J. Phys. Chem. 1984, 88, 1420-1425
1420
(2) The value of the apparent charge is in agreement with the Manning theory and confirms a priori the monochain conformation. (3) In water without added salt, the KC is extended and the presence of some "blobs" with four or five individual chains of KC is possible. This conformation changes and is destroyed with increasing salt concentration. Further study with q dependent (light or neutron scattering) should allow a more accurate estimation of the long-range electrostatic interactions in aqueous solution and give some information on the transition "2" coil dimer as a function of the salt and polymer concentrations, the nature of the counterion, and the temperature. Some other polysaccharides like chondroitin sulfate or heparin show very different dynamic behavior from KC. For these macromolecules the extrapolation of Deffto C, 0 gives the same value of Do for all salt concentrations. This could indicate no
-
-
modification of conformation with salt concentration. For KC, the extrapolations of Deffin water or in 0.1 M NaCl are very different and seem to show a change of conformation with added salt. We believe that there are two types of electrostatic interactions in salt-free aqueous solutions of polyelectrolytes that depend on the structural characteristics of the charged macromolecules (molecular weight, shape, and charge). If C < C*, the interactions polyion-counterion and also polyion-polyion modify the dynamics of polyelectrolytes. With C > C*, only polyion-polyion interactions are present and can give some interchain conformation type "blobs". Finally, the scaling relations as applied to polyelectrolyte solutions are in good agreement with some experimental r e s ~ l t s ~ ' , ~ ~ and seem to give a qualitative estimation of the dynamic and structural properties of aqueous solutions of K-carrageenan. Registry No. Carrageenan sodium salt, 37359-47-0.
Interaction Potentials and the Properties of Xenon Overlayers Physissrbed on the Graphite Basal Plane Michael L. Klein,* Chemistry Division, National Research Council of Canada, Ottawa, Canada K I A OR6
Seamus O'Shea, Chemistry Department, University of Lethbridge, Lethbridge, Alberta, Canada TI K 3M4
and Yoshiaki Ozaki? Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada L8S 4Ml (Received: June 1 , 1983; In Final Form: August 3, 1983)
Monte Carlo calculations have been carried out for monolayers and bilayers of xenon physisorbed on the graphite basal plane. The temperature dependence of the adatom vibrational amplitudes derived from hydrogen atomic beam and electron diffraction data is used along with thermodynamic data to obtain new parameters for the xenon-graphite interaction. These data suggest that the xenon-graphite interaction is somewhat stronger than previously supposed.
1. Introduction For a number of reasons there is much current interest in overlayers of rare gases physisorbed on the graphite basal plane. First, there is the question of the nature of the adatom-surface' and adatom-adatom* potentials. While the asympototic behavior of such interactions is now well ~haracterized,"-~ stringent tests have yet to be applied that probe the detailed functional form at shorter separation^.^ In addition, physisorbed layers are by their very nature quasi-two-dimensional (2D) and, as such, have attracted attention as model systems to study 2D melting.5 However, in reality, the graphite basal plane is not completely flat, and fascinating questions concerning epitaxy and commensurate-incommensurate transitions also arise.6-10 Our interest here is mainly in the nature of the interactions that are present in physisorbed overlayers, and, in particular, we will focus on the xenon-graphite system. The detailed phase diagram of xenon, physisorbed on graphite, has been constructed by Thomy and D ~ v a lit; ~resembles that of a strictly two-dimensional system interacting with a Lennard-Jones p~tential.~ At low temperatures one can prepare solid expitaxial monolayers ((v'3Xv'3)R30) in which the xenon atoms are in registry with the underlying graphite
