Langmuir 1986, 2, 606-612
606
Dynamics of Nitrogen Molecules Adsorbed on Graphite by Computer Simulation Alexei V. Vernov" and William A. Steele Department of Chemistry, 152 Dauey Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802 Received May 2, 1986 As part of a computer study of the properties of nitrogen adsorbed on the basal plane of graphite, the time-dependent behavior of these molecules was simulated at 73.6 K and at a number of coverages ranging from the commensurate monolayer to roughly 2l/, layers. The properties reported in this paper include translational velocity time correlations, angular velocity time correlations, mean-square molecular displacements and time-dependent orientational changes. Changes in these variables are given separately for in-plane and for out-of-plane motion and for the various adsorbed layers, when more than one layer is present. The physical picture produced by these calculations is one where the out-of-plane motions of the molecules are oscillatory and librational in both the first and second layers, the in-plane reorientations are relatively free, and the in-plane translations are highly hindered. Calculations of the mean-squareforces and torques are presented in order to characterize the relative importance of the molecule-solid and molecule-molecule interactions in determining the molecular dynamics. The implications of these findings are discussed, with particular attention given to the nature of the in-plane translational motions. 1. Introduction In order to learn more about the molecular properties of simple nonspherical molecules physisorbed on a wellcharacterized solid surface, molecular dynamics simulations were carried out for model N2-graphite ~ystems.l-~ Surface coverages in these studies range up to 2 I I 2 monolayers and temperatures have ranged from 5 to 75 K. One virtue of the molecular dynamics algorithms is that they allow one to determine time-dependent molecular properties as well as equilibrium thermodynamics quantities. In previous papers2i3dynamical properties were reported for monolayers at low temperatures. These properties were also generated at 73.6 K as part of a study4p5a t varying coverage. These results are reported in the present paper. The overall motion of a nitrogen molecule conventionally is decomposed into translation and rotation. (We have neglected internal vibrations in all simulations, primarily because the computations are based on classical equations of motion and are thus unable to deal with quantized intramolecular vibrational dispacements.) The presence of the molecule-solid interaction causes these motions to be quite anisotropic, so that quantities such as translational velocity and/or displacement must be decomposed into their components parallel to and perpendicular to the surface if one is to gain an insight into the dynamical behavior. We here report translational velocity correlation functions C,,(t), where q = z (Ito the surface) or x,y, and the analogous angular velocity correlation functions C,,(t), where q here denotes a rotation axis oriented in the z direction or in the plane of the surface (11 ). These functions are normalized to unity a t t = 0 and thus are defined as (v,(t ?v&' + t )) C",(t) = (1.1) (U q 2 )
where the bracket denote an average over molecules and over different time origins t'in the simulation. Similar equations can be written for the angular velocity correlations. In principle, the fact that the molecule-solid potential depends upon x and y (the position variables in the plane of the surface) as well as z means that C,,(t) and C,,(t) may differ from one another. In fact, the graphite
* Permanent address: Department of Physical and Colloid Chemistry People's Friendship University, 117302 Moscow, USSR. 0743-7463/86/2402-0606$01.50/0
surface has a sufficiently high symmetry and a sufficiently weak periodic component in the molecule-solid potential to eliminate such dependence a t 73.6 K, the temperature of this study. Of course, an important variable remains: the molecule-solid separation distance. One expects that all molecular properties, equilibrium and dynamic, will be quite sensitive to the initial value of this distance. Since we have found that the molecules in these physisorbed phases are readily assigned to layers, each of which is characterized by a narrow range of molecule-solid distances, it is reasonable and convenient to handle this distance dependence by reporting properties for the molecules in various layers, which thus involve averages over all distances associated with a given layer. A second group of calculations are the time-dependent orientational and translational displacement functions. In particular, the mean-square displacements of a molecule in a given layer in the directions paralllel and perpendicular to