摘要:

Date 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 1980 1981 1981 1982 1982 1983 1983 I984 1984 1985 FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Subject Inelastic Collisions of Atoms and Simple Molecules High Resolution Nulcear Magnetic Resonance The Structure of Electronically Excited Species in the Gas Phase Fundamental Processes in Radiation Chemistry Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-eff ects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion- Ion and Ion-Solvent Interactions Colloid Stability Structure and Motion- in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids Photoelectrochemistry High Resolution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Electron and Proton Transfer Intramolecular Kinetics Concentrated Colloidal Dispersions Interfacial Kinetics in Solution Radicals in Condensed Phases Polymer Liquid Crystals Oxidation 3 13 Volume 33* 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 .58 59 60 61 62 63 64 65* 66 67 68 69 70 71 72 73 74 75 76 77 78 79 * Not available; for current information on prices, etc., of available volumes, please contact the Marketing Oficer, Royal Society of Chemistry, Burlington House, London W1 V OBN stating whether or not you are a member of the Society.Date 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 1980 1981 1981 1982 1982 1983 1983 I984 1984 1985 FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY Subject Inelastic Collisions of Atoms and Simple Molecules High Resolution Nulcear Magnetic Resonance The Structure of Electronically Excited Species in the Gas Phase Fundamental Processes in Radiation Chemistry Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-eff ects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion- Ion and Ion-Solvent Interactions Colloid Stability Structure and Motion- in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids Photoelectrochemistry High Resolution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Electron and Proton Transfer Intramolecular Kinetics Concentrated Colloidal Dispersions Interfacial Kinetics in Solution Radicals in Condensed Phases Polymer Liquid Crystals Oxidation 3 13 Volume 33* 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 .58 59 60 61 62 63 64 65* 66 67 68 69 70 71 72 73 74 75 76 77 78 79 * Not available; for current information on prices, etc., of available volumes, please contact the Marketing Oficer, Royal Society of Chemistry, Burlington House, London W1 V OBN stating whether or not you are a member of the Society.

ISSN:0301-7249    

DOI:10.1039/DC98580FX001

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1985, 80, 7-15 Twenty-second Spiers Memorial Lecture The Gas-Solid Interface BY JAMES A. MORRISON McMaster University, Hamilton, Ontario, Canada Received 20th September, 1985 It is a sobering and humbling experience to be asked to give this 22nd Spiers Lecture in memory of the first secretary of the Faraday Society. The high standard which has been set by preceding Lecturers is nothing short of intimidating. Thus, Mr President, I should like at the outset to express my deep gratitude for the confidence which has been expressed in me by the Council of the Faraday Division. My involvement with the former Faraday Society began at an early stage in my career when the Society held Discussion No. 14 in Canada in September, 1952. Several of us were ‘requisitioned’ by the late President of the National Research Council of Canada, Dr E.W. R. Steacie, to look after the forty-five members of the Society who came from Europe to attend the Discussion, and who visited a number of Canadian universities and industries. I believe that this was the first occasion when the Society held one of its meetings outside of the United Kingdom. The expedition was supported by a grant of S250 from the Royal Society, which was a generous offering for the time! The main reason for the Discussion being held in Canada will not appear in any of the records. It was, in brief, that Dr Steacie wanted Canadian societies to aspire to higher standards for their scientific meetings, and he invited the Faraday Society to set an example. There is much evidence that the strategy worked and, moreover, a tradition was established. We are now beginning the fifth Faraday Discussion to be staged in Canada. I hope you will not find it too peculiar if I approach the subject of the gas-solid interface from a rather distant and superficially unrelated point.One of my col- leagues in the Department of Music, William Wallace, who is a composer, used to have lunch regularly with a group from the Department of Physics. I was not privy to the conversations which were held over lunch, but I should like to describe one of their consequences for you briefly. At some point, Wallace was introduced to the idea of random number selection and was challenged to use it to compose a piece of music. He accepted the challenge and produced a composition with the title Ancaster Serenade, named after the village where I live on the western edge of Hamilton.In constructing his composition, he assigned notes to particular digits, e.g. C to 1, D to 2, E to 3 etc., and then drew numbers at random from the Ancaster section of the local telephone directory. The digit 9 was taken to be wild and could be used to satisfy any needs of harmony. All of the numbers in the Ancaster section have seven digits and. are of the form: 648-abcd. Thus, a theme or motif was automatically established by the recurring sequence 648. When the music received its first public performance, residents of Ancaster who were in the audience were advised to listen carefully because their numbers might come up. 78 SPIERS MEMORIAL LECTURE Table 1.Preceding related Discussions no. date topic 40 1965 Intermolecular Forces 52 1971 Surface Chemistry of Oxides 55 1973 Molecular Beam Scattering 58 1974 Photoeffects in Adsorbed Species 59 1975 Physical Adsorption in Condensed Phases 66 1978 Structure and Motion in Molecular Liquids What has this odd tale got to do with a discussion of physical interactions and energy exchange at the gas-solid interface? We shall soon see that there is a repetitive theme: the accurate prediction of adsorptive behaviour depends critically on a detailed knowledge of adsorbate-substrate and adsorbate-adsorbate interaction potentials. Professor Steele may be excused if he develops a feeling of d&h uu. His fine and useful book' of ten years ago is constructed around the same theme.BACKGROUND The Faraday Society, and now the Faraday Division of the Royal Society of Chemistry, has held a series of Discussion on surfaces and intermolecular forces over the past twenty years. Their titles and dates are summarized in table 1. Several of the contributors to the present Discussion participated in one or more of the earlier ones. In his Spiers Memorial Lecture at Discussion No. 40, Professor H. C. Longuet-Higgins2 presented a thoughtful account of the fundamental origin of intermolecular interactions in terms of basic forces. Several of the accompanying papers were concerned with the main themes of the present Discussion: physical interactions and energy exchange at the gas-solid interface. For instance, Professor Douglas Everett3 dealt with the determination of the second virial coefficient for the two-dimensional adsorbed gas and with the restriction it placed on the form of the interaction potential between two adsorbed molecules.The data which were analysed (for noble gases and CH, on graphitized carbon black) bear a haunting relation to our consideration of the structure of overlayers of simple molecules, their mobility and the scattering of atomic and molecular beams from them as they are held in the vicinity of the surface of graphite. Does this mean that our understand- ing of these simple adsorbing systems has advanced rather little over a period of twenty years? The answer to that question is unequivocally no, as the papers to follow will clearly demonstrate. In the first place, powerful new experimental techniques now give us exquisitely detailed information about the structure of overlayers and the dynamics of adsorption at solid surfaces.Quite generally, experiments with beams and jets are having a profound effect on many areas of physical chemistry/chemical physics. Surface studies are among those being greatly affected. In the second place, we can deal successfully theoretically with interactions between spherical atoms and we know precisely the form of long-range interactions for more complex particles. However, as will come out in the course of the Discussion, it is still troublesome for us to deal adequately with short-range interac- tions between molecules and a surface. To complete these general introductory remarks, I should like to draw your attention to the founding of the Faraday Society in 1903 'to promote the study of electrochemistry, electrometallurgy, chemical physics, metallography and kindredJ. A.MORRISON 9 Table 2. Approximate energies (in kJ mol-’) for the krypton-graphite system energy Kr-graphite 11.5 Kr-Kr 1.7 graphite corrugation 0.3 subjects’. That, incidentally, could easily be a description of much of modern materials research. The tradition it established and which carries on is to encourage multidisciplinary studies. Thus, as a group, we are physicists and chemists, theorists and experimentalists who share a deep interest in the gas-solid interface. To affirm the merit of this kind of multidisciplinary interaction, we need only think of how rapidly the innovative theory of Kosterlitz and Thouless4 on two-dimensional melting (the purest of pure physics) has led us to accept that phase diagrams for physisorbed layers can be extraordinarily complex. While the subject of phase transitions in adsorbed layers is the theme of the next Faraday Symposium to be held in December 1985, we shall be discussing here the equilibrium structure of certain overlayers and the consequences of that on the scattering of atomic beams.Throughout, we shall be conscious of the pivotal role now being played by molecular simulations in mediating the results of theory and experiment. INTERACTIONS At the beginning of a closer look at the substance of this Discussion, we should note the relative magnitudes of the interaction energies for our two-dimensional systems.To take the particular example of the krypton-graphite system, approximate values are listed in table 2. Here, the value given for the Kr-graphite interaction corresponds to a surface coverage of about a monolayer. It would be larger for a single atom on a bare surface. The Kr-Kr interaction is estimated as the heat of fusion. In the adsorbing system, a balance of interactions is struck which, as we now know, is often delicate and leads to subtle phase changes, partial wetting and other fascinating phenomena. The variation of the interaction energy with the amount adsorbed can be consider- able even in simple cases such as in the one illustrated in fig. 1. Here, the isosteric heat of adsorption of xenon on graphite, as determined ~alorimetrically,~ is plotted against the amount adsorbed, n,. These results were used to refine the well depth for a xenon-carbon interaction potential.6 The graphite surface is obviously not perfectly homogeneous (regularly periodic) as the higher heats of adsorption at the lowest surface coverages indicate.Nevertheless, an accurate value of the interaction energy of a xenon atom with a carbon surface can be obtained by extrapolation to n,=0, as is indicated by the dashed line in fig. 1. The contribution of the Xe-Xe interactions is given by the excess of qst over that for n , = 0 . The monolayer is marked by the abrupt fall in qst at higher surface coverages. Had the experiments been carried far enough, qst would have fallen to approximately the heat of vaporiz- ation of liquid xenon: 12.5 kJ mol-’.This example is an uncomplicated one from the points of view of both theory and experiment. An encouraging contribution to this Discussion takes us a step further. Talbot et u1.’ have investigated N2 absorbed on graphite by means of10 SPIERS MEMORIAL LECTURE 24 22 - I d : 20 \ c": 18 16 0 0.5 1.0 1.5 2.0 2.5 n,/ lo-) moI Fig. 1. Isosteric heat of adsorption of xenon on Grafoil at T = 195.5 K5 (with permission of Chemical Physics Letters). molecular dynamics simulation with Lennard-Jones-type potentials for both molecule-surface and molecule-molecule interactions. A gratifying degree of agree- ment is achieved with the experimental values of the heat of adsorption* over a range of surface coverages less than a monolayer. We shall see soon that measure- ments of heats of adsorption (and of adsorption isotherms) can possibly give us more than just numbers for theory to shoot at.In particular, effects observed in the course of the measurements can alert us to subtleties of the balance of interactions. The first three contributions to the programme of the Discussion are concerned with light-atom scattering from surfaces. Danielson et 0 1 . ~ have investigated a prototypic system with a minimum number of electrons from which the scattering of the He atoms takes place. By using very low-energy beams, they are able to make a close comparison with a semiempirical pairwise potential which accounts for nearly all of the observations. By contrast, the attempt by Jonsson and Weare" to refine elastic scattering from physisorbed overlayers ends on an uncertain note. The logical introduction of a long-range triple-dipole correction for non-additive effects seems to worsen the correlation between theory and experiment.We are thus given both good news and bad news about our progress in evolving satisfactory potentials. It will be surprising indeed if the role of many-body interactions does not become a strong element in the scientific debate to follow. MOBILITY OF ADSORBED LAYERS AND WETTING OF SOLID SURFACES Let us now shift our attention to other aspects of physisorption which give us bits of information to be used for refining interaction potentials. The pioneering studies of Thorny, Duval and their collaborators" delineated phase diagrams for sub-monolayer physisorbed films on graphite and stimulated theoretical efforts to characterise the two-dimensional liquid-vapour critical point.One such calculation is described in a contribution by Klein and Cole to this Discussion,'* in which potentials of the Lennard-Jones form are used with modifications such as theJ. A. MORRISON 11 introduction of a triple-dipole three-body contribution. The essential result is that the reduced liquid-vapour critical temperature for two dimensions should be close to 0.50 for some rare gases and light methane. The agreement for the rare gases is satisfactory, but TZ for adsorbed CH,, as estimated from heat-capacity experi- m e n t ~ , ~ ~ is appreciably smaller than calculated. Some recent measurements of heats of adsorption of CH, on graphite indicate', either that unexpected clustering of adsorbed molecules is taking place at low surface coverages or that the two-phase critical temperature is higher than has been estimated.Diffusion of atoms or molecules over a solid surface must obviously depend upon the corrugation of the surface, and this is either the main or the subsidiary concern of several contributions to the Discussion. The experimental approaches, e.g. quasielastic neutron scattering or diffraction of atomic or molecular beams, yield well defined results, but their analysis to provide quantitative descriptions of the interaction potentials is proving to be difficult to accomplish. Here, the repulsive part of the potential is the crucial element, and we have no general theory on which to build its description, as we have for long-range attractive interaction.This is perhaps an appropriate place to digress briefly to consider physisorbed multilayers and the attempts which are being made to estimate their stability in relation to that of the bulk solid adsorbate. The point at issue is whether adsorption on well defined surfaces of graphite or simple metals proceeds relentlessly with increasing pressure unless it is limited by geometric factors such as a finite pore volume of a substrate. In other words, are the surfaces always wetted completely by the adsorbed layers? It is now known that several systems display partial or incomplete wetting below a characteristic wetting temperature, and one of them, ethylene-graphite, has been investigated exhaustively by X-ray and neutron diff rac- tion.15 The adsorption isotherms for this system show16 layer-by-layer adsorption up to the equivalent of about three layers. Beyond that, the vapour pressure becomes that of bulk solid ethylene.Recently, similar types of adsorption isotherms have been observed17 for the CH,-graphite system. The excess surface density of the adsorbate seems to have a temperature dependence of the form ( Tw- T)-' with /3 = and the wetting tem- perature Tw = 75.5 K, which is much below the melting temperature of bulk CH4 (90.7 K). Bruch and Nil8 report calculations of the stability of trilayers of rare gases on graphite and the ( 1 11) face of Ag, and thereby illustrate the delicacy of the balance of the different interaction energies involved.Several general treatments of the phenomenon have been a t t e m ~ t e d , ' ~ - ~ ~ but the latest is again that interaction potentials need further refinement before reliable predictions of wetting behaviour can be made. STRUCTURES OF OVERLAYERS Elements of structure determination are contained in several of the contributions which deal with the sensitive and elegant experimental methods that are now available to us: atom and molecule scattering and diffraction, and neutron scattering. To them, we can add LEED and X-ray scattering. Particularly interesting experiments and calculations have been performed on overlayers of the simple diatomic molecules N2 and CO adsorbed on graphite. Hoydoo You and Fain2, have uncovered evidence for the existence of a triangular incommensurate structure for N2 under compression.Existing models for the system25 predict a pinwheel structure, and so the new results provide a severe test of the assumed N2-N2 and N2-C interaction potentials.12 SPIERS MEMORIAL LECTURE 0 0.1 0.2 0.3 0.4 0.5 P I Po Fig. 2. Adsorption/desorption of CH4 on graphite at T = 84.5 K: (1) adsorption, (2) desorp- tion from 4.8 x mol, (4) desorption from 17.4 x mol, (3) desorption from 7.4 x mol and (5) desorption from 17.8 x mol. Refined optical spectra of .heteronuclear diatomics adsorbed on oxides and halides obtained by Plater0 et u Z . ~ ~ also yield information about the structure of adlayers and of lateral interactions of the dipole-dipole type. However, for one system at least (CO on NiO), the interaction with the solid surface is rather large and may lie beyond the range commonly associated with physisorption. Both structure and dynamics enter in the studies by Gibson and Sibener27 of the scattering of helium from overlayers of various thicknesses for rare gases adsorbed on Ag( 11 1).It is interesting and satisfying that the experimental dispersion curves can be accounted for satisfactorily, as Cardini et ~ 1 . ~ ~ have shown, through lattice dynamics and computer-simulation calculations using reasonable Morse and Lennard-Jones potentials. Despite this success, it is perhaps not inappropriate to raise a question about the reversibility of multilayer adsorption on these well defined surfaces which we are discussing. We were much surprised recently to observe an odd type of hysteresis in the adsorption of CH4 on gra?hite,’4”7 and some results are illustrated in fig.2. The monolayer capacity for the particular system is ca. 2.5 x lop3 mol, and so the segment illustrated corresponds to surface coverages between 0.8 and 2.2 layers. The remarkable features are (i) that the magnitude of the hysteresis depends uponJ. A. MORRISON 13 18 16 e I d z 3 14 -2 c;: 12 IC Fig. 3. Isosteric heat of adsorption of CH4 on Grafoil MAT at T = 84.5 K for low surface coverages. the amount adsorbed before desorption, (ii) that there are no boundary curves such as are found for adsorption/desorption in porous and (iii) that the hysteresis disappears abruptly at around P / Po = 0.2. It is very unlikely that the hysteresis is caused by experimental pathology because similar effects have been observed at other temperatures and for another system (krypton-graphite). Moreover, hysteresis is detected in the dependence of qst upon the amount adsorbed for a surface coverage greater than the equivalent of a monolayer. Fig.3 shows qst for CH, on a particular type of exfoliated graphite at T = 84.5 K and, except for the 'spike', the form is similar to that displayed for Xe-graphite in fig. 1. The 'spike' marks the region where fluid and incommensurate solid surface phases coexist. The behaviour of qst at higher surface coverages is illustrated in fig. 4. We see that, after desorption from a high surface coverage, the variation in qst is the same, but shifted to higher coverages. Some additional details can be found elsewhere.It is hard to reach a conclusion from these experiments other than that physi- sorbed multilayers get irreversibly compressed during the adsorption process. Neutron diffraction studies of several years ago for the CD,-graphite system3' showed that the lattice parameter for the adsorbate decreased in the region immedi- ately above monolayer coverage. It would be valuable at this stage if such studies could be extended to much thicker overlayers. Finally, extraordinary sensitivity to structure is demonstrated in the experiments of Poelsema and Comsa3' which capitalise on the large cross-section of isolated atoms and surface imperfections for diffuse scattering of low-energy helium beams. It is especially satisfying to see that the basis of the phenomenon is being investigated theoretically, as described in the contribution by L ~ u .~ ~14 SPIERS MEMORIAL LECTURE Fig. 4. Isosteric heat of adsorption of CH4 on Grafoil MAT at T = 84.5 K for surface coverages greater than the equivalent of a monolayer: 0, initial series; 0, after desorption from n, = 15 x mol. ENERGY EXCHANGE I have left to last any mention of one of the main themes of the Discussion, viz. energy exchange at the gas-solid interface. In part, this is because anything that I might try to contribute to the subject would certainly be overwhelmed quickly by the elegant contributions of Kreuzer, Barker, Auerbach and Tully . Here, especially in the work of Barker and A ~ e r b a c h , ~ ~ we begin to see some of the microscopic detail of the dynamics of collisions between gas atoms or molecules and solid surfaces.It is again very satisfying that these theoretical developments are proceed- ing in close connection with refined experiments. Those who would like to assimilate background in the topic could hardly do better than to consult a recently published review.34 I should like to express appreciation to Dr A. Inaba and Dr M. L. Klein for sharing many happy hours in discussing gas-solid interactions. W. A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon Press, Oxford, 1974). H. C. Longuet-Higgins, Discuss. Furuduy Soc., 1965, 40, 7. D. H. Everett, Discuss. Furuduy Soc., 1965, 40, 177. J. M. Kosterlitz and D. J. Thouless, J. Phys. C , 1973, 6, 1181. J.Piper and J. A. Morrison, Chem. Phys. Lett., 1984, 103, 323. S. F. O’Shea, Y . Ozaki and M. L. Klein, Chem. Phys. Lett., 1983, 94, 355. J. Talbot, D. J. Tildesley and W. A. Steele, Furuduy Discuss. Chem. Soc., 1985, 80,91. * J. Piper, J. A. Morrison, C. Peters and Y. Ozaki, J. Chem. SOC., Furuduy Trans. I , 1983,79, 2863. L. Danielson, J-C. Ruiz, C. Schwartz, G. Scoles and J. M. Hutson, Furuduy Discuss. Chem. Soc., 1985, 80, 47. For example, see A. Thorny, X. Duval and J. Reginer, Surf: Sci. Rep., 1981, 1, 1. A. D. Migone, Z . R. Li and M. H. W. Chan, Phys. Rev. Lett., 1984, 53, 810. lo H. Jonsson and J. H. Weare, Furuduy Discuss. Chem. Soc., 1985, 80, 29. *’ J. R. Klein and M. W. Cole, Furuduy Discuss. Chem. Soc., 1985, 80,71. 11 13J. A. MORRISON 15 A. Inaba, Y.Koga and J. A. Morrison, Faraday Symp. Chem. SOC., 1985, 20, in press. See, for example, M. Sutton, S. G. J. Mochrie and R. J. Birgeneau, Phys. Rev. Lett., 1983, 51, 407; S. G. J. Mochrie, M. Sutton, R. J. Birgeneau, D. E. Moncton and P. M. Horn, Phys. Rev. B, 1984, 30, 263. A. Inaba and J. A. Morrison, Chem. Phys. Lett., to be published. L. W. Bruch and X-Z. Ni, Faraday Discuss. Chem. SOC., 1985, 80, 217. 14 15 l6 J. Menaucourt, A. Thomy and X. Duval, J. Phys. (Paris), 1977, 38, C4-195. 17 18 l 9 D. E. Sullivan, Phys. Rev. B, 1979, 20, 3991. 2o R. Pandit, M. Schick and M. Wortis, Phys. Rev. B, 1982, 26, 5112. 21 R. Pandit and M. E. Fisher, Phys. Rev. Lett., 1983, 51, 1772. 22 M. P. Nightingale, W. F. Saam and M. Schick, Phys. Rev. Lett., 1983, 51, 1275; Phys. Rev. B, 23 S. Dietrich and M. Schick, Phys. Rev. B, 1985, 31, 4718. 25 C. Peters and M. L. Klein, Mol. Phys., 1985, 54, 895. 26 E. E. Platero, D. Scarano, G. Spoto and A. Zecchina, Faraday Discuss. Chem. SOC., 1985,80,183. 27 K. D. Gibson and S. J. Sibener, Faraday Discuss. Chem. Soc., 1985, 80, 203. 28 G. Cardini, S. F. O’Shea and M. L. Klein, Faraday Discuss. Chem. SOC., 1985, SQ, 227. 29 D. H. Everett, in The Gas-Solid Interface, ed. E. A. Flood (Marcel Dekker, New York, 1967), 30 P. Vora, S. K. Sinha and R. K. Crawford, Phys. Rev. Lett., 1979, 43, 704. 31 B. Poelsema and G. Comsa, Faraday Discuss. Chem. SOC., 1985,80, 247. 32 W-K. Liu, Faraday Discuss. Chem. SOC., 1985, 80, 257. 33 J. A. Barker and D. Auerbach, Faraday Discuss, Chem. SOC., 1985, 80, 277. 1984, 30, 3830. H. You and S. C. Fain, Jr., Faraday Discuss. Chem. SOC., 1985, 80, 159. 24 p. 1055. J. A. Barker and D. Auerbach, Surf: Sci. Rep., 1984, 4, 1. 3 4

