Langmuir 1989,5,600-607
600
features are comparable, but the EELS intensity of P-3MT on the 3MT preadsorbed Pt surface is less than that observed on the clean, annealed Pt(ll1) surface. EELS spectra of polyimide films have also been shown to be very dependent on the nature of the underlying s u p p ~ r t This .~ is not surprising considering our previous studies, which have shown that the nature of the electrode surface can have profound effects on the voltammetric and UHV electron spectroscopic behavior of adsorbed layers,%as well as on the electrochemistry of unadsorbed reagents.26 It is apparent that electrode surface structure and compo(24) (a) Gui, J. Y.; Kahn,B. E.; Lin, C.-H.; Lu, F.; Salaita, G. N.; Stern, D. A,; Hubbard, A. T. Langmuir, in press. (b) Gui, J. Y.; Kahn, B. E.; Lin, C.-H.; Lu, F.; Salaita, G. N.; Stern, D. A.; Zapien, D. C.; Hubbard, A. T. J. Electroanal. Chem. 1988,252, 169. (25) Batina, N.; Kahn, B. E.; Lin, C.-H.; McCargar, J. W.; Salaita, G. N.; Hubbard, A. T. Electroanalysis 1989, 1, in press.
sition affect polymerization reactions conducted a t electrodes. The polymers studied on Pt(ll1) certainly contain more than a monomolecular layer of P-3MT. The Auger spectra of P-3MT show no peaks attributable to Pt, indicating that the polymer film is thick enough (and sufficiently uniform) to attenuate the Pt Auger signal completely. Acknowledgment. This work was supported by the Edison Sensors Technology Center, administered by the Cleveland Advanced Manufacturing Program. Instrumentation was provided by the National Science Foundation, the Air Force Office of Scientific Research, and the University of Cincinnati. Registry No. 3TCA, 88-13-1; BTCA, 527-72-0; BTAA, 696421-2; ZTAA, 1918-77-0; WE, 110-02-1;3MT, 616-44-4;33'DMBT, 67984-20-7; 44'DMBT, 111372-97-5; P-3MT, 84928-92-7; Pt, 7440-06-4.
Computer Simulations of Mono- and Trilayer Films of Argon on Graphite? Ai-Lan Cheng* and William A. Steele*Ps Department of Physics and Department of Chemistry, The Pennsylvania State University, 152 Davey Laboratory, University Park, Pennsylvania 16802 Received October 27, 1988. In Final Form: February 1 , 1989 The structures and thermodynamic energies of tri- and monolayer films of argon adsorbed on the basal plane of graphite have been studied at temperatures ranging from 50 to 110 K. Within this range, layer-by-layermelting was found for each of the three layers and for the single layer simulated. We surmise that these processes are first-order transitions observed under conditions such that the process is continuous. Energies and areas of melting are estimated. Roughening of the simulated trilayer film occurs gradually as the temperature increases. 1. Introduction
Computer simulation studies of model physisorption systems are proving to be a valuable addition to experiment and theory,'P2 especially if the simulation employs realistic models for the adsorbate-adsorbate and the adsorbate-solid interactions. For example, one can generate information about the temperature and coverage dependence of the orientational behavior of nonspherical adsorbate molecule^,^ which is difficult if not impossible to obtain experimentally. Similarly, computer simulation of multilayer films can give local densities and energies, which are essentially inaccessible to experimental measurements (of course, careful analysis of diffraction studies of solid multilayer films could yield local crystal structures,4 but this technique loses its power as the layer liquefies). Furthermore, questions concerning the nature of the thermodynamic phases and phase transitions within the layers of a multilayer film are hard to answer accurately by either experiment or theory. In this paper, we report a molecular dynamics computer simulation study of argon monolayer and multilayer films Presented at 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. Department of Physics. 8 Department of Chemistry.