the surface were evaluated, as were several functions of the change in the orientation of the molecular axis in time t (specifically, the changes in functions of the direction cosines of the axis were averaged). One difficulty that must still be dealt with is the effect of interlayer transfer upon the time-correlation functions for such a system. Although these transfers certainly do occur for the multilayer systems studied here, it was shown previously5 that they are not frequent on the time scale of the calculations. We have chosen merely to eliminate any molecule undergoing interlayer transfer from the computation; thus, when a simulation for nth layer molecules is reported, it describes the dynamics of only those molecules that remain in the layer for the duration of the calculation. Previous simulations of nitrogen monolayers at lower temperatures indicate that the in-plane reorientation (1) Talbot, J.; Tildesley, D. J.; Steele, W. A. Mol. Phys. 1984,51,1331. (2)Vernov, A. V.; Steele, W. A. Surf. Sci. 1986,171,83. (3) Lynden-Bell, R.M.; Talbot, J.; Tildesley, D. J.; Steele, W. A. Mol. Phys. 1985,4183.Talbot, J.; Tildesley, D. J.; Steele, W. A. Surf. Sci. 1986, 769 71 _"", ._. (4) Talbot, J.; Tildesley, D. J.; Steele, W. A. Faraday Discuss., Chen. SOC.,in press. (5) Vernov, A. V.; Steele, W. A. Longmuir 1986,2,219. ( 6 ) Joshi, Y.P.; Tildesley, D. J. Mol. Phys. 1985,55,999. (7)Peters, C.;Klein, M. Mol. Phys. 1985,54, 895.
0 1986 American Chemical Society
Langmuir, Vol. 2, No. 5, 1986 607
Dynamics of Nitrogen Adsorbed on Graphite Table I. Parameters for the Simulation Runs system 1 2
3 3a
4 5
no. of molecules in simulation box 96 120 130 130 192 240
N,* 1.00 1.25 1.35 1.35 2.00 2.5
Nl* 1.00 1.08 1.08 1.06 1.06 1.06
should be moderately free at temperatures as high as 70 K, and this was confirmed in the calculations reported here. Furthermore, rotational or translation motion of the first-layer molecules in the direction perpendicular to the surface is dominated by nearly harmonic oscillations in the deep moleculesolid potential well. Several interesting and important questions remain to be answered concerning, for instance, the nature of translation in the plane parallel to the surface: is it diffusional, as expected for a liquidlike layer, or is it oscillatory, as expected for a solidlike layer? Also, what is the nature of the dynamics of molecules in the second (or higher) layers of a multilayer system? How is the strongly anisotropic motion of a first-layer molecule modified as the molecule moves away from the surface? We will provide at least partial answers to these questions, among others, in the results to be presented below. 2. Simulation and Model As reported previ~usly,~ we found that the usual molecular dynamics algorithms, which describe a system of molecules a t constant total number N t in a constant volume V and with constant total energy E, are not convenient for multilayer physisorption systems. A problem arises when a molecule undergoes interlayer transfer, which happens on a time scale of -100 ps. Since the molecule-surface potential energy o i such a molecule changes greatly when it leaves or enters the first layer, energy conservation requires an equal and opposite jump in kinetic energy. This causes significant, long-lived fluctuations in the temperature, which are proportional to the mean kinetic energy. To eliminate these variations in temperature, a constant-temperature molecular dynamics algorithm from Evans et al. and Hoover et al? was adapted to the present problem. Although the resulting system is not exactly equivalent to a canonical ensemble: the resemblance is sufficiently close to allow us to evaluate time-dependent averages from the configurations generated by the algorithm. Details of the simulation procedure and of the intermolecular potential energies utilized are given elsewhere.'8 It is worth noting that the graphitic solid is considered to be an inert, rigid source of potential energy for the adsorbed nitrogen molecules. The molecule-solid potential depends upon all three center-of-mass molecular coordinates x , y , and z and upon molecular orientation as well. The N2-Nz interactions are taken to be given by a pairwise additive function that contains both the electrostatic quadrupole-quadrupole terms and a site-site representation of the shape-dependent dispersion-repulsion energy. The parameters of the Nz-N2 potential have been given by Murthy et a1.,lo who showed that the resulting model nicely reproduces the properties of the bulk phases of nitrogen. The simulation itself is a full three-dimensional (8) See, for example: Evans, D. J.; Morris, G. P. Comput. Phys. Rep. 1984, 1, 297. (9) NmC, S. J . Chem. Phys. 1984,81, 511. (10) Murthy, C. S.; Singer, K.; Klein, M. L.; McDonald, 1. R. Mol. Phys. 1980,41, 1387.