ISSN:0301-7249    

DOI:10.1039/DC9858000007

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1985, 80, 17-27 Helium Diffraction as a Probe of Surface Topography BY MARK J. CARDILLO AT&T Bell Laboratories, Murray Hill, New Jersey 07974, U.S.A. Received 22nd April, 1985 A brief commentary is presented on the strengths and limitations of various probes of surface structure presently in use. The sensitivity of He diffraction to structure is examined in more detail. The requirements for an open lattice, compared with the scattering cross- section of He, is displayed graphically based on analytical expressions for the He surface potential. The sensitivity argument is highlighted with experimental data as examples. Recent He diffraction studies of open semiconductor surfaces are then specifically discussed to demonstrate the nature of the topographical information which can be extracted.A knowledge of the structure of surfaces is essential to an understanding of all surface phenomena both static and dynamic, yet the structures of clean and ad- sorbate-covered surfaces continue to remain distant from an experimental technique which is of widespread applicability and free from major uncertainties. Con- sequently there is continued research and progress with all of the techniques of surface structural analysis regardless of their sometimes severe limitations. Recently dramatic progress has been made in the development of the scanning tunneling microscope' (STM) as a probe of surface structure. Direct images of the topographies of the often-studied Si( 11 1)7 x 7, Au( 110)2 x 1 and Ge( 11 l)c(2 x 8) surfaces have been reported.'>2 Although the details of the surface structure cannot be extracted from these measurements, the real-space topographies are immediately evident.In view of this exciting progress it is appropriate to review the capabilities and limitations of the various probes of surface structure. In this paper some aspects of the capabilities of He diffraction will be discussed including a qualitative assess- ment of the sensitivity of He diffraction to surface structure. The application of He diffraction to open semiconductor surfaces is then illustrated to demonstrate the capability of this technique to yield important structural information in the regime where it is sensitive. Only a few low-Miller-index surfaces have structures which are close to the ideal termination of the bulk.Most semiconductor surfaces and some metal surfaces, such as the (1 10) faces of noble metals, reconstruct, Le. undergo dramatic changes in surface atomic positions and thus bonding in order to lower the surface free energy. Many surfaces, such as the (001) faces of W and Mo, undergo further structural perturbations driven by weak electronic instabilities. In all of these surfaces strain plays an important role in determining the equilibrium periodicity and in the details of the surface structure. Atomic and molecular adsorption on surfaces is a problem of additional complexity, and for the most part it is only for simple atomic adsorbates that structures have been clearly resolved. Many approaches have been applied to structures of surface^.^ Important pro- gress continues to be made in the application of surface extended X-ray adsorption fine structure (SEXAFS) and high-energy ion scattering to complement the widely applied technique of low-energy electron diffraction (LEED).X-ray diffraction at glancing incidence and photoelectron diffraction continue to be developed as probes 1718 He DIFFRACTION AS A SURFACE PROBE of surface structure along with surface vibrational spectroscopy and stimulated- desorption ion angular distiibutions. The latter are primarily of value as measures of molecular configurations. Much of the problem for scattering probes is the small total number of atomic species which constitute a sample. For typical areas probed there are only 10'3-1014 atoms of the surface which are sampled.This is 105-106 less than typically probed by X-ray or neutron diffraction in the bulk. Thus scatterers with large interaction cross-sections are required, such as low-energy electrons. This strong interaction, which provides measurable signal levels, eliminates the great simplification of scattering-data inversion permitted for interference spectra from weak or 'single scattering' interactions. In general, model calculations based on complex theory must be fitted to scattering data, so that uniqueness and sensitivity are greater and more complex concerns. A complication for electron-based structural determina- tions is the penetration of the probe into several layers of the sample. For many surfaces subsurface strain or relaxation prevents the structure from being periodic in the surface normal direction until up to five atomic layers deep, adding a very large number of coordinates to the problem.Each experimental technique presently in use has specific strengths or advantages, which suggests that a consensus analysis may be the resolution to most problems. For example, for the technique of SEXAFS some of the above complications are avoided as the short-range interference scattering between unlike species, one of which resides only at the surface, is monitored. Although in SEXAFS considerable precision is claimed, experiments are hampered by a small number of observable interference oscillations, raising the question of uniqueness for all but most simple or highly symmetric geometries.High-energy ion scattering (channeling) enjoys the significant advantage of a simple, classical mechanical interpretation. The number of atoms per string is reasonably straightforward to obtain for several crystal directions, yielding strict requirements for possible structural models in a manner similar to triangulation. However, the numbers obtained are averages over the unit cell, which for some periodicities can involve a large number of atoms. Thus ion scattering serves more often to test rigorously structural models than to derive them. Glancing incidence X-ray diffraction is also capable of precise coordinates of surface species, but only in the two dimensions of the surface plane. The scattering data contain restricted information about normal di,splacements.The scanning tunneling microscope images tunneling current ca. 4-6 A above the surface. The contours of constant tunneling current reveal many detailed features of the topography of the underlying surface. However, the interpretation of these features in terms of actual atomic configurations is not straightforward. The restrictions and sensitivity of He diffraction are difficult to formulate in a simple manner. At the He atom energies typically emplcyed (10-100meV) large-angle atomic scattering cross-sections are larger than 10 A2, greater than the area per atom on most surfaces. Thus He diffraction is completely non-penetrating, which gives rise to a high degree of surface sensitivity. Small deviations from complete order are easily discerned with He diffraction.For high surface atomic densities the He scattering pattern is completely insensitive to the second layer, which immediately suggests severe restrictions in the structural information which can be derived. It is clear simply from the scattering cross-section that structural sensitivity for He diffraction should be greatest wgen the lattice is relatively open, i.e. for top-layer nearest-neighbour spacings of 4 A or larger. In the following sections the structural sensitivity of He diffraction is qualitatively addressed based on recently developed analytical expressions for the relationship between the principal features of the scattering potential and surface geometry.M. J. CARDILLO 19 These expressions give rise to a graphical interpretation which is highlighted by comparison with experimental data.The specific cases of the open semiconductor surfaces GaAs(llO), Si(lOO), and Ge(100) are then discussed to provide a direct illustration of the sensitivity of He diffraction to surface order and topography. THE HELIUM-SURFACE POTENTIAL The helium-surface potential may be qualitatively characterized by a weak long-range attractive region which attains a typical depth of 5-2cmeV, and which changes to a repulsive corrugated potential at distances of 2-3 A measured from the outermost plane of nuclei. The structural information arises predominantly from this corrugated repulsion, as it dominates the scattered angular distribution. Thus for simplicity the attractive part of the helium-surface potential may be ignored in this structural-sensitivity analysis.An important and simple result has now been established for the repulsive component of the helium-surface potential. It has been that the repulsive energy of an He atom immersed in a dilute electron density, or interacting with another rare-gas atom of local density p, is linear in p. Specifically, E = Ap, where A = 500 eV au3. Thus thermal-energy He atoms probe the surface potential only up to distances at which the target charge density is of the order of au, and contours of constant charge density are approximately isopotential contours. Charge profiles at surfaces believed to be accurate at very low densities were first calculated and compared with He diffraction experiments using the linear augmented plane-wave (LAPW) approach.' These self-consistent calculations com- pared well with the experimentally derived potential parameters ; the calculations confirmed the tilt angle of the surface bond in the GaAs( 110) surface reconstruction, and identified the appropriate binding site and probable bond length of H on the Ni( 110) surface.However, a definite discrepancy with the small corrugation of the clean Ni(ll0) and Cu(ll0) surfaces has been noted. The discrepancies are for close-packed metal surfaces with small corrugations, and the differences are also numerically small (but significant). There is essentially no structural information content in the diffraction data because of the small corrugations. Regardless of the source of this discrepancy, for systems with large corrugations in the scattering potential, it is likely that absolute LAPW charge density errors will not be significant.The shape of the charge density in these cases is predominantly determined by the strong electron potential in the region near the surface atoms, and only weakly affected by the shape of that potential in the low density region. Thus LAPW calculations can be taken as a standard which may be used to assess simpler schemes, such as the superposition of atomic charge densities. Atomic charge superposition has been tested and found acceptable (within ca. 10%) for the surfaces of homogeneous materials. However, for surfaces with chemical heterogeneity or where charge-transfer occurs, errors in generating scattering potentials can be very large. The specific example of semiconductor surfaces, where superposition does not work, is discussed below.Similarly charge transfer between a homogeneous absorbate and the substrate can also lead to significant errors in structural conclusions. STRUCTURAL SENSITIVITY Despite the restrictions on the application of atomic charge superposition it can be used to derive an analytic expression permitting qualitative conclusions about20 He DIFFRACTION AS A SURFACE PROBE structural sensitivity. Following Tersoff et aL7 the surface charge density is approxi- mated by the superposition of spherical atom charge densities: p ( r ) = c d4r - R ) 4 ( r ) = 4% [exp ( - K r ) l / K r ( 1 ) ( 2 ) R where R is a surface lattice vector and C$( r ) is a convenient analytic approximation which models the atom charge density far from the nucleus.Since C$( r ) and hence p( r ) are solutions of V2 - K~ = 0, eqn (1) can be expanded for z > 0 as p ( r ) = C aG exp [ - ( K ~ + G2)1'2z] cos ( G r ) (3 1 G where the G are the (two-dimensional) surface reciprocal lattice vectors. A two- dimensional rectangular lattice with one atom per unit cell is assumed for simplicity. The effect of the second layer can be included approximately by adding its G = 0 component to eqn (3). This extra contribution is p2W = n2ao exp [ - K W + 4 1 d and n2 being the interlayer distance and the number second layer. The coefficients aG are evaluated as a, = ~ T C $ O ( ~ ~ O K ~ ) - ~ aG = 2ao( 1 + G ~ / K ~ ) - ' / ~ (GZO (4) of atoms per cell in the where flo is the unit cell area.In general, the corrugation is greatest along the direction of the smallest reciprocal lattice vector G,, which permits a simplification to one dimension. For the purpose of this paper the potential is assumed to be described adequately by the surface of constant potential ( i e . constant charge density), which gives the classical turning point of the He atoms. This surface zc(x, y ) is implicitly defined by p ( r ) = p,. Ap, = Ei, where Ei is the incident He kinetic energy and pc is the charge density which, in conjunction with the linear charge-potential relationship, gives the He classical turning point. The charge perpendicular to G, is thus averaged and the extrema of the surface p ( r ) = p, are found, which occur where cos (G,x) = *l. Labelling the extremal z as z+ and z- and combining eqn (3)-(5) we have and qn = ( K~ + n2G:)1/2 - K .The peak-to-trough corrugation is denoted as A = z+ - z-. A is taken as the most significant indicator of the surface structure, although it is desirable experimentally to determine 3 or 4 Fourier coefficients with an accuracy of 610%. In fig. 1 the solid lines represent the values of A, as the lattice spacing is increased for a series of second layer heights, d, taken from eqn ( 6 ) . The value K = 1.4 au-', appropriate for Au, was used for these curves, and pc was set at au. For latticeM. J. CARDILLO 21 Fig. 1. Peak-to-trough corrugation amplitude, A, of the surface charge density at p = au plotted against the (001) surface atomic spacing Q for an f.c.c.(llO) surface.Each line corresponds to a different interlayer spacing d as indicated. The points are experimental values for the principal corrugation of a variety of systems plotted against the surface atomic spacing in the direction of that corrugation. spacings <ca. 5 the corrugation is insensitive to the height of the second layer, i.e. the surfaces have become close packed. He diffraction in this regime is not structurally informative, as it is sensitive only to the surface lattice constant, which is in general a known quantity. Studies in this area have been useful primarily as part o[ the development of an understanding of the helium-surface potential. Above a =r 6 A the curves are sensitive to the second-layer spacing, a manifestation of the surfaces being 'open' compared with the helium-surface large-angle scattering cross-section.These results are in general insensitive to 4 and Z,, and K does not vary strongly with the elements. Thus these plots may be considered to be qualita- tively general. Note that the sensitivity to d at any lattice spaFing can be visualized by taking a vertical cut at any value of a. Below a 5 A there is essentially no difference in corrugation between the second layer 1 A below the surface and the surface isolated in the vacuum, i.e. d + a. Experimental points have been plotted on the curves of fig. 1. These points represent an estimate of the experimental peak-trough corrugation in the direction of maximum corrugation for several surfaces taken from the literature and are plotted for illustration.The points have error bars which are a rough estimate of the experimental uncertainty, including the scattering theory used to derive the values of A. The sequence of metals Ni,* C U , ~ Ag" and W" follow the trend of A with lattice spacing as given by eqn (6) except that the values are low. The discrepancy remains, even in an accurate comparison of theory and experiment, and is not presently understood. It is clear, however, that no new structural information from He diffraction can be derived from these experiments. In contrast, the corrugation of the Au(ll0) ( A = 1.4A)12 is clearly consistent with the missing-row model, as has recently been confirmed by other techniques.22 He DIFFRACTION AS A SURFACE PROBE +=180" ei =65" d 81 155" i; I I I I I I I 60 ' 80 fAI0 r14 x 10-3 i.Fig. 2. Scattered He signal from GaAs( 110) plotted against reflected angle for a sequence of incident angles (in "). Results for two azimuths 180" apart are plotted [ ( a ) c$ = 180°,0(b) c$ = 0'1 and correspond to projections across the surface ridges and troughs. A = 0.98 A. For the adsorbate systems O-Ni,13 O-Cui4 and Cl-Ag" the Fourier coefficients of the surface potential must be determined with considerable accuracy to permit more than a lower bound for the height of the adsorbate. Although no upper limit to the adatom height can be reliably extracted from the data for these systems, they have been effectively utilized to choose between proposed structural models of very different topographies based on other surface-structural techniques.For the semiconductor series GaAs( 1 lo), Si( 100) and Ge( 100) the surface recon- structions significantly change the spacings of the surface layer atoms so that the surface are effectively 'open'. In these cases He diffraction can be quite sensitive to the surface structure. SEMICONDUCTOR SURFACES: GaAs( 1 lo), Si( 100) AND Ge( 100) GaAs( 110) The three semiconductor surfaces can all be viewed as variations of a 'tilted dimer' surface reconstruction but with different surface atomic spacings. They areM. J. CARDILLO 23 GaAs relaxed ( 1 10) surface-valence charge density (a.u.) 0 2 4 6 6 6 0 2 4 6 8 1 0 distance along (001) (a.u.1 Fig. 3. Valence charge densities for the GaAs( 110) surface calculated by the self-consistent LAPW technique. good examples of the sensitivity of He diffraction to the surface spacing and to surface topography in general when the surface is 'open' in the sense of fig.1. For each of these surfaces simple spherical atomic charge density superposition fails to generate the principal features of the scatterin8 potential. He diffraction data at a wavelength h =: 1 A are shown in fig. 2 for scans taken across the dimer ridges of the GaAs( 110) surface.'6 Rainbow maxima and super- numerary rainbow maxima are evident at incident angles of &=55" and 35", respectively, for each of the azimuthal orientations shown. From classical,'6 eikonal and rigorous scattering theory' thcse data were shown to be consistent with a principal Fourier coefficient of ca. 1 A across the troughs, with only a slight additional asymmetry.Analysis of the data in the smoother perpendicular directio?, along the ridges and troughs, indicated a principal Fourier coefficient of ca. 0.3 A. Laughlin ~howed,~ for the known structure of GaAs( 1 lo), that simple atomic charge superposi- tion could not account for this corrugation. In free space the Ga atom is larger than the As atom. However, at the crystal surface there is a substantial charge transfer from Ga to As, resulting in the As charge completely dominating the He scattering potential. An 'effective atom' charge superposition that could roughly describe this scattering potential essentially eliminates Ga from the superposition and adds just negatively charged As atom densities. This argument was confirmed with self-consistent LAtW calculations6 which obtained principal Fourier coefficients of 0.9 and 0.3 A for the corrugation of the appropriate charge densities of closest approach in the two perpendicular directions.These charge densities are shown in fig. 3 and may be considered as isopotential contours for He scattering. A direct measure of structural sensitivity was carried out in this calculation by forcing the Ga and As into the same plane and recalculating the LAPW charge contours. This constraint is expected to reduce the charge transfer, but more importantly it reduces the maximum surface atomic spacing. The resulting corruga- tion across the trough is reduced from ca. 0.9 to ca. 0.5 A for the coplanar surface.24 He DIFFRACTION AS A SURFACE PROBE A - I % C Fig.4. He diffraction scans for Ge(100) plotted against AKII = ( 2 ~ / h ) ( s i n &-sin Or), the parallel momentum transfer. The scans are rotated to correspond to the azimuthal angle allowing a vertical correspondence with the c(4 x 2) reciprocal net plotted above. A composite reciprocal net including the broad cross representing the shaded extra intensity is shown in the upper left corner. The largest surface atomic spacing for this hypothetical surface is ca. 4 & which when compared with fig. 1 confirms the general trends predicted by eqn (6). Si(100) AND Ge(100) For both Si(100) and Ge(100) the equilibrium periodi ity and structure remain LEED patterns at room temperature. Ge(100) has been observed to order into a c(4 x 2) periodicity at lower temperatures" and Si( 100) has been observed in two reports as c(4 x 2) at room He diffraction also shows a sharp (2 x 1) two-domain periodicity at room temperature for both surfaces but with substantial controversial.They are most often reported to have sharp 2 4 X 1 two-domain (square)M. J. CARDILLO I I ' l ' l ' l l l ' l 1 25 8 6 4 2 0 - 50 -30 - 10 10 30 50 70 90 o r / O Fig. 5. He diffraction scans for Ge(100) and Si(100) taken for the same azimuth at similar incident angles and wavelengths. The rainbows and supernumerary rainbows are shaded to highlight the comparison. additional intensity streaked across the reciprocal net at the half-order positions. 17120 A representation of the periodicity observed with He diffraction for Ge( 100) is shown in fig.4. This is the same reciprocal net, including disorder, previously observed for Si( 100) with He diffraction. In addition to the presence of nearly identical reciprocal nets, the diffraction scans, i.e. the angular distributions of He diffracted intensity, for both surfaces are also quite similar. The angular distributions are most easily related to the shape of the scattering potential surface and thus to the topography of the surface charge d_ensity at the lov4 au level. In fig. 5 angular scans for Si( 100) and Ge( 100) in the [lo] direction are shown for the same He wavelength and nearly equivalent angles of incidence. Both show rainbow maxima displaced far from the specular beam and supernumerary maxima for these near-grazing angles of incidence. For GaAs(ll0) at Bi=65", the rainbow structure is collapsed into the specular beam owing to shadowing and the softness of the scattering potential.From fig. 5 it can be concluded that both Si(100) and Ge(100) are more open and corrugated than26 He DIFFRACTION AS A SURFACE PROBE LO 30 Y %I 20 - modified charge ---- SLAPW superposition SLA PW -_-- I I I I L I I I I I 0 20 LO 60 0 20 LO 60 80 Fig. 6. Valence charge densities for Si( 100) for a tilted dimer configuration with (2 x 1 ) periodicity. Dotted lines represent the self-consistent LAPW charge density. The solid lines are for the simple spherical atomic superposition ( a ) or the modified superposition of charge described in the text ( b ) . The curves A and B are for cuts over the top of the dimers and between the dimers, respectively.x l A GaAs( 110) and that they have very similar topographies and reconstruction ener- getics. A wide range of He scattering data confirm the extent of this basic similarity. In order to compare structural models with the He scattering data, spherical atomic charge densities were summed for dimer and tilted dimer models to generate the He surface potential.21 To check the accuracy of this summation, LAPW charge densities for the 2 X 1 periodicity were calculated and showed relatively poor agree- ment with the atomic summation for both Si and Ge. The comparison for Si( 100)2 x 1 is shown in fig. 6. The c(4 x 2) unit cell is too large for a fully converged self-consistent LAPW calculation, and thus some simplification scheme is required to proceed on this problem.Sakai et aL21 have developed a charge superposition procedure from ‘effective’ atoms, termed a modified atomic charge superposition (MACS). Two parameters representing an exponential decay of atomic charge for each of the dimer atoms are introduced and fitted to the LAPW charge densities for the (2 x 1) tilted dimer. In addition an anisotropy parameter was found necessary to describe the effective atomic charge distribution accurately. The transferability of the para- meters was checked against the LAPW contours for the c(2 ~ 2 ) dimer array. The parameters of the fit clearly show the importance of charge transfer on both Si( 100) and to a lesser but significant extent on Ge( loo), as well as the asymmetric contribu- tion to the He potential of the vertical p orbitals of the dimer pair.Using the simple eikonal scattering theory, a comparison of the angular positions of the rainbow and supernumerary rainbows maxima, it appears that the Si(100) diffraction data are consistent with the topography of a tilted dimer only if the surface is in the c(4 x 2) periodicity. CONCLUSION At present a variety of surface structural experiments forming a consensus are needed to resolve surface structural problems. The development of the STM will very likely provide the impetus and basis for resolution of many long-standing complex structural problems based on the power of its direct images. He diffraction has particular advantages as a probe of surface structure and severe restrictions as well. These have been illustrated by the discussion of theM.J. CARDILLO 27 sensitivity to surface structure, summarized in fig. 1. The surface spacing of top-layer atoms is the key to structural sensitivity for He diffr$ction. For open surfaces, compared with an effective size of the He atom (ca. 3 A), He diffraction can give important topographical information. This is illustrated for the semiconductor surfaces. Even for open surfaces and qualitative structural conclusions, however, care must be taken in the generation of model potentials from structures. In summary, valuable structural conclusions can be derived from He diffraction, if there is a choice between given structural models, provided these models have substantially different topographies in the dilute change densities ( au) extending out from the surface.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 I8 19 20 21 G. Binning, H. Rohrer, Ch. Gerber and E. Weibel, Phys. Rev. Lett., 1982, 49, 57. J. Golovchenko, Bull. Am. Phys. SOC., 1985, 30, in press. Proc. 1st Int. Con$ Surface Structure, Berkeley, August 1984, ed. M. van Hove (Springer Verlag, Berlin, in press). N. Esbjerg and J. K. Norskov, Phys. Rev. Lett., 1980, 45, 807. R. B. Laughlin, Phys. Rev. B, 1982, 25, 2222. D. R. Hamann, Phys. Rev. Lett., 1981, 46, 1227. J. Tersoff, M. J. Cardillo and D. R. Hamann, Phys. Rev. B, 1985, 32, in press. K. H. Rieder and T. Engel, Phys. Rev. Lett., 1979, 43, 373. A. Liebsch, J. Harris, B. Salanon and J. Lapujoulade, Surf: Sci., 1982, 123, 338. A. Luntz, L. Mattera, M. Rocca, S. TerennL F. Tommasini and U. Valbusa, Surf: Sci., 1983, 126, 695. R. I. Masel, R. P. Merrill and W. H. Miller, Surf: Sci., 1974, 46, 681. K. H. Rieder, T. Engel and N. Garcia, Proc. 4th Int. Con$ on Solid Surfaces and 3rd European Con$ on Surjiuce Science, Cannes, France, Sept. 1980, Suppl. la Revue Le Vide, les Couches Minces, no. 201, p. 861. K. H. Rieder, Phys. Rev. B, 1983, 27, 6978. J. Lapujoulade, Y. LeCruer, M. Lefort, Y. Lejay and E. Maurel, Phys. Rev. B, 1980,22, 5740. M. J. Cardillo, G. E. Becker, D. R. Hamann, J. A. Serri, L. Whitman and L. F. Mattheiss, Phys. Rev. B, 1983, 28, 494. M. J. Cardillo, G. E. Becker, S. J. Sibener and D. R. Miller, Surf: Sci., 1981, 107, 469. W. R. Lambert, P. L. Trevor, M. T. Schalberg, M. J. Cardillo and J. C. Tully, Proc. Muter. Res. SOC. Vol. 3 5 , ed. D. K. Biegelsen, G. A. Zozgonyi and C. V. Shank (Mater. Res. SOC., Boston, 1984); Y. Kuk, personal communication; S. Kevan, J. Vuc. Scz. Technol., in press. J. J. Lander and J. Morrison, J. Chem. Phys., 1962, 37, 729. T. D. Poppendieck, T. C. Ngoc and M. B. Webb, Surg. Sci., 1978, 75, 287. M. J. Cardillo and G. E. Becker, Phys. Rev. B, 1980, 21, 1497. A. Sakai. M. J. Cardillo and D. R. Hamann. to be Dublished.