*
0743-7463/89/2405-0600$01.50/0
adsorbed on a graphite substrate. The temperature of these simulations ranged from -50 K, where the films were completely solid, up to 110 K, where the films are completely liquid. We will show that melting occurs layer-by-layer for a trilayer film. The thermodynamic melting parameters were estimated for each layer in these films. In addition, the melting of a single monolayer of adsorbed argon was studied. We should note a t the outset that the solidified Argraphite system presents a particularly difficult problem for computer simulation. It is well-known that solid argon layers form an incommensurate lattice on graphite.6 Diffraction data indicate that the monolayer a t very low temperature is triangular with a nearest-neighbor spacing of -3.8 A,compared to 4.26 A for a commensurate film.6 This spacing changes with the thermal expansion of the layer, and furthermore, one might guess that the spacing in a low-temperature trilayer film might not be exactly the same as that in the monolayer. In our simulation, the periodic variation in the Ar-graphite interaction energy was included in an attempt to be as realistic as possible.
-
(1) Nicholson, D.;Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: New York, 1982. (2) Abraham, F. F. Adu. Phys. 1986,36, 1-111. (3) Talbot, 3.; Tildesley, D. J.; Steele, W. A. Surf. Sei. 1986, I69,71. (4) Larese, J.; Passell, L., private communication. ( 5 ) Shaw, C. G.; Fain, S. C., Jr.; Chinn, M.D. Phys. Reu. Lett. 1978, 41,955. (6) Hanson, F.; McTague, J. P. J. Chem. Phys. 1980, 72, 6363.
0 1989 American Chemical Society
Langmuir, Vol. 5, No. 3, 1989 601
Computer Simulations of Ar Films This produces problems in setting the proper periodic boundary conditions for the solid layers. As is well-known,7 a periodic boundary condition is indispensable in making a simulation of a few hundred particles mimic the much larger thermodynamic system. This condition requires that one set up the computer algorithm so that any atom leaving the computer box on one side is replaced by an image atom that enters on the opposite side. To have energy conservation, the image must be a t precisely the same position as the real atom relative to both the argon (adsorbate) lattice and to the periodically varying graphite substrate. Matching of these two conditions puts significant constraints on the two-dimensional argon lattice constants that can be studied and, in addition, does not allow thermal expansion to occur except by defect formation or by liquefaction. Consequently, a necessary preliminary to the actual simulation was a series of studies that allowed us to find the minimum energy configuration for an argon layer on the graphite surface. The system size was then set to be as close to this condition as possible and the simulations run from this starting point. Aside from the fact that the thermal expansion of the solid layers is not taken into account, we believe that this approach should give results in agreement with experiment. In particular, we see layer-by-layer melting in the three-layer system. Simulation values obtained for the energies, density changes, and temperatures for the beginning and end of the melting processes should be reasonably close to the actual experimental values. However, experimental data that are directly comparable to these simulations of one and three layers of argon on graphite do not seem to be available at present. 2. Details of the Simulation The simulation algorithm used was a straightforward solution of Newton’s equations for a collection of classical point particles moving in the potential produced by a rigid solid and by the interatomic interactions. A constanttemperature constraint? was applied in order to prevent the large excursions in temperature that occur in a constant-energy system when an atom jumps from one layer to another. The Ar-Ar interaction was taken to be Lennard-Jones 12-6 functions with potential well depth c k h / k = 120 K and size parameter u A - ~ = 3.405 A. Substrate-mediated changes in this function were roughly included by reducing the well depth by 15% whenever both atoms in a pair were in the first adsorbed layer. The argon-graphite potential was assumed to be given by a pairwise sum over all C sites in the solid. This function was transformed to the well-known Fourier series expressions for atom-graphite energy, which can be written as UB(z,T) = Eo(z) + Ei(z)f(x,Y) (2.1) with the periodic term part given by f(x,y) = -2.0[ cos 2* ( x :
-
+
hY) + )+
cos 2a ( x -
-y 3112
cos -y
$ 2 1
(2.2)
Eo(%) and E,(z) are given by expressions which will not be repeated here;gthe constants of the Ar-C site pair potential (7) Vernov, A,; Steele, W. A. Surf. Sci. 1986, 171, 83. (8) Hoover, W. G.; Ladd, A. J. C.; Moran, B.Phys. Rev. Lett. 1982,48, 1818. (9) Steele, W. A. Surf. Sci. 1973, 36, 317.