Nz*
Lx*
0 0.17 0.27 0.30 0.76, 0.18 0.98, 0.46
18
16 16 16 18 18
L,*/31/2 8 9 9 9 8 8
no. of time steps used for calculations 9775 10925 7676 11250 8700 5550
calculation, using periodic boundary conditions in the x , y plane and a reflecting boundary that reverses the z component of velocity of any molecule that leaves the surface region. Although the complexity of such a model means that the simulations require a relatively large amount of computer time, it was felt that the results are sufficiently realistic to allow one to draw conclusions concerning the behavior of the real N2-graphite system with some confidence. In fact, the most ambiguous aspect of the model used here is to be found in the periodic part of the moleculesolid potential. This is derived by summing site-site Lennard-Jones 12-6 interactions between the N atom sites and the carbon atoms of the solid. The transformation of the pairwise sums over the solid atoms into surface-periodic functions is well-known and the resulting potential has been widely used in simulations. However, there are theoretical" and experimental6indications that this model is overly simple and that the magnitude of the periodic terms may be considerably larger than that given by the sum over (spherically symmetric) N-C interactionspossibly by as much as a factor of 2. Thus, one simulation was run using a molecule-solid potential with periodic terms that were arbitrarily made twice as large as those for the spherical N,C-site model. (It seemed of particular interest to see if the motion of N2 molecules parallel to the surface was significantly affected by this modification of the potential.) Table I contains a summary of several relevant properties of the simulated systems to be discussed here. All simulations were performed at T = 73.6 K with varying numbers of molecules on the surface and, in the case of system 3a, the modified periodic potential. The equilibrium layer densities are shown as Ni*, which is defined to be the ratio of the number of molecules in layer i to that for the commensurate lattice ( d 3 X d3,R30°). The size of the simulation box is indicated by the reduced edge lengths L,* and Ly*, with L,*= LJ2.46 A. In evaluating the single-molecule dynamical properties to be reported here, one averages first over all particles in the system. Quantities evaluated generally involve a molecular property (position or velocity) at two different times. This information was obtained from sets of data stored at a series of times separated by 0.0505 ps (25 timesteps). Thus, values of a time correlation function (A(O)A(t)) were evaluated for times equal to multiples of 0.0505 ps. Improved statistics were obtained by averaging over different time origins. Technically, these origins should be sufficiently separated so that no correlations in position and velocity persist between one origin and the next. Although the separation of 0.30 ps (150 timesteps) utilized did not completely satisfy this criterion, no errors are introduced except in the statistical uncertainties, which we do not report. Table I indicates the total number of (11)Vidali, G.; Cole, M. W. Surf. Sci. 1980,91,339. Bonino, G.; Pisani, C.; Ricca, F.; Revetti, C. Surf. Sci. 1975,50,379. Nicholson, D. Surf. Sci. 1984,146, 480.
608 Langmuir, Vol. 2, No. 5 , 1986
Vernou and Steele 1.0
0.5
C
“n 0
-0.5
time (picosecl
Figure 1. Normalized translational velocity correlation functions for N2 molecules in the first adsorbed layer at several surface coverages. The numbers indicate the system studied as described in Table I. Correlations for the component of velocity perpendicular to the surface are shown here.