Permanent address: Chemistry Department, Nagoya Technical University, Nagoya 466, Japan. 0022-3654/84/2088-1420$01.50/0
basal plane; every third graphite hexagon being occupied by a xenon atom.I0 At higher temperatures (60-70 K) a commensurate-incommensurate transition arises'O that involves a misfit in (1) M. W. Cole and J. R. Klein, Surf.Sci., 124, 547 (1983). (2) S. Rauber, J. R. Klein, and M. W. Cole, Phys. Rev. E: Condens. Matter, 27, 1314 (1983). (3) A. D. McLachlan, Mol. Phys., 7, 381 (1964). (4) L. W. Bruch and H. Watanabe, Surf.Sci., 65, 619 (1977); L. W. Bruch, J. Unguris, and M. B. Webb, Surf. Sci. 87,437 (1979); L. W. Bruch and M. S.Wei, ibid., 100,481 (1980); J. Urguris, L. W. Bruch, E. R. Moog, and M. B. Webb, ibid., 109, 522 (1981); L. W. Bruch, ibid., 125, 194 (1983). (5) J. M. Phillips, L. W. Bruch, and R. D. Murphy, J . Chem. Phys., 75, 5097 (1981); S. Toxvaerd, Phys. Reu. Lett., 44, 1002 (1980). (6) D. E. Moncton, P. W. Stephens, R. J. Birgeneau, P. M. Horn, and G. S. Brown, Phys. Reu. Lett. 46, 1533 (1981); P. A. Heiney, R. J. Birgenau, G. S. Brown, P. M. Horn, D. E. Moncton, and P.W. Stephens, ibid. 48, 104 (1982). (7) J. P. McTague, J. Als-Nielsen, J. Bohr, and M. Nielsen, Phys. Rev. V: Condens. Matter, 25, 7765 (1982). (8) F. F. Abraham, S. W. Koch, and W. E. Rudge, Phys. Reu. Lett., 49, 1830 (1982); F. F. Abraham, ibid., 50, 978 (1983); F. F. Abraham, W. E. Rudge, D. J. Auerbach, and S.W. Koch, ibid., 52, 445 (1984). (9) A. Thomy and X . Duval, J . Chim. Phys. Phys.-Chim. Biol. 67, 1101 (1970). (10) J. A. Venables, H. M. Kramer, and G. L. Price, Surf. Sci., 55, 373 (1976); 7, 782 (1976); P. S. Schabes-Retchkiman and J. A. Venables, ibid., 105, 536 (1981); C. W. Mowforth, T. Rayment, and R. K. Thomas, private
communication.
0 1984 American Chemical Society
Xenon Physisorbed on the Graphite Basal Plane
The Journal of Physical Chemistry, Vol. 88, No. 7, 1984 1421
Xe-C
lattice constants of up to 6%. For the case of a registered monolayer, extensive lattice dynamical studies have been carried out."-13 Such calculations have revealedI3 that, for a heavy absorbate like xenon, the modes of vibration of the adatom overlayer are localized below the band of the graphite bulk modes. Accordingly, it should be a fair approximation to consider the graphite basal plane as a rigid substrate when discussing the dynamics of the xenon overlayer. Recent molecular beam experiments using hydrogen atoms have probed the temperature dependence of the Debye-Waller factors for solid xenon monolayers and bi1a~ers.I~This data, taken together with earlier low-energy electron diffraction (LEED) measurements for a monolayer," will be used to test present models of the adatom-surface and adatom-adatom interact i o n ~ . ' ~ JSuch ~ models can be further tested against information from Henry's law constants" and the measured latent heat of transformation between the 2D crystal and the 3D vapor phase. In the next section we briefly review the Crowell-Stele rnodell5 for surface-adatom interactions16 and draw attention to the work of Cole and co-workers' that suggests the use of much smaller surface-adatom separations than used hitherto. We also discuss the adatom-adatom interactions, the effect of the substrate being incorporated via McLachlan's t h e ~ r y . ~In. ~section 3, we present the results of our canonical ensemble Monte Carlo calculations, which have been employed to evaluate the mean potential energy ( U ) and the adatom mean-square displacements ( uz2) for registered (v'3Xv'3)R30 monolayers and bilayers, using various interaction potentials. Although we are unable to find a totally consistent model, the weight of evidence suggests that the well depth of the xenon-graphite interaction is considerably deeper than proposed recently in the work of Cole and Klein.' Our suggestion could be tested by careful measurements of the isosteric heat of adsorption, at low coverage, for well-characterized samples of graphite.