-5*60* -5.70
I
00
0
0O
-5.90 -00.6-
1.50
1
a
0
1.52
i
1.56 1.58 1.60 a* Figure 1. Lennard-Jones interaction of Ar-Ar in a triangularpacked monolayer plotted as a function of the lattice parameter. Reduced units are Uo*= U o / c wand a* = a12.46 8, The squares are for 0 K (no thermal motion), and the circles show values for simulations at 40 K-in this case, the harmonic oscillator thermal energy of T* has been subtracted from Lennard-Jones interaction to make the results comparable to the 0 K calculation. 1.54
Table I. Heats of Adsomtion of Ar on Graohite at 77.4 K slope of qu, kJ/mol q,(O), kJ/mol experiment 9.66 3.60 simulation 9.62 3.58
were taken to be c M / k = 57.9 K and ~ h - c= 3.402 A. (These values were obtained from earlier worklo in which experimental Henry’s law adsorption data were fitted to theory.) A check on their accuracy was run by simulating submonolayer coverages of Ar on graphite a t 77.4 K and calculating the average energies of adsorption as a function of coverage. A linear dependence on coverage was observed, which allows one to compare simulated values of the intercept in qst(8=0) and the slope dq,,/de with experiment.’l Table I shows that the simulations are in satisfactory agreement with experiment. The next step is to determine the “best” lattice structures for low-temperature simulations. Thus, a number of monolayer simulations were run a t 40 K using various numbers of atoms and lattice constants. It was observed that the argon lattice tended to rotate relative to the graphite and to distort from perfect triangular symmetry if the lattice constant had been badly chosen. In order to find the most realistic constant, two calculations were undertaken: the Lennard-Jones energies of a perfect two-dimensional triangular lattice were calculated (at 0 K) as a function of the lattice parameter. Values obtained are plotted in Figure 1 together with a similar set given by simulations of these solids a t 40 K. The plots show Uo* (=Vu*- T*), where Vu* is the average potential energy per particle and T* is the thermal vibrational energy of a classical two-dimensional harmonic oscillator, in reduced units. It is evident that finite temperature has only a minor effect on this system up to 40 K and that a nearest-neighbor distance a = (1.56)(2.46) = 3.84 8, nearly corresponds to the most stable structure, especially if one admits to some thermal expansion at T > 40 K. Note that one can simulate only at discrete values of the lattice constant because of the requirements of matching the periodically varying surface and argon lattice a t the boundary of the computer box, for a finite number of atoms in the simulation. Table 11gives a detailed summary of the box dimensions, number of argon atoms, and in(10) Steele, W. A. J.Phys. Chem. 1978,82, 817. (11) Grillet, Y.; Rouquerol, F.; Rouquerol, J. J. Colloid Interface Sci. 1979, 70, 239.
602 Langmuir, Vol. 5, No. 3, 1989 Table 11. Parameters for the Monolayer Simulations at 40 K cell dimensions atomic distance (in reduced (in reduced number of atoms units) unit+' 1.516 256 21 X 14(3)'12 1.540 432 32 X 16(3)'/2 1.55gb 400 27 X 18(3)'/2 1.575 484 30 X 20(3)'12 1.588 192 22 X 11(3)1/2 a Dimension = dimension (&/2.46 A. lations.