0.5 1.0 time (picosecl
1.5
Figure 2. Same as Figure 1but functions for the velocities parallel to the plane of the surface are plotted.
data sets stored and used in the averaging for each system considered. A crude estimate of the uncertainty in the velocity autocorrelation functions reported below is i0.03. 3. Results We begin with the most understandable time-dependent property of these adsorbed films, which is the translation velocity correlation function for motion perpendicular to the surface. For molecules in the first layer, motion in this degree of freedom is dominated by the large moleculesurface force which, to first order, gives rise to harmonic oscillation. Figure 1 shows the results obtined for several coverages (see Table I). All curves exhibit the oscillatory behavior expected for a particle moving near the minimum of the nearly harmonic molecule-solid potential; the frequency of the motion increases somewhat as the number of molecules in the second layers increases (curve 5 relative to curve 1 or 3). These changes are due to the added interaction with the overlayer molecules. A change in the barrier to translation across the surface has little effect on the frequency of oscillation perpendicular to the surface but does cause a decrease in the damping of the motion (curves 3 and 3a)-this is most likely a secondary effect of the slight shifts in first- and second-layer density produced by the change in barrier height-see Table I. Indeed, the most obvious differences between the various curves of Figure 1are in the damping of the motion, which can be seen to increase as the coverage in the second-layer increases. The time dependence of the components of translational velocity parallel to the surface is much different from that perpendicular to the surface, as is illustrated in Figure 2. The rapid decay of these correlations is characteristic of the Brownian motion found in simulation of bulk liquids. Note that the initial decay hardly changes as the coverage changes. As is well-known, the short-time behavior of this function is given by (3.1)
where Fllis the component of force on a molecule parallel to the surface and m is the molecular mass. Evidently, Fll for first-layer molecules hardly changes in this range of coverage. This point w ill be discussed in more detail below. This apparent liquidlike behavior in the plane parallel to the surface can be confirmed by evaluating mean-square displacements ( d ? ) in the direction denoted by 1 (either
O’OT
1
Y
0
0.5
1.0
1.5
time ( picosecl
Figure 3. Mean-squaredisplacements in the plane of the surface for first-layermolecules are shown for various systems, as described in Table I. Displacements are in reduced units obtained by dividing distance by 2.46 A. The initial curvatures are a remnant of free translation, and the nearly linear time-dependence at long times is characteristic of diffusional motion. (Extensionof these curves to longer times confirms the approach to linearity indicated
here.)
z or 11). Although these can be calculated from the time integrals of the velocity correlations, more precise values can conveniently be obtained directly from the simulation data. Displacements parallel to the surface are shown in Figure 3 for molecules in the first layers. (Remember that any molecule that changes layers in the time span of the calculation is excluded from the calculation.) Mean-square displacements perpendicular to the surface are not shown but quickly reach a constant value -0.02 in all cases. Although the parallel displacements suggest that these two-dimensional films are liquidlike, there are some problems with this interpretation having to do with the curvature of the plots in Figure 3. If the molecules are undergoing simple self-diffusion, a linear plot of mean square displacement vs. time is predicted, for times long compared to the decay time of the corresponding velocity correlation function. On the other hand, theory12 for strictly two-dimensional fluids indicates that the velocity correlation functions should exhibit a t-’I2 decay at long (12) See, for example: Fox, R. 118A, 383.
F. Physica A: (Amsterdam) 1983,
Dynamics of Nitrogen Adsorbed on Graphite
Langmuir, Vol. 2, No. 5, 1986 609
2
0
0.5
1.0
1.5
time [picorer.)
Figure 4. Same as Figure 1,but correlation functions for velocities perpendicular to the surface are shown for second-layer and, in one case, third-layer molecules.