pot entia1
18319
11. Potential Models A . Adatom-Surface Interactions. It has been customary in studies of physisorption on graphite to utilize the Crowell-Steele mode1.'5J6 The basic postulate is that the adatom-surface interaction is represented by the sum of Lennard-Jones (12-6) adatom (A)-carbon atom (C) potentials, uAC(S) = 4cAC[(UAC/S)'* - ( u ~ ~ / Swhere )~]S , is the A-C separation. The evaluation of the total adatom-graphite substrate potential, V, is then handled in one of two ways; either by direct summation, V = &uAC, with an efficient algorithm20 or, since the surface potential is weakly varying across the basal plane, by a truncated Fourier series expansion.16 We have chosen to use the latter procedure and the adatom-substrate potential, V, then takes the formI6
where Z = r, is the distance of the adatom above the basal plane and a is the distance between the centers of the graphite hexagons. (1 1) J. P. Coulomb and P. Masri, Solid State Commun., 15, 1623 (1974);
J. P. Coulomb, J. Suzanne, M. Biefait, and P. Masri, ibid., 15, 1585 (1974). (12) k. K. Agrawal, Phys. Rev. B: Condens. Mutter, 23, 1778 (1981). (13) E. de Rouffignac, G. P. Alldredge, and F. W. de Wette, Phys. Rev. B: Condens. Matter, 24, 6050 (1981); F. W. de Wette, B. Firey, E. de Rouffignac, and G. P. Alldredge, Phys. Rev. B: Condens. Mutter, 28,4744 (1983). ' (14) T. H. Ellis, G. Scoles, and U. Valbusa, Chem. Phys. Lett., 94, 247 (1983). (15) A. D. Crowell and R. B. Steele, J. Chem. Phys., 34, 1347 (1961). (16) W. A. Steele, Surf. Sci., 36, 317 (1973); "The Interaction of Gases with Solid Surfaces", Pergamon Press, Oxford, 1974. (17) T. R. Rybolt and R. A. Pierotti, J . Chem. Phys., 70,4413 (1979); W. A. Steele, J . Phys. Chem., 82, 817 (1978). (18) J. Suzanne, J. P. Coulomb, and M. Bienfait, Surf. Sci., 44, 141 (1974); 47, 204 (1975); J. Suzanne and M. Bienfait, J . Phys. (Orsay, Fr.), 38, C4-93 (1977). (19) J. Regnier, J. Rouquerol, and A. Thomy, J . Chim. Phys., 72, 327 (1975). (20) D. R. Dion, J. A. Barker, R. P. Merrill, Chem. Phys. Lett., 57, 298 (1978).
+ I
3.5
0
4.0
z/A
Figure 1. The leading term V,(Z) in the adatom-surface potential for xenon on graphite. The dotted curve is derived from traditional parameterization of the Crowell-Steele model,I6 the dashed curve is derived from the paramaters of Cole and Klein,l and the full curve is the present model (excc = 107 K, uxec = 3.40 A). The crosses indicate the effect of including the second term in the adatom-surface potential (see text).
The functions Voand VI involve summations over substrate layers (n). For example,I6 n=O
+
where Zn = Z nd; d = 3.35 A is the graphite interplanar separation, and p is the surface density of carbon atoms. The function fi expresses the surface periodicity16 fi = -2[cos 27rs, + cos 2aSz + cos 2n(S, S,)]
+
here a s , = 2X/3'I2 and as2 = Y - X/3'I2. The minimum of f l occurs at the center of a graphite hexagon. Figure 1 shows the variation of Vo(Z)with Z for three sets of parameters exec and cXeC, to be explained below. As noted by previous investigators, the minimum energy is always obtained for 2 uAC. The additional contribution arising from the term VJl is small, being about 2% at the minimum of V,. The effect of including this term is indicated by the crosses in Figure 1; it increases for the models with smaller gAC. A common route to the potential parameters tACand gACis to utilize combining rules.16 Starting from Lennard-Jones (1 2-6) potentials for Xe-Xe (e 226 K, uXeXe= 3.98 A) and C-C (ecC = 28 K, uCC = 3.4 xy:Je have derived the dotted curve for Vo(Z),shown in Figure 1 . The resulting potential energy of a xenon atom interacting with the graphite basal plane, about 1810 K, is considerably less than indicated by analysis" of experimental isotherm data in the Henry's law region (1928 K) or analysis16 of the isosteric enthalpy of adsorption (1920 K). This is not the only shortcoming of the traditional approach to potential param-
1422 The Journal of Physical Chemistry, Vol. 88, No. 7, 1984
Klein et al.