T,K 50.0 60.0
65.0 71.0 75.0 80.0
85.1 88.9 92.7 96.5 100.0 103.0 105.4 112.0
Value used in the simu-
Table 111. Summary of Simulation time steps time steps for after initial configurequilibration equilibration ation 600 Atoms in Three Layers fcc 40 000 5000 a 7 000 5000 a 8000 5000 a 8000 5 000 a 8000 5 000 fcc 30 000 5 000 fcc 18 000 5000 a 5000 5000 T = 85.1b 6000 5000 a 5000 15 000 T = 92.7b 10 000 5000 a 15 000 10 000 T = 100.Ob 10 000 5000 a 5000 10 000
92.7 96.5 100.0
1200 Atoms in Three Layers 5 000 15 OOO 5 000 15 OOO 5 000 15 000
85.14 88.96 92.67 93.59 94.58 96.49 104.25
400 Atoms in One Layer 5000 15 000 5000 5000 5000 15 000 5000 5000 5000 5 000 5000 5000 5 000 5000
fcc fcc fcc fcc a fcc c c c C
configuration is the last configuration of previous run. *Initial configuration is the last configuration of given T run. cInitial configuration is the last configuration of T = 92.67 run. a Initial
teratomic distances chosen for all simulation runs reported here. 3. Results Having selected the system with reduced lattice constant = 1.56 as a realistic argon solid, molecular dynamics simulations were carried out for a three-layer system over a range of temperatures. Note that there is a choice of packing for three triangular-packed layers, namely, aba or abc. We were unable to distinguish between the average energies obtained for the two structures and thus arbitrarily chose the abc packing as initial lattices for the great majority of the simulation runs. Briefly, the technique used was to simulate the three layer system at a series of increasing temperatures. A very long equilibration run was carried out initially-4OOOO time steps a t 50 K (equal to 134 ps). Subsequent runs at higher temperatures were started from the atomic configurations generated in the previous run, and thus, shorter equilibration periods were taken. A summary of the run times and temperatures is given in Table 111. Note that the number of layers is correct for the initial configuration only-at higher temperatures, thermal expansion and melting cause some atoms to be promoted to the fourth
2.50 2.00 I .50
1.00
0.50 0.00 0
1
2
3
4
5
0
1
2
3
4
5
6
7
Z*
2"
T= 96.5 K
T=112.0K
4
* 2.00 2 1.50
:1.00
0.50 0.00
0
1
2
3
4 Z*
5
01
1
2
3
-*4
5
6
7
L
Figure 2. Local density in (atoms/A3)(2.46 A)3 plotted as a function of the Ar-surface distance z* = 212.46 A. It is evident that the film was three layers thick at low temperature, but thermal motion and melting cause it to be become thicker and more diffuse at high T.
and fifth layers. In an attempt to determine whether finite size effects were significant, a few runs were carried out using 1200 atoms in three layers. Since the results were not significantly different from those obtained for 600 atoms, the bulk of the simulations was carried out with the smaller number. One final technical point: for the largest system, the efficiency of the simulation was improved by using a link-cell algorithm for locating the neighbors of a given atom in the layer.I2 This proved to give a worthwhile reduction in computation time (slightly more than a factor of 4). Three Layers of Argon. As mentioned above, one of the strengths of computer simulation is that it produces a detailed molecular picture that gives useful physical insights as well as quantitative values of the energy and density of the multilayer films. Figure 2 shows the local density in units of atoms per reduced volume, where the reduced volume is 2.46 A.3 Curves are shown for a number of temperatures, ranging from 50 K, where the trilayer is completely solid, to 112 K, where it is completely liquid. It is evident that the first two layers maintain their identity even though the density in these layers decreases significantly (due primarily to melting-see below). The thirdlayer densities also decrease with increasing temperature, but more importantly, atoms promoted to the fourth layer at 75 K and above tend to blend in with this layer a t the highest temperatures to give a region where layering is no longer easily discernible. In other studies, we have found that computer-generated plots of the trajectories of the adsorbed atoms provide helpful insights into the structures and dynamics of adsorbed 1 a ~ e r s . l ~Sets of such trajectories are shown in (12) Smith, W., Information Quarterly for Computer Simulation of Condensed Phases, No. 20, 1986, p 62. (13) Bhethanabotla,V.;Steele, W. A. J.Phys. Chem. 1988,92,3285; Langmuir 1987, 3,581.
Langmuir, Vol. 5, No. 3, 1989 603
Computer Simulations of Ar Films
T=65.0 (K)
T=50.O(K)
T = 71.0 (K)
Second layer
First layer
1
1
i i
Side view
.................... ...................
LuL@beIl*Iba..ab4b#.
Figure 3. Computer-generatedargon atom trajectories. Top views of the atoms in each layer and a side view of all the atoms in the film are shown. One can see quite clearly the start of melting as the temperature increases.