times, which then yields no diffusional behavior when substituted into the (rigorous) relation:
Figure 2 shows that the parallel velocity correlations for lower coverages are still nonzero and slowly decaying at t > 1.0 ps. As a consequence, the observed negative curvatures of the (dIi2)plots are not unexpected. For the highest coverage shown, the velocity correlation is much closer to zero for t > 1ps and the (d:) plot is more nearly linear. Diffraction e~periments’~ indicate that the real N2 on graphite monolayer at densities equal to or slightly greater than commensurate is solidlike (yielding a sharp set of X-ray spots, for example). Although our simulations do not give clear evidence of a solid, the molecular motions parallel to the surface in the first layer are not those of a conventional fluid either. The translational motion of particles in the second and third layer of nitrogen is similarly difficult to characterize. Correlation functions for the z component of velocity are shown in Figure 4 and indicate that this motion is still oscillatory for molecules in the second and third layers but has a much lower frequency and stronger damping than for first-layer molecules. This is reflected in the meansquare displacements shown in Figure 5 for both the z and the surface-parallel coordinates. The displacements in the z direction, while much larger than those for the first-layer molecules, are still those for solidlike or oscillatory motion. On the other hand, the mean-square displacementsparallel to the surface are relatively large and appear to be diffusional &e., proportional to time) at times longer than 0.5 ps. The strong dependence of this quantity upon total coverage merely reflects the fact that the density within the second (and third) layer is changing considerably from one run to another-see Table I. As usual, the “diffusion constants” for motion parallel to the surface, which are proportional to the slopes of the linear part of the meansquare displacement curves, decrease rapidly with increasing density, indicating some kind of collision-limited motion within each layer. In an attempt to better understand the forces that are controlling the velocities of these molecules, time-correlation functions of the forces were also evaluated. This (13) Kjems, J. K.; Passell, L.; Taub, H.; Dash, J. G.; Novaco, A. D. Phys. Reu. B 1976,13,1446. Eckert, J.; Ellenson, W. D> Hastings, J. B.; Passell, L. Phys. Reu. Lett. 1979, 43, 1329. Nielsen, M.; Kjar,K.; Bohr, J. J . Electron Spectrosc. Relat. Phenom. 1983, 30, 111. Morishige, K.; Mowforth, C.; Thomas, R. K.Surf. Sci. 1985, 151, 289.
5
time (picosec)
Figure 5. Mean-square translational displacements for molecules in the second (and for system 5, the third) layer are given here in the reduced units used in Figure 3. Displacements in the direction perpendicular to the surface (denoted by 2 ) are shown as well as displacements in the plane of the surface. Displacements in the z direction have been artificially limited by excluding molecules undergoing layer changes from the calculation.
has the advantage that it is easy to divide the total force on a molecule into the molecule-surface fi,, and the molecule-molecule ti,, part; here, i denotes the ith Cartesian component relative to surface-defined coordinates. As is well-known, (3.3)
Our primary interest here is in the partial correlations which contribute to the force-force correlation. These are defined by J’i,ss
J’i,mm
J’i,ms
= (fi,s(o)fi,s(t))
(3.4)
= (fi,m(O)fi,m(t))
(3.5)
= (fi,m(O)fi,s(t))
(3.6)
We do not normalize these functions to unity at t = 0 because we are interested in their relative magnitudes. In particular, note that the time correlation Fi(t) for the ith component of the total force on a molecule is given by J’i(t) = (fi,s(o)fi,s(t)) + (fi,m(O)fi,m(t))+ 2(fi,s(o)fi,m(t)) (3.6)
A number of the partial force-force correlation functions are shown in Figure 6 for a molecule in the first layer and in Figure 7 for the second layer. In those cases where the mean-square force is large, the curves exhibit a damped oscillation. The oscillatory behavior is somewhat less obvious for the smaller terms, but one must remember that the time dependences are determined by the molecular motion, which is strongly oscillatory in the z direction and, depending upon density, somewhat less so in the direction parallel to the surface. Of greater interest are the initial values of these functions, since these are a measure of the relative importance of the various forces operative in the films. Table I1 shows the various force-force averages computed for two of the systems studied. As expected, the z component of the molecule-solid force is quite large while the parallel components are essentially negligible. The decrease in this quantity as one goes from layer 1 to
610 Langmuir, Vol. 2, No. 5, 1986
Vernov and Steele Table 11. Force-Force Averages' S-M -
_ _ _ l _ l _ _
s-s
M-M
_ l _ _ _ _ l _
Y
z
X
2.1 5.7
641.8 1828.4 35.0
4.5 7.3
system
X
1 (comm.) 4 (layer 1, 0 = 2) 4 (layer 2, 8 = 2)
2.1 5.2 0.0
0.0
Y
z
X
Y
z
3.5
-48.3 -106.9 -55.9
646.6 1310.1 688.9
625.0 1428.9 693.3
487.9 520.3 823.5
4.7 0.0
0.0
aValues are given in reduced units obtained by dividing force by cgs/a,where egs/k = 31.9', a = 2.46 A.