I
I
2400
X e Graphite
I
26
28
30
I
,
I
32
34
36
T T
1
a ~ s[i] Figure 2. Plot of gas-phase potential minima, R,, vs. the adatom-surface u parameter, am. The traditional Lennard-Jones parametersI6yield the open circles and bold line on the right-hand side of the figure. The bold line on the left is suggested by Cole and Klein.’ The circles with error bars are experimental adatom-surface separation^.^'^^ The triangle is the value adopted for uxCcin the present work.
eters for adatom-graphite interactions. Carlos and Cole2’pointed out that, in order to interpret data derived from scattering experiments, the helium-graphite potential had to be anisotropic and, in addition, the optimal choice of u parameter (uHec= 2.74 A) was different from that indicated by the use of combining rulesI6 (uHeC= 2.98 A). In the case of xenon, the effect of surface anisotropy is perhaps less important, but the possible need for a considerably smaller uAcis a significant result. This theme was taken up by Cole and Klein,’ who pointed out that experimental diffraction data for overlayers of Ar,23and Kr24also indicate the need for smaller values of UAC. They therefore proposed a new set of parameters for C-C interactions (ecc = 25.7 K, ucc = 2.84 A) that are consistent with the He-C parameter cited above. These new C-C parameters taken together with realistic potential parameters for eAA and,a formed the basis for a new set adatom-surface potentials.’ Cole and Klein1 also pointed out that the combination of realistic gas-phase parameters25and their new C-C potential yields the correct value4 for the long-range adatom-surface interaction coefficient C, ( V ( 2 ) C 3 / Z 3as Z a). However, the zero-coverage isosteric enthalpy for their model (the dashed curve in Figure 1) lies at least 20% above the experimental value. A similar situation pertains to Ar and to a lesser extent Kr, but this will not concern us here. This disappointing result probably arises because the traditional choice of adatom-substrate interaction derived from a Lennard-Jones potential is not sufficiently flexible to simultaneously describe the long-range asymptotic behavior and the region near the minimum of Vo(Z). The failure of the combining rules may also be important. Although the above discussion points to the need for a new functional form for adatom-graphite potentials, in this paper we will not pursue such a goal. Instead, we will stay within the framework of the Crowell-Steele model15 and seek a new set of parameters. As a starting point, we note that absorption data suggest17the use of a larger value for exec than adopted by either Steele16 or Cole and Klein’ and the experimental diffraction data22-24suggest the use of a smaller uxeCthan that of Steele.l6
-
-
(21) W. E. Carlos and M. W. Cole, Phys. Rev. Lett., 43,697 (1979); Surf. Sci.. 91. 339 (1980). (22) K. Carneiro, L. Passell, W. Thomlinson, and H. Taub, Phys. Reo. B Condens. Mutter, 24, 1170 (1981). (23) C. G. Shaw, S. C. Fain, M. D. Chinn, and M. F. Toney, Surf. Sci., 97, 128 (1980). (24) C. Bouldin and E. A. Stern, Phys. Rev. B Condens. Mutter, 25,3462 (1982). (25) J. A. Barker, M. V. Bobetic, and M. L. Klein, IEM J . Res. Deu., 20, 222 (1976).
30Oo-
20
60
40
I
,
80
!
100
T [KI
Figure 3. Temperature dependence of the average potential energy, (U), and mean adatom-surface energy, ( V ) , for a xenon monolayer. The experimental value for (U)is indicated by the squarel8and for ( V ) by the triangle.’6J7The labels A, B, C, and D refer to (U)for four models described in the text. The labels E and F refer to the old and new mean adatom-surface energies, (V)(see text).