T = 75.0 (K)
T = 92.7 (K)
T = 88.9 ( K)
Third layer
Second layer
First layer
Side view
1 . '
/
Figure 4. Same as Figure 3, but for higher temperatures. (Some vaporization of the adsorbed atoms is also present in this temperature regime.) Figures 3-5; these extend over 5000 time steps and are shown both as side views and as top views of the motions of atoms in a given layer. (At high temperatures, the third layer was defined to be those atoms for which 4.6 > z* > 3.3, as given by the low-temperature local density plots of (14) McTague, J. P.; Frenkel, D.; Allen, M. P. Ordering in TUJO Di-
mensions; Sinha, Ed.; North Holland Amsterdam, 1980; p 147.
Figure 2.) These trajectories show quite clearly that disordering and, thus melting, occurs layer-by-layer; this is in good agreement with experiment^'^ and simulation result~.'~Also, significant numbers of atoms are driven into (15) Zhu, D. M.; Dash, J. G. Phys. Reu. Lett. 1986,57, 2959. (16) Larher, Y.; Millot, F. J. Phys. (Paris) Colloq. 1977, 38, (24-189. Saam, W. F. Surf. Sei. 1983,125, 253. (17) Pandit, R.; Schick, M.; Wortis, M. Phys. Reu. B 1982, 26, 5112.
604 Langmuir, Vol. 5, No. 3, 1989
Cheng and Steele
T 96.5( K)
T =lOO.O(K)
T =105.4(K)
Third layer
1
I
I
Figure 5. Same as Figure 4,but at higher temperatures.
0.80-
-
0.70
..
-
0.60
-d
i
-
:i
0.50-
-
0.40-
-
b
0.20-
0.30
0.10
T* Pair correlation function, projected into a plane parallel to the surface, is shown for the atoms in each layer at 5" = 50 K. The peaks for pairs of atoms separated by T* = r/2.46 A (and the areas under these peaks) correspond well to a triangular lattice. Layer two is slightly more ordered than either layer one or three. Figure 6.
regions more distant than z* = 4.5 at temperatures of 75 K and higher. One can even see the anisotropic vibrational amplitudes for first-layer atoms as well as the occasional vaporization that occurs at the highest temperatures. (The top of the computer box is taken to be a specularly reflective surface, as can be seen by the gas-phase trajectories.) There are several ways of characterizing order within the solid Mlayer f i . We show two such calculations here. The fmt is a conventional two-dimensional pair correlation function g(7). If playeris the number of atoms/A2 in a layer, (18) Phillips, J. M., to be published.
.........
-
-
0.00
0
Layer 3 Layer I , 2
-
'
1
1
T* Figure 7. Bond-orientationalcorrelation function (defined in
the text) for pairs of atoms in a given layer plotted as a function of the pair separation
T.
The results for a trilayer film at 50 K
and indicate that this type of order is stronger in the interior layers than in the outermost third layer. pleyelg(7) is the density of neighbors a t a distance T away from a central atom. Calculations of g(7) at 50 K for each of the three layers are shown in Figure 6. The peaks occur at the expected distances for a triangular lattice; the fact that the peaks and valleys are slightly higher for the second layer than for the first or third indicates that the thirdlayer atoms are less tightly bound, as one might expect, and that the decrease in Ar-Ar interaction taken for the first layer causes these atoms also to be less tightly bound than in the second layer. A second measure of order is the
Langmuir, Vol. 5, No. 3, 1989 605
Computer Simulations of Ar Films
:I
4 h
c.
Y
(3
43-
2-
OO I
I-
OF T
L
0.600.50-
-
0.40 h
-
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-
0.20
0.10-0. I -0.10;
I
'
2
I
' ' 6' '
4
O
0 J
8
T Figure 8, Pair correlations (top panel) and bond-order correlations (bottom)are shown here for atoms in the first layer of a three-layer film. Curves are shown for a series of increasing temperatures ranging from a solid to liquid layer. From the top the plots are for temperatures of 85.1, 96.5, 103.0, and 105.4 K.