Table 111. Torque-Torque Averages'
s-s system 1 (layer 1) 4 (layer 1) 4 (layer 2)
Y
2
X
Y
12.9 38.1 0.0
12.4 37.9
0.1 0.2
0.0
0.0
-3.6 -11.7 -0.1
-3.2 -12.1 -0.1
Values are given in reduced units obtained by dividing torque by I
I
I
I
-
S-M
X
I
I
M-M
z
0.0
-0.1
0.0
cgs.
I
0
0.1
X
1'
z
15.5 27.1 19.8
13.9 27.4 17.5
22.1 45.0 22.5
0.3 0.4 Q5
0.2
0.6
7
time (picosec)
Figure 7. Same notation as in Figure 6, but curves are shown for second-layer molecules. time (picosecl
Figure 6. Force-force time correlations for the component parts of the force on first-layer molecules. Autocorrelations are shown in all cases, with sa denoting the correlation for the moleculeaolid force and mm denoting the molecule-molecule function; directional components of the force are denoted by z and by /I, and numbers indicate the systems described in Table I.
layer 2 is also anticipated, but the rather large change in the first-layer z component when significant second-layer coverage is added is not. It is interesting to note that the Nz-Nz interactions give rise to surprisingly large z component of the force even within the monolayer. Also noteworthy is the large jump in parallel component of the first-layer Nz-N2 force in comparing run 1and run 4-this is due primarily to a small increase in first-layer density, as indicated in Table I. An analogous calculation of the component terms in the torques on these adsorbed molecules was also carried out. Values for various mean-square torques are shown in Table I11 and, for the most part, corroborate the physical picture already developed. Note that a torque around the x or y axis is associated with libration perpendicular to the surface. For first-layer molecules, these torques are dominant, arising from both Nz-surface and Nz-Nz interactions. Mean-square torques hindering in-plane motion (z component) are not overly large and are due almost en-
~
i\
\
z l a y e,r I
'\' '\
0.31
0.1
---_ 0.5
1.0
1.5
time f picosec)
Figure 8. Time dependence of direction cosines of the molecular axis. The curves denoted z show reorientation relative to the surface plane, and those denoted I/ show how reorientation within the plane rapidly comes to a random distribution in which the average direction cosine is zero.
Langmuir, Vol. 2, No. 5, 1986 611
Dynamics of Nitrogen Adsorbed on Graphite
Table IV. Self-Diffusion Constant for Motion Parallel to the Surface system D X lo5, cm2/s from bulk N2, 75 K 4,5 (layer 1) 3 (layer 2) 4 (layer 2) 5 (layer 2) 5 (layer 3)
0.I
:::;I 0,060
,
,
,
0.5
1.0
1.5
time
(picosec)
Figure 9. Logarithmic plob of the in-planeangular displacement correlation functions defined in eq 3.7. Systems are described in Table I. The exponential decays expected for the simple
random walk reorientation model are not found.