ISSN:0301-7249    

DOI:10.1039/DC9858000017

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1985,80, 29-45 Elastic Scattering of Light Atoms from Physisorbed Overlayers What are the Non-additive Many-body Corrections ? BY HANNES JONSSON AND JOHN H. WEARE Department of Chemistry B-014, University of California at San Diego, La Jolla, California 92093, U.S.A. Received 18th April, 1985 Atom scattering from physisorbed overlayers is first briefly reviewed. While good agree- ment has been obtained between measurements and calculations both for diff raction scans and selective adsorption [H. J6nsson, J. H. Weare, T. H. Ellis, G. Scoles and U. Valbusa, Phys. Rev. B, 1984,30,4203; T. H. Ellis, G. Scoles, U. Valbusa, H. J6nsson and J. H. Weare, Surf. Sci., 1985, 155, 499; H. Jhsson, J. H. Weare, T. H. Ellis and G. Scoles, in preparation; J. M.Hutson and C. Schwartz, J. Chem. Phys., 1983, 79, 51791, detailed information on the scattering potential has not been inferred from the experimental data because of three highly correlated unknowns in the theoretical potential: (1) uncertainty in the probe-adatom pair potential, (2) uncertainty in the probe-substrate potential and (3) non-additive many-body corrections. New close-coupling calculations are presented here for the scattering of He from a commensurate overlayer of Kr on graphite and compared with recent selective adsorption data of Larese et al. (J. Z. Larese, W. Y. Leung, D. R. Frankl, N. Holter, S. Chung and M. W. Cole, preprint). This is a favourable system in that the He-Kr pair potential is well known and a semiempirical helium-graphite potential consistent with extensive experimental data on clean graphite has recently been constructed (H.J6nsson and J. H. Weare, in preparation). Good agreement is obtained when the theoretical potential only contains two-body contribu- tions, but it is significantly improved by shifting the lowest bound-state energy by -0.12 meV. The long-range triple-dipole correction alone is frequently used to represent the total non- additive correction, e.g. in calculations of the binding energy of rare-gas solids. When applied to the present problem, the triple-dipole correction shifts the lowest bound state in the wrong direction by +0.16meV. A correction due to thermal vibration further shift this level by +0.07 meV. There is, therefore, a 0.35 meV difference between the lowest bound-state energy of this theoretical potential and the experimental value.We estimate the uncertainty in the two-body terms [ ( l ) and (2)] to be kO.1 meV. Typically the uncertainty due to experimental parameters is estimated to be kO.1 meV. The data therefore suggest that the total non-additive correction is not well represented by the long-range triple-dipole correction alone and that other many-body corrections of opposite sign need to be included. Three-body corrections to the short-range repulsive interaction have not been quantitatively estimated but are known to be attractive [W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225). The scattering of light atoms (H, He) from physisorbed overlayers is interesting for several reasons. Since the probe-adatom interaction can be quite well known from crossed-beam and dilute-gas experiments, a good surface potential for the overlayer can be constructed by summation over two-body interactions.This allows the comparison of calculated atom-surface scattering with measurements without adjusting parameters of the surface potential. There are, however, small corrections to the sum of two-body interactions.' These non-additive corrections are not well understood and could possibly be measured with the atom-surface scattering. The structure and dynamics of physisorbed layers is furthermore a problem of 2930 ATOM SCATTERING FROM PHYSISORBED OVERLAYERS 0 0 polar angle/" Fig. 1. Measured and calculated scattering from a monolayer of Xe on graphite [reproduced from ref.D(8)] ( a ) using best-fit pair potential to crossed-beam rneasureqents, lattice constant a = 4.26 A, ( b ) using same pair potential as in ( a ) , but with a = $34 A and ( c ) using pair potential with (T reduced by 2%, a = 4.26 A.considerable interest and is currently investigated with many techniques, primarily X-ray scattering. Atom scattering can make a contribution here because of its high sensitivity to the overlayer. This, for example, allows the use of single-crystal substrates instead of the polycrystalline substrates necessary in X-ray measurements. This paper will only be concerned with elastic atom scattering from a full monolayer. Inelastic He scattering has been measured by Mason and Williams for various adsorbates on Cu and Ag substrates.2 Very recently Gibson et al.followed the dispersion curve as successive layers were adsorbed on Ag( 11 l).' Atom scattering has also provided information about physisorbed . overlayers in the low-coverage regime (0-5% The first measurements of diffraction6 and selective adsorption7 on physisorbed overlayers were reported by Ellis et al. They measured the scattering of H atoms from Xe and Kr overlayers on graphite. These data were analysed with close-H. JONSSON AND J. H. WEARE 31 coupling calculations and a theoretical potential constructed from two-body terms only. Very good agreement was found between the experiment and theory both for diffraction intensities and selective adsorption.* A remarkable sensitivity to the probe-adatom pair potential was found. Small changes in the pair potential sig- nificantly affect the surface scattering.Fig. 1 is reproduced from ref. (8) and shows a diffraction pattern for hydrogen-atom scattering from a Xe monolayer. In fig. 1 ( a ) the pair potential used is a best fit6 to crossed-beam results.' In fig. 1( c) a pair potential with a 2% smaller CT is used (u is the distance to the zero crossing of the pair potential), which is within the limits of uncertainty. In theke calculations it was assumed that the Xe layer was commensurate with the graphite substrate. The position of the diffraction peaks, however, was consistently better fitted by using a slightly larger lattice constant [fig. 1 ( b)].* Although very good agreement was obtained between the calculation and the experiment, it can only be concluded that non- additive corrections to the corrugation, which were not included, are of similar magnitude or smaller than the uncertainty coming from the pair potential.An additional uncertainty affects the interpretation of the selective adsorption data because the probe-substrate potential (here hydrogen-graphite) gives a sizeable contribution to the laterally averaged potential and shifts the bound states con- siderably. Bracco et aZ." measured the scattering of He atoms from a Xe overlayer on graphite. Close-coupling calculations were performed by Hutson and Schwartz," who found good overall agreement with the selective-adsorption data. They demon- strated that shifts in the bound-state energies due to the triple-dipole non-additive term are of similar magnitude as the uncertainty arising from the pair potential and the uncertainty arising from the helium-graphite potential. Very recently, Larese et uZ.'~ and Frankl13 measured the selective adsorption of He atoms scattered from a commensurate monolayer of Kr on graphite using a very monochromatic and well collimated beam.Since the He-Kr pair potential is better known than either He-Xe or H-Xe and an improved helium-graphite potential has recently been constr~cted,'~ we have analysed these data with close coupling calcula- tions in order to evaluate the non-additive corrections to the bound-state energies of the He-Kr/graphite potential. In the following sections we first describe briefly the two-body terms in the theoretical potential. The calculation is then compared with the experimental data.Finally we discuss estimates of the non-additive correc- tions. TWO-BODY TERMS THE HELIUM-GRAPHITE POTENTIAL We assume that the helium-graphite interaction can be written as a sum over an effective He-C pair potential. This potential is chosen so that the surface potential is consistent with some theoretical and experimental data on the laterally averaged surface potential. The theoretical information comes from a calculation of Liebsch et 4'' who found that the lateral average of the repulsive part of the potential is well approxi- mated by an exponential VGP( 2') = Vo exp (-a?). Their theory determines the softness parameter, a, within narrow bounds from the work function, but the pre-exponential factor, V,, is sensitive to details and is not well determined.In order for the sum over pair potentials to have the exponential32 ATOM SCATTERING FROM PHYSISORBED OVERLAYERS form, we choose the repulsive part of our pair potential to be exp (-ar) r dep( r ) = A We take a = 3.57 &', as determined by Liebsch et al." The parameter A is treated as an adjustable parameter. In the attractive part of the pair potential we include the dipole-dipole term Vatt'( r ) = -f6( r ) ( p / r6) (3) where f 6 ( r ) is a damping function to correct for charge overlap. Experience with the form of &(r) has been obtained from studies of atom-atom interaction~.'~~'~ We use a form recently proposed by Tang and Toennie~,~' which only depends on the softness, a, of the repulsive part: Combined with theoretical calculations of the long-range dispersion energy and the short-range repulsion, this damping function and other similar ones have led to quite accurate theoretical pair potentials.A similar approach was recently used successfully to construct a semiempirical He-LiF potential.'* We do not include higher-order dispersion terms, such as dipole-quadrupole, and simply adjust the parameters p and A to fit the bound-state data for the helium-graphite potential. The bound-state energies of the laterally averaged helium-graphite potential have been determined to high precision by Derry et all9 both for 4He and 3He. A total of eight bound states are known in the range -0.17 to -12.06 meV with estimated error of *O.l2meV.'' (The shifts in position of isolated resonances due to the corrugation are ca.0.01 meV and can therefore be negle~ted.'~) Only a very narrow range in the v$ue of p gives a fit within the estimated error bars, p = 1750-185!meVA3. Choosing the average, the best-fit value of A is 1.119 x lo5 meV A. The helium-graphite surface potential is the sum over all He-C pair potential^.^" The lateral average is14 a4 6! - - ( 1 + az') exp (- az') where a: is the area of thc graphite surface unit cell and d is the spacing between graphite layers (d = 3.37 A). The functions E , ( x ) are the exponential integrals2' and c ( n , t ) is the Riemann zeta function.? The potential 'is shown in fig. 2. The higher Fourier components (G # 0) are given in ref. (4). In the asymptotic region, z' --.* 00, the leading term of this surface potential has the correct form: Voo( z') = - c3/ z'3.( 6 ) The asymptotic coefficient, c3, can be calculated given the dielectric function for graphite and tbe polarizability function of He. Watanabe2' and B r u ~ h ~ ~ obtained C3 = 170 meV A3 f 10%. Vidali et u Z . , ~ ~ using a simpler He polarizability function, t This function is simply related to the polygamma function +;,S' = (-l)"(n - l ) ! c(n, x), which is discussed and tabulated in ref. (22).H. JONSSON AND J. H. WEARE 33 4 0 -4 a 1 0 4 ' 8 -12 -16 1 0 1 2 3 4 5 6 +I0 Fig. 2. ( a ) The He-graphite potential and ( b ) the He-Kr/graphite potential given by the sum of all two-body terms (potential 1). The bound-state energy levels for 3He- and 4He-graphite are shown [ (- - -) measured levels; (-) calculated] and the lowest level for 4He-Kr/graphite.The origin is at a Kr atom. The figure shows that the classical turning points for the lowest He-Kr/graphite bound state are close enough to the graphite surface so that the He-graphite interaction in that range is determined by the measured He-graphite bound-state energy levels. obtained the value 188 meV A3. Expanding our He-graphite potential [ eqn (5)], the valueo of p that fits the bound-state energy levels, 1800 meV A6, gives C;ff = 110 meV A3, i.e. ca. 35% lower. We tried avalue o f p consistent with c3 = 170 meV A3, but the calculated bound-state energy levels did not agree with the measured ones, no matter what softness a was used. Adding the dipole-quadrupole term, fs( r ) q / r8, to the pair potential [eqn (3)] and adjusting q also did not lead to agreement with the experimental data when p was fixed at the higher value.In particular, the highest measured bound state, E ~ = -0.17 meV, is always calculated to be too low when the maximum difference between measured and calculated energy levels is minimized. We note that in the case of He-LiF a similar discrepancy was found.'* It is possible that, although our potential gives a good description of the region of the attractive well, it is too shallow further away from the surface, where the magnitude has dropped below 0.2 meV (z' > 10 A). Fortunately, this uncertainty does not affect the conclusions of this paper, because non-additive forces are expected to be appreciable only in the lowest bound state of the He-&-/graphite potential.Between the classical turning points associated with this level, the helium- graphite potential is --$ to -1 meV and is therefore in the range probed by the He scattering from clean graphite (see fig. 2). More important for the present discussion is the position of the helium-graphite potential with respect to the He-Kr adlayer potential. Using a measured value for the distance between the Kr and C planes of nuclei, the problem is reduced to finding the position of the helium-graphite potential with respect to the topmost C plane. He scattering from clean graphite is insensitive to this. However, because34 ATOM SCATTERING FROM PHYSISORBED OVERLAYERS we assume that the helium-graphite potential is a sum of isotropic pair potentials, this distance is fixed in our model.Fortunately there are some experimental data on this. Carneiro et a1.26 scattered neutrons from a monolayer of He adsorbed on Grafoil and determined the helium-graphite distance to be 2.85 A. Our potential gives the expectation value ( z ) = 2.7 A, which is in quite good agreement. The corrugation of our helium-graphite potential is also determined by the assumption of isotropic pair potentials, because all parameters are determined from properties of the laterally averaged potential. The corrugation is directly probed by the He diffraction peak intensities, which have been measured by Boato et at two beam energies. The Debye-Waller inelastic correction for these data is very and the peak heights have been deconvoluted to give the relative i n t e n ~ i t i e s .~ ~ A close-coupling calculation using our potentail agrees well with the experimental data. The average difference between measured and calculated diffraction intensity is 5% of the specular intensity and maximum difference 15%. When a Kr monojayer is adsorbed on the graphite surface, the turning point of the He atom is > 5 A away from the topmost plane of C nuclei. It is therefore sufficient to include only the second term in eqn ( 5 ) as the helium-graphite contribu- tion. It is, however, not sufficient for the present discussion to use the simple approximate form Voo( 2') = - c,/ ( Z' - z ~ ) ~ (7) with zo chosen so that the first two terms in the asymptotic expansion of eqn (7) and ( 5 ) agree (zo= d/2).698711 Using this approximation the lowest bound state is shifted by ca.0.15 meV, which is too crude here as will be shown later. THE He-Kr POTENTIAL The He-Kr pair potential has been determined from a variety of theoretical and experimental information. In a recent review,29 the potential of Watanabe et aL3' was found to be the best in describing a variety of properties: differential scattering, virial coefficients and transport properties. This pair potential has the form (atomic units) v( r ) = [ y( 1 + 0.1 r ) - 13 exp aor + a , + a2 + ") -f( r)( $+ cs + 3) (8) r r2 r8 where f ( r ) = [exp [ - 0 . 4 ( F - 1 ) 2 ] if r<1.28rm I 1 otherwise and r, is the position of the minimum. The helium-adlayer potential is obtained by summing over the 8 ~8 overlayer lattice.We assume here that the Kr atoms are not vibrating. The summation can be carried out more efficiently if the exponen- tials of inverse powers of r are expanded: where C = exp ( a , ) in hartrees. Keeping here terms up to k = 5 , the He-Kr potential given by eqn (9) deviates from that in eqn (8) by 3% when 1 s Iu(r)l/meVs 300.H. JONSSON AND J. H. WEARE 35 Using eqn (9) rather than eqn (8) only shifts the lowest bound-state energy level of He-Kr/graphite by 0.006 meV. The Fourier components of the surface potential are2' V,(Z) = - 2.rrpG a s ~ o ~ J o ( G R ) ~ [ ( R 2 + z 2 ) 1 ~ 2 ] R dR (10) where /3 is the structure factor and as is the area of the adlayer unit cell. R is the surface projection of the He atom position vector and G is the surface reciprocal lattice vector.Choosing the origin at a Kr atom, PG = 1. The laterally averaged potential is For z > 1 .28rm, HG( z ) = 0. Otherwise where u( z ) = [( ~ 2 8 r , ) ~ - 22]1/2 and r = ( R2+ z2)1/2. This function is easily evaluated numerically. The higher Fourier components are where s = (a2 + G2)lj2 and K is a modified Bessel function of the second kind. The functions I"')(a, b ) are given in ref. (14). The total He/&--graphite potential is, to a first approximation, simply the sum of these two-body terms. The helium-graphite term only contributes to the laterally averaged potential: zc is the distance between Kr and the topmost C plane. We take zc = 3.3 A as suggesttd by Chung et ~ 1 . ~ ' The minimum of this potential is D = -6.90 meV at z = 3.3 A, with the contribution from the helium-graphite term being -0.8 meV.The bound states of this potential, potential 1, are given in table 1. The higher Fourier components are given by zqn (12) and (13). The corrugation of the equipotential surface at 20 meV is 1.05 A, which is similar to the H-Xe/graphite corrugation at 80 meV.636 ATOM SCATTERING FROM PHYSISORBED OVERLAYERS Table 1. Bound states of different potentials (in meV) 1 2 3 4 EO -4.58 -4.83 -4.42 -4.7 f 0.1 El -1.67 -1.85 -1.59 -1.7 E2 -0.46 -0.56 -0.43 -0.5 E 3 -0.086 -0.13 -0.08 1 -0.09 ( 1 ) Best two-body t e p s . (2) Deeper helium-graphite potential C3 = 150 meV A3. (3) Triple-dipole terms added to 1. (4) Best fit to experiment. COMPARISON OF THEORETICAL CALCULATIONS WITH EXPERIMENT The close-coupling calculations are converged, with 127 channels included. It is important to use converged calculations here because we are looking for detailed agreement between calculation and experiment.With fewer channels some reson- ance features can be shifted or lost entirely. The calculation has been carried out at the most probable energy only. As the velocity distribution in the beam is quite narrow, this should prove sufficient for the analysis of the larger features but very small features in the calculated spectrum are probably averaged out. The possible effect of inelasticity on the shape of the resonances has not been included. Each point takes ca. 2 h on a VAX 11/780 computer. Fig. 3. He scattering from a commensurate Kr layer on graphite; variation of sp cular intensif with azimuthal angle at Bi = 59.65' and E = 17.2 meV.Solid line: experimental data.12. Dotted line: calculation for a potential made up of two-body terms only, potential 1 (see table 1). The bars on top show the position of resonances as predicted by the zeroth-order theory. In the top row, A, are resonances associated with the largest higher-order (G # 0) Fourier component, G = ( 1 , O ) and equivalent, in the second row, B, the second largest G = (1, 1) and equivalent etc. The D resonances associated with the lowest bound-state energy level are marked with 0. Dotted bars indicate thresholds.H. JONSSON AND J. H. WEARE 37 Fig. 3, 4 and 5 show the measured (solid line) and calculated (dashed line) variations in specular intensity as the incident azimuthal angle is varied at three different polar angles.The calculation is for potential 1, i.e. best estimate of two-body terms only. The agreement is quite good, with clear correspondence between features in the calculated and measured spectra. Note the large changes Fig. 4. As fig. 3, but for Oi = 60.65'. A I I e I11 I I l i I D 8 1 I I I I l l I I I I I I 1 , I I I l I -8 -4 0 4 8 4P Fig. 5. As fig. 3 and 4 but for Oi = 61.65'38 ATOM SCATTERING FROM PHYSISORBED OVERLAYERS as the polar angle is changed by only 1". In principle a smoothly varying background signal should be added to the calculation,' but we simply shift the experimental intensity to allow the best comparison with the azimuthal position of the oscillations. The position and shape of the oscillations are especially good close to the symmetry line, 4 = 0.This is a very complicated region, where many resonances are predicted from the zeroth-order approximation (see marks on top of the spectra). The top To*, A, marks the position of resonances associated with the largest Fourier component [ G = (1,O) and equivalent], row B the second largest [ G = (1, l)] etc. (There is a 60" angle between our reciprocal lattice vectors.) At the symmetry line most resonances are crossing their symmetric counterpart and in highly corrugated systems this can be quite different from what is expected from first-order perturbation theory.8 The major discrepancy between this calculation and the experimental data is the position of the oscillations at 141 = 5-9".These are consistently calculated to be too close to the symmetry line. (Note that the rise in the signal in fig. 3 as 4 increases from 4 = 0" is not reproduced at negative 4.) We have analysed this region at Bi = 59.65" by gradually reducing the corrugation (multiplying all the higher-order Fourier components by th? same factor). The calculated results are shown in fig. 6. This shows that at 0.z A the B resonance, 1(-1, 2), is a maximum, but already at aocorrugation of 0.3 A it is a mixed maximum-minimum. At a corrugation of 0.5 A it is clearly a minimum. The A resonance, 2(1, - l ) , is a minimum at all 0.8 0.6 5 04 2 0 02 I ' I ' I ' 1 1 ' I ' 1 0.2 - 0 3 - 05 - 1 O 7 1 I \ I 0.051 \ 4 / O Fig. 6. Calculated intensity at Oi = 59.65" using potential 1 (as in fig.3) and potentials with smaller corrugation but the same lateral average, Voo. All higher order (G # 0) Fourier components are multiplied with the same factor. This shows quite clearly that the B resonance gives a minimum in the specular intensity at a corrugation of 0.5 A, while at 0.2 8, it is a maximum. The corrugation (A) is shown next to the curves.H. JONSSON AND J. H. WEARE 39 corrugations. As the corrugation is increased from 0.5 to 1.1 A the two minima are pushed away from each other. The reason seems to be that the coupling becomes strong enough to give rise to a maximum associated with the D resonance 0(-2,3). In fig. 3 and 5 the resonances of this order seem to consistently give rise to a maximum. This sugggsts that the position of the oscillations at 141 = 5-9' can be improved by shifting the D resonance away from the symmetry line.This can be done by lowering the lowest bound state. The close-coupling calculations were repeated with the attractive helium-graphite potential increase by a factor of 15/11. This corresponds to C3 = 150 meV A3, which is closer to the asymptotic value predicted from the polarizability and dielectric f u n ~ t i o n s . ~ ~ - ~ ~ The bound-state energy levels of the Voo component of this potential, called potential 2, are given in table 1. Note that all levels, not only the lowest, are now deeper. Fig. 7-9 show the results. Indeed the oscillations at 1u1= 5-9" are moved out, but too far. The agreement with the experimental data closer to the symmetry line is, however, considerably worse, especially at Oi = 61.65'. Since none of the resonances in this region is associated with the lowest level, we conclude that potential 1 has the right n = 1-3 levels, but the lowest, n = 0, should be lower.The bound states of potential 2 are very similar to those of Chung et aL,31 who used the asymptotic value C3 = 180 meVA3 for He-graphite term and also included a net repulsive correction due to non-additivity and vibrations. Excellent agreement between the calculation and experiment is obtained when a small, short-range attraction of the form -S/(z+ y ) 6 is added to the two-body terms ( i e . added to potential l), taking S = 8.5 x lo3 meV A6 and j = 3.04 A. Fig. 10 shows the close-coupling calculation for this potential at Oi = 60.65'.The bound-state energies of this potential are given in column 4 of table 1. The lowest bound state is shifted -0.12 meV by this arbitrary term, but the higher bound states are almost unchanged. Preliminary results show that this potential is also in excellent agreement with the experimental data taken at Oi=59.65 and 61.65'. By adding this small 0.05 z 4 - 1 0 0.00 l ' l ' l ' l ' l ' l ' l I I ' I !I' I I ,! I , A I 8 1 0 I ! I I I , l r l , l , l r l r l -8 -4 0 4 8 4/a Fig. 7. Same experimental data as in fig. 3 (solid line), but the potential used in this calculation, potential 2, has a larger helium-graphite term (corresponding to C3 = 150 meV A3). All bound states are lower than for potential 1 (see table 1).40 )I 2 0.2 0 4 0.1 0.0 ATOM SCATTERING FROM PHYSISORBED OVERLAYERS l ' l ' l ' l ' l ' l f l A I II I 8 I ill Ill I D I i l l 1 1 1 1 : I - - - - - - I 1 1 1 I 1 1 1 1 I 1 I I .-8 -4 0 4 8 Fig. 8. Same experimental data as in fig. 4 (solid line), but a calculation using potential 2 (dashed line). 0.3 & - 2 0.2 W a 0.1 0.0 I l l l l l l l l l l l l l l -12 -8 -4 0 4 8 12 +I0 Fig. 9. Same experimental data as in fig. 5 (solid line), but a calculation using potential 2 (dashed line). The poor agreement here at 141 < 4", as compared to fig. 5, indicates that the higher bound-state energy levels ( n = 1,2,3) of a potential z are too deep.H. JONSSON AND J. H. WEARE 3 41 Fig. 10. Same experimental data as in fig. 4 and 8 (solid line), but the potential used in the calculation (dashed line) is potential 1 with the addition of a short-range attractive term.The bound-state energy levels are given in column 4 of table 1 as best estimates of the true energy levels. attractive term, we have fitted the experimental data and have obtained a very good estimate of the true bound-state energy levels. Despite the good agreement, there is still uncertainty in this estimate due to possible errors in the incident angles, Oi and or the incident wavevector, k. A typical estimate of this uncertainty is k0.1 meV." It is remarkable in view of the complexity of the spectra, which even makes the calculation hard to interpret, that Larese et all2 came up with a very similar number for the lowest bound state without doing any close-coupling calculations. They simply associated minima with the B resonances and followed their trajectory in the (Kx, K y ) plane.While our calculations show that isolated B resonances would give rise to minima, the actual position of dips in the intensity, if they occur at all, can be influenced by other resonances, either because of coupling between them or simply because there is a balance between a minimum and a nearby maximum (as for example at 1#1= 5-9"). Even if the trajectory of the assigned feature (minimum, maximum) falls close to a circular arc in the ( K , K y ) plane as predicted by the zeroth-order approximation, this does not convincingly prove that the right choice was made when the intensity is oscillating continuously, as in the present case. Most likely the next feature (maximum, minimum) follows a similar trajectory corresponding, however, to a different bound-state energy.Indeed, the higher energy levels were determined by Larese et al. to be too deep (a little deeper than those of potential 2). NON-ADDITIVE MANY-BODY CORRECTIONS The two-body terms described earlier can be considered as the leading terms in a many-body expansion of the atom-surface potential. This approach has generally42 ATOM SCATTERING FROM PHYSISORBED OVERLAYERS been taken, for example, in studies of rare-gas solids. The convergence of this expansion has been questioned.’ The leading non-additive correction term to the attractive dispersion energy is the triple-dipole term. Axilrod and Teller32 and M ~ t o ~ ~ derived an expression for this in third-order perturbation theory for three atoms well separated so that charge overlap can be neglected: (15) 1 +3 cos O1 cos O2 cos O3 rir:ri Z)DDD(rl, r2, r 3 ) = y123 The coefficient ~ 1 2 3 is easily evaluated from the atomic p~larizabilities.~~ Using the (1.31 f 0.03) x lo4 meV A9.Klein and Cole,35 using a slightly more appcoximate polarizability function, obtained a similar value: vHeKrKr = 1.35 x lo4 meV A9. The sum over all Kr pairs in the adlayer is very well approximated by upper and lower bcunds given by Tang et aZ.34 we obtain vHe=Kr- - vg;D( z) = c:-a/ ( z + Z:-a)6 (16) with c : - ~ = 1.67 x lo4 meV A6 and = 3.04 A. This simQle expression agrees with the exact sum to within 2% in the relevant region (z > 2.5 A and I Vg&,I > 0.05 meV). At the well minimum, z = 3.3 & the correction is VgAD= 0.26 meV.The higher Fourier components are negligible compared with those of eqn ( 13).11731 In principle some damping function should be included in eqn (15). As an estimate we simply take with f 6 given by eqn (4). At 2.5 A above a Kr adatom this reduces the sum by 14% but at z = 3.0 A only by 5 % . We therefore neglect this effect. A term similar to eqn (15) but involving He-Kr/graphite has been estimated by Klein et aZ.36 using the theory of M~Lachlan.~’ The result is given in ref. (31) as with Cs2 = 1760 meV A6 and b = 1.9 A. Note that thisJerm has opposite sign to eqn (16) but is smaller: V;AD(z) = -0.07 meV at z = 3.3 A. Other corrections to the dispersion energy are, for example, the quadrupole- dipole-dipole terms, quadrupole-quadrupole-dipole terms etc.in third-order per- turbation theory and the dipole-dipole-dipole interaction to fourth and higher orders. It has been that these higher-order terms tend to cancel in the binding energy of rare-gas solids. When only the Axilrod-Teller-Muto ( ATM) term is included, the calculated binding energy agrees well with measurement (the correction amounts to 10% for Kr and Xe). This has led to the conclusion that the total many-body correction is effectively represented by the ATM term alone. [See for example ref. (40).] It has been suggested that this is also the case in atom-surface s ~ a t t e r i n g . ~ ~ , ~ ~ Table 1, column 3, gives the bound states of a potential containing the two-body terms (potential 1 ) plus the triple-dipole corrections given by eqn (16) and (18).Because the triple-dipole correction is fairly short range, only the lowest bound state is appreciably affected. It is shifted by +0.16 meV. This shift is in the wrong direction in comparison with the best fit to the experimental data (column 4). We note that because the higher bound states are unaffected, good agreement is not obtained by simply deepening the helium-graphite potential. For example, addingH. JONSSON AND J. H. WEARE 43 the triple-dipole terms to potential 2, the lowest bound state is in agreement with experiment, but the higher bound states are too deep. Other three-body corrections are expected to be of shorter range3* and we therefore feel that the good agreement between the higher bound state energies of potential 1 and experiment gives support to our two-body terms.Non-additive corrections to the repulsive part of the potential have been esti- mated with SCF calculations of He, Ne and Ar t r i m e r ~ . ~ l - ~ In all cases the correction is found to be attractive in the equilateral-triangle configuration and therefore tends to cancel the ATM correction. At the van der Waals distance, r = r,, for He3 this correction is actually larger than the ATM term, for Ne, it is ca. $ and for Ar3 the two are equal. Meath and A Z ~ Z , ~ calculated the binding energies of rare-gas crystals including this correction as well as corrections to the dispersion energy. There is significant cancellation between these corrections, so the good agreement obtained when only the ATM term was included appears to be fortuitous. CONCLUSIONS Although the non-additive corrections as estimated in the previous section are certainly small, there is, in view of the excellent agreement obtained here, sufficient sensitivity in this experiment.The bound-state energies of the potential are well determined by fitting the selective adsorption spectrum with close coupling calcula- tions. However, clear conclusions can not be drawn because of possible errors in the experimental parameters (&, Oi and ki), which are typically estimated to be 3tO.1 meV, and because of possible errors in the two-body terms. We now estimate roughly the latter of these two. There are three principal sources of error in the two-body terms: (1) the He-Kr pair potential, (2) the distance between Kr and the topmost C plane and (3) the position of the helium-graphite potential with respect to the topmost C plane.Since we are only concerned here with the lowest bound state of the He-&-/graphite potential, the shape of the helium-graphite potential is quite well known in the relevant region from the measured bound-state energy levels of He on clean graphite. (1) The first error we estimate by choosing an alternative pair potential. The second best He-Kr pair potential reviewed in ref. (24) is ‘the Maitland-Smith type of potential parametrized by Smith et u Z . ~ ~ Using this pair potential instead of the WAM potential, the lowest bound-state energy is shifted by +!.05 meV. (2) The Kr-C distance was measurcd by EXAFS46 to be zc = 3.35 f 0.1 A.As suggestFd by Chung et al. we used 3.30 A. If instead we take the Iargest value, zc = 3.45 A, the lowest bound-state energy is shifted by +0.05 meV. (3) Information on the position of the helium-graphite potential with respect to the plane of the C nuclei is provided by the measurement of neutron scattering fJom a He monolayer.26 As mentioned earlier, this value was determined to be 0.12 A larger than predicted by our potential. Shifting the potential outwards by 0.15 A lowers the bound state by -0.06 meV. Adding the squares of these three independent sources of uncertainty gives the error bar *O.l meV. The effect of thermal vibrations of the adlayer on the bound-state energy levels has been estimated by Chung et aL31 They calculate a +0.07 meV shift in the lowest bound-state energy for the present case.The results of the present data analysis are therefore the following. The lowest bound-state energy of the true potential is -4.70 * 0.1 meV. The theoretical potential, which only includes two-body terms, has a lowest bound state at -4.51 hO.1 meV (including the thermal correction). When the long range triple-dipole corrections44 ATOM SCATTERING FROM PHYSISORBED OVERLAYERS are added, this level is shifted in the wrong direction by +0.16meV. The data therefore suggest that the total non-additive correction is smaller than the triple- dipole terms and possibly of opposite sign. We cannot, however, exclude the hypothesis that the triple-dipole terms alone are a valid representation of the total correction, because our error bars are perhaps too optimistic.By addressing each factor contributing to the uncertainty, this experiment can in principle be made more conclusive. It is unfortunate that in this system, where the two-body terms are well known, the ATM correction is estimated to be rather small. 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Glachant, Surf: Sci., 1984, 136, 169.lo G. Bracco, P. Cantini, A. Glachant and R. Tatarek, Surf: Sci., 1983, 125, L81; G. Bracco, P. l 1 J. M. Hutson and C. Schwartz, J. Chem. Phys., 1983, 79, 5179. l 2 J. Z. Larese, W. Y. Leung, D. R. Frankl, N. Holter, S. Chung and M. W. Cole, preprint. l4 H. J6nsson and J. H. Weare, in preparation. l5 A. Liebsch, J. Harris and M. Weinert, Surf: Sci., 1984, 145, 207. l6 C. Douketis, G. Scoles, S. Marchetti, M. Zen and A. J. Thakkar, J. Chem. Phys., 1982, 76, 3057. l7 K. T. Tang and J. P. Toennies, 3. Chem. Phys., 1984, 80, 3726. l8 V. Celli, D. Eichenauer, A. Kaufhold and J. P. Toennies, Phys. Rev. B, to be published. D. R. Frankl, personal communication. 13 G. Derry, D. Wesner, W. Carlos and D. R. Frankl, Surf: Sci., 1979, 87, 629. W. A. Steele, Surf Sci., 1973, 36, 317; W. E. Carlos and M. W. Cole, Surf Sci.., 1980, 91, 339. *’ M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). 22 H. T. Davis, Tables of the Higher Mathematical Functions (The Principia Press, Bloomington, 23 H. Wanatabe, personal communication referred to in ref. (25). 24 L. W. Bruch, Sut$ Sci., 1983, 125, 194. ’’ G. Vidali, M. W. Cole and C. Schwartz, Surf: Sci., 1979, 87, L273. 19 20 1935); see also ref. (21).H. JONSSON AND J. H. WEARE 45 26 K. Carneiro, L. Passell, W. Thomlinson and H. Taub, Phys. Rev. B, 1981, 24, 1170. 27 G. Boato, P. Cantini, C. Guidi, R. Tatarek and G. P. Felcher, Phys. Rev. B, 1979, 20, 3957. 28 G. Boato, P. Cantini, C. Salvo, R. Tatarek and S. Terreni, Surf: Sci., 1982, 114, 485. 29 R. A. Aziz, in Inert Gases, ed. M. L. Klein (Springer, Berlin, 1984). 30 K. Watanabe, A. R. Allnatt and W. J. Meath, Chem. Phys., 1982, 68, 423. 31 S. Chung, N. Holter and M. W. Cole, preprint. 32 B. Axilrod and E. Teller, J. Chem. Phys., 1943, 11, 299. 33 Y. Muto, Proc. Phys. Math. SOC. Jpn, 1943, 17, 629. 3s J. R. Klein, M. W. Cole, Su$ Sci., 1983, 134, 722. 36 J. R. Klein, M. W. Cole and L. W. Bruch, unpublished results. 37 A. D. McLachlan, Mol. Phys., 1964, 7, 381. 38 M. B. Doran and I. J. Zucker, J. Phys. C, 1971, 4, 307. 39 W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. 40 J. A. Barker, in Rare Gas Solids, ed. M. L. Klein and J. A. Venables (Academic Press, London, 1976). 41 0. Novaro and V. Beltran-Lopez, J. Chem. Phys., 1972, 56, 815. 42 0. Novaro and F. Nieve, J. Chem. Phys., 1976, 65, 1109. 43 M. Bulski, Chem. Phys. Lett., 1981, 78, 361. 44 J. P. Daudey, 0. Novaro and M. Berrondo, Chem. Phys. Lett., 1979, 62, 26. 45 K. M. Smith, A. M. Rulis, G. Scoles, R. A. Aziz and V. Nain, J. Chem. Phys., 1977, 67, 152. 4h C. Bouldin and E. A. Stern, Phys. Rev. B, 1982, 25, 3462. K. T. Tang, J. M. Norbeck and P. R. Certain, J. Chem. Phys., 1976, 64, 3063. 34