s o - d e d bond-orientational function &( 7 ) , which is defined as14
where m,k, and j,l denote pairs of atoms separated by a distance 7 and Bmk is the angle made by the vector from atom rn to atom k , relative to some arbitrary coordinate axis. If the lattice has perfect hexagonal symmetry, &?e)(. will be unity a t each 7 corresponding to a peak in g(7) (see Figure 6). Figure 7 shows that thermal vibration causes each of the peaks in & ( 7 ) to be smaller than unity. Perhaps the most interesting feature of this curve is that the peaks do not decay noticeably with distance even for the third layer. Thus, both g(7) and & ( 7 ) indicate that the range of the order is a t least longer than the box size at a temperature of 50 K. The temperature variation in these two pair correlation functions as one passes from solid to liquid is of some interest, since one might imagine that these functions would be sufficiently different in the two phases to allow them to be used to determine whether a liquid or a solid is present. Thus, both g(7) and &(7) are plotted for various temperatures in Figure 8 for layer one and in Figure 9 for layer two. Evidently, the range of the order in both g(7) and & ( 7 ) is a rapidly decaying function of 7 in the liquid which is present a t the highest temperature shown. The
'
2
4
W
J
6
8
T Figure 9. Same as Figure 8, but for the second layer and for temperatures of 85.1, 89.0, 92.7, 96.5, and 111.9 K.
curves change gradually from solidlike with long-range order to short-ranged liquidlike as the temperature increases through the two-phase region, as one might expect. The primary thermodynamic quantity calculated for these systems was the average potential energy of the film. A plot of this energy versus temperature is shown in Figure 10. The straight line is a "base line", corresponding to a constant heat capacity. The wavy line given by the simulations evidently will produce a heat capacity curve that has three pronounced peaks at temperatures of -73, -93, and -99 K. The origin of these peaks is clarified if one plots Vu, the average Lennard-Jones interaction of the atoms in a layer, as a function of temperature. A plot of Vu* - P ,where the reduced variables are given by the unreduced variables divided by tM for the layer in question, is shown in Figure 11. These energies, which should be constant for harmonic oscillator layers, show rises that, when taken together with the trajectory plots, can be interpreted as layer melting. The center points of the increases correspond reasonably well with the estimated heat capacity maxima. Note that the fact that the melting occurs over ranges of temperature does not exclude first-order transitions. The problem here is that the conditions for a sharp first-order change are not satisfied. This can be seen by viewing the temperature dependence of the layer densities in Figure 12. It is evident that layer melting is accompanied by a significant change in density. However, an expansion in layer one or two causes the atoms expelled to migrate to the outer regions of the film. This changes the film thickness and is a process that cannot proceed at constant spreading pressure. We believe that layer melting in this system is thermodynamically
606 Langmuir, Vol. 5, No. 3, 1989
Cheng and Steele
-I3*
0.251
io 60 I
I
1
I
I
I
70
80
90
100
110
T(K)
Fi ure 12. Density (in reduced units defined as (atom/A2)(2.46 A) ) is shown as a function of temperature for each layer in the trilayer film. The sharply changing regions are associated with melting and correlate well with the analogous temperature ranges in Figure 11.
f
Table IV. Energies and Areas of Laver Melting AQayOrIRn
from total potential energies, K energies, K 600 Atoms in Three Layers 59 46 65 94 112 88 from layer potential
layer 1 2 3
1
T (K)
Figure 10. Total energy per particle in a trilayer film is plotted as a function of temperature. Reduced units are used for energy, and the dashed line is for reference-it corresponds to a film with
400 Atoms in One Layer 24 65
AA-,
A2/atom
0.6 1.9 4.0
0.88
Where R is gas constant, = 8.314 J mol-' K-l.
Figure 11. Intralayer energies minus the harmonic oscillator thermal energy of T* are plotted for the trilayer film. Points for a 1200-atom simulation (open circles) are essentially the same as those for the 400-atom simulations (filled circles and crosses). The sharply rising portions of these curves are associated with layer melting.