This latter point is illustrated in Figure 2 where we display a correlation between the location of the gas-phase-pair potential minima R , and the distance uAC. It should be noted that for a Lennard-Jones potential R, = 21/6uAA,and this is also a fair approximation for realistic potentials.26 Also shown in Figure 2 are the experimental values22-24for ( Z ) ,the overlayer-graphite separation. With the exception of He, which is influenced by large zero-point energy effects,’ (2)= u On the basis of Figure 2, we adopted the value uXeC= 3.40 g a n d , for reasons that will become apparent below, we chose exec = 107 K. These parameters yield a long-range dispersion coefficient C3about 40% too large.4 However, in view of the fact that higher order dispersion coefficients are omitted from our model, this is perhaps a necessary result. The bold curve in Figure 1 shows V,(Z) for our choice of parameters. It should be noted that for our model the potential energy of xenon on the graphite surface is about 5% greater than suggested by low-pressure adsorption data on carbon b l a ~ k s . ~ ~ J ~ B. Adatom-Adatom Interactions. Interatomic interactions between rare-gas atoms are now well characteri~ed?~J~ but in the presence of a surface these interactions become modified. The effect on the long-range dispersion interactions for a pair of adatoms was discussed first by Sinanoglu and P i t ~ e r . In ~ ~addition, there is also the question of the role played by a surface field that can polarize the adatoms and hence induce an adatom-adatom repulsion.z8J9 In keeping with other recent investigators2,28we will ignore such surface field effects. Moreover, for the formulation of the dispersion forces, we follow the work of McLachlan, which utilizes a continuum model for the substrate and the notion of image forcesS3 In McLachlan’s theory the dispersion interaction for a pair of adatoms, separated by a distance R, take the form Udlsp(R) = -c,5/R6 - C S ~ / ( R ’+) ~ CS,(2 3 cos 20 + 3 cos 2 4 ) / 6 R 3 ( R q 3
+
(26) J. A. Barker in “Rare Gas Solids”, M. L. Klein and J. A. Venables, Eds., Academic Press, New York, 1976. (27) 0. Sinanoglu and K. S . Pitzer, J . Chem. Phys., 32, 1279 (1960). (28) G. L. Price and J. A. Venables, Surf. Sci., 59, 509 (1976); J. A. Venables and P. S.Schabes-Retchkiman, J. Phys. (Orsay, Fr.), 38, C4-105 I1 977) .,. \ - -
(29) D. H. Everett, Discuss. Faraday Sot., 40, 177 (1965)
The Journal of Physical Chemistry, Vol. 88, No. 7, 1984 1423
Xenon Physisorbed on the Graphite Basal Plane
2
Xe on g r a p h i t e
M.C.
M.C.
A
Xe on g r a p h i t e
= =Monolayer
=Monolayer =lst Layer of Bilayer =2nd Layer of Bilayer
Layer of Bilayer =2nd Layer of Bilayer
a=lst 0
0
0
0
0 0 0
00
0 00
00
00 00
do 0
oa
0
&
0 0 0
CB 0 0
ooo x O0 00 &?oE@Oo
x x x
do
0
0
n
0
c@ 0
x x x
0
0
x x
A
a
A
doa 0
20
80
80
0
20
80
T;oK Figure 4. Mean-square vibration amplitudes ( uz2) for xenon overlayer calculated from traditional Lennard-Jones adatomsurface and adatomadatom parameters.16 The full squares refer to the monolayer and the open squares and triangles to the upper and lower layers of the bilayer. The experimental results, which are normalized to the calculations at 30 K, are indicated by circles14and crosses.I1
Figure 5. Mean-squarevibration amplitudes ( uzz) for xenon overlayer calculated from revised adatom-surface parameters (exec= 107 K,uXec = 3.40A) and a Lennard-Jones Xe-Xe potential. The symbols have the same meaning as in Figure 4.