ISSN:0301-7249    

DOI:10.1039/DC9858000029

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1985, 80, 47-56 Very Low-energy Scattering of Helium Atoms from Crystal Surfaces A Quantitative Comparison between Experiment and Theory BY LAURIE DANIELSON,? JESUS-CARLOS RUIZ SUAREZ, CAREY SCHWARTZ~ AND GIANCINTO SCOLES Centre for Molecular Beams and Laser Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada AND JEREMY M. HUTSON*§ University Chemical Laboratory, Lensfield Road, Cambridge, CB2 1 EW Received 25th March, 1985 Very low-energy scattering experiments are described for He atoms colliding with an LiF(001) surface. The apparatus is capable of performing experiments for beam energies down to 4meV. The He-LiF results are compared with close-coupling calculations on the interaction potential of Celli et al. ( J . Chern. Phys., 1985, 83, 2504).Very good agreement between experiment and theory is found, but there are small discrepancies which should allow the potential to be refined. It is well known that the diffractive scattering of atoms from solid surfaces is sensitive to details of the atom-surface interaction potential. 1*2 In particular, the phenomenon of selective adsorption provides information on the potential in the chemically interesting region around the minimum. Numerous experimental studies of selective adsorption of helium atoms have been carried out, for a wide variety of surfaces; in most cases the positions of the selective adsorption resonances have simply been interpreted in terms of the bound-state energies of an atom vibrating in the surface-averaged potential well.* In a zeroth-order picture (neglecting corruga- tion), a resonance occurs when k2=(K+G)*+2rnEn/h2 ( 1 ) where k is the wavevector of the incident beam, K is its projection onto the surface plane, G is a surface reciprocal lattice vector, m is the atomic mass and En is the (negative) binding energy of the atom in the surface-averaged potential Voo( z), obtained from the one-dimensional Schrodinger equation: [(-ZZ2/2m) d2/dz2+ Voo(z) - En]xn(z) = 0 where z is the atom-surface distance normal to the surface. However, analysing data in this way takes account only of resonance positions and ignores the informa- tion contained in intensities and lineshapes. In order to extract this information it is necessary to perform close-coupling calculations using the full corrugated interac- tion potential.Such comparisons have been carried out in a few case^^-^ but are computationally very expensive, particularly for strongly corrugated surfaces. It t Present address: Chemistry Department, University of Alberta, Edmonton, Alberta, Canada. $ Present address: Physics Division, Naval Weapons Center, China Lake, California 93555, U.S.A. $ Stokes Research Fellow, Pembroke College, Cambridge. 4748 VERY LOW-ENERGY He-LiF SCATTERING has not yet been possible to determine a complete atom-surface potential for a strongly corrugated system from an iterative comparison of diff ractive scattering experiments and close-coupling calculations. From a theoretical point of view, data obtained at very low scattering energies E offer several advantages.First, and most important, the experimental data at such energies are much more sensitive to small changes in the attractive part of the interaction potential than are high-energy data. For the He-Xe/C system,' for example, it has been shown that selective adsorption resonances for k = 4.0 A-1 ( E = 8.4 meV) can distinguish clearly between potentials which appear very similar for k = 6.5 A-' ( E = 22.1 meV). In addition, close-coupling calculations are much cheaper for low scattering energies, since the computer time required is proportional to the cube of the number of channels carried, and the number of open channels is roughly proportional to E. The He-LiF system is one of the basic prototypes for atom-surface scattering experiments. Accurate measurements of high-energy diff ractive scattering7 and selective adsorption resonances8y9 are available, and inelastic effects have also been investigated in detail." Recently, Celli et aL6 have obtained a realistic semiempirical potential for this system by assuming pairwise additivity of intermolecular forces, and have compared the results of close-coupling calculations on this potential with the experiments of Boato et a1.' at E = 63 meV and Frank1 et aL8 at E = 17 meV.They obtained quite reasonable agreement between experiment and theory. Their He-LiF potential is strongly corrugated, with a corrugation amplitude (peak to trough) of ca. 0.57 A in the repulsive region and a well depth of ca. 8.6 meV. The purpose of the present paper is to describe very low-energy scattering experiments for He-LiF and to compare these with close-coupling calculations on a realistic atom-surface interaction potential.Our original intention was to use the data to optimise parameters of the potential, but that has not yet been possible because we have only recently obtained data of the necessary quality. Nevertheless, the availability of the potential of Celli et aL6 was very important, because we were able to use calculations on this potential to guide our experimental programme. EXPERIMENTAL The apparatus employed in the measurements is a modified version of the atomic-beam diffractometer described in ref. ( 1 l), which consists basically of two vacuum chambers. The first chamber contains the source, while the second houses the crystal manipulator and beam detector.The most important modifications involve the addition of a supersonic atomic-beam source which can be cooled to temperatures in the region of 20 K, and the alteration of both beam cross-section and detector collimation geometries, which are now rectangular (instead of circular) and permit better angular resolution at the same beam intensity levels or better sensitivity at the same resolution. Fig. 1 shows a schematic view of the experimental apparatus. The new variable-energy beam source was constructed using a 20 pm molybdenum electron-microscope aperture, situated at the tip of the coldest stage of a CTI model 22 cryocooler mounted on a flange capable of small x and y displacements. With these movements the position of the nozzle can be optimized in front of a 290 p m diameter skimmer, which can in turn be moved in the z direction to optimize its distance from the nozzle.The nozzle is mounted on a small hollow copper block, filled with copper wire, where the Hegas becomes accommodated to the temperature of the cryotip before expanding into vacuum. After purification in a liquid- nitrogen-coofed trap, the He gas is fed to the source through a stainless-steel tube of small diameter, a heat exchanger located at the 77 K station of the cryocooler and a final section of thin-walled stainless-steel capillary. In order to maintain the source at temperatures higher than the operating temperature of the cryotip (ca. 15 K under our conditions), a heater andL. DANIELSON et al. 49 I I 1500 dm3 s-' I I I LI I 4600 dm3 s-' I ' " ' I I I Fig.1. Schematic view of the experimental apparatus, showing: n, nozzle; sk, skimmer; ch, chopper; Cz, beam-defining collimator and b, bolometer. a temperature sensor have been installed; with the help of a feedback system the source can be maintained at any temperature between 20 and 300 K for several hours, with a stability estimated to be of the order of 0.5 K. After the skimmer, the beam passes a shutter, a chopper, a coarse collimator C1 (3 mm in diameter, not shown in fig. 1) and finally a beam-defining collimator C2, whose dimensions are 0.2 mm ~ 2 . 0 mm (the longer dimension being in the vertical direction). The distance between the nozzle and the beam-defining collimator is 400 mm, so that the angular divergence of the beam is ca.0.3" in polar angle 8 and 0.03" in azimuthal angle 4. The beam is detected by a semiconductor bolometer of dimensions 0.3 mm x 3.0 mm, located 30 mm from the crystal. The bolometer has an operating temperature of 1.7 K, a responsivity of 3 x lo5 V W-I and a noise-equivalent power of W Hz-''*. A standard lock-in amplifier and microcomputer sequence take care of signal amplification, integration and storage. In this way we have obtained beam intensity levels sufficient for diffraction measurements down to beam energies of ca. 4meV. For the experiments described here, the dispersion in beam velocity is CQ. 2% (f.w.h.m.). The LiF crystal used in our study (obtained from Harshaw Chemicals) was first hardened by y-irradiation, then cleaved in air and finally mounted on the crystal manipulator.After the system had been evacuated in the shortest possible time and the cryostat shields had been cooled to liquid-nitrogen temperatures, the crystal was typically heated to 550 K overnight while the vacuum outside the liquid-nitrogen shields was in the Pa range. The crystal was heated to 700 K two hours before the experiment and then cooled to 77 K just before transferring liquid He into the cryostat that cools the detector and cryopumps the environment around the manipulator. RESULTS The observed peak intensity of the specular beam for He-LiF as a function of polar angle 8 for k = 4.01 A-' ( E = 8.40 meV) is shown in the upper parts of fig. 2, 3 and 4 for azimuthal angles 4 = 45, 46 and 47", respectively. These results were obtained by scanning through 4 for each value of 8 (varied in ca.0.5" steps) and identifying 4 = 45" as the point about which the scans showed reflection symmetry.'*50 VERY LOW-ENERGY He-LiF SCATTERING 0.8 0.6 : 0 . 4 0 * 0.2 0.0 0.8 5 CI 0.6 0.4 0.2 0 0.0 D I 20 40 60 80 O / O Fig. 2 Comparison of experimental scattering intensities (upper) with close-coupling calcula- tions (lower) for k = 4.01 A-' and 6 = 45". The measured and calculated points are denoted by dots; the lines are simply interpolating cubic splines. The positions of zeroth-order resonances calculated from eqn (1) and (2) are shown above the calculated spectrum, labelled with the resonant G vector. It should be emphasised that a very small change in 4 ( < 0.5") can cause considerable changes in intensity (>50'/0), so that it is very important to calibrate 4 accurately for each value of 8.Preliminary scans gave resonance spectra quite different from those shown in the figures because of drift in the actual value of 4 from one value of 8 to the next.L. DANIELSON et al. 51 0.6 0.4 4 -.. + 0.2 0.0 0.8 0.6 0-4 0.2 - +- \ s, 0 0.0 20 40 60 80 o/ O Fig. 3. Comparison of experimental scattering intensities (upper) with close-coupling calcula- tions (lower) for k = 4.01 A-' and 4 = 46". CLOSE-COUPLING CALCULATIONS The formalism required for performing close-coupling calculations of atom- surface scattering has been described by W01ken.I~ In the present work, all open channels and all closed channels with IK + GI < 8.4 &' (3) were included in the basis set.The size of this basis set varies slightly with 8, but it typically consists of 43-50 channels. It includes all channels which can give rise to selective adsorption resonances and gives specular intensities converged to f For 4 = 45" the reflection symmetry allowed this basis set to be reduced to 24-27 channels. The coupled equations were integrated from z = 1.5 to 12.0 8, with a grid spacing of 0.08 A using the log-derivative method.14 The calculations used ca. 1.3 s per angle on a CRAY 1s computer (0.4 s per angle for 4 = 45").52 0.5 0.4 20.3 0.2 0.1 0.0 0.8 0.6 0.4 0.2 ---. 'r 4" --. 4 VERY LOW-ENERGY He-LiF SCAT-I-ERING 20 40 60 80 01 O Fig. 4. Comparison of experimental scattering intensities (upper) with close-coupling calcula- tions (lower) for k = 4.01 k1 and = 47".The He-LiF potential of Celli et aL6 is written as a pairwise additive sum of helium-ion potentials v(R 2) = Knd(R, z ) + c y(pi) (4) 1 where K,d(R, z ) is the induction energy, which may be calculated analytically as described by Steele.15 The summation runs over all lattice ions, with pi denoting the distance between the He atom and ion i. The individual helium-ion potentials V , (pi) are of the semiempirical 'Hartree-Fock + damped dispersion' form:16 K ( p i ) = A i exp (-bipi) - (c6i/p?)f(pi) ( 5 ) where the A and b coefficients for He-F- and He-Li' were taken from first-order SCF calculations and the He-F- C6 coefficient was determined empirically fromL. DANIELSON et al. Table 1. Parameters of the pair potentials for helium-ion interactions [eqn (5)] ~~~ parameter Celli er aL6 modified AdeV 4408 3500 me^ A6 4425 4134 ALiIeV 650.9 650.9 b L i / k ' 5.092 5.092 bF/ A-' 4.435 4.435 ~ , , , / r n e ~ A" 120 120 the requirement that the resulting potential should reproduce the bound-state ener- gies of Derry et aL9 The damping function f( p i ) is as given by Tang and Toennies.16 The potential parameters obtained by Celli et al.are given in table 1. In the present work the nearest-neighbour F-F distance in the crystal was taken to be 2.84 A and the potential was expanded as a Fourier series, retaining all terms up to V,~(Z). The first 6 Fourier components of this potential have been plotted by Celli et aL6 DISCUSSION Specular intensities calculated for the potential of Celli et al.for k = 4.01 A-' and 4 =45, 46 and 47" are shown beneath the corresponding experimental results in fig. 2, 3 and 4, respectively. The quantity plotted is the square modulus of the appropriate S matrix element, corrected for inelastic scattering using a simple Debye- Waller factor including the Beeby correction." The agreement between experiment and theory is very good, except that the theory overestimates the relative intensity of the peak just above 8 = 40" in all three cases. The absolute intensities are in quite good agreement for 4 = 45", but increasingly poorer agreement for 4-46 and 47". The peak positions obtained for He-LiF in the close-coupling calculations are very close to the resonance angles predicted by the zeroth-order picture of eqn (1) and (2).The positions of the zeroth-order resonances in the K plane are shown in fig. 5 and are indicated for 4 = 45" above the close-coupling results in fig. 2. This agreement is in marked contrast to the situation in the more strongly corrugated system He-Xe/C,3 where it was difficult even to establish an unequivocal assignment of most of the resonant features. Fig. 5 immediately explains the pronounced changes in resonance structure for small changes in 4 : for 4 = 45" the resonances due to G = (gl, g,) and (g,, 8,) are degenerate and so give rise to a single peak in the spectrum; however, as 4 alters from the symmetry direction the degeneracy is lost and the peaks split and shift. The only peaks which do not change greatly with 4 are those for G = (1, l), which are non-degenerate because g, = g,.As mentioned above, the beam-velocity dispersion for the experiments described here is ca. 2% f.w.h.m. In view of the extreme dependence of the calculated intensities on the incident angles 8 and 4, it is worthwhile to investigate the velocity dependence of the resonance structure. We have therefore performed calculations for k = 4.04 and 4.08 A-1 for 4 = 45". The results showed the expected small shifts in resonance positions, but no marked change in the overall pattern of intensities and lineshapes. We may thus conclude that averaging over the experimental velocity distribution is not of primary importance for the present experimental conditions.54 90 60 50 40 ' 30 0 20 10 0 VERY LOW-ENERGY He-LiF SCATTERING r 1 I I I I I I I I 0 10 20 30 40 50 70 90 OJ" Fig.5. Positions of zeroth-order resonances in the K plane for k = 4.01 A-' using bound-state energies calculated from eqn (2) for the potential of Celli et al. The solid arcs are resonance energies and the dashed arcs are thresholds. The straight dotted lines indicate the sections of the plot cut by resonance scans for 4 = 45, 46 and 47". The most obvious likely deficiency of the interaction potential of Celli et ai. is in the repulsive part. They used He-F- potentials calculated for the free F- ion, but it is known that the effect of the ionic lattice is to cause a shrinkage of negative ions.'8 We have therefore considered a modified potential in which the parameters of the He-F- repulsion were altered as given in table 1, which results in a shrinkage of the F- ions by ca.0.05 A. The He-F- C, coefficient was then reoptimised to reproduce the bound-state energies determined by Derry et aL9 The results of close-coupling calculations on this potential are shown in fig. 6; there is no marked change in the resonance spectrum between these two potentials. We have not yet carried out a systematic study of the sensitivity of the resonance spectra to potential parameters, but this is clearly the next step. It will be necessary to investigate the sensitivity of the resonance scattering to features such as anisotropy of the pair potentials and many-body contributions, as well as to the parameters of the isotropic pair potentials. The low-energy scattering data are the easiest to1 .o 0.8 0.6 0.4 P --.r L. DANIELSON et al. 0.2 0.0 20 40 60 80 0 55 e l O Fig. 6. Close-coupling calculations for k = 4.01 A-' and 4 = 45", using the modified potential with smaller F- ions. calculate, but it will probably be necessary to perform a simultaneous fit to high- energy apd low-energy data to determine a reliable interaction potential. Additional information is also provided by the intensities of non-specular diffracted beams. Our early attempts to optimize potential parameters for He-LiF from low-energy scattering data led us down several blind alleys. It is perhaps useful to mention one approach in particular which did not work, so that other investigators may avoid the same problem. Initially, we attempted to perform least-squares fits of potential parameters directly to diffraction intensities at particular values of 8, 4 and k In retrospect this was never likely to be successful, for two reasons.First, measurement errors of a fraction of a degree in 8 or 4 can have drastic effects on the intensities, which completely mask any effects due to changes in the interaction potential. Secondly, the scattering intensities are very fast-varying non-linear func- tions of the potential parameters, so that least-squares fitting algorithms do not converge. It is absolutely essential to perform scans of scattering intensities as a function of 8 and 4, and to fit to the positions and intensities of peaks rather than to intensities at particular angles. CONCLUSION We have performed very low-energy scattering experiments for the system He-LiF, and have compared the results with close-coupling calculations on the potential of Celli et aL6 The agreement is very good, but there are small discrepancies which which should allow refinement of the interaction potential.The scattering intensities are dominated by selective adsorption effects and are very fast functions of both polar and azimuthal angles. It is a pleasure to thank Daniel Mitchell and Tom Ellis for their help during the course of the experiments. One of us (J-C.R.S.) would like to acknowledge support from La Universidad Autonoma de Puebla and a scholarship from CONACyT, Mexico.56 VERY LOW-ENERGY He-LiF SCATTERING H. Hoinkes, Rev. Mod. Phys., 1980, 52, 933. Verlag, Berlin, 1982), p- 61. J. M. Hutson and C. Schwartz, J. Chem. Phys., 1983, 79, 5179. R. Schinke and A. C. Luntz, Surf: Sci., 1983, 124, L60. H. J6nsson, J. H. Weare, T. H. Ellis, G. Scoles and U. Valbusa, Phys. Rev. B, 1984, 30, 4203. V. Celli, D. Eichenauer, A. Kaufhold and J. P. Toennies, J. Chem. Phys., 1985, 83, 2504. G. Boato, P. Cantini and L. Mattera, Surf: Sci., 1976, 55, 141. 41, 60. G. Derry, D. Wesner, S. V. Krishnaswamy and D. R. Frankl, Surf: Sci., 1978, 74, 245. lo G. Brusdeylins, R. B. Doak and J. P. Toennies, Phys. Rev. Lett., 1981, 46, 437; R. B. Doak and J. P. Toennies, Surf: Sci., 1982, 117, 1 ; G. Brusdeylins, R. B. Doak and J. P. Toennies, Phys. Rev. B, 1983, 27, 3662; G. Lilienkamp and J. P. Toennies, J. Chem. Phys., 1983, 78, 5210. G. Caracciolo, S. Iannotta, G. Scoles and U. Valbusa, J. Chem. Phys., 1980, 72, 4491. * K. H. Rieder, in Dynamics ofGas-Surface Interaction, ed. G. Benedek and U. Valbusa (Springer- * D. R. Frankl, D. Wesner, S. V. Krishnaswamy, G. Derry and T. O’Gorman, Phys. Rev. Lett., 1978, l2 G. Boato, P. Cantini, C. Guidi, R. Tatarek and G. P. Felcher, Phys. Rev. B, 1979, 20, 3957. l 3 G. Wolken, J. Chem. Phys., 1973, 58, 3047. B. R. Johnson, J. Comput. Phys., 1973, 13, 445. W. A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1974). 14 l6 K. T. Tang and J. P. Toennies, J. Chem. Phys., 1984, 80, 3726. l7 J. L. Beeby, J. Phys. C, 1971, 4, L359. R. P. Hurst, Phys. Rev., 1959, 114,746; G. D. Mahan, Solid State Ionics, 1980, 1,29; P . W. Fowler and P. A. Madden, Mol. Phys., 1983, 49, 913.