tirely to N2-Nz interactions. Finally, Figure 8 shows the time dependence of the direction cosines of the orientation of the N2molecular axis for two of the systems studied. Specifically, C&t) = (cos n [ W - N3l)
(3.7)
where 4 denotes the angle between the molecular axis and either the normal to the surface ( 2 ) or the surface plane ([I). Figure 8 is yet another way of showing the reorientation perpendicular to the surface is restricted to libration (on the time scale of the figure). In contrast, in-plane molecular orientation rapidly becomes random. If the molecules were undergoing rotational diffusion in a plane, theory yields14 C,(t) = (cos n[@(t)- 4(0)]) = exp[-n2Rllt] (3.8) where R,,is a rotational diffusion constant. Logarithmic plots of these functions for first-layer molecules are shown in Figure 9 for n = 1and 2. The significant deviations from linearity and the fact that the long-time decays of these correlations do not vary as n2 are a clear indication that random walk in angle space is not a good model for these motions. In fact, behavior such as that shown in Figure 9 is generally ascribed to the presence of memory (or dynamical coherence) over the time scale of the simulations. 4. Discussion
The physical picture that emerges for the dynamics of adsorbed N2 molecules is not totally unexpected. For first-layer molecules, both translation and rotation in the plane perpendicular to the surface are strongly hindered and can best be described as damped vibration and libration. Although the frequencies of these motions are only slightly coverage-dependent, one finds that the damping becomes much more pronounced when secondlayer molecules are added. Even in the second layer where the N2-surface forces and torques are much weaker, oscillatory motion persists. The in-plane dynamics are characterized by moderately free rotation under the influence of relatively weak but coverage-dependent intermolecular torques. As found in studies of bulk liquids,16 dynamical coherence or memory effects are present when the in-plane torques are as small as in these systems. Only (14)See, for example: Steele, W. A. Adu. Chem. Phys. 1976,34, Sec.
I11 B.
(15)Steele, W.A. J.Mol. Liq. 1984, 29, 209.
2/3(3.3) = 2.2 2.3 11.5 4.7
3.1 7.7
in the case of the in-plane translational dynamics is there an element of uncertainty arising from the well-known neutron diffraction experiments which are interpreted as showing a solidlike, translationally ordered monolayer at temperatures and coverages similar to those considered in the present work. Thermodynamic measurements16have confirmed this argument, but these simulations do not clearly indicate solid formation. Although it is not easy to distinguish between liquid and solid in a simulation of a few hundred molecules, the large in-plane mobilities observed here are indicative of a liquidlike monolayer. Indeed, one can estimate a diffusion constant 2 0 for inplane motion from the long-time slopes of the plots of mean-square displacement vs. temperature. A comparable figure for the bulk fluid can be obtained from the simulations of Cheung and P0w1es.l~ The result for two dimensions would just be two-thirds of the bulk value. Table IV compares this value with diffusion constants calculated from the simulations presented here, and one sees that the monolayer diffusion constants are quite liquidlike. Furthermore, motion in the low-density bilayer is less hindered than in the monolayer, as expected. Since one possible explanation for the discrepancy between simulation and experiment is that the periodic barriers to in-plane translational motion are not high enough in our model to lock the Nzmolecules into a solidlike registered layer, the potential was modified by arbitrarily doubling the periodic terms (systems 3 and 3a). In fact, Figure 3 shows that an increase in the barrier height has the unexpected effect of causing an increase in the nitrogen mobility parallel to the surface. In fact, the motion shows a strong correlation with first-layer density (see Table I), with the smallest mobilities observed for the systems with the largest areal densities (systems 2 and 3). Judged by this criterion, the high-barrier system exhibits a slightly smaller mobility than the other two cases with the same first-layer density (systems 4 and 5). Another method often used to distinguish between liquid and solid is to look for long-range order in the radial distribution functions. Several of these distributions have been simulated and published el~ewhere.~ Unfortunately, it is not possible to determine translational order for separations larger than the simulation box dimensions. At the allowable distances, thermal disorder is sufficient to almost completely destroy the distinction between fluid and solid in these two-dimensional systems. This is the basic reason for placing such emphasis in this work on mobility as a criterion for solidlike behavior. In order to resolve the difficulties encountered in this attempt to determine whether the N2 monolayer at commensurate or higher density is solidlike at this temperature, computer graphics were used to generate a direct view of the in-plane translational motion by plotting out the trajectories of the molecular centers-of-massfor a relatively (16)Butler, D.M.; Huff, G. B.; Toth, R. W.; Stewart, G. A. Phys. Reu. Lett. 1975,35, 1718. Chan, M. H. W.; Migone, A. D.; Miner, K. D., Jr.; Li, Z. R. Phys. Reu. B 1984, 30, 2681. (17)Cheung, P.S.Y.; Powles, J. G . Mol. Phys. 1976, 32, 1383;1975, 30, 921.