where the leading term is the usual London contribution and the remaining terms arise because of the presence of the s ~ b s t r a t e . ~ Here R'is the distance between_oneof ;he adatoms and the image of the other in the substrate; R and R' subtend angles 0 and 4 with respect to the surface. McLachlan pointed out3 that when two adatoms are far apart (R = R', 0 = $C = 0) on a surface with high dielectric constant, vdisp(R)= -2C6/3R6; Le., there can be a one-third reduction in the dispersion forces. The coefficients Csl and Cszthat enter into the McLachlan theory are now known for a number of adatom-substrate combination^,^ and hence a parameter-free evaluation is now possible.* For the case of xenon on graphite there is about a 16% reduction in the well depth for a pair of adatoms.2 For two atoms in the top layer of a bilayer the McLachlan terms are very small. However, when one atom is in the top layer and the other in the bottom layer (or vice versa), the McLachlan terms can be attractive. The maximum effect for a registered bilayer increases the pair interaction by 4%. The calculations to be described in the next section will utilize a realistic xenon potentialz5to examine the properties of registered monolayers and bilayers. Thus, as for the bulk solid, it is formally necessary to also include the three-body Axilrod-Teller interact i ~ n , 'in~ addition to the McLachlan terms. By direct summation we have confirmedzs that, for a monolayer, the Axilrod-Teller three-body energy is small (25 K), amounting to less than 1% of the total potential energy. The contribution of the McLachlan terms is 7 times larger than this. For a bilayer, the Axilrod-Teller energy is more important than in the monolayer and is now comparable in magnitude to the contribution from the McLachlan
111. Monte Carlo Calculations A . Motivation. We mentioned in the Introduction that the teinperature dependence of mean-square amplitudes of vibration ( uzz) for xenon overlayers have been derived from LEED data" and by using atomic hydrogen beams.14 In addition, the heat of transition between an epitaxial 2D crystal and the vapor phase, AH, has been measured by using a graphite single-crystal and Auger electron spectroscopy (AES).'* The value AH = 2768 f 50 K was reported for the temperature range 73-100 K; calorimetric datalg yield 2868 f 150 K. The heat of transition from monolayer to bilayer can be estimated from experimental data.'* In the range 55-62 K we estimate AHz = 1910 A 150 K, which is almost identical with AH for the bulk solid.z5 To a good approximation, AH can be related to the average potential energy ( U )of the 2D condensed phase, via the equation A H = RT - ( U ) . Thus, we derive ( U ) = -2782 A 50 K, at a mean temperature of 86 K, for the monolayer. Using the analysis of monolayerbilayer equilibria due to Bruch and Wei4 and ignoring spreading pressure effects, we estimate that the potential energy difference between the monolayer and the bilayer of Xe to be -1850 A 150 K at about 60 K. B. Outline of the Calculations. In order to study the properties of a monolayer, either 64 or 144 xenon atoms were initially disposed as an 8 X 8 or 12 X 12 array with the ( d 3 X d 3 ) R 3 0 structure. Oblique axes were used along with periodic boundary conditions to simulate an infinite system. For the bilayer, twice the number of particles were employed. Our calculations followed
T/K
terms, which are dominated by the contribution from the first layer.
1424
The Journal of Physical Chemistry, Vol. 88, No. 7, 1984
2
Klein et al.
z-D is t r ibution Function
Xe o n graphite
N =
Xe on graphite
M.C.
=Monolayer =lst Layer of Bilayer 0 =2nd Layer of Bilayer
128
T = 4 0 K fl
A
0
"1
0
0
0 0 0 0
00
0
0
do
s"
0
mo 8 0
H
0
8 8
X
o
EP0 00
k
5
x x x
n
n n
n
A 0
2
8
4
10
Z?A 0
20
T;z;
80
80
Figure 6. Mean-square vibration amplitudes ( uz2) for xenon overlayer calculated from revised adatomsurface parameters (exec = 107 K, uxec = 3.40 A) and a realistic XeXe p0tential.2~The symbols have the same meaning as in Figure 4.
standard canonical Monte Carlo procedures? Typically, runs were started at 20 K and, after about 5000 moves per particle, the temperature was increased, by 10 K, and a new calculation was started from the final configuration of the previous run. The density profile normal to the basal plane, g ( Z ) , was monitored throughout the run as was the average potential energy, ( U ) ,and the mean-square vibration amplitude, (u:) = ( ( Z - ( Z ) ) 2 ) .For the 8 X 8 system the Xe-Xe interactions were truncated at 14 A while for the 12 X 12 system a 21-A cutoff was employed. For the latter system the truncation error in ( U ) amounted to only about 1 K and hence was essentially negligible. With due allowance for this truncation effect, the results for ( U ) and ( uzz) were independent of system size.