ISSN:0301-7249    

DOI:10.1039/DC9858000047

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. E. Zaremba (Queen’s University, Kingston, Ontario) said: I would like to comment on the relationship between the repulsive part of the helium surface potential and the electronic charge density, specifically for metallic surfaces. It is frequently stated that Esbjerg and Norskov have ‘established’ the proportionality of the potential to the charge density, an appealing result from the point of view of surface structure determinations. That there is a correspondence between the potential and charge density is obviously true, but a strict proportionality is not. The original Esbjerg-Norskov suggestion was based on calculations of the embed- ding energy of helium in a uniform electron gas for which a linear dependence on the mean electron density is indeed found.Even if such a result were to apply to the surface potential problem, the coefficient of proportionality would be modified by the fact that the surface electron density is inhomogeneous. However, the true situation is more complex in that the repulsive potential is actually determined by an energy average of I-dependent scattering amplitudes weighted by an appropriately defined local density of states.’ Only in the asymptotic region beyond that of physical interest is a proportionality to density valid (as first pointed out by Zaremba and Kohn), but the constant of proportionality is substrate-dependent rather than a universal constant. Although the Esbjerg-Norskov suggestion can provide a useful guide, it cannot be assumed to be quantitatively reliable. E.Zaremba and W. Kohn, Phys. Rev. B, 1977,15, 1769; J. Harris and L. Liebsch, J. Phys. C, 1982, 15, 2275. Dr J. E. Black (Brock University, St. Catharine’s, Ontario) said: Dr Cardillo remarked informally that: ‘only altering force constants, when the surfaces also relax and reconstruct, may not be appropriate’, to which I would comment: The models used in the present lattice dynamical studies are admittedly simple, but they are the best available. While it is true that the gold(ll1) surface reconstructs, evidence is that others, such as silver( 11 1) undergo no reconstruction or relaxation. Prof. M. W. Cole (Pennsylvania State University, U.S.A.) said: I would like to make several comments concerning the interaction V ( r ) between a He atom and the system consisting of a Kr layer adsorbed on graphite.Before doing so, let me make two remarks. One is that there is, overall, rather gratifying consistency between the alternative potentials being discussed and hence with the experimental data of Larese et aL;’ the remaining discussion pertains to 10% effects. The second comment is that we have not performed close coupling calculations, such as those of J6nsson and Weare (henceforth denoted JW), which are necessary ultimately for resolving differences between ‘competing’ potential models. Nevertheless, we believe that our conclusion’ that three-body effects are present may well be correct, in disagreement with that of JW. When we began our study, we enquired about the role of three-body effects. John Barker suggested that we adopt the Axilrod-Teller-Muto (ATM) triple-dipole potential; this term alone is found to be successful in predicting in detail the behaviour of bulk noble-gas system^.^ A recent review by Meath and Aziz4 found use of the ATM term to be ‘the method of choice for representing non-additive 5758 GENERAL DISCUSSION effects, with due caution'.In the surface domain, the ATM term has been found to be appropriate in calculating both the surface tension of liquid Ar5 (J. Miyazaki, J. A. Barker and G. M. Pound, J. Chem. Phys., 1976,64, 3364); and in treating the statistical mechanics of adsorbed gases,6 where the three-body interaction is the ca. 15% 'McLachlan correction' due to the substrate. Based on this experience, we included the ATM term in our He potential.The net difference between our potential (CHC) and that of JW has several contributions, two of which are nearly negligible; this leaves A V V&, - VJW = VAT,( r ) - ( C:Hc - Ciw)( z + b)-3 where the choice b = 1.9 A is derived by Chung et al. as an accurate approximation to the sum over graphite planes. We refer here to the potential model 1 of JW for specificity. Here is the important point: we chose CFHC = 180 meV - A3, the theoretical value of Vidali et al.,' while JW used C:" = 110 meV - A3 semiempirically. One can discuss at length the merits of either choice. Instead, I prefer to discuss the relevant issue of uniqueness. To be specific, A V is -0.22 It 0.02 meV in the region 3 < z / A < 4 above the Kr layer. This is a very small magnitude (<20% of the total graphite contribution and 5% of the total potential); it is so small because the larger attraction of the CHC potential is partially cancelled by the repulsive ATM term. Thus the presence (or absence) of the latter is detectable only if the C3 disagreement is resolved.This requires, in part, a more precise assessment of the shallowest bound states for graphite. If the previous data' are confirmed, interesting questions will be raised about the theoretical value of C,. In summary, quite similar potentials have been proposed by CHC and JW. The difference between them is smaller than or comparable to both the triple-dipole term and the uncertainty in the graphite potential. Thus I believe that it is premature to draw any conclusion about the importance of the various terms contributing to these potentials.Note added in proof: New measurements by Chung et aL9 revise the situation for He reacting with bare graphite. The data are consistent with predictions of Toigo and Cole" based on the theoretical value of C3; this confirms, in my opinion, the analysis of Chung et aL2 of the adlayer problem. ' J. Z. Laresse, W. Y. Leung, D. R. Frankl, N. Holter, S. Chung and M. W. Cole, Phys. Rev. Lett., ' S. Chung, N. Holter and M. W. Cole, Phys. Rev. B, 1985, 31, 6660. 1985, 54, 2533. J. A. Barker, in Rare-gas Solids, ed. M. L. Klein and J. A. Venables (Academic Press, New York, 1976), vol. 1. W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. J. Miyazaki, J. A. Barker and G. M. Pound, J. Chem. Phys., 1976, 64, 3364; J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Oxford University Press, Oxford, 1982), p. 184. L. W. Bruch, Surf: Sci., 1983, 125, 194. G. Derry, D. Wesner, W. Carlos and D. R. Frankl, Surf: Sci., 1979, 87, 629. S. Chung, A. Kara and D. R. Frankl, SurJ Sci., in press. l o M. Toigo and M. W. Cole, Phys. Rev. B, 1985, 32, 6989. ' G. Vidali, M . W. Cole and C. Schwartz, Surf: Sci., 1979,87, L273. Mr H. Jonsson and Prof. J. H. Weare ( University of California, San Diego, U.S.A.) (communicated): The potential used by Chung, Holter and Cole' is sufficiently different from our best-fit potential to give intensities in clear disagreement with the experimental data. It is furthermore qualitatively different from our theoretical potential in the approach to the He-graphite contribution.Fig. 7-9 in our paperGENERAL DISCUSSION 59 show the results of close-coupling calculations using a potential (potential 2) very similar to that of Cole and coworkers. The difference in the bound-state energy levels is only *0.02 meV, except for the n = 2 level where the potential used in the calculation is 0.05 meV closer to the best-fit value. There is very clear disagreement between this calculation and the experimental data, whereas our best-fit potential gives extremely close agreement with experimental data (fig. 10 in our paper). The selective absorption spectrum of highly corrugated surfaces is a remarkably sensitive probe of the bound-state energy levels of the atom-surface potential. It is precisely this high sensitivity which makes it possible to see the many-body correction. In our view there is no reason to assume the C3 coefficient calculated using a dielectric function fitted to reflectivity data, is highly accurate (not to mention C,, C5 etc.).Our approach is to use experimental data on the He-graphite interaction itself to construct a complete surface potential. Unless the experimental data are in error, this approach determines the He-graphite potential in the relevant region quite well. In ref. (2) Cole and coworkers presented these scattering data in support of the empirical rule of only including long-range three-body corrections. We agree with Prof. Cole’s current position that it is premature to draw such conclusions. ’ S. Chung, N. Holter and M. W. Cole, Phys.Rev. B, 1985, 31, 6660. J. Z. Larese, W. Y. Leung, D. R. Frankl, N. Holter, S. Chungand M. W. Cole, Phys. Rev. Let?., 1985,54, 2533. Dr J. A. Barker (IBM, Sun Jose, U.S.A.) said: I want to comment on the question raised in the subtitle of the paper by Jonsson and Weare,’ ‘What are the Non-additive Many-body Corrections,’ and in particular on the comment ccncerning earlier work by myself and my colleagues on rare gases, that ‘the good agreement (with experi- ment) obtained when only the Axilrod-Teller-Muto term was included appears to be fortuitous’. This latter statement is based on and echoes the conclusion of a paper by Meath and Aziz.’ That paper exhibited excellent agreement between ‘experimental’ many-body energies of rare-gas crystals at zero temperature and pressure and theoretical values calculated using only long-range many-body interac- tions (see table l ) , but concluded that this must be regarded as fortuitous since the long-range many-body interactions appeared to be substantially cancelled by first- order exchange three-body interactions.The agreement shown in table 1 confirms extensive earlier work, reviewed for example in ref. (3). Given that there is no adjustable parameter in the many-body calculation it is unlikely to be fortuitous. If this is coincidence it could more aptly Table 1. Total binding energies and long-range non-additive many-body energies of rare-gas crystals, from Meath and Aziz’ (units are J mol-’) Ne Ar Kr Xe total binding -1927*6 -7675*52 -11 190*84 -15902*89 total binding -1933*8 -7722I-t 11 -11 148* 13 -15 784*35 energy, calculated energy, experimental energy, calculated energy, experimental long-range many-body 81 * 6 663 f 52 1042*84 1397*89 total many-body 75*8 616* 11 1084* 13 1515*3523 22 21 20 19 18 17 16 15 vo~ume/cm~ mol-' Fig.1. Pressure-volume relation for argon at 0 K [from ref. (3), chap. 41. The solid curve includes pair potential and long-range many-body interactions, the dash-dotted curve includes also the nearest-neighbour first-order three-body exchange interaction; the dashed curve uses pair interactions alone. 20.0 22.0 21.0 volume/cm3 rnol-' Fig. 2. Pressure-volume relation for argon on the melting line in fluid (upper curve) and solid (lower curve) phases. From data in ref. (1 1). The curves are Monte Carlo results using pair interactions and long-range many-body interactions; the circles are experimental data.GENERAL DISCUSSION 61 3000 L 2000 z t' E 1000 0 I I I I I I I 1 100 200 300 400 T / K 10 Fig.3. Third virial coefficients for argon [from ref. (3), chap, 41. The dashed curve uses pair interactions alone, the dotted curve includes triple-dipole interaction, the dashed curve also includes third-order dipole quadrupole interactions and the solid curve also includes fourth- order dipole interactions. The circles are experimental data. be called conspiracy on the part of nature. Furthermore, the agreement is not confined to zero temperature and pressure or to the solid phase, as shown by comparison of calculated and experimental pressures for solid and fluid argon in fig. 1 and 2.Similar results are given in ref. (3) for neon, krypton and xenon. Independent evidence is provided by third virial coefficients as shown in fig. 3; values calculated with pair interactions alone are much too small, while those calculated with long-range three-body interactions agree well with experiment. Also shown in fig. 1 are pressures for solid argon calculated by including the first-order three-body exchange energy for nearest-neighbour triangles as calculated by Bulski and Chala~inski.~ These values are in disagreement with experiment. If this interaction is to be cancelled by some dther many-body interaction so as to give agreement with experiment, then this hypothetical interaction must have essen- tially the same density dependence (and opposite sign). This seems unlikely; it is almost equally likely that the calculated values represent an overestimate of the first-order three-body exchange energy.It is noteworthy that Wells and Wilson' find, for the Hartree-Fock contribution to the three-body energy of three helium atoms in an equilateral triangle of side 5.6a, a value which is smaller by a factor of 4.7 than the value of the first-order three-body exchange energy found by Novaro and Beltran-Lopez,6 who used methods roughly comparable with those of ref. (4). It is clear that overlap-dependent many-body interactions are much more difficult to calculate ab initio than the long-range interactions which can be related to oscillator strengths, and I suggest that the problems with many-body interactions raised by Meath and Aziz,* by Jonsson and Weare,' and in a later paper in this Discussion by Bruch and Ni7 are really problems of ab initio theory and computation; the experimental evidence that the long-range many-body interactions give the dominant many-body contribution is extremely strong.I would also like to suggest that the difficulty experienced by Jonsson and Weare in fitting the data for He scattering from Kr overlayers may be due at least in part to the He-Kr pair interaction, whi'ch I believe has larger uncertainty than they62 GENERAL DISCUSS I 0 N Table 2. He-Kr pair potentials r.m.s. deviation of scattering data r.m.s. deviation of depth, room second virial potential ( € / k ) / K liquid N2 temperature coefficients ESMSV~ U ( 0, K)'O HFD-19 HFD-29 24.7 0.6 1.6 29.1 0.6 1.7 33.7 1.2 1.5 30.8 0.4 2.2 0.7 suggest, especially with respect to the attractive well to which the selective-adsorption data are most sensitive.This is illustrated by the data in table 2, taken from ref. (9) and (lo), which suggest that potentials with depths as low as 24.7 K and as high as 33.7 K can fit the differential scattering cross-section data and second virial coefficients, which are the data most sensitive to the attractive well. The fourth potential in table 2 is that adopted by Jonsson and Weare.' The ESMSV potential does not fit gas-transport data, but these are not sensitive to the attractive well. Clearly these are extreme variations, but it does not seem unreasonable to suppose that the depth of the attractive well is uncertain by ca.5% or perhaps even 10%. The spectroscopic bound-state data which determine the attractive well very closely for like-atom pairs are not available for unlike pairs. ' H. Jonsson and J. H. Weare, Furuduy Discuss. Chem. Soc., 1985, 80, 29. W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. Rare-gas Solids, ed. M. L. Klein and J. A. Venables (Academic Press, New York, 1976), vol. 1. M. Bulski and G. Chalasinki, Chem. Phys. Lett., 1982, 89, 450. V. A. Novaro and V. Beltran-Lopez, J. Chern. Phys., 1972, 56, 815. L. W. Bruch and X-Z. Ni, Furuduy Discuss. Chem. SOC., 1985, 85, 217. C. H. Chen, P. E. Siska and Y . T. Lee, J. Chem. Phys., 1973, 59, 601. K. M. Smith, A. M. Rulis, G. Scoles, R. A. Aziz and V. Nain, J. Chem. Phys. 1977, 67, 152.' B. H. Wells and S. Wilson, Mol. Phys., 1985, 55, 199. lo K. Watanabe, A. R. Allnatt and W. J. Meath, Chem. Phys., 1982, 68, 423. " J. A. Barker and M. L. Klein, Phys. Rev. B, 1973, 7 , 4707. Prof. W. J. Meath (University of Western Ontario, London, Ontario) said: In reply to Dr Barker's comments, I would like to begin by reviewing, briefly, the state of affairs regarding various non-additive interactidn energies that can have the same magnitude as the triple-dipole energy for relevant interatomic or intermolecular configurations. Of particular interest are configurations of importance in the evalu- ation of the binding energy of rare-gas crystals. For details, including many refer- ences, see Meath and Aziz,' ref. (39) of Mr J6nsson and Prof. Weare's paper.Using Ne, Ar, Kr and Xe as examples, it is clear that the non-additive contribu- tion, AE(O), to the crystal binding energy E ( 0 ) of the rare-gas solids at OK is substantial, i.e. much larger than the experimental error in E ( 0 ) ; AE(0) is ca. 10% of E ( 0 ) for Ar, Kr and Xe and is evaluated' by subtracting the additive contribution to E(O), as calculated using reliable two-body potentials, from the experimental results for E ( 0 ) . Often the non-additive part of the many-body potential is repre- sented by the triple-dipole dispersion energy; this is the three-body analogue of the dipole-dipole dispersion energy occurring in the interaction between two atoms or molecules. The results so obtained for AE(0) are in good agreement with the known values of the non-additive contribution to the crystal binding energy.ForGENERAL DISCUSSION 63 what follows it is important to realize that 60% of AE(0) is from the first-shell (nearest-neighbour) lattice sum. This contribution arises from equilateral triangular configurations of the rare-gas atoms of side equal to the nearest-neighbour distance Ro in the crystal; Ro == R,, the position of the van der Waals minimum in the two-body potential. There are other terms in the non-additive part of the many-body potential: the question is, are there any having the same magnitude as the triple-dipole dispersion energy for important atomic configurations? Relatively speaking much less is known about these non-additive energies than is known about the energies in the two-body potential that have the same magnitude as the (analogous) two-body dipole-dipole dispersion energy.However, what is known suggests that there are non-additive energies that can compete with the triple-dipole energy. The best understood ‘competitor’ is the three-body first-order exchange energy Ex(’) (NA) for which results are available for (He),, (Ne),, (Ar), and the quartet spin state of H3. The quartet state of H3 can be regarded as a model for rare-gas trimers that is analogous to using H2(3Z3 to model rare-gas dimers.* Exact results for E,“’(NA) for H3 have been available for some time. Reliable results for the rare-gas trimers, for relevant interatomic distances, have been available only recently; they have been computed’ in the Heitler-London method using quality SCF wavefunctions to represent the isolated atoms and in the SCF trimer scheme with corrections for superposition errors (both techniques agree well with each other).These are difficult and expensive calculations, and data are sparse, but fortunately results for the first-order three-body exchange energy are available for the atomic configurations of most importance to the crystal binding energy calculations. In all cases Ex(’)( NA) is negative for equilateral triangular configurations of side R , and cancels part, all, or more than all, of the triple-dipole contribution to AE(0) leading to a disagreement with experiment if the non-additive part of the many-body potential is represented by the sum of the triple-dipole and the first-order three-body exchange energies. There is evidence to support the view that the SCF atom is smaller than the real atom and therefore the magnitude of E,‘”(NA) is underesti- mated, not overestimated, by using SCF wavefunctions to represent the isolated atoms; for example dipole polarizabilities evaluated at the SCF level of calculation are smaller than the exact polarizabilities for the type of species of interest to this discu~sion.~ Further the recent calculations of Wells and Wilson’ support the results of Jeziorski et aL3 for the first-order three-body exchange energies of (He), in equilateral triangular configurations and not those of Novaro and Beltran-Lopez.6 This can be seen by plotting In lEx(l)(NA)l, as a function of the side of the equilaterial triangle, from the three sources of the calculations (fig.4). The calculations in ref. (6) take no account of superposition effects and their reliability decreases rapidly as the side of the equilaterial triangle increases past 5 bohr. I would like to emphasize that Dr Aziz and I in no way suggested using the sum of the triple-dipole and first-order three-body exchange energies to represent the non-additive part of the interaction energy. Rather we used this sum of two energies to give an example of a cancellation effect between two non-additive energies of the same magnitude that occurs for relevant interatomic separations in a well defined application. We then went on to discuss other possible non-additive interaction energies, or other effects, that could be significant in helping to supply the ‘missing’ contribution to AE (0).Dr Aziz and I tried very hard to provide an unbiased presentation of the difficulties associated with the representation of many-body interaction energies.64 GENERAL DISCUSSION 3 4 5 6 Fig. 4. Plot of In lEx(’)(NA)/ against the side of the equilateral triangle for (He), from the three calculations discussed in the text: x, Polish group; A, Wells and Wilson; 0, Novaro and Beltram-Lopez. side We have apparently succeeded in this since, for example, at this Discussion three papers used our work to support three different points of view, i.e. (1) that the triple-dipole energy is not adequate, (2) that it is adequate and (3) that the representa- tion of many-body potentials is not well understood. Perhaps the best way to end this comment is to provide a few quotations’ from ref.(39) of Prof. Weare’s paper. ‘What is needed are reliable calculations of the many-body potential that are consistent through all terms of the same order of magnitude.. .’; ‘The authors hope this paper will serve as a stimulus for further a6 initio calculations of many-body interaction energies that are designed to help resolve this difficult problem.. .’; ‘An important point must be made clear. There is little doubt that the total many-body interaction often seems to be effectively represented by the triple-dipole interaction energy especially when coupled to a reliable two-body potential. The evidence for this can be found in papers by Barker and coworkers, and others, that are concerned with the evaluation of third virial coefficients as well as liquid and crystal properties of the rare gases; the references in ref.(1) provide details. Since, at present, very little is known about the other relevant non-additive forces needed to counter-balance the cancellation between the non-additive dispersion and the three-body first-order exchange energies, this approach is still probably the method of choice for representing non-additive effects,GENERAL DISCUSSION 65 with due caution, in the immediate future. However, there are excellent reasons for concern and a recent example augments the earlier discussions on this point’. (This is discussed briefly, with references, in my next comment.) ’ W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. These model dimer potentials have been quite successful; see references in ref.(1) and, for example, the detailed review by R. A. Aziz, Springer Ser. Chem. Phys., vol. 34, ed. M. L. Klein (Springer, Berlin, 1984), pp. 5-86. Of particular importance in this context are the relatively recent calculations of G. Chalasinski, M. Bulski, B. Jeziorski and L. Piela, Chem. Phys. Lett. 1982, 89, 450; 1981, 78, 361; Theor. Chim. Am, 1980, 56, 199; Znt. J. Quantum Chem., 1976, 10, 281. For example see H. J. Werner and W. Meyer, Phys. Rev. A, 1976, 13 and Mol. Phys., 1976,31,855. B. H. Wells and S. Wilson, MoZ. Phys., 1985, 55, 199. U. A. Novaro and V. Beltran-Lopez, J. Chem. Phys., 1972, 56, 815. Prof. S. J. Sibener ( University of Chicago, U.S.A.) said: I would like to mention that the well regions of the pair potentials between helium and heavy rare-gas pair potentials (Ne, Ar, Kr, Xe) may still be a little uncertain. The last experimental differential scattering cross-sections were taken at collision energies that were, I believe, larger than the well depths of these systems.It would be really nice if data were available from experiments which used cryogenically cooled He expansions operating at $10 K. Such measurements would be expected to confirm the accuracy of the current potentials, or lead to further (slight) refinements. It is obviously of critical importance that the most accurate pair potentials be used as input to gas-surface scattering calculations which seek quantitatively to assess higher-order contributions to the interaction.Prof. D. R. Frank1 (Pennsylvania State University, U.S.A.) said: Much of the discussion of this paper has hinged on the accuracy of the experimental energy levels. This is very difficult to assess. In addition to the statistical error limits, there may be systematic errors due to various misalignments and to lack of perfect planarity of the crystal surface. These problems are particularly severe for the higher-lying levels, which are the most important for the determination of C,. In addition to the measurement errors, there are the further problems of identifying the resonance signatures and of relating their energies to the actual bound-state energies. For these reasons, the kinematically determined energies should be regarded as, at best, a starting point for a complete scattering calculation.Agreement of the calculated beam intensity pattern is, as Prof. Weare and others have emphasized, the critical test. Mr H. J6nsson and Prof. J. H. Weare ( University of California, San Diego, U.S.A.) (communicated): The evidence, Dr Barker points out, for the validity of only including long-range many-body corrections, is certainly very impressive. There remains, however, the theoretical problem of explaining why this rule works. Theoretical calculations of three-body corrections to the exchange repulsion do not support it. The scattering experiment is in many ways different from the condensed matter experiments discussed by Barker (quasibound state us. bound, heteronuclear us. homonuclear) and applying this empirical rule to the scattering experiment is an extrapolation.Our tentative conclusion is that a significant discrepancy exists between the experimental bound states of He-Kr/graphite and the theoretical potential constructed from best estimates of two-body potentials and Barker’s rule for many-body corrections. However, as we point out in our paper, this experiment can not66 GENERAL [>ISCUSSION yet be considered conclusive. If, as Prof. Frank1 suggests, there is a systematic error in the experimental bound states reported for He-graphite, or if, as Dr Barker suggests, the well depth of the He-Kr pair potential is off by 10°/O, the discrepancy could be removed. By presenting our numbers at this discussion with quantitative estimates of error bars based on some literature values, we hoped to stimulate critical reevaluation of these factors.Prof. Scoles and coworkers are currently remeasuring the He-graphite scattering with very high resolution, and Prof. Aziz er al. have adjusted the He-Kr pair potential simultaneously to all available experimental (two-body) data (seven different properties).’ This new He- Kr pair potential gives larger discrepancy between the measured and calculated bound state of He-Kr/graphite (0.44 meV instead of 0.35 meV). Possibly, additional high-resolution experimental data which particularly probe the well depth of the pair potential are needed, as Prof. Sibener points out. Dr Barker points out a discrepancy between the a6 initio calculations performed by Wells and Wilson’and those performed by Novaro and Beltran-Lopez3 for He, at R = 5 .6 ~ . We have recently been doing first-order calculations following the formalism of Bulski and c o ~ o r k e r s . ~ We find quite good agreement with the calculations of Wells and Wilson, with difference of only 8%. For smaller distances, the restricted Hartree-Fock calculations of Novaro et al. give values that are close to the first-order calculation. At this large distance (5.6u), however, there is a discrepancy, but we note that Novaro et al. only report one significant figure in that case. We calculated the He- Kr-Kr three-body correction to the first-order exchange repulsion. When the Kr atoms are placed one lattice spacing apart (4.26 A) and the He atom placed at the van der Waals distance away from each Kr atom (3.7 A) to form an isosceles triangle, this correction is larger in magnitude by 10% and opposite in sign to the ATM correction.However, the scattering experiment is sensitive to the bound states of the surface potential averaged over the unit cell. Because of the large corrugation and rapid drop in the three-body correction to the exchange repulsion, the shift in the lowest bound state energy level of the average potential is very small, only 10% of the triple-dipole correction. Theoretically, Barker’s rule is therefore better justified for the scattering experiment than the condensed phase experiments. We are grateful to Prof. Meath for pointing out typographical errors in the values given for the ATM coefficients in the preprint of our paper. ’ R. A. Aziz and M.J. Slaman, 9th S>.nip. on 7hermopli,t,qiccrl Propcwies, June 23-27, 1985, University ’ B. H . Wells and S. Wilson, Mol. PI1y.s.. 1965, 55, 199. of Colorado. Prof. G . Scoles (Uniueryity oj’ Waterloo, Ontario) said: ( i ) With respect to the magnitude of the error in the two-body tcrms of the He-Kr-graphite potential the figures quoted by Dr Barker are definitely too large. Indeed for the case of He interacting with Ar it has been established beyond any doubt’ that the well-depth values obtained in Berkeley, quoted by Barker as possible numbers, were much too low, while the results obtained pleviousi): in Waterloo, with well depths around 30 K, were correct ( i i ) While it is true thdt high-resolution differential collision cross-sections similar to those csed to seti’n the He-Ar case, which were obtained in the laboratory of Dr LJ.Ruck in Gottingeil, have not yet been measured for the He-Kr case, it is also true that there is absolutely no reason to believe that the latter system would behave any dittetently from the former one. (iii) Moreover. the recent work of Aziz and coworkers has substantially increased the level of precision with which two-body intermolecular forces are being studied. Since many properties which depend on the two-body interaction only are being fitted at the same time, the resulting potentials have been proved to be as accurate, if not better, than thoseGENERAL DISCUSSION 67 obtained via the highest-resolution scattering and spectroscopic experiments. Since the recent He-Kr potential of Watanabe, Allnatt and Meath, used by Jonsson and Weare, has also passed through the same filter and is in substantial agreement with the Waterloo collision cross-section data, we conclude that the He-Kr interaction is presently known with a very good degree of confidence. Minor adjustments (< 1 '/o on the R axis and <4% on the energy axis) may be still possible but larger ones are quite unlikely.In summary, while Jonsson and Weare, quite correctly, call our attention to all possible sources of error that could invalidate their conclusions, their tentative conclusion about the inadequacy of the triple-dipole correction to account for all many-body effects is not likely to require drastic changes owing to future increases in the precision with which the two-body interaction is known.On the other hand, and for completeness, it should be noted here that our present knowledge of the He-graphite and Kr-graphite interactions, while rather good, is far from perfect and that more work in this area would be very useful. ' R. A. Aziz, P. W. Riley, U. Buck, G. Maneka, J. Schleusenes, G. Scoles and U. Valbusa, J. Chem. Phys., 1979, 71, 2637. Prof. W. J. Meath (University of Western Ontario, London, Ontario) said: The evaluation of the bound states of He interacting with a Kr overlayer on graphite involves the use of He-Kr and He-graphite two-body potentials. The He-Kr potential' is apparently a very good two-body potential which reproduces2 most of the experimental gas-phase properties for HeKr to within experimental error.However, it is based on the simplest form of the XC (exchange-coulomb) potential model, which involves only a single overall damping-corrector function to convert the long-range dispersion energy series into a representation of the second- and higher-order coulomb energies.lv3 The parameter in the representation of the exchange energy was adjusted to fit experimental data for the second virial coefficient, as were the dispersion-energy coefficients within their estimated uncertainty. This simple model is adequate for two-body rare-gas potentials, since the uncertainties in the higher-order dispersion energy coefficients are relatively large. An individually damped dispersion energy version4 of the model is necessary for other types of interactions and will also probably be needed for rare-gas interactions when the dispersion energy coefficients are known more accurately.It is to be emphasized that the adjustments in the He-Kr potential were made to best-fit mixed second virial coefficients for He-Kr. The fact that the potential reproduces most of the other gas-phase properties for the He-Kr system as well gives credence to the model, but it is important to realize that no attempt was made to improve the fit to all the available experimental data. Given the errors inherent in the gas-phase data used to construct and test the XC potential, which are generally larger for mixtures than for pure gases, and the possibility that gas-phase and gas-surface properties may well be sensitive to different ranges of interatomic separations, there is a distinct possibility of small but significant errors being introduced into the calculation of the He-surface binding energies through the use of the XC ;potential.More impor- tantly it seems that the He-graphite potential may well be less well understood than the He-Kr potential. The binding energy co evaluated using the two-body potentials is 2.5% higher than the estimated true energy, which has an error of *2.1% associated with it. On the other hand E~ evaluated using the two-body potentials plus the triple-dipole energy is 6% higher than the estimated true energy, with the contribution due to the triple-dipole energy being positive and only 3.4% of the magnitude of the true68 G E N E RAL r) I sc us s I o N value for e0. The authors are correct in stating that the shift due to the addition of the triple-dipole term is in the wrong direction to yield an overall calculated result value for eo.The authors are correct in stating that the shift due to the addition of the triple-dipole term is in the wrong direction to yield an overall calculated result, in agreement with the estimated correct value of E ~ . This is suggestive of a problem in using the triple-dipole energy to represent the many-body effects in this calculation. However, as much as I would like it to be so, 1 do not think this is a definitive example showing the inadequacy of the triple-dipole energy as a representation of non-additive interaction energies. For example, taking the most unfavourable inter- pretation of the error in the correct result for E~ leaves a discrepamy of 3.9% between calculation and the estimated correct value of E ~ , a difference that could be accounted for by the combined errors in the two-body potentials.The authors do suggest the possibility of their error bars being too optimistic and also discuss another contribution to eO, namely that due to thermal vibrations. There is a more definitive example in the literature that does strongly suggest that there are problems in using the triple-dipole energy to represent non-additive energies in the calculation of bulk properties.556 It involves the analysis, by Teitsma and Egelstaff,6 of the virial expansion of the direct correlation function for Kr gas using measurements of the structure factor of the gas determined by neutron- diffraction techniques.Their analysis of the data gave results for a generalization C(q, T ) of the usual third virial coefficient C(0, T ) , which differs significantly, for q # 0, from results obtained by using the triple-dipole energy as a representation of the many-body forces. C(q, T ) is quite sensitive to many-body effects. K. Wsanabe, A. R. Allnatt and W. J. Meath, Chem. Phys., 1982, 68, 423. ’ R. A. Aziz, Springer Ser. Chem. Phys., vol. 34, ed. M. L. Klein (Springer, Berlin, 1984), pp. 5-86. See, for example, A. Koide, W. J. Meath and A. R. Allnatt, Mol. Phys., 1980, 39, 895; R. A. Aziz, W. J. Meath and A. R. Allnatt, Chem. Phys., 1983, 78, 295 (erratum, 1984, 85, 491). W. J. Meath, D. J. Margoliash, B. L. Jhanwar, A. Koide and G. D. Zeiss, in Intermolecular Forces, ed B.Pullman (Reidel, Dordrecht, 1981), pp. 101-115. W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. A. Teitsma and P. A. Egelstaff, Phys. Rev. A, 1980, 21, 367; P. A. Egelstaff, Adv. Chem. Phys., 1983, 53, 1. Prof. G. Scoles (University of Waterloo, Ontario) said: The very high sensitivity of the atom-surface scattering data to errors in the two-body term of the interaction, which is a prerequisite for safe conclusions about the contributions due to three-body forces should not, in my opinion, be a deterrent to seeking progress in this direction. This is because of three reasons. First, as I mentioned in the previous remark, with little extra work the two-body terms may indeed be obtained with sufficient accuracy. Second, there is, in my opinion, sufficient information in the gas-surface scattering data to extract from them, with a careful analysis, both a better two-body interaction and the three-body contributions.Third, because of the broken symmetry at the surface, three-body forces at the gas-solid interface are bound to be different from their counterparts in thF oulk and to teach us a lot about their importance in liquid cavities, solid defects and similar situations. This makes the problem much more interesting and should represent an extra incentive for working toward its solution. Prof. M. W. Cole (Pennsylvania State University, U.S.A.) said: Could Dr Hutson describe to us the procedure he mentioned for evaluating a two-body’ dispersion interaction for the case of an ionic crystal?GENERAL DISCUSSION 69 Dr J.M. Hutson and Dr P. W. Fowler (Cambridge University) replied: We construct the atom-surface potential using a procedure similar in spirit to that of Celli et aZ.' in that we start with a pairwise additive sum over He-F- and He-Li+ potentials, and represent the pair potentials using an SCF repulsive part and a damped dispersion series for the attractive part. However, there are a number of points of difference. ( 1 ) We calculate SCF repulsive energies involving anions in the crystalline environment, rather than free anions. The crystalline environment is simulated by performing calculations on a cluster made up of a central F- ion at the surface of the crystal and its five nearest-neighbour cations. The entire cluster is embedded in a point charge lattice, so that the effects of the Madelung field are correctly represented. The resulting SCF repulsion energies are considerably smaller than those used by Celli et al. This reflects two effects. First, that an in-crystal anion is considerably smaller than a free anion; and secondly, that first-order SCF calculations significantly overestimate interaction energies involving anions. (2) Instead of fitting an empirical He-F- c6 coefficient to the measured bound states,' we compute c6 and C, coefficients for He-F- and He-Li' interactions using coupled Hartree-Fock calculations, with the in-crystal ions modelled as described above. The polarizabilities and dispersion coefficients of surface F- ions are found to be ca. 10% larger than those of ions in the bulk, but very much smaller than those of free F- ions. The detailed manner in which we include non-additive terms and construct the full He-LiF interaction potential is too complicated to give in full here, but will be described in a forthcoming paper.2 With no adjustable parameters, we obtain a He-LiF potential which has a well depth of 8.11 meV; this may be compared with the value of ca. 8.7 meV required to reproduce the measured resonance energies. It should be possible to bring the well depth into agreement with experiment with fairly small modifications of the a6 initio parameters. We hope that the resulting potential will give as good an account of the experimental data as that of Celli et aZ., but will have much more realistic asymptotic behaviour at both long and short range. ' V. Celli, D. Eichenauer, A. Kaufhold and J. P. Toennies, J. Chem. Phys., 1985,83, 2504. P. W. Fowler and J. M. Hutson, Phys. Rev. B, in press.

