simple phenomenological potentials. Given this infor- mation, the goal then is to gain some insight3nto the structure, dynamics, and phase transitions in these systems and compare them with the properties of idealized two-dimensional ones.Molecular dynamics (MD) and Monte Carlo (MC) techniques have been applied with great success t o three-dimensional liquids, high temperature solids, and the solid-liquid transition, and some effort in this direction has begun [l, 2, 3, 4, 5, 6, 71 for two-dimensional and adsorbed phases. Particularly
(*) Supported in part by U.S. NSF Grant CHE76-21293.
interesting is the early work of Alder and Wainwright [3] o n hard disks, which showed a n apparently first order solid-fluid transition. Likewise, two-dimensional pressure-area isotherms of Lennard-Jones systems appear t o have van der Waals loops in the region of density and temperature where solid-liquid and liquid-gas transitions might be expected. Such loops are t o be expected in finite systems whose infinite equivalents show first order transitions [3]. The relationship of these results t o physical adsorption is subject t o a t least two uncertainties ; namely, the role of layer promotion a t higher temperatures and coverages, which introduces a third dimension, and the difficulty of drawing unambigous conclusions about phase transitions from MD and MC simulations, due t o the long time constants and system size effects in transition regions.
Despite the considerable amount of progress in understanding and classifying magnetic transitions in two dimensions there is a s yet no firm theoretical basis for predicting the order of 2-D phase transitions in atomic and molecular systems. As a
first step in this direction, then, we here attempt t o compare available experimental and simulation data
STRUCTURE AND DYNAMICS OF ADSORBED FILMS C4-77
for a particularly simple, well-studied system, argon approximations to 2-D spatial systems to be found in on the (001) plane of graphite. nature.
We thus represent the lateral Ar-C potential by the 2. Potential models. - It is well-established that lowest fourier components of eq. (3) which preserve the Lennard-Jones potential the symmetry of the graphite surface :
~ r o v i d e s a rather good re~resentation of the where G, and G, are the reciprocal lattice basis effective potential for the condensed rare gases, and vectors, and is measured from the center of a
has been in MC and MD graphite hexagon. This simplified 2-D potential simulations of bulk, adsorbed, and 2-D systems. In
reproduces quite well the overall form of the eq. is the depth the pair complete sum of eq. (3) The 2-D lateral potential is
conventionally taken as e
/
k,
= 120 K for Ar, and ashown in figure is the diameter at which the attractive and repulsive
contributions to the pair energy cancel (a,, = 3.405
A).
Using this potential form, we have made three distinct types of MD calculations : (i) a purely 2-D calculation ; (ii) a 2-D calculation in an external 2-D potential representing the lowest fourier components of the argon-graphite interaction ; and (iii) a 3-D calculation having a smooth substrate represented by a Lennard-Jones continuum, yielding a 9-3 potential in the vertical direction :Steele [8] has estimated that
It is to be expected that this continuum form for the interaction of adsorbed atoms with a graphite surface will not be quantitatively accurate for Z of order of the equilibrium distance between the graphite surface and the first adsorbed layer, but it should be sufficient to qualitatively reproduce the behavior of incommensurate adsorbed films. The constant @ ( = 33) has been adjusted in our calculations to reproduce the experimental heat of adsorption.
For an atom interacting with a rigid surface the potential may be in general written as
where {G) is the set of reciprocal lattice vectors appropriate to the surface. Steele [8] has calculated the V ( G , 2 ) for the Lennard-Jones model of the Ar-Carbon interaction. R is a 2-D vector parallel to the surface. The resultant barriers to free translation on the graphite surface are of order 30 K for Ar. A greatly simplyfying result is that the vertical locations of the local potential minima vary by less than a few percent across the surface, making it effectively quite flat. Because of this flatness, graphite surfaces are probably the closest
VSUB
-
A S A S PFIG. 1. - The graphite basal plane structure is shown with reciprocal lattice vectors G , and G2. Below, the model lateral
substrate potential is plotted along the line A-S-A-SP.
3. Results. - Much of our effort has been concentrated on densities below and near monolayer coverage. For all cases (smooth, rough, 2-D, 3-D) in this regime there is a rapid change in the energy as a function of temperature around T = 45 K .
Figures 2 and 3 show the energy per particle plotted as E* - d T * (the difference in energy between the
system and a d-dimensional harmonic lattice at the same temperature). The curves should have limiting slopes of 0 and - d /2 at low and high temperatures,
respectively. [E
*
is the reduced energy EI E ,
andT
*
the reduced temperature k , T / e . Each pointC4-78 F. E. HANSON, M. 3. MANDELL AND J. P. McTAGUE
FIG. 2.