IV. Results We have examined four basic models derived from different combinations of adatomsurface and adatomadatom interactions. The first is the Crowell-Steele model15 derived from Lennard= 226 K, ccc Jones potentials and traditional parametersI6 (cxxexe = 28 K, uXexe= 3.98 A, cCc = 3.4 A). The temperature dependence of the monolayer average potential energy ( U ) for this model (labeled A in Figure 3) is in poor agreement with the experiment. Also shown in Figure 3 is ( V ) ,the mean adatomsurface energy; again the Crowell-Steele model (labeled E) is in poor agreement with experiment. Elsewhere30 we have presented detailed comparisons of our results for this model with lattice dynamics calculations;"-13 hence, such comparisons will not be repeated here. The main finding was that anharmonic effects are important, particularly at higher temperatures.12 For completeness, Figure 4 shows our Monte Carlo results for ( uz2) for
Figure 7. The density profile g ( Z ) normal to the basal plane ( Z direction) for a xenon bilayer calculated from revised adatom-surface parameters (exec = 107 K, oxXec = 3.40 A) and a realistic Xe-Xe potentialaZ5
the Crowell-Steele model and compares these with beam datal4 (circles) and LEED datal' (crosses). Since the diffraction data only produce relative values, we have normalized the experimental data to the calculations at 30 K. As ndted p r e v i o ~ s l y the ,~~ Crowell-Steele m ~ d e l ' ~shows J ~ a temperature dependence that is too strong. Figure 5 shows the vibration amplitues for our second model, which employed our revised adatom-surface parameters (eXeC = 107 K, uxec = 3.4 A) but retained the same Lennard-Jones Xe-Xe potential. It can be seen that for this model the monolayer vibration amplitudes are in excellent agreement with experiment, as is ( U ) (labeled as B in Figure 3). However, Figure 5 shows that the bilayer vibration amplitudes increase too rapidly as the temperature is raised. Results for the vibration amplitudes of our third model are shown in Figure 5. This model, which retains the adatomsurface interaction of the second model, employs a realistic Xe-Xe potentialZ5but neither Axilrod-Teller nor McLachlan nonadditivity corrections. There is now excellent agreement with the diffraction data, but, due to the lack of repulsive nonadditive terms, the (U) (labeled C in Figure 3) is too attractive. Figure 7 shows a typical density profile g(Z) for this model. It should be noted that, in the case of a bilayer, the lower layer suffers a strong compressive effect from the upper layer. This in turn is responsible for the sharp first peak in g(Z)and the low vibration amplitudes indicated for the first layer of the bilayer in Figures 4-6. Our final model (whose ( U ) is labeled D in Figure 3) is the same as the third model except that it includes the McLachlan (30) S.F. OShea, Y.Ozaki, and M. L. Klein, Chem. Phys. Lett., 94,355 (1983).
J. Phys. Chem. 1984, 88, 1425-1436 terms in the Monte Carlo calculations, but not the Axilrod-Teller ones. The values of ( uz2) derived for our final model are virtually identical with those of Figure 6 and hence are not reproduced. It should be noted that this model gives not only an excellent account of the diffraction but also the value of (U)derived from the AES data.'* However, in keeping with the second and third models, which share the same adatomsurface potential, the calculated value of ( V ) (labeled F in Figure 4) is about 100 K larger in magnitude than is indicated for adssorptionon graphitized carbon black. l6,I7 At 60 K our Monte Carlo calculations yield a difference in potential energy between the bilayer and monolayer of about 1900 K for the first two models and 2170 K for the last two. The former value agrees rather nicely with the experimental value 1860 f 150 K that can be derived from AES data1*whereas the latter value is only barely consistent once allowance is made for the Axilrod-Teller terms.