ISSN:0301-7249    

DOI:10.1039/DC9858000057

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1985, 80, 71-78 Adsorption Interactions and the Two-dimensional Critical Temperature BY JAMES R. KLEIN* Department of Physics, The Pennsylvania State University, Worthington Scranton Campus, Dunmore, Pennsylvania 18512, U.S.A. AND MILTON W. COLE Department of Physics, The Pennsylvania State University, State College, University Park, Pennsylvania 16802 U.S.A. Received 6th March, 1985 Thermodynamic properties of films depend on both the interactions within the adsorbate and those between the adsorbate and the substrate. We investigate these in relation to the problem of the two-dimensional (2D) liquid-vapour critical point. Our focus is the value of the critical temperature T, for physically adsorbed atoms and methane molecules. The calculations employ perturbation theory, using the equation of state of a 2D Lennard-Jones (LJ) fluid as a reference.Explicitly included effects are deviations of the two-body potential from LJ shape, substrate modification of the interaction (leading to different 2D and 3D potentials) and three-body interactions within the adsorbate. Each of these contributes a measurable shift of T, from the value computed with LJ theory and the 3D potential. The final predictions are in semiquantitative agreement with experimental data for gases on graphite. Interactions between atoms in three dimensions (3D) have been studied by measurements of scattering, thermodynamic, transport and spectroscopic properties, as well as by a variety of theoretical techniq~es.'-~ This has led to the accurate knowledge of a number of interparticle potentials, especially in the cases of homopolar and heteropolar noble-gas diatomics.By comparision, the interactions between adsorbed atoms or molecules have not been investigated very e~tensively.~-~ The most relevant data came from LEED, Auger and thermodynamic measurements of along with direct imaging of dimers with the field-ion microscope.'o Depending on the parrticular situation, the substrate affects the interaction to a greater or lesser extent. This influence arises from dynamical polarization (disper- sion) forces and/or static polarization of the ~ubstrate."-'~ Thus a very diverse spectrum of behaviour can occur. This paper considers this interaction in relation to the problem of the critical temperature, T,, of submonolayer films.The solution has two parts: ( a ) determina- tion of the interaction between adsorbed atoms, including the effect of the substrate, and (b) evaluation of T, from this interaction. Here we examine the case of noble gases and methane adsorbed on graphite, for which we have considered the two-body interactions previ~usly.'~-'~ The same basic ideas should be applicable to adsorption on lamellar halides, for which extensive 2D critical data exist but where the interactions are less well understood.'8-20 The thermodynamic properties of such systems can be substantially affected by commensurability, associated with epitaxy and the strength of the lateral variation of the p ~ t e n t i a l . ' ~ - ~ ~ In this paper we make the simplistic assumption that the motion 7172 THE TWO-DIMENSIONAL CRITICAL TEMPERATURE is two-dimensional.Our results are then appropriate, at best, for those instances where there is a poor matching of substrate lattice spacing and overlayer atomic size. Thus our study excludes Kr, N2 and He, for which the commensurate phase dominates the low-temperature properties on graphite.7917 The calculations presented here are based on perturbation theory. The unpertur- bed system consists of particles interacting only via a Lennard-Jones (LJ) potential: Vo(r) = ~ E [ ( c T / ~ ) ' ~ - ( u / ~ ) ~ ] . ( 1 ) We treat three effects: ( a ) substrate modification of the gas-phase potential, (6) deviation of the net potential from LJ form and ( c ) three-body interactions. The last two of these are familiar aspects of 3D calculations.The substrate-modified 2D potential is calc~lated'~ from the theory of McLachlan;" at a lateral separation r, atoms at a distance L from the substrate have a potential R2= r2+4L2. The coefficients C,, and C,, have been evaluated for a variety of adsorption v3D is taken from reliable gas-phase potentials. The effect of V, is to reduce the attractive well depth in 2D by 210% from its 3D value (cf: tables 1 and 2). The next section describes our method of determining T,; we then test it by calculating T, for noble gases in 3D and address the 2D adsorption problem. A comparison is made between our predictions and the data for a variety of systems. 20326-34 METHOD Perhaps the most widely studied model of a system of iqteracting spherical particles assumes that the total potential energy is a sum of two-body LJ interactions.We take this model to be the reference system upon which our perturbation theory is based. For a given system we define the parameters in eqn (1) by setting E equal to the actual potential well depth and setting ~ = 2 - ' / ~ r ~ , where r, is the position of the potential minimum.* The critical parameters of the LJ model have been evaluated by a variety of numerical techniques. In particular, the reduced liquid- vapour critical temperature T,* = kT,/E: ( 5 ) has been found for a LJ system to be ca. 1.3 in 3D and CQ. 0.53 in 2D.35136 These predictions are not well satisfied by real systems in either dimensionality. As seen in tables 1 and 2, experimental r e s ~ l t s ~ ~ - ~ ~ for T,* lie cc.15% below these values. We note and emphasize that the discrepancy in 2D would be even greater were it not for the fact that we are there using smaller well depths than in 3D, as discussed above. In the following we describe a simple, but plausible, calculation which explains these deviations semiquantitatively. We take into account two defects of the W * Alternative definitions are possible [e.g. ref. (15)], but the results are independent of this choice if the deviation from LJ shape is small.J. R. KLEIN AND M. W. COLE 73 Table 1. 3D values of parameters and predictions, compared with experiment. For the Lennard-Jones system T: = 1.3 [ref. (35)] Ne 40.6 3.12 1.18 1.09 44.48 Ar 142.1 3.76 1.11 1.06 150.9 Kr 201.9 4.01 1.08 1.04 209.4 1.03 289.8 Xe 281.0 4.36 1.07 CH4 217 3.88 1.05 0.88 190.6 a Two-body parameters obtained from ref.(39) (Ar, Kr, Xe), ref. (40) (Ne) and ref. (41) (CH,). From ref. (37), except for Ne [ref. (38)]. Table 2. 2D values of parameters and predictions, compared with experiment. Both interac- tions and data are given for gases on graphite. For an LJ system 2 = 0.53 [ref. (36)] gas E K" r, Aa rb V* T,* (calc.) T,* (exptl)d Ne 34.6 3.15 0.165 0.035 0.50 0.46 Ar 120 3.80 0.253 0.077 0.50 0.46 Xe 236 4.41 0.320 0.115 0.49 0.50 CH4 177 3.91 0.482 0.130 0.48 0.39 From eqn (2), as evaluated in ref. (15). Defined by eqn (10). Defined in eqn (17), Uses E from first column. Data using v values calculated with the method of ref. (45). from ref. (29) and (30) (Ne), ref.(31) (Ar), ref. (27) (Xe) and ref. (32) (CH4 and Ar). model: inaccurate potential shape and the absence of non-pairwise-additive many- body forces.42 Of the latter, we expect that the triple-dipole three-body term should dominate, and we restrict our attention to its contribution. Our calculation is based on the virial expansion P p / n = 1 + Bn 4- Cn2 + - - - (6) where P-' is Boltzmann's constant times the temperature T, n is the density and p is the pressure; in 2D, n and p are the areal density and spreading pressure, respectively. The quantity (7) B = -1 2 dr{exp [-PV(r)I- 1) is the second virial coefficient and C is the third coefficient. We shall assume eqn (6) to be qualitatively reliable for calculating the effect of a perturbation. In particular, it is used to compute the difference in pressure between the true system and the Lennard-Jones version thereof AP=P-P* P A p / n = nAB+ n2AC. Here A B and A C will be evaluated to first order in each of the perturbations considered.Thus the two-body change is A B zE \ d r exp(-PVo)( V- Vo) 274 THE TWO-DIMENSIONAL CRITICAL TEMPERATURE 00 As-! 1 dr(V-V,) AB = a d T / 7'" (104 where d is the dimensionality. The approximation (lob) is based on the fact that the magnitude of Vo is small except at small separation [where the integrand of approximation (10a) is small, so that approximation (106) omits this region] and near the well minimum [which does not contribute significantly to approximation (lOa), since V = Vo at the minimum]. The dimensionless parameter r is a measure of the deviation from LJ shape.The correction due to the three-body interaction V, is43 where The triple-dipole three-body interaction isu v3 = ~ ( i + 3 cos el cos e2 cos e3)r;23r;3;: (14) where the angles Oi and lengths riy are the interior angles and sides, respectively, of a triangle joining the trio of atoms. The coefficient u is known to sufficient accuracy for our purposes.45 Our procedure is to evaluate Ap from eqn (9), (10) and (12). Combining this with pu values according to eqn (8), we obtain a prediction for the pressure itself. The critical properties may then be evaluated. We concentrate on T,, since the n, prediction is extremely uncertain owing to the divergent compressibility at T,, and pc is difficult to measure in 2D. RESULTS We consider first the model's accuracy in 3D.In this case the virial expansion represents rather well the properties of the LJ system.35 Also, A C contributions from the three-body interaction are known.46 Fig. 1 illustrates the role of the various correction terms in the case of Ar at T" = 1.15, considerably below the critical value of the LJ system. The AB and AC terms make comparable positive shifts in Ap; the resulting isotherm is slightly supercritical. The eventual Tz = 1.11 is ca. 5% higher than the experimental value for Ar. As indicated in table 1, this is a typical situation, i.e. the theory yields a large fraction of the experimental reduction of Tz. Methane is exceptional insofar as the LJ representation is particularly crude.41 Encouraged by the 3D results, we now proceed to discuss the 2D case.Here the calculation suffers from the fact that the five known LJ virial coefficients4' are not sufficient, because the series converges more slowly in 2D than in 3D.48 Barker et aL36 have obtained some Monte Carlo LJ data to supplement this series in the critical region. However, since their principal concern was the region T* > 0.5, they provide only limited results for lower T". This deficiency adds a non-negligible uncertainty to our final results for p ( n, T); the eventual uncertainty in T,* owing toJ. R. KLEIN AND M. W. COLE 75 0.3 8 P 0.2 0.1 0.0 0.2 0.4 0.6 0.8 n* Fig. 1. 3 D values of reduced pressure p*=2.rrpu3/3& as a function of reduced density n* = 2.rrnu3/3 for Ar at T* = 1.15. The curve marked LJ is obtained with the Lennard-Jones potential.35 The results marked LJ+AB and LJ+AB+AC are obtained from eqn (8) with the indicated term(s) present in eqn (9).this is ca. 0.01. Further uncertainty in the 2D case arises from the absence of previous determinations of AC. To remedy this we evaluate eqn (12); a step-function appr~ximation~~ is used: g( r ) = [ 1 + 4 ~ p ( a/ r)']e( r - 0 ) (15) where 8 is the unit step function. The associated inaccuracy in the final answer should be small relative to other uncertainties in this calculation. Using eqn (15) some algebraic manipulations described in the Appendix enable one to evaluate eqn (12) ; the result is AC = a4u* 7'"- 4[0.850+ 1.331 T* + 0.813 T*2+0.225 T*3] (14) u* = U / ( & 0 9 ) . (17) This approximation was found to provide an accurate estimate of the two-body contribution to the LJ third virial coefficient.Our 2D results are exemplified in fig. 2. As in 3D, both contributions to Ap are positive, so that T, is predicted to be smaller than for an LJ system. The LJ system is nearly critical at T: = 0.55. In contrast, the computed isotherm for methane at T* = 0.45 is slightly subcritical; we infer that T: = 0.48, ca. 10% less than the LJ value. Also apparent from fig. 2 is that Tz is successively higher for members of the sequence CH4, Xe, Ar and Ne; see table 2 for the results. DISCUSSION We have developed a theory of the deviations of real systems from the behaviour predicted with a reference model based on LJ interactions. We find that the two contributions to this deviation are of comparable importance.In both 3D and 2D the theory comes close to explaining the reduction in T,. Possible explanations of the remaining discrepancy in 3D include unreliability of the perturbation theory and the role of higher-order many-body interactions. We surmise that the former76 0 . i 0 . 1 * Q 0.0 -0.1 THE TWO-DIMENSIONAL CRITICAL TEMPERATURE I I I Ne - 0.0 0.1 0.2 0.3 0.L n* Fig. 2. 2D values of reduced pressure p* = p u 2 / & as a function of reduced density. The dashed curve is for an LJ system at T" = 0.55. The solid curves are for LJ, Ne, Ar, Xe and CH4 particles at T* = 0.45. The LJ results are from ref. (36). is responsible, since the triple-dipole three-body term seems to be adequate in other 3D Alternative approaches capable of resolving this issue are simulation studies with realistic interactions and the use of a LJ reference system with modified parameters, Le.a more reliable basis for the unperturbed theory. In 2D we find that T: is reduced from the LJ value, but not sufficiently to agree with the film data. There are several possible reasons in addition to those cited above. One is that the McLachlan theory relies on a continuum model of the substrate. The 2D potentials used here place the substrate's boundary one-half a lattice spacing outward from the top carbon layer, an uncertain as~umption;~~ a further shift outward would increase the substrate repulsion and depress T,. We note, however, that contrary evidence exists elsewhere for a partial reduction in the McLachlan repu1sion.'2.'6 Other possible sources of difficulty are associated with inaccuracy of the 2D approximation, e.g.the effects of commensurability and out-of-plane motion. 1 7 ~ 1 8 9 5 0 9 5 ' In the case of Xe, for example, there is a close correspondence between the value39 r, = 4.36 A and the spacing at commensurate coverage on graphite, 4.26A. This should lead to a value of T, for Xe which is higher than the value predicted by our ZD calculations.50 Indeed, this expectation is borne out by the Xe value appearing in table 2. Our study has led us to conclude that the subtrate-modified interaction is appropriate to the 2D problem and that three-body effects play a non-negligible role in determining T,. We are limited to such qualitative conclusions because of the approximations in our theory of T,. Computer simulations involving these realistic interactions are appropriate if greater accuracy is desired.However, even these become inaccurate very close to the critical point. On balance one should recognize that the theory of the 2D liquid-vapour critical temperature is rather successful. Without any free parameters, a straightforward calculation predicts T, within ca. 10%. This error is systematic and has a probable origin in several approximations which may be remedied in future work.J. R. KLEIN AND M. W. COLE 77 We are grateful to Moses Chan and Bill Steele for extensive discussions. This research was supported in part by NSF grant DMR-8419261. APPENDIX The integral in eqn (12) can be put into a more manageable form as follows.Let G( r12, r13, r23) be a symmetric function of its variables. It can be shown that47bi52 ldr12dr13 G=24v lo dx lo dy lo dRR3G{R, R(x2+y2)'12, R [ ( l - ~ ) ~ + y ~ ] " ~ } . In our case G = V3g,,g1,g2,; the three-body term can be expressed in terms of r12, r13 and r23, and hence in terms of R, x and y, as 1/2 [x(2+x)l"2 00 (A1 1 Using the approximation in eqn (15), one finds that the integrand in eqn ( A l ) vanishes for R < a / ( x 2 + y2)l12 and that the integral over R can be carried out analytically to yield an expression for AC that involves integrations over a finite region in the xy plane. After some analytic simplifications, these were done numerically to obtain eqn (17). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 J.A. Barker, in Rare Gas Solids, ed. M. L. Klein and J. A. Venables (Academic Press, New York, 1976), vol. 1, chap. 4. G. Scoles, Annu. Rev. Phys. Chem., 1980, 31, 81. R. A. Aziz, in Inert Gases, ed. M. L. Klein, Springer Series in Chemical Physics (Springer-Verlag, Berlin, 1984), vol. 34, chap. 2. L. W. Bruch, Surf: Sci., 1983, 125, 194. D. Nicholson, personal communication. T. Takaishi, Progr. Surf. Sci., 1975, 6, 43. L. W. Bruch and M. B. Webb, in Interfacial Aspecrs of Phase Transformations, ed. B. Mutaftschiev (Reidei, Dordrecht, 1982); 0. E. Vilches, Annu. Rev. Phys. Chem., 1980, 31, 463. A. Thomy, X. Duval and J. Regnier, Surf: Sci. Rep., 1981, 1, 1. J. Suzanne, J. P. Coulomb and M. Bienfait, Surf: Sci., 1973, 40, 414; 1974, 44, 141. R.Casanova and T. T. Tsong, Phys. Rev. B, 1980, 22, 5590; 1981, 24, 3063. A. D. McLachlan, Mol. Phys., 1964, 7 , 381. J. Unguris, L. W. Bruch, M. B. Webb and J. M. Phillips, Surf: Sci., 1982, 115, 219. E. R. Moog and M. B. Webb, Surf: Sci., 1984, 148, 338. G. Vidali and M. W. Cole, Phys. Rev. B, 1980, 22, 4661. S. Rauber, J. R. Klein and M. W. Cole, Phys. Rev. B, 1983, 27, 1314. M. W- Cole and J. R. Klein, Sur$ Sci., 1983, 124, 547, partially revised in G. Vidali, M. W. Cole and J. R. Klein, Phys. Rev. B, 1983, 28, 3064. J. R. Klein, M. H. W. Chan and M. W. Cole, Surf. Sci., 1984, 148, 200. F. Millot, Y. Larher and C. Tessier, J. Chem. Phys., 1980, 76, 3327. C. Tessier, Thesis (University of Nancy, 1983), unpublished. Y. Larher and B. Gilquin, Phys. Rev. A, 1979, 20, 1599.D. K. Fairobent, W. F. Saam and L. M. Sander, Phys. Rev. B, 1982, 26, 179. L. M. Sander and J. Hautman, Phys. Rev. B, 1984, 29, 2171. R. J. Gooding, B. Joos and B. Bergerson, Phys. Rev. B, 1983,27,7669; F. F. Abraham, Phys. Rev. B, 1983, 28, 7338. J. J. Rehr and M. Tejwani, Phys. kev. B, 1979, 19, 345. S. Rauber, J. R. Klein, M. W. Cole and L. W. Bruch, Surf: Sci., 1982, 123, 173. H. K. Kim and M. H. W. Chan, Phys. Rev. Lett., 1984,53, 170. A. Thomy and X. Duval, J. Chim. Phys., 1970, 67, 1101.78 THE TWO-DIMENSIONAL CRITICAL TEMPERATURE 0. Ferreira, C. C. Colucci, E. Lerner and 0. E. Vilches, Surf: Sci., 1984, 146, 309. 28 29 G. B. Huff and J. G. Dash, J. Low Temp. Phys., 1976, 24, 155. 30 R. E. Rapp, E. P. de Souza, and E. Lerner, Phys. Rev. B, 1981, 24, 2196.31 F. Millot, J. Phys. Lett., 1979, 40, 9. A. D. Migone, 2. R. Li and M. H. W. Chan, Phys. Rev. Lett., 1984, 53, 810. 33 J H. Quateman and M. Bretz, Phys. Rev. B, 1984, 29, 1159. J. P. Coulomb, M. Bienfait and P. Thorel, J. Phys. (Paris), 1981, 42, 293. 35 J. A. Barker, P. J. Leonard and A. Pompe, J. Chem. Phys., 1966, 44, 4206. 36 J. A. Barker, D. Henderson and F. F. Abraham, Physica 106A, 1981, 226. 37 J. F. Mathews, Chem. Rev. 1972, 72, 71. 38 M. W. Pestak and M. H. W. Chan, Phys. Rev. B, 1984, 30, 274. 39 J. A. Barker, R. A. Fisher and R. 0. Watts, Muf. Phjjs., 1971, 21, 657; J. A. Barker, R. 0. Watts, 32 34 J. K. Lee, T. P. Schafer and Y. T. Lee, J. Chem. Phys., 1974, 61, 3081. B. Brunetti. R. Cambi, F. Pirani and F. Vecchiocattivi, Chem. Phys., 1979, 42, 397. G. P. Mathews and E. B. Smith, Moi. Phys., 1976, 32, 1719. 40 41 42 W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. 43 E. A. Mason and T. H. Spurling, The Virial Equation of State (Pergamon, Oxford, 1969). B. Axilrod and E. Teller, J. Chem. Phys., 1943, 11, 299. 45 J. R. Klein and M. W. Cole, Surf: Sci., 1983, 134, 722. C. H. J. Johnson and T. H. Spurling, Aust. J. Chem., 1974, 27, 241. ( a ) I. D. Morrison and S. Ross, Surf: Sci., 1973, 39, 21; (b) B. C. Kreimer, B. K. Oh and S. K. Kim, Mol. Phys., 1973,26,297; (c) E. D. Glandt, J. Chem. Phys., 1978,68,29_52; ( d ) J. A. Barker, Roc. R. SOC. London, Ser. A, 1981, 377, 425. C. Bruin, A. F. Bakker and M. Bishop, J. Chem. Phys., 1984, 80, 5859. 46 47 48 49 H. W. Graben and R. D. Present, Phys. Rev. Lett., 1962, 9, 247. 50 P. A. Monson, W. A. Steele and D. Henderson, J. Chem. Phys., 1981, 74, 6431. 51 P. A. Monson, M. W. Cole, F. Toigo and W. A. Steele, Surf: Sci., 1982, 122, 401. 52 T. Kihara, J. Phys. Soc. Jpn, 1948, 3, 265.