-
E * - 2 T* plotted for three coverages (A-15%,13-64 %, C-85 %) from molecular dynamics of two dimensional systems. 13 shows a comparison of lateral substrate effects. Filled circles in A, B and C are from runs with substrate and the open circles in B without. D is a constant volume heat capacity measurement of T. T. Chung for argon adsorbed on Grafoil at what would be a registered coverage (80%). The solid line superimposed is the heat capacity from an approximate fit to the
run at 64 %with substrate (B-filled circles).
The MD calculations employed the usual periodic boundary conditions in the lateral directions.]
The filled circles in figures 2a, b, and c represent points calculated for a 2-D system with the lateral substrate potential, eq. (3). Coverages are listed as fractions of the close-packed equilibrium structure at T = 0 and zero pressure. The curves for 15 %
(100 particles) and 64 % (240 particles) are practically superimposable, suggesting that neither particle number nor pressure effects varying with coverage are responsible for the observed broad feature centered at T* =. 0.38, or T = 45 K for Ar.
Furthermore, a 2-D system without~any substrate potential; shown as open circles in figure 26, has a slightly narrower transition the integrated specific heat being approximately the same. Above the main transition region there is some tendency for small clusters to orient in order ,to lower their potential energy in the substrate field when present. This is a small effect, however, and a smooth substrate potential appears to be a sufficiently good qualitative and quantitative model for reproducing the main features of the transition in this density range.
We saw no evidence of a registered phase in any of our runs, but an interesting effect of the substrate was observed in the low temperature frozen
FIG. 3. -The temperature dependence of some properties of the three-dimensional system at N I A = 0.8 ab2. (Top left scale) - the full width at half maximum of the two-dimensional structure factors. Filled circles are molecular dynamics results in the fluid temperature region. The squares are neutron scattering results for argon on Grafoil. The vertical lines show approximate error limits (for the neutron data) and the low temperature points are due to finite system size. (Middle right scale) - Two dimensional diffusion constant through the transition region. (Bottom left scale) - E* - 3 T* exhibits a relatively abrupt rise at about
T* = 0.37. It also shows effects of second layer population at
T* = 0.7.
configurations in all systems examined with a lateral potential. For these cases the triangular crystalline adsorbate lattice was aligned with its lattice vec- tors approximately parallel to the substrate carbon-carbon bonds. This is the same direction as the
fi
x
d 3
- 30°,registered structure observed formany adsorbed systems. However, for both our simulated system and for Ar, the adsorbed layer is incommensurate with the registered structure, being some 10 % smaller in linear dimension. This orientational phenomenon did not occur for the run a t 6 4 % coverage in the absence of substrate potential, nor in the 3-D smooth substrate simulation. The origin of this orientational epitaxy is discussed by Novaco and McTague [9]. There is preliminary experimental evidence for this behavior in high precision LEED measurements (M. Chinn and S. C. Fain, Jr., private communication) of Ar on graphite.
STRUCTURE AND DYNAMICS OF ADSORBED FILMS C4-79
the transition region and above. They do, however, appear to adequately simulate the behavior of Ar on graphite at submonolayer coverage, at least up to
T = 60 K. Figure 2d shows an experimental heat capacity of Ar on grafoil for an 80 % coverage (T. T. Chung, unpublished). The heat capacity maximum at
T I 50 K has a magnitude of 8 k , per atom, a factor
of 5lower than that observed for the registered-nonregistered transition in N,-grafoil. Although we did not perform MD calculations at this coverage, the heat capacity derived from the 64 %
run with substrate field is shown as the solid line in the figure. Clearly the magnitudes of the anomalies are similar, and agree with earlier calculations of Cotterill [I]. No adjustment of the scale was made, the gas phase Ar-Ar values being used. This comparison strongly suggests that the 2 D MD simulation provides a realistic picture of the experimental system up to T = 60 K for submonolayer coverages. In other words, the observed transition has the same character as that of a 2 D Lennard-Jones system.
A test of the significance of the third dimension has been made by including a Lennard-Jones interaction with a smooth continuum substrate below and a reflecting wall at Z = 10 u. Results for a coverage of 87 % are shown in figures 3 and 4. The energy-temperature curve at the bottom of the figure shows a transition around T* = 0.37, as in the 2 D cases. The relative sharpness of the transition at this high coverage is consistent with the density trend noticed for 2 D systems above. As the density is
A Second Layer A Population A .20 C
2
bFIG. 4. - Diffusion constants and second-layer promotion for the three-dimensional system. Note the sharp increase in the Super-Burnett coefficient, DZ, at the onset of significant
second-layer population.