612 Langmuir, Vol. 2, No. 5, 1986 8
Vernou and Steele 2
Figure 10. Computer-generatedmolecular trajectories are shown here for an initially commensurate monolayer containing 192 N2 molecules at 76.7 K. Trajectories are plotted by connecting center-of-maw positions ( x , y ) evaluated a t intervals of 0.0606 ps for a total of 300 times (Le., for an 18.2-ps interval). Part a shows only those molecules that have spontaneouslyjumped up into the second layer. (Trajectories are terminated when they return to the first layer.) Note particularly the solidlike oscillatory trajectories in the upper-middle part of Figure lob, far from the defecta created when molecules leave the monolayer as indicated in Figure loa. Also, remember that the periodic boundary conditions cause a disturbance that is near an edge to propagate via the image molecules to the opposite side of the sample.
long time span of the simulation. For this purpose, a sample of 192 molecules was simulated at a temperature of 76.7 K for a time interval of 18.0 ps. The first point t o note is that not all molecules remain in the monolayer for this period of time. Thus, trajectories for first- and second-layer molecules are shown separately in Figure 10, parts a and b. The specific molecules shown in Figure 10b and the times of second-layer existence (relative to the starting time) are molecule 42, from 15.2 to 16.0 ps; molecule 126, from 8.0 to 8.9 ps; and molecule 192, from 2.3 to 6.1 ps. Although the statistics here are obviously very poor, this corresponds only to 0.1-0.2% second-layer promotion. Nevertheless, Figure 10a shows that this has
a very important effect on first-layer translations, since significant displacements are found primarily in the neighborhoods of the defects in the solidlike layer that are formed when a molecule is promoted to layer 2. Interestingly, much of the observed first-layer motion is in directions parallel to the rows of molecules and may reasonably be interpreted as hopping to an adjacent vacant site. The trajectory plot leads to the conclusion that the simulations produce commensurate layers that are defect solids at T N 77 K. Furthermore, the large mean-square displacements observed for these monolayers are not signatures of a fluid phase but of jump diffusion in the neighborhood of the defect sites in the solid. It also appears that the height of the barrier to translation plays a very minor role here, in contrast to the situation at temperatures near the melting point for a patch of molecules on a partly filled surface.6 Nevertheless, the effect of alterations in the molecular potentials upon the structural and dynamical properties of these films is worthy of more study. For example, there are theoretical18 and experimentallggrounds for belief that the value of the N2-Nz site-site well-depth used here (ess/k= 36 K) may be roughly 20% too large. A weakened attractive potential could stabilize the registered density relative to the denser monolayers observed here in systems 2-4; or, more generally, it could significantly alter the first-second-layer equilibrium densities. Another possibility for future work would be the extension of these simulations to higher temperatures to ascertain whether they are capable of yielding a “critical point” for the solid (commensurate) monolayer, as is suggested by experiment.
Acknowledgment. Support for this study provided by the Division of Materials Research of the NSF, the US. State Department, and the Materials Research Division of the National Science Foundation is gratefully acknowledged. This work was done while A.V.V. was an exchange visitor at Penn State under the auspices of the International Research and Exchange Board and the Ministry of Higher Education of the U.S.S.R. Helpful discussions with Professors S. Fain, M. Chan, and D. J. Tildesley are acknowledged. Registry No. N P ,7727-37-9; graphite, 7782-42-5. (18) Sinanoglu, 0.;Pitzer, K. S. J. Chem. Phys. 1964,41,1322.Bruch,
L.J. Chem. Phys. 1983,79,3148. McLachlan, A.D.Mol. Phys. 1964,7, 381. (19)Bojan, M. J. Ph.D. Thesis, The Pennsylvania State University, 1986.