ISSN:0301-7249    

DOI:10.1039/DC9858000071

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1985, 80, 79-90 Mobility of Two-dimensional Ethane adsorbed on Graphite A Quasielastic Neutron-scattering Study BY JEAN-PAUL COULOMB AND MICHEL BIENFAIT* Departement de Physique,'! Facult6 des Sciences de Luminy, Case 901, 13288 Marseille, France AND PIERRE THOREL Institut Laue Langevin, 38042 Grenoble, France Received 19th March, 1985 The diffusive mobility of a rod-like molecule, ethane, adsorbed on the (0001) surface of graphite has been studied by incoherent quasielastic neutron scattering as a function of temperature. The measurements have been limited to the submonolayer domain ( 8 < 0.7 monolayer) and to the 10-122 K temperature range. At 10 K ethane molecules do not display any diffusive motion ; at 53.5 K the herringbone solid structure exhibits reorientational motion of the methyl groups about the threefold carbon-carbon axis.Above the two- dimensional melting point (65 K) ethane molecules perform an isotropic rotational motion plus a diffusive translation which can be described, at 87 K, as a jump diffusion between equivalent surface sites (lattice liquid). At 122 K the adsorbed molecules display brownian motion (isotropic liquid). The first condensed layer of ethane adsorbed on graphite exhibits a very rich phase diagram at low temperatures. Several solid phases, termed S,, S2 and S3, have been observed. The S, solid is stable at low coverage and displays a herringbone structure with C2H6 molecules lying on the graphite substrate.' On the other hand the molecules of the S3 solid are erect tetrahedra.As for the S2 solid phase, the unit cell has been determined but the molecular structure remains unknown. The melting of S, occurs at 65 K via an intermediate phase I, which has been successively proposed as a short-range ordered solid' or a hexagonal lattice fluid.2 At a higher temperature (ca. 95 K) the I1 state transforms continuously into a fluid-like phase. The aim of this study was to characterize by quasielastic neutron scattering the mobility of the C2H6 molecules in the coverage range ( 6 < 0.7)2 where the S, solid and its associated 'melted' phases are stable. The ethane molecule can exhibit several types of mobility. In the solid state it can undergo rotation of the methyl groups around the C-C axis or isotropic rotation around its centre of mass. In the liquid state translational brownian mobility is added to the preceding motions.In the intermediate state, 11, the rod-like ethane molecules can progressively lose their positional and orientational order and form a strongly correlated lattice liquid.2 Here we wished to study the change in the degrees of freedom of the C2H6 molecules adsorbed in the submonolayer range ( 6 < 0.7) as a function of temperature and, in particular, to focus attention on the controversial question of the existence of a two-dimensional lattice liquid. The problem of molecular mobility at higher coverage, i e . in the solids S2 and S3 and their associated fluid(s), is left to a forthcoming paper.3 t Unit associated with the C.N.R.S. 7980 MOBILITY OF ADSORBED 2D ETHANE As stated above, we have employed the quasielastic neutron-scattering technique, which is currently used to study molecular mobility in bulk matter.4-6 This technique has been successfully adapted to surface studies in the case of molecules adsorbed on homogeneous powders with large specific areas.’-*’ NEUTRON-SCATTERING LAWS FOR SIMPLE MODELS Neutrons can exchange momentum and energy with moving molecules.Accord- ingly, a monochromatic incident beam interacts with condensed matter and is scattered quasielastically. As ethane molecules have a strong incoherent cross- section, we have access only to the self-correlation functions, i.e. information concerning individual molecular motions. The main features of incoherent quasielastic neutron spectra are usually repro- duced by scattering laws deduced for simple models reflecting isotropic or uniaxial rotation, or brownian or jump diffusion.The most currently used models are described below and their corresponding scattering laws are given below in eqn (1)-(4). In each case the classical expression for bulk matter is given together with its adaptation to surface studies. In these expressions the Debye-Waller factor is set at unity. This simplification yields an overestimation of the calculated scattered intensity that is less than a few percent, a value comparable with the experimental uncertainty. brownian translational diffusion: jump translational diffusion (see text): isotropic rotational diffusion: jump rotational diffusion (see text): In these equations Q is the scattering vector, Ao = E - Eo is the gain or loss of energy with respect to the incident energy E,, Dt is the translational diffusion coefficient and7 The meaning of angles p, Q and 7 is given in fig.l(b). The experiment have been performed on an oriented graphite powder (Papyex) whose crystallites obey an angular distribution function g ( P ) . f? is the angle between an individual (0001)J. P. COULOMB, M. BIENFAIT AND P. THOREL 81 I I I p v 4 /2 0 Fig. 1. ( a ) Orientational distribution g ( p ) of the (0001) adsorbing area. ( b ) Geometrical relationships of G, the normal oto the foil, No, the normal to an individual (0001) surface, and Q, the scattering vector. surface No and the normal zo to the Papyex foil. j o is the spherical Bessel function of zero order, a is the radius of gyration of a hydrogen atom, f ( Q ) is a periodic function depending on the crystal lattice,4’12 S ( w ) is the Dirac function, D, is the rotational diffusion coefficient, j are the spherical Bessel functions andl3?l4 BROWNIAN TRANSLATIONAL DIFFUSION The scattering law describing the interaction of neutrons with bulk fluids is a Lorentzian [ eqn ( l)] which depends on the translational self-diff usion coefficient Dt.4-6 It has been adapted to adsorbed-layer studies by taking into account the preferential orientation of the adsorbing powder [ eqn ( l’)].’ This preferential orientation is defined by the angular distribution function g ( p ) represented in fig.1 ( 4 . JUMP TRANSLATIONAL DIFFUSION In this model molecules are allowed to jump between well defined lattice sites. The corresponding scattering law is given by eqn (2),4’12 where f( Q ) depends on the crystal lattice.For small Q eqn (2) reduces to eqn (1). In this case the diffusion coefficient Dt is related to the mean jump time r between adjacent sites by the relation Dt = (d2)/67, where d is the distance between sites. Moreover, the width Aw of the scattering function is proportional to DtQ2 for small Q. At larger Q the widths of the Lorentzian laws (1) and (2) no longer obey the same quadratic variation with Q. For eqn (1) Am is still proportional to DtQ2, but for eqn (2) Aw drops continuously and displays a minimum when Q=2rr/d (see fig. 5 later). This interesting behaviour is used later to interpret the experimental data and differentiate the mode of translation between jump and brownian models.The jump-diffusion model has been adapted to surface studies for a hexagonal powder in which all the adsorption planes are parallel but are suffering rotational82 MOBILITY OF ADSORBED 2D ETHANE disorder about their axis of 0rientati0n.l~ Although it does not correspond exactly to our problem, we have used it to analyse our results as a first approach because our graphite powder is strongly oriented. ISOTROPIC ROTATIONAL DIFFUSION In this model hydrogen atoms undergo free rotations limited by random collisions with nearest-neighbour molecules. The corresponding scattering law is given by eqn (3), which is the superposition of a &function centred at o = 0 and a sum of Lorentzian curves.The rotational properties are characterized by the rotational- diffusion coefficient 0,. Because of the spherical symmetry of the rotation, expression (3) is not modified by the existence of a preferential orientation of the crystal powder and is valid for bulk and surface studies. JUMP ROTATIONAL DIFFUSION Expression (4) gives the incoherent scattering law when the methyl groups of an ethane molecule perform rotational jumps of 120" about the C-C In this reorientational model the centre-of-mass of the molecule does not move but the molecule is assumed to librate about an equilibrium orientation during an average residence time r and to jump between different equilibrium orientations in an infinitely short time. Expression (4) can be deduced from more general equation representing the scattering law of uniaxial rotational motion14 if an isotropic distribu- tion of the crystal powder is assumed.We use eqn (4) to reduce our data as a first approach, although we should take into account the preferential orientation of our graphite( 0001) surfaces. This simplification offers the advantages of handling an easily tractable expression and of restoring the main features of the incoherent experimental spectra. It leads to an uncertainty in the determination of the residence time T that will be discussed later. SUPERPOSITION OF ROTATION AND TRANSLATION If the molecules are assumed to undergo self-diffusion and reorientation and if these motions are considered to be independent, the total scattering law is the convolution of the translational and rotational scattering laws.MEASUREMENTS The experiments were carried out on the IN 5 time-of-flight spectrometer at the Institute Laue Langevin, Grenoble, with an incident wavelength of 8 8, (Eo = 1.278 meV). Twelve detectors were used at various scattering vectors, Q, ranging from 0.34 to 1.44 A-'. The instrumental resolution had a triangular shape with an f.w.h.m. of 35 peV and the wavelength was larger than the cut-off wavelength of the (hkO) diffracted graphite beams. The substrate is a recompressed exfoliated graphite called Papyex. It exhibits a large specific area (ca. 20 m2 cm-') mainly made up of basal-plane (0001) surfaces. Its characterization has been carried out carefully and it has been shown that this powder is suitable for neutron-scattering studies on two-dimensiooal(2D) phase^.^-^ As stated above, Papyex has a preferred alignment of the basal planes parallel to the plane of the foil, with an angular distribution g ( p ) shown in fig.l(a). AllJ. P. COULOMB, M. BIENFAIT AND P. THOREL 83 the neutron experiments were performed in the in-plane geometry, i.e. with Q parallel to the plane of the foiL7 The coverage calibration was carried out in the neutron-scattering cell by record- ing adsorption-isotherm measurement^.^ Diffraction experiments performed on a different neutron spectr~meter'~'~ have located the stability domains of the 2D solids S1, S2 and S3 on Papyex and allowed one to identify parts of the adsorption isotherm, i.e. the coverage 8 corresponding to the existence (or coexistence) of the different solids.Following ref. (2), we set the monolayer completion (8 = 1) to the densest monolayer phase, S3. This differs from previous definitions of 8, i e . 8 = 1 correspond- ing to the pure S2 solid, by a factor of ca. 1 . l . We report here experiments performed for 8 < 0.7, a coverage range for which the solid S, is stable at low temperatures. Several points (8, T) in the phase diagram have been explored on both sides of the S, melting temperature (65 K): 8 = 0.4, T = 10,87 and 122 K; 8 = 0.54, T = 53.5,66.4,71.4,76.2 and 84.1 K; 8 = 0.63, T = 87 and 122 K. The adsorbed-layer spectra are obtained as the difference between neutron- scattering data obtained for graphite plus the surface layer and for bare graphite. They are corrected for the scattering of the sample holder, absorption and self- shielding and normalized with respect to each other by comparison with a vanadium standard.DATA ANALYSIS The various scattering laws of eqn (1)-(4) are correlated with the instrumental resolution and compared with the experimental data. Four sets of spectra are presented in detail; two are chosen in the fluid-like phase(s) (8 = 0.4; T = 122 and 87 K) and the others in the S1 solid state (8 = 0.54; T = 53.5 K and 8 = 0.4; T = 10 K). The remaining physical conditions have also been analysed and the conclusions obtained are reported briefly in this section. ISOTROPIC LIQUID The incoherent neutron spectra recorded at 122 K (8 = 0.4) exhibit large broaden- ings increasing with Q. These features are the signature of a strong translational mobility.Above Q = 1 A-' the broadening is so large that the resulting spectra are reduced to a flat background with very poor statistics. In other words it is useless to analyse the data above Q == 1 A-'. We know from a previous section that the brownian and jump translational models [eqn ( 1 ) and (2)] have the same asymptotic behaviour at small Q. In fact, eqn (2) reduces to eqn (1) and it is impossible to distinguish between the two models experimentally. Accordingly we used eqn (l'), the adaptation of eqn (1) to surface studies, to interpret our data and thus obtained the self-diff usion coefficient Dt. However, the analysis is more complicated because, at 122 K, it is reasonable to assume that the ethane molecules also perform isotropic rotational diffusion.One must combine eqn (1') with eqn (3). The convoluted expression is made of two terms, one containing j i and the other a sum of the j : . The first term: oto2 sin r j:( Qa) 1; ' 2 g ( / 3 ) sin r d r dq *TT ( DtQ2 sin2 T ) ~ + o is sufficient to fit our experimental spectra. The contribution due to D, is negligible84 MOBILITY OF ADSORBED 2D ETHANE + - 1 p . 2 , , 1 I * 0 0.2 0.6 hw/meV 1 .o Fig. 2. Set of measured cross-sections of ethane adsorbed on graphite for various scattering vectors Q. T = 122 K; 8 = 0.4 layer; Q = 0.34, 0.61 and 0.96 &’. The full line represents the best fit using eqn ( 5 ) [D, = ( 5 * 1) x lo5 cm2 s-l] convoluted with the instrumental resolution. (Brownian motion of molecules performing isotropic rotation.) in that case. This means that translational broadening is always larger than rotational broadening; in other words, energy transfer due to translation is always larger at 122 K than energy transfer due to rotation.Three neutron spectra interpreted by eqn ( 5 ) are represented in fig. 2. The radius of gyration, a, of the ethane molecule is 1.55 A.18 Two adjustable parameters are used to fit the data. One is the proportionality factor between the experimental curve and eqn ( 5 ) . It is kept constant for all spectra. The second is the translational diffusion coefficient D,. We found D, = ( 5 f 1) X cm2 s-l for the three spectra and for four more spectra not represented on the figure. This value is typical for the mobility of a bulk liquid and has the same order of magnitude as the diffusion coefficient of CH, adsorbed on graphite.’ Here we use the word ‘liquid’ and not fluid on purpose because we know that the 2D critical temperature is 130K.19 Unlike a hypercritical fluid that covers the whole surface offered to it, a 2D self-bounded liquid [below Tc(2D)] is made up of 2D droplets separated by a 2D gas.The density of the 2D gas is too small to provide contribution to the neutron spectrum. Hence our scattering data at 122K give information on the densest phase, the 2D liquid, whose local density (and thus mobility) must stay constant as the coverage changes. We checked this point by measuring Dt at 122 K for 0 = 0.63. We found D, = ( 5 f 1) X cm2 s-I, as for 8=0.4 at the same temperature. Furthermore, the integrated intensity of the quasielastic peak is proportional to coverage, as expected from the above discussion.The same behaviour has been found for the 2D liquid phase of CH, adsorbed on graphite.’J. P. COULOMB, M. BIENFAIT AND P. THOREL 85 LATTICE LIQUID The LEED pattern of the 2D liquid at 122 K displays a diffracted ring typical of a classical isotropic liquid with molecules performing brownian motion.2 Below 95 K the ring becomes modulated, and at a lower temperature it is resolved into six angularly elongated spots. The structure is then commensurate with the graphite substrate. This diffraction pattern observed between 95 K and the melting tem- perature 65 K has been interpreted2 as being due to a 2D liquid strongly correlated with the graphite potential wells, i.e.a lattice liquid. The aim of this section is to confirm this interpretation. The quasielastic spectra measured at 66.4, 71.4, 76.2 and 84.1 K (8 = 0.54) and 87 K (8 = 0.4 and 0.63) cannot be interpreted using a rotational model [eqn (3) or (4)] or a translational model [eqn ( l ) , (1') or (2)] only. We have to combine translation and rotation to fit the data. In this case the broadening resulting from translation is smaller than that due to rotation. In fact, the wings of all the spectra quoted above can be interpreted with a single rotational-diffusion coefficient [eqn D, = ( 5 f 1) x 1o'O s-l. On the other hand, the central part of the quasielastic peak cannot be interpreted with a single translational coefficient, Dt. This rules out the brownian model described by eqn (1) or (1').The best fit is obtained by convoluting eqn (2) and (3), i.e. a jump translational motion and an isotropic rotational diffusion. Typical neutron spectra and their fits are represented in fig. 3 for T = 87 K and 8 = 0.4. The central full-line curve corresponds to whereas the dotted line corresponds to the contribution of the second part of the above equation only. This justifies our claim that the broadening due to rotation is much larger than that caused by translation. In eqn (6) we know that the radius of gyration a = 1.55 A'9 and D, = 5 X 10 s-l. The only adjustable parameters are a proportionality factor, which is kept constant for all spectra, and the width 2 f ( Q ) of the translational Lorentzian. The values of 2f(Q) obtained are shown in fig.4 as a function of Q. f ( Q ) first increases with Q and then drops above 1.1 A-'. This behaviour is typical of a translational diffusion by jumps on equidistant lattice sites." As mentioned earlier, the bulk model has been adapted to molecule motion on hexagonal surfaces exhibiting a perfect preferential ~rientation'~ [ i.e. for a very narrow g ( p ) distribution function, cf: fig. 11. The theoretical width obtained, which depends on the periodicity d of the 2D lattice and of the mean jump time r, is represented in fig. 4 for d = 2.46 A, the distance between nearest, neighbour-graphite potential wells and for r = 5 x lo-*' s. The only adjustable parameter is r. The jump time r is determined from the low-Q part of the data because the experimental width 2f( Q) obeys a 2DQ2 law with D = 5 x Although the general features of the experimental data are represented by the model, there is a disagreement as to the position of the maximum in f(Q) which occurs at ca.1.1 and 1.5 k' for experiment and theory, respectively. This disagree- ment is probably related to the roughness of the model, which does not take into account the true angular distribution g ( p ) of our graphite sample. Nevertheless, it can be concluded that the main features of fig. 4 argue strongly in favour of the existence of a lattice liquid at 87 K. cm2 s ' (D = d 2 / 4 r on a surface).86 MOBILITY OF ADSORBED 2D ETHANE Fig. line the A 8 x lo3- 6x103- h rn c) ._ E: 4 6 5 Lx103- 3 v m 0 0.2 0.6 hw/meV 1.0 1 3. As fig.2. T=87K; 8=0.4 layer; Q=O.34, 0.61, 0.83, 1.11 and 1.36A-'. The full represents the best fit using eqn (6). The dotted line corresponds to the contribution of second part of eqn (6). All the equations have been convoluted with the instrumental resolution. (Jump translation of molecules performing isotropic rotation.) 0 0.5 1 1 .S 2 O W ' Fig. 4. Experimental determination of the translational broadening at 87 K as a function of Q, compared with the theoretical width obtained for a 2D jump-translational model.J. P. COULOMB,\M. BIENFAIT AND P. THOREL 87 2x1 m c, ." c 0 0.2 0.6 1 .o Aw/meV Fig. 5. As fig. 2. T = 53.5 K; 8 = 0.54 layer; Q = 0.34, 0.61, 0.82, 1.23 and 1.43 A-'. The full line represents the best fit using eqn (4). The dotted line corresponds to the second part of eqn (4).The scattering laws have been convoluted with the instrumental resolution. (Methyl groups performing reorientation about the C-C axis.) Finally, we also checked that the width of the neutron spectra obtained at T = 87 K is the same for 8 = 0.4 and 0.63 and that the intensity of the integrated quasielastic peak is proportional to coverage. Hence, according to our earlier discussion, the mobile phase at 87 K is also a self-bounded liquid. UNIAXIAL ROTATION OF THE METHYL GROUPS At 53.5 K ( 8 = 0.54) the solid S, is and the spectra, some of which are represented in fig. 5, exhibit broad wings at large Q. These wings indicate the existence of some motion of the C2H6 molecules. The isotropic rotational model [eqn (3)] is unable to account for the experimental data.However, eqn (4) fits very well the quasielastic peaks. This means that the Sl molecules, which are stacked in a herringbone structure, undergo a jump rotation about their C-C axis. The methyl groups perform 120" jumps between indistinguishable orientations as in the bulk crystal. l6 In eqn (4) the radius of rotation of the proton, a, is 1.019A.'6918 The only adjustable parameters are a proportionality factor (the same for all spectra) and the average residence time r. The whole set of data is fitted with a single value of r, namely r = ( 5 f 2) x lo-" s. Hence the residence time is ca. times shorter for adsorbed molecules than for molecules in the bulk solid at the same temperature.16 This is probably related to the fact that surface molecules have fewer nearest neighbours.88 MOBILITY OF ADSORBED 2D ETHANE 0 0.5 1 1.5 QIA-' Fig.6. Elastic incoherent scattering factor (EISF) plotted against scattering vector Q for three temperatures. Full line: EISF for the isotropic rotational model,& Qu) with a = 1.55 A. Dotted line: EISF for the uniaxial rotational model, f[ 1 + 2j0( Q u a ) ] with a = 1.019 A. A, 10; a, 53.5 and 0,87 K. FROZEN ROTATIONS At 10 K (8 = 0.4) the quasielastic peaks display a triangular shape representing the instrumental resolution. No wing can be detected at large o whatever the value of Q. At this temperature diffusive molecular motion is frozen. An orientational transition occurs between 10 and 53.5 K but no temperature-variation study has been carried out to locate the transition temperature.ELASTIC INCOHERENT STRUCTURE FACTOR (EISF) The elastic incoherent structure factor is the fraction of the total quasielastic intensity contained in the purely translationally broadened term?-6 When molecules undergo no translation, the EISF is the fraction of the intensity contained in the central peak, which is broadened by the instrumental resolution. The EISF rep- resentation is a simple test of the scattering models. The experimental EISF is obtained easily by measuring the ratio between the integrated experimental data above the dotted lines in fig. 3 and 5 and the total observed intensity. When no broadening is observed, as at 10 K, the EISF is the relative integrated intensity of the central peak. In fig. 6 the experimental EISF for T = 87 K (8 = 0.4), T = 53.5 K (8 = 0.54) and T = 10 K ( 8 = 0.4) are represented together with the theoretical EISF for eqn (6) [j:( Qa)] and for eqn (4) {$l + 2j0( Qad?)]}.The agreement between experiment and theory is fairly good and justifies again the conclusions drawn in previous sections. Fig. 6 is a very telling picture because it shows clearly that at 10K the rotational mobility is frozen and that two kinds of motions are performed by the ethane molecules at 53.5 and 87 K.J. P. COULOMB, M. BIENFAIT AND P. THOREL 89 DISCUSSION The main features of the neutron quasielastic spectra have been interpreted in this paper. However, one may question whether the obtained values of the transla- tional and rotational diffusion coefficients Dt and 0, and of the different residence times r have been determined with good accuracy.The determinations of Dt and D, have been performed with scattering laws [eqn (1’), (3), (5) and (ti)] valid for surface studies. More than ten spectra have been used to deduce these diffusion coefficients and the corresponding estimated unaccuracy in Dt and D, is ca. 20%. As for the mean jump time r in the lattice liquid, its value is probably quite inaccurate because it has been determined from an oversimplified model. Neverthe- less, one must accept the order of magnitude of the value of r obtained. The other time r related to jumps of the methyl groups in their reorientational motion about the C-C axis has been measured with the help of relation (4), valid for an isotropic distribution of a crystal powder.We did not take into account the preferential orientation of our graphite crystallites. We can estimate the error in the value of r because we compared in a previous paper7 data reduction carried out with models including or omitting a preferntial orientation of the powder. From these previous tests we can conclude that we may be wrong by a factor of two in the determination of the average residence time r between the reorientation jumps. Nevertheless, whatever the inaccuracy of the mobility parameters, we can con- clude from our study that the molecules of a submonolayer of ethane adsorbed on graphite ( 8 < 0.7 layer) display no detectable diffusive molecular mobility at 10 K, rotate about the carbon-carbon axis at 53.5 K, perform a translational motion by jumps on the graphite sites at 87 K (lattice liquid) and obey brownian motion (isotropic liquid) at 122 K.The experiments were performed at the neutron high-flux reactor, ILL, Grenoble. We than the IN5 staff for technical assistance. Part of the interpretation was carried out by M.B. during a visit to the University of Missouri-Columbia. M.B. also acknowledges financial support via a N.S.F.-C.N.R.S. grant (no. 032) and he expresses gratitude to Prof. H. Taub for his kind hospitality. ’ J. Suzanne, J. L. Seguin, H. Taub and J. P. Biberian, Surf: Sci., 1983, 125, 153. J. M. Gay, J. Suzanne and R. Wang, J. Phys. Lett., accepted for publication. J. P. Coulomb and M. Bienfait, J. Phys., to be published. T. Springer, in Springer Tracts in Modem Physics, ed. G . Hohler (Springer-Verlag, Berlin, 1972), ’ F. Volino and A. J. Dianoux, in Proc. Euchem. ConJ on Organic Liquids: Structure, Dynamics and Chemical Properties (Wiley, New York, 1978), chap. 2, pp. 17-47. F. Volino, in Microscopic Structure and Dynamics of Liquids, ed. J. Dupuy and A. J. Dianoux, NATO Advanced Study Institutes Ser. (Plenum Press, New York, 1979), vol. 33, pp. 221-300. ’ J. P. Coulomb, M. Bienfait and P. Thorel, J. Phys. C, 1977, 38, 4 31; 1981, 42, 293; Phys. Rev. Lett., 1979, 42, 733. * P. Thorel, J. P. Coulomb and M. Bienfait, Surf: Sci, 1982, 114, L 43. M. Bienfait, P. Thorel and J. P. Coulomb, in Surjhce Mobilities on Solid Material, ed. Vu Thien Binh, NATO Advanced Study Institutes Ser. (Plenum Press, New York, 1983). lo M. V. Smalley, A. Huller, R. K. Thomas and J. W. White, MoZ. Phys., 1981, 11, 1. R. K. Thomas, Prog. Solid State Chem., 1982, 14, , 1-93. C. T. Chudley and R. J. Elliott, Proc. Phys. Soc., 1961, 77, 353. l 3 C. Steenbergen and L. A. de Graaf, Physica, 1978, 94B, 228. l4 A. J. Dianoux, F. Volino and H. Hervet, MoZ. Phys., 1975, 30, 1181. l6 L. A. de Graaf, C. Steenbergen and A. Heidemann, Physica, 1980, 101B, 209. V O ~ . 64, pp. 1-100. C. Riekel, A. Heidemann, B. E. F. Fender and G. C. Stirling, J. Chem. Phys., 1974, 71, 530.90 MOBILITY OF ADSORBED 2D ETHANE '' J. P. Coulomb, J. P. Biberian, J. Suzanne, A. Thorny, G. J. Trott, H. Taub, H. R. Danner and l 8 G. J. H. van Nes and A. Vos, Acta Crystallogr., Sect. B, 1978, 34, 1947. F. Y. Hansen, Phys. Rev. Lett., 1979, 43, 1878. J. RCgnier, J. Menaucourt, A. Thorny and X. Duval, J. Chirn. Phys., 1981, 78, 629. 19