.
-
.
.
Diffusion A
increased we also note that the heat of transition decreases. This is presumably caused by the constraint of constant density.
Some further information about the nature of the transition is provided by studying the structure factor of the adsorbed Ar layer. Neutron scattering measurements at low temperature [lo] indicate well-developed Bragg peaks, characteristic of a
crystalline 2-D triangular array. The widths of the Bragg peaks are proportional to the linear dimension
L for which coherent order persists. Imperfections in the grafoil substrate appear to limit the range of order at low temperature to a value of about 100
A.
Figure 3 (top) shows the temperature dependence of the [lo] Bragg peak width (Squares). The continuous broadening from a crystalline value to one characteristic of a liquid is in contrast to the abrupt change in width which occurs in macroscopic. 3 D systems upon melting (3-I3 melting is a strong first order transition, of course). Computer simulations of the width of this peak obtained from the calculation are shown a s filled circles in the figure. The Lennard-Jones system also appears to show a continuous melting transition, characterized by a gradual decrease in the value of L, the correlation length. (It is known that, at any finite temperature, 2-D solids cannot have the infinite (L = a) rangeorder of 3-D crystals, so a continuous transition is not ruled out by symmetry). Note that the correlation length decrease continues well above the region of the energy anomaly.
The single particle lateral diffusion constant
L
.
0
A
b ADl P = . 9 d 2 A.
:
1 d
D , =
-
- ( ( r ( t ) - r(0))' ) also changes rapidly in 4 d tthe transition region. At higher temperatures it is liquid-like, but in the transition region and below it is rapidly decreasing in a manner one might expect for a solid with a high defect and/or dislocation density. Influences of the third dimension begin to appear at
T*
-
0.6, as shown in figure 4. The lower portion shows the temperature dependence of second layer population for two coverages, N / A = 0.8 and 0.9 u-', corresponding to 0.87 and 0.98 T = 0 monolayers, respectively. Layer promotion has a significant influence on lateral diffusion, as might be expected. In the upper portion the diffusion coefficient D, is shown. It is apparent that the two quantities are approximately linearly related under the observed conditions.The super-Burnett diffusion coefficient [I 11
provides a further characterization of the translational processes. In principle, D, is divergent in 2 D , but in practice a meaningful asymptotic number can be obtained by evaluating the derivative at times T .= 10-20. For a single gaussian diffusion
C4-80 F. E. HANSON, M. J. MANDELL AND J. P. McTAGUE
as layer promotion provides a further diffusion path. We have also observed that the ordinary self-diffusion constant D l shows a minimum as a function of density at monolayer coverage and a maximum at about 1 1/2 monolayers. At monolayer density Dl increases approximately linearly in the range 0.5
<
T *<
1.0. There is little density dependence of Dl in the neighborhood of one monolayer ; apparently the increased second layer population compensates for the loss of vacancies in the first.The dynamic structure factor
S ( K , o ) = (2 a ) - ' l d t cos o f ( p ( K , O ) p ( - K, t ) ) has been calculated for the in-plane directions on the p * = 0.8 sample at T* = 0.227 ( = 27 K for Ar). Here p(K, t ) is the spatial fourier transform of the areal density. This temperature is well below the transition region. The resultant longitudinal dispersion curve along the T-M-T direction is shown in figure 5. The frequencies here are significantly less than those determined by neutron scattering at T = 5 K, and also below the T = 0 K
harmonic Lennard-Jones phonon dispersion curves. Our results show a low frequency slope of 6.7 2 0.5 a / r , or about 1 100 meters s-' for Ar. The corresponding T = 0 lattice dynamics result is about 1 500 meters s-'. There thus appears to be significant renormalization due to thermally induced anharmonicity, even below the transition region.
FIG. 5 . - S ( k , o ) for the three-dimensional system a t T* = 0.227 (27 K), calculated from
r
tor
in the Brillouin zoneshown in the upper right corner. The frequency transform was taken over a time interval of 12 T. The total run time for these
points was about 200 r.
4. Discussion.
-
Taken as a whole, the intercomparison of the MD calculations and experiment indicate that a 2-D Lennard-Jones model on a flat substrate is adequate to represent the structural, dynamic, and thermodynamic properties of Ar on graphite at submonolayer coverages. There is some indication from the temperature of the heat capacity anomaly that the effective Ar-Ar interaction parameter E on the graphite surface maybe a few percent different from bulk values. This is in qualitative agreement with earlier estimates [12] based on adsorption isotherm data. It is remarkable, however, how well the unadjusted standard Ar-Ar potential parameters reproduce the observed data.