layers one and two but not for the third layer. (Of course, this assumes that the density changes in 3D and 2D are comparable.) Experimentally, there appear to be no data that are directly relevant to these simulations. However, studies of thicker argon films15show that as coverage increases, the melting temperature approaches the bulk triple point (83.8 K). In addition to the main peak in the heat capacity of a thick film, a small peak a t higher temperature is observed. We interpret this as the melting of the layer close to the substrate, since we find that both the first layer and second layer melt above the bulk triple point. Indeed, we surmise that the experimental peak is due to second-layer melting and that the first layer melts at a temperature higher than4 that reached in the experiments. We have also carried out a simulation of the melting of one complete monolayer of argon. The trajectory plots (Figure 13), the layer energy vs temperature plot (Figure 14), and the layer density versus temperature plot (Figure 15) all indicated a sharp, probably first-order melting a t 93 K. Estimated values of AU,,,, and AA,,, for this transition are also listed in Table IV. Again, experimental data suitable for comparison with this simulation do not seem to be available.
first-order. This view is supported by the trajectory plots, which seem to show two-phase equilibria (of microphases, to be sure) a t temperatures in the melting ranges. One can estimate AUb and AAh, the energy and area changes for melting in a given layer. Values of these parameters are shown in Table IV. Values of AUhp, should be roughly one-half the bulk value of AU, (=71R) based on a reduction in the number of nearest neighbors from 12 to 6 in changing from 3D to 2D. This holds true for
4. Discussion There are several theoretical concepts that are tested by these simulations. In the monolayer case, one can ask whether the melting transition is first order and whether the nature of the transition is sensitive to total coverage. The answer to both is a tentative yes. Thermodynamic order is always difficult to determine in simulations, but the transition appears to be abrupt, and its seems possible that layer promotion during the melting would slightly
constant heat capacity.
/
-6.0
-7.51 I 1
50
I 60
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I
80 T(K)
I
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1 100
I 110
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Langmuir, Vol. 5, No. 3, 1989 607
Computer Simulations of Ar Films
T = 85.I4 ( K 1 L
Second layer
T 96.49 (K)
T = 92.67M) I
1
1
I
I
Figure 13. Computer-generated atomic trajectories for the nominal monolayer films studied. Top views for the atoms in each layer are given together with a side view of the trajectories for all atoms in the layer. 0.5(
*II
-5.5
t
O
f + - 1
N*1200
I
80
I
100
90
I I I IO
T (K)
Figure 14. Temperature dependence of the average Ar-Ar energy per particle in the monolayer film. The harmonic oscillator thermal energy has been subtracted off to give emphasis to the sharply rising portion associated with melting. A simulation for 1200 atoms gave wentially the same resulta as the 6oo-atom runs shown by the filled circles.
smear out the process in any case. We find that a coverage change from one to three layers alters the nature of first layer melting appreciably. Since layer promotion is absent for the "patch" melting that occurs at less than monolayer coverage, it seems likely that this process will exhibit different characteristics than those for the full monolayer. It has been suggested16 that a sharp roughening transition may exist for thick films. We can use the p ( z ) in the neighborhood of the outer edge of the trilayer system as a measure of the roughness of the solid-vapor interface. It is reasonable to suppose that a decrease to zero occurring over a distance small compared to the atomic diameter would be a flat interface and that the decrease for a rough interface would occur over a distance large compared to the argon diameter. By these criteria, we observe a gradual roughening as the outermost two layers merge into a single liquidlike region. In part, this is due to the melting (and density change) of the third layer, which begins a t -65 K. This value is reasonably close to the theoretical esti-
O.4OL
io
I 90
I 100
I
]
I IO
T (K)
Figure 15. Temperature dependence of the layer density (in the same reduced units as in Figure 12) shown for the monolayer film. The temperatures for the sharply decreasing portion of this curve correlate well with the sharp energy changes of Figure 14.
matels for roughening temperature of 0.8Ttriple= 67 K. Roughening continues as more atoms are added to this outermost region due to the initiation of second layer melting (at -85 K) and then of first layer melting (at -89 K). It is also evident that layer-by-layer models" are not a good representation of the entire multilayer film, even though they may be capable of describing the innermost layers. It appears that simulations of even thicker films would be of interest, since they should be capable of telling whether the observation of liquidlike or roughened behavior is limited t o t h e two outermost layers regardless of thickness. In addition, it would be interesting to see how first- and second-layer properties depend upon film thickness.
-
Acknowledgment. This work was supported by N.S.F. Grant DMR-8718771. Ragistry No. Ar, 7440-37-1; graphite, 7782-42-5.