V. Discussion The properties of physisorbed overlays aire determined by both adatomsurface and adatom-adatom interactions, and in the past such interactions have often been modeled1 with Lennard-Jones In the case of xenon we have shown that the traditional parameterization of the adatom-surface interactionI6 yields insufficient binding to the graphite surface and, if the diffraction data are taken at face value, adlatom vibrational amplitudes that are too large.30 Since within the framework of the Crowell-Steele model, the former result is directly proportional to exec and the latter inversely proportional to the same quantity: these deficiencies can be simultaneously rectified simply by increasing exec. However, while we have confirmed that such a procedure does indeed work, the resulting zero-coverage (isosteric) heat of adsorption is about 5% too large and the long-range interaction coefficient, C,, about 40% too large." This latter finding is reminiscent of the situation for gas phase, where the use of a Lennard-Jones (1 2-6) potential invariably leads to long-range
1425
coefficients, C,, that are too large.26 The time is thus ripe to seek a more rigorous and systematic approach to adatom-surface potentials, much as was done for the rare gases a decade ago.26 We note that the vibrational amplitudes measured for the bilayerI4 have been crucial in identifying the need for a realistic Xe-Xe potential rather than the traditional Lennard-Jones form. Once allowance is made for the many-body Axilrod-Teller and McLachlan contributions, a consistent model for both the monolayer and the bilayer appears to emerge. Finally, the use of a smaller value for uASis consistent with recent theoretical studies of Kr and Xe over layer^.^^ There it was found that the traditional Steele parameters16 lead to a surface potential that is too weakly modulated. The differences between the crosses and the minima of the curves in Figure 1 is a measure of this effect. As a direct consequence of using a smaller uAS,our revised parameters for the Crowell-Steele model yield a surface potential with the required modulation. Note Added in Proof. A surface adatom potential for Xe on graphite, with parameters similar to those suggested here, has been proposed recently.32 In addition, new isosteric heat measurements confirm the need for a somewhat deeper p ~ t e n t i a l . , ~ Acknowledgment. We thank Louis Bruch, Milton Cole, Tom Ellis, Sam Fain, Jim Phillips Giacinto Scoles, Bill Steele, and Bob Thomas for their interest and helpful comments. Jennifer Piper and Jim Morrison kindly provided details of their experiments prior to publication. This work was supported in part by NSERC and is issued as NRCC No. 22622. Registry No. Xenon, 7440-63-3;graphite, 7782-42-5. (31) R. J. Gooding, B. Joos, and B. Bergerson, Phys. Reu. E Condens. Matter, 27, 7669 (1983); B. Joos, B. Bergersen, and M. L. Klein, Phys. Reu. B: Condens. Matter, 28, 7219 (1983). (32) G. Vidali, M. W. Cole, and J. R. Klein, Phys. Reu. B: Condens. Matter, 28, 3064 (1983). (33) J. Piper and J. A. Morrison, Chem. Phys. Lett., 103, 323 (1984).
Silver Clusters in Rare Gas Matrices. Thermal and Photochemical Silver Atom Aggregation Reactions Steven A. Mitchell? arid Geoffrey A. Ozin* Lash Miller Chemistry .Laboratories, University of Toronto, Toronto, Ontario, Canada M5S 1Al (Received: June 6, 1983; In Final Form: August 10, 1983)
5p1, 2p 5s1, ?3 photoexcitation of silver atoms in rare gas matrices at 12 K promotes diffusion and aggregation of the silver atoms, resulting in the formation of discrete molecular silver clusters Ag, (where n = 2-6). In this study the kinetics of photoinduced silver atom dimerization in rare gas solids are investigated in detail. To determine the role of diffusion effects on the overall rate of this reaction, experiments have been conducted which explore the dependence on silver atom concentration, the sensitivity with respect to the nature, microscopic structure, and pretreatment of the matrix, the contribution of different silver atom trapping sites, and the consequences of silver concentration and light beam intensity inhomogeneities throughout the volume of the matrix sample. The kinetics of photoinduced silver atom dimerization are complex but the overall aggregation process is well described by a simple statistical cluster growth model. Relative molar extinction coefficients for Ag, Ag,, and Ag, are (determinedby using both photoinduced diffusion and silver concentration techniques and allow silver cluster distributions to be characterized by UV-visible absorption measurements. +-
Introduction A remarkable feature of the structure of metals is the role of the extended crystal lattice in determining the overall electronic the limit of very properties and stability of the bulk phase, small particle sizes or very thin films one may expect to find variations in certain of the metallic propertie;, as a result of surface Present address: Division of Chemistry, National Research Council of Canada, Ottawa, Canada K1A OR6.
0022-3654/84/2088-1425$01.50/0
effects arising from the imposition of finite dimensions. The manifestations of these effects in experimentally observable Properties of finely divided metals have been extensively studied, and a voluminous literature of experimental results and theoretical Metal clusters containing only a few metal models now (1) Proceedings of the Second International Meeting on the Small Particles and Inorganic Clusters, Surf. Sci., 106 (1981). (2) Faraday Symposia of the Chemical Society No. 14, Diatomic Metals and Metallic Clusters, 1980.
0 1984 American Chemical Society