Given this confidence in the microscopic dynamical model, the more important issue is the nature of the broad transition observed. The structure factor and diffusion coefficient indicate that the low temperature limit is solid-like, while above the heat capacity anomaly the systems are fluid-like. Although melting in 2-D Lennard-Jones systems has not been systematically studied to date, there is information on other 2-D melting transitions. As noted earlier, the hard disk system has a first order melting transition, whereas the melting of the 2-D electron Wigner lattice is characterized by a Lambda-type anomaly [13].
There is MC and MD [4, 51 evidence that, at high coverages, the L-J melting transition is first order. This is inferred from the observation of van der Waals loops in calculated isotherms, a known feature of finite systems [3] whose infinite analogues show first order transitions. The submonolayer coverages explored here differ in an essential way, since there is free area on the surface. Although we do not yet have a complete systematic survey of the influences of system size and coverage on the transition, comparison of figure 2a, a run involving 100 particles at 15 % monolayer coverage with figure 2b with 240 particles and 64 % coverage, suggests that our observations represent properties of the systems rather than artifacts due to system size. Only as monolayer coverage is approached does the transition sharpen, presumably eventually becoming first order [4,5]. The intriguing possibility of a 2-D solid-fluid critical point is not without precedent. Recently, Mihuru and Landau [14] have made MC calculations on a triangular lattice gas model which shows a similar result.
STRUCTURE AND DYNAMICS OF ADSORBED FILMS
References
[I] COTTERILL, R. M., JENSEN, E. J., KRISTENSEN, W. D., [8] STEELE, W. A., Surf. Sci. 36 (1973) 317.
Anharmonic Lattices, structural transitions and Mel- [9] N o v ~ c o , A. D. and MCTAGUE, J. P., J. Physique Colloq. 38 ting, edited by T. Riste Noordhoff, Leiden (1974) 405. (1977) C4-116.
[2] DE WETTE, F. W., ALLEN, R. E., HUGHES, D. S. and [lo] TAUB, H., KJEMS, J., PASSELL, L., CARNEIRO, K., McTA- RAHMAN, A., Phys. Lett. 29A (1969) 548. GUE, J. P. and DASH, J. G., Phys. Rev. Lett. 34 (1975)
[31 ALDER, B. J. and WAINWRIGHT, T. E., Phys. Rev. 127 (1962) 654.
359. [ l l ] MCLENNAN, J. A., Phys. Rev. A 8 (1973) 1479. 141 TSIEN, F. and VALLEAU, J. p., Mof. phvs. 27 (1974) 177. 1121: WOLFE, R. and SAMS, J. R., J. Chem. Phys. 44 (1966) 2181. [51 TOXVAERD,
s.,
MOI. ~ h y s . 29 (1975) 3j3. [I31 HOCKNEY, R. W. and BROWN, T. R., 3. Phys. C . : Solid[61 LANE, J. E. and SPURLING, T. H., Aust. J. Chem. 29 (1976) State Physics 8 (1975) 1813.
2103. 1141 MIHURA, 3. and LANDAU, D. P., Phys. Rev. Lett. 38 (1977) [71 ROWLEY, L. A., NICHOLSON, D. and PARSONAGE, N. G., 977.
Mol. Phys. 31 (1976) 365 ; 31 (1976) 389.
DISCUSSION
R. K. THOMAS. - Diffusion coefficients are a space-time average of the dynamics and in many cases may not be suitable for characterizing surface motions. In addition, the experimental techniques for studying surface dynamics do not measure diffusion coefficients directly. Neutron quasielastic scattering, for example, resolves the motion in frequency and momentum transfer space. It would therefore be more useful to present dynarnical information in the form of the appropriate correlation functions which may be more fully related to the experimental data.
J. P. MCTAGUE.
-
F ( K , t ) is certainly a moreinformative quantity than D, and we have calculated it in a few cases. D is, however, much easier (and cheaper !) to compute. We are severely limited by the capacity and speed of our small PDP-11/45 computer.
A. STEWART.
-
D. Butler and I have studied the heat capacity of submonolayer krypton below registered coverages. As in Chung's N, and Ar data, we see only a. single line of heat capacity peaks,whose width and position are coverage dependent. The neutron data show that a solid-fluid transition occurs in the region of our anomalies.
F. PUTNAM.
-
Your results show no liquid-gas transition or critical point. This is a t variance with earlier interpretation of some adsorption isotherms and Monte Carlo simulations. Do you have any explanation for these discrepancies ?J. P. MCTAGUE.
-
Our runs were done for very long times, of order 10, 0002-20,000 time steps(