ISSN:0301-7249    

DOI:10.1039/DC9858000079

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1985, 80, 91-105 Molecular-dynamics Simulation of Fluid N, Adsorbed on a Graphite Surface BY J. TALBOT? AND D. J. TILDESLEY" Department of Chemistry, The University, Southampton SO9 5NH AND W. A. STEELE Department of Chemistry, Pennsylvania State University, State College, University Park, Pennsylvania 16802, U.S.A. Received 13th March, 1985 A sub-monolayer film of fluid nitrogen adsorbed on a graphite surface has been studied using the molecular-dynamics technique. The molecule-molecule potential consists of a two-centre Lennard-Jones interaction with additional coulombic interactions between partial charges. The surface is modelled as a static external field using a Fourier expansion in the reciprocal lattice vectors of the graphite basal plane.Simulations are performed at a variety of coverages at a temperature close to 75 K. The isosteric enthalpy of the model is in good agreement with experiment. The underlying surface lattice modifies the structure of the adsorbed liquid and there is significant out-of-plane orientational ordering at all the coverages studied. At temperatures < 100 K and coverages below one monolayer (0.064 nitrogen adsorbed on graphite forms a variety of solid and fluid phases. At low temperatures the molecules form a (a x a ) R 3 0 ° commensurate lattice,' with orientational ordering in a two-sublattice herringbone structure.293 As the tem- perature is raised above 27 K, the in-plane herringbone orientational ordering is lost and the molecules form an adsorbed plastic ~ r y s t a l .~ - ~ The solid melts to an adsorbed fluid at a transition temperature which is dependent on coverage. At approximately one-half monolayer the melting transition occurs at ca. 48 K, while at coverages close to one commensurate monolayer the transition temperature shifts to ca. 80K.496 The solid-fluid transition is sharp and first order4 and NZ, in the same fashion as Kr on graphite, melts directly to a hypercritical fluid without passing through a liquid phase.' A molecular-dynamics (MD) simulation has been performed on adsorbed solid N2 at temperatures <45 K.8 The model gave a reasonable account of the orientational-disordering transition. The study has been extended to consider the incommensurate uniaxial phase of adsorbed N2 and an analysis of the simulated reorientational correlation functions shows that they can be modelled using a first-order cumulant expansion at low temperatures.'' In the present study the simulations are extended to model the adsorbed fluid at ca.75 K. We have performed eight MD simulations in the microcanonical ensemble at coverages between 0.21 and 1.0. Our results can be compared with the thermodynamic measurements of Piper et aL" Surprisingly, there have been few experimental studies of the structure of adsorbed fluids. In principle, coherent X-ray or neutron scattering of adsorbed fluid N2 can give the site-site radial distribution t Present address: Mathematics Department, Rutgers University, New Brunswick, New Jersey 08903, U.S.A. 9192 MOLECULAR DYNAMICS OF ADSORBED N2 function of the fluid, but there are technical difficulties associated with the low intensity of the signal from the small quantity of adsorbed N2.Moreh and ShahalI2 have performed y-ray resonance measurements which give information on the out-of-plane ordering of the molecules. The simulation can give detailed information on the translational and orientational ordering parallel and perpendicular to the surface, as well as yielding thermodynamic data. In the next section we present a brief discussion of the potential model and the molecular-dynamics technique used in this paper, and then present discussions of the isosteric enthalpy and of the structure of the adsorbate. POTENTIAL MODEL AND SIMULATION The N2 molecule is represented by 3 colinear sites at constant equal separations.The two outer sites, separated by I, are Lennard-Jones centres interacting with similar sites on different molecules through the potential u a y ( r ~ y ) = 4ENN[(uNN/ r a y ) 1 2 - (‘NN/ (1) The quadrupole moment of molecular N2 is modelled by placing three partial charges on the axis. A charge of -q at each of the Lennard-Jones sites and a charge of +2q at the third site in the centre of the bond q is related to the quadrupole moment, 8, by e = fz2q (2) and there are nine site-site coulombic interactions between a pair of N2 molecules. In this study we have used model X1 of Murthy et aL,” which accurately reproduces the thermodynamic and structural properties of bulk solid and liquid nitrogen. (The main weakness of the model is that it overestimates the librational frequencies in the a-phase of solid N2.) In this model E”/R = 36.4 K, (7“ = 3.318 8, and I = 1.098 A.The partial charge q = 6.49 x 10-20C (0.411el) gives a quadrupole moment of 3.91 X C m2, which is ca. 15% less than an experimental estimate of 4.7 x C m2.14 The low value of 8 probably compensates for higher moments in multipole expansion that are neglected in the model.” We stress that the partial charge representation theoretically includes all higher moments, i.e. the hexadecapole @ is given by and in this model is -1.55 X lop6’ C m4. However, this hexadecapole is .x times smaller than the best ab initio estimates16 and the partial charges used in this work are best thought of as a convenient representation of the quadrupole rather than an attempt to model accurately the complete charge distribution.[Fuller discussion of this point is given in ref. (15) and (17).] The molecule-surface potential energy is also a sum of site-site potentials. Both atoms in the N2 molecule interact with each carbon atom in the graphite through a Lennard-Jones potential, uNc( rNc), which is characterized by energy and length parameters E~~ and uNC. The details of the expansion and the formulae for calculating the molecule-surface forces are given in appendix A of ref. (18). The parameters E~~ and uNC are calculated from the usual mixing rules, which work well when the component parameters are similar: a) = ( 1/8)qZ4 (3) ENC = (ECCENN)”~, u N C = % c N N + act). (4) Values of E ~ ~ / k = 28 K and ucc = 3.40 are obtained from compressibility measure- ments on graphiteIg and give values of ~~,--k = 31.92 K and uNC = 3.36 8, whenJ. TALBOT, D.J. TILDESLEY AND W. A. STEELE 93 Table 1. Simulations of the fluid phase" 0.0136 0.0219 0.02 19 0.0307 0.0409 0.0480 0.0545 0.0636 42 33 33 28 24 22 21 18 24 19 19 16 14 13 12 12 75.9 74.7 54.3 74.5 72.3 73.6 75.2 75.9 34.48 34.36 35.21 34.1 1 34.02 33.62 33.36 32.95 1.79 2.70 3.14 3.95 4.92 5.91 6.78 8.38 36.26 3 7 -06 38.36 38.05 38.94 39.53 40.14 41.33 0.357 0.343 0.398 0.3 10 0.304 0.265 0.235 0.157 a The dimensions of the simulation cell are in reduced units. Urn, is the molecule-surface configurational energy, Urn, is the N2-N2 configurational energy and U is the total configur- ational energy. Energies are per molecule and reduced with respect to E ~ ~ ; no long-range corrections have been added.substituted in eqn (4). The energies calculated in the simulation are reduced by cNC and distances are reduced with respect to a, the lattice spacing of the substrate, which has the value 2.46 A. The molecular-dynamics simulations are performed at constant energy. At each coverage, p = N / A , the number of molecules N is fixed at 144 and the dimensions of the basic cell chosen so that cell is as square as possible. The dimensions L, and L,, of the cell are given in table 1. The starting configuration for each simulation is a square lattice with the molecules at a height of z = 3.32 A. The molecules are parallel to the surface and have random in-plane orientations. The equations of motion are solved using a fifth-order predictor corrector technique, and the orienta- tional motion is described in terms of the quaternion parameters. Each run is equilibrated for 2000 timesteps and averages are taken over a production phase of 4000 timesteps.A timestep of 8.2 x lo-'' s leads to fluctuations in the fifth decimal place of the total energy and is used in all the simulations reported in this paper. Periodic boundary conditions are implemented in the x and y directions and the molecules are free to move in the z direction perpendicular to the surface. The forces and configurational energy are calculated using the minimum-image conven- tion and no spherical cut-off is applied. Since the adsorbate is in a static external field the linear momentum is not necessarily conserved and in the presence of periodic boundary conditions the total angular momentum is not conserved.ISOSTERIC ENTHALPY The configurational energies calculated in the simulations are given in table 1. In fig. 1 the total (U) and the molecule-surfdce (Urns) contributions are plotted as a function of coverage (for this series of runs the average value of the temperature is reasonably close to 75 K):94 MOLECULAR DYNAMICS OF ADSORBED N2 341 - + + + + I f + I + 1 32 1 I I I I I 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 P W 2 Fig. 1. Configurational energy per molecule in units of E ~ ~ : 0, U ; +, Urns; (-) least- squares fit to the first five points; (- - -) h x h commensurate density of 0.0636 A-2. N is the number of molecules and urn, and urn, are the molecule-molecule and molecule-surface pair potentials discussed in the previous section.Interestingly, although the total configurational energy increases with coverage (in a negative sense), the molecule-surface contribution decreases. The reason for this is because as the coverage is increased the molecules are forced to tilt away from the surface plane, which lowers the molecule-surface energy. A discussion of out-of-plane ordering is given later. The isosteric enthalpy of adsorption, qst, is given by qst=RT- U (8) where U is the partial molar configurational energy Fig. 1 shows that, to a good approximation, the variation of U with N is linear (although beyond p = 0.0545 k2 there is some divergence). Treating the simulation data as isothermal and assuming U = a + bp, then the partial molar U is u=a+2bp.(10) A least-squares fit to the first five points gives a = -34.99 and b = -95.92 A2. The isosteric enthalpy of the simulation model at ca. 75 K is qst/ kJ mol-' = (RT/ 1000) + 9.29 + 50.93~ (11) with p in A-2. Piper et aL" used an adiabatic calorimeter to measure the heat of adsorption of N2 on graphoil at 79.3 K and obtained q,,/kJ mol-' = (RT/ 1000) + 9.34( k0.2) + 48.7( k3)p. (12) In fitting this equation to their data we have assumed that one monolayer corre- sponds to a density of 0.0636 The agreement between eqn ( 1 1 ) and (12) isJ. TALBOT, D. J. TILDESLEY AND W. A. STEELE 95 excellent, particularly considering the differences in temperature, and confirms that the potential model is a useful representation of the system.TRANSLATIONAL STRUCTURE In the absence of an external field all the properties of a fluid are translationally invariant. However, in an adsorption system the periodic structure of the adsorbate perturbs the structure of the fluid. This effect is monitored by calculating the function F(x,y)=cos27T x-- +COS27T x+- +cos - ( 2 ( 2 (2) where x and y are the position of the centre of mass of the N2 molecule in units of a. This function, introduced by Whitehouse et aL20 is simply related to the functionf,(x, y ) , which is responsible for the lateral variation in the molecule surface potential [ F ( x , y ) = -fi(x, y ) / 2 ] . For a molecule over an adsorption site F = 3 while for a molecule over a carbon atom F = -1.5. In the simulation we calculate P(F) = n ( F ) / a ( F ) (14) where n ( F ) is the average number of molecules in the range from F - iAF to F + iAF and a( F ) corresponds to the number in a random fluid.a( F ) , which is a complicated function of F, has been calculated at 20 equally spaced points ( A F = 0.225) using a Monte Carlo technique;21 we have verified these results. The results for four simulations are shown in fig. 2. Below a coverage of 0.0636 the functions are remarkably insensitive to density, showing a small preference for adsorption sites over carbon atoms. [In an isotropic fluid p ( F ) should be 1.0 for all F.] In marked contrast at p = 0.0636 A-2, which is very close to the freezing density, there is sharp rise, indicating preferential adsorption above the centres of hexagons.p( Fj can be modelled approximately by assuming that the distribution is unaffec- ted by the presence of other N2 molecules. In this approximation where E ( F ) is the minimum energy averaged over in-plane orientations at the average height of the molecules. (Strictly speaking E should also be averaged over z and out-of-plane orientations, but these distributions are sharp and have been ignored for the purposes of this calculation.) E ( F ) has been calculated for the 20 points between -1.5 and 3.0. The variation is slight with a difference between the maximum (at F = -1.5) and the minimum (at F = 3.0) of 19.5 K. E ( F ) is linear to a good approximation. Fig. 3 shows p ( F ) at two temperatures for a density of 0.0219 A-2. The solid lines are the zero-density approximation of eqn ( 1 9 , which gives a good account of the behaviour.The oscillations in p ( F ) are almost certainly due to neighbouring molecules, and the smaller slope of p( F ) at higher temperatures indicates that the liquid is less affected by lateral variations in the surface potential as the temperature is increased. The simplest method of monitoring the pair structure of the adsorbed fluid is to calculate the atom-atom or site-site distribution function for the nitrogen atoms. gay( r ) = p -2( N ( N - 1 ) 8 ( r ; ) S ( r; - r ) ) (16)96 MOLECULAR DYNAMICS OF ADSORBED N2 4, v Q 2- 21 T o ' v i' - 3 ' ' 5 ' I 21 5 0 1 3 0 1 -1.5 0 1.5 3 c2 v1 3 E O-0 -1.0 '1 0' -1-0 0.0 1.0 '1 Fig. 2. Site-site distribution function ga7(r), p ( F ) [eqn (14)] and n (cos p ) as a function of coverage: (a) 0.0636, ( b ) 0.0480, (c) 0.0307 and (d) 0.0136 A-2.The temperatures are given in table 1 and are close to 75 K. is the normalized probability of finding nitrogen atom a on molecule 1 a distance r from atom y on molecule 2. Fig. 4 shows a gay( r ) for the fluid at p = 0.0545 k2. ga,(r) rises sharply at r = u = 3.32 A because of the contact between atoms or different molecules. The cusp-like feature at 4.42 A is due to a correlation between a nitrogen atom and a second (non-contacting) nitrogen atom in a neighbouring molecule. These features are well known in the structure of three-dimensional liquids22 and should be seen in the Fourier transforms of neutron- and X-ray- diffraction measurements on adsorbed molecular fluids.g,,(r) is also shown as a function of density in fig. 2. The distribution changes from the structureless curve, related to the exponential of the site-site potential, to a curve typical of an adsorbed solid with long-range translational structure at a high density. This radial distribution function is averaged over all the in-plane directions of the vector ray Since the fluid is anisotropic, owing to the surface periodicity, we have averaged out any azimuthal structure in the calculation of this function.J. TALBOT, D. J. TILDESLEY AND W. A. STEELE 97 Fig. 3. Observed p( F ) (+), compared with the zero density distribution, po( F ) (-), for a density of p = 0.0219 A-2 and for two temperatures: (a) 74.7 and (b) 54.3 K. u- F+l 2 . 0 2.51 ""1 0.0- 1 I I I 0.0 1.0 2.0 3.0 L.0 5.0 r l a Fig.4. Simulated site-site radial distribution function for the fluid at p = 0.0545 A-2 and T = 75.2 K.98 MOLECULAR DYNAMICS OF ADSORBED N, ORIENTATIONAL ORDERING IN-PLANE ORIENTATIONAL ORDERING At low temperatures adsorbed solid N2 forms a two-sublattice herringbone structure which is determined, primarily, by the electric quadrupole-quadrupole interaction. The two-sublattice structure disappears 20 K below the melting tem- perature, but the quadrupole is still expected to play a r5le in determining orienta- tional correlations in the adsorbed fluid. A method of describing the in-plane orientational ordering is to treat the fluid as two-dimensional and to expand the pair correlation function in circular harmonics.The coefficients in the expansion are one-dimensional functions of the intermolecular separation, which are easy to display and store and for which it is possible to build simple theoretical models. The extent to which the fluids are two-dimensional is discussed in the next section. The normalized probability of finding two N2 molecules with a separation r12 making angles 8, and O2 with respect to the intermolecular vector is g(rI2, el, 0,) dr,, dol d8,/(27~)~. In the circular harmonic expansion g(r12:, e2) = 1 gr1m(~12) exp ( i n e l ) exp (irne2) (17) n. m where the sum is over all positive and negative integer values of n and rn. The coefficients can be obtained by using the orthogonality of the circular harmonics: grim = (1/r2) 1; do1 j: do2 g(r12,81,e2) exp (-inel> exp (-ime2) (18) where the symmetry of the N2 molecules has been used to restrict the range of the integration.In the simulation the coefficients are calculated from grim( rI2) = goo( rI2)(cos ne, cos mi?, -sin no, sin me,) (19) where goo( r12) is the pair distribution function of molecular centres and the average is taken in a shell of thickness dr,, centred on r12. Symmetry considerations show that grim( rI2) = g-n-m(r12) and that for pairs of equivalent homonuclear diatomic molecules gmn( r12) = gnm( r12) if n and rn are even. With these simplifications the total pair distribution function can be reconstructed as follows: g(r12, 81, 62) = goo(r12) +2[g2o(r12)(cos 261 +cos 282) + g22(r12) cos 2(&+ 0 2 ) + g2-2(r12) cos 2( 6 , - e,) + - * -1. (20) The expansion coefficients have the following asymptotic behaviour gflm ( r12) 09 r12 < (T goo(r12) = 1, r12 >> f7 g n m ( r l 2 ) ‘ 0 7 r12>>0- (21) In the simulation the coefficients are computed by sorting into intervals of thickness 0 .0 5 ~ (= 0.123 A) u to r12 = 5a. Typical coefficients are shown in fig. 5( a ) - ( e ) for a density of 0.0307 I-,:, with T = 74.5 K. The series is rapidly convergent with the g4, term close to zero over the complete range. The qualitative features of each of the curves do not change with density, but the magnitudes of the peaks and troughs increase slightly with increasing density. These conclusions do not apply to the solid structure at 0.0636A-2, where many of the molecules are forced to tilt awayJ . TALBOT, D.J. TILDESLEY AND W. A. STEELE 99 from the surface. A simple model for the coefficients can be calculated from the low-density form of the pair correlation function: x (COS no, cos me, - sin no, sin me2) (22) 3 . 0 - 2.5- 2 . 0 - 0 1 . 5 - G 1 . o - 3.0 2.5 2.0 - - - . . - ( a ) 1 . 5 - . . .&': : 0 '. 0 0: : 6.. 0 ' - 1.0 0 . 5 - 0.0 - I 0 1 2 3 4 5 0 . 5 0 .o 0.0 1 .o 2.0 3.0 L . 0 5 .O r / a 1 .o 1 .o I I I I 0.6 0 . 2 - - 0.6 - - 0.2 N G - 0 . 2 - - 0 . 6 -1.0 0.0 1.0 2 .o 3.0 4.0 5 . 0 r / a Fig. 5. Comparison of the simulated coefficient in the circular harmonic expansion (0) with that calculated from eqn (22) ( - - -). The inset shows the approximation eqn (22) with the quadrupole-quadrupole interaction included in Urn, (-) and without the quadrupole- quadrupole interaction (+).p = 0.0307 A-' and T = 74.5 K. ( a ) goo( r ) , ( b ) go,( r ) = g20( r ) , (4 g2Arh (4 g*-2(r) and (4 g44(r).100 1 0 0 N G - 0 O I 6 2 -I 2i 9 0 0. -0.64 s: -1 .o+ 1 1 1 1 1 0.0 1.0 2 .O A .O 5.0 3.0 r i a - 1 . o j , 0.0 1 . 0 2 .O 3.0 4.0 5.0 r l a 0 .o 0.6 f 0.2 @4 - 0 . 2 -0e61 - 1.0 0 .o 1.0 2 .o 3.0 4 . 0 5.0 Fig. 5. (continued).J. TALBOT, D. J. TILDESLEY AND W. A. STEELE 101 where ~ m m ( ~ 1 2 , el, 0,) is the energy of a pair of N2 molecules held parallel to the plane and excluding any contribution from the surface. The coefficients calculated from eqn (22) are shown in fig. 5 ( a ) - ( e ) . At moderate densities the goo coefficient is not predicted accurately. This is not surprising, and there are adequate theories such as RISM21 and RAM22 which will handle the distribution of molecular centres. However, the principal features of the high coefficients are predicted by eqn (22), indicating that the two-dimensional approximation is reasonable and that the orientational ordering is determined by the low-density form of g(r12, el, OJ.In the insets to fig. 5 ( a ) - ( e ) there is a comparison of eqn (22) for ~mm(r12, el, 8,) calculated with and without the quadrupole interaction. The major difference is in the g,, coefficient, where the simulated coefficient is in closer agreement with the low-density g:o( r12) calculated with the quadrupole than with the gE0( r12) calculated using only the site-site core and dispersion interaction. We can conclude that in the simulation model the quadrupole-quadrupole interaction affects the fluid structure of the adsorbate.OUT-OF-PLANE ORDERING Molecules in the simulation are not restricted to the xy plane. The strong molecule-surface potential confines the centre .of mass of the. N2 molecules to a band between z = 2.98 and 4.78 A. The distribution of the molecules as a function of z is shown in fig. 6 for three coverages. The density, p(z), is Calculated by taking the average number of molecules, N ( z ) , in a rectangular prism with the surface as a base of area A and a height of 0 . 0 2 ~ centred at z. Then p ( z ) = N ( z)/0.02A (23 1 and the integral of p ( z ) with respect to z gives the coverage. The distribution p ( z ) is not symmetric around the maximum and reflects the asymmetry in the molecule- surface potential as a function of z.The average height of the adsorbed layer increases slightly with coverage while the width of the layer increases significantly. The molecules are also free to take up any orientation of the molecular axis with respect to the surface. This distribution is monitored in the simulation by taking calculation n (cos P ) , where n (cos P ) d (cos p ) is the probability of finding a molecular axis which makes an angle 0 with the normal to the surface such that cos P lies between cos P and cos P + d (cos p). This distribution is shown in fig. 2 as a function of coverage. At low coverages most molecules prefer to be parallel to the surface and the distribution is sharply peaked around cos p = 0 ( i e . P = 7~/2).(The distribution should be symmetrical around cos P = 0 because of the symmetry of the N2 molecules but we have shown it over the full range from -1 to +1 in fig. 2.) At the lowest coverage a small fraction of the molecules are perpendicular to the surface, as evidenced by the non-zero n(k1). This is due to the thermal librations of the molecules at 75 K. As the coverage is increased the fraction of molecules perpendicular to the surface is increased. At this temperature for the coverages we have studied the maximum in n (cos P ) is at cos P = 0. We also find that n (cos p ) is not uniform until temperatures close to 120 K. The actual magnitude of the out-of-plane ordering can be estimated from the order parameter OP,, which is the average value of the second Legendre polynomial, P2 (cos P ) :102 MOLECULAR DYNAMICS OF ADSORBED Nz Fig.6. p ( z ) at three coverages: (- - - - .) p = 0.0136 A-*, (- - - -) p = 0.0301 A-* and p = 0.0636 A-2. The definition of p ( z ) is given in eqn (23). 7 0.2 Q Q I 1 O”’ 0.b2 I 0.04 0.06 Fig. 7. Order parameter OP, as a function of coverage p. plmolecule A-’ OP, has a value of -0.5 when the molecules are parallel to the surface and is zero if the out-of-plane distribution is isotropic. The value of OP, is given as a function of coverage in table 1 and fig. 7. The error bars are calculated from subaverages over 200 timesteps. The third point at p = 0.0307 k2 appears to depart from the general trend. In fact, it is likely that fourth point at p = 0.0409 k’ is too high since the temperature in that simulation falls to 72.3 K.(Some estimate of the magnitude of the temperature dependence of OP, can be gained by examining rows 2 and 3 of table 1 for the same coverage at two temperatures.) Despite the temperature differences between the simulations there is significant structure in the curve of OP, against p around p = 0.03 k2. This is the density region where packing effects caused by the molecule-molecule potential become important and may explain the curve.J. TALBOT, D. J. TILDESLEY AND W. A. STEELE 103 Finally we note that at the highest coverage there is an upturn in n (cos p ) at cos p = kl, indicating some favouring of the perpendicular orientation. The easiest way to understand this behaviour is by examining a plan of the liquid constructed from the simulation data.In fig. 8 we show three such instantaneous configurations. The projection of the molecules has been drawn onto the surface. At medium to low densities a preponderance of parallel orientations is visible, while at p = 0.0636 we can see the formation of a more ordered structure with a substantial number of atoms perpendicular to the surface. Although this density is appropriate for the commensurate structure, the actual structure produced by freezing in the simulation with the periodic boundary conditions may well resemble the triangular incommensurate structure found above one monolayer. Fig. 8. Three plans of the instantaneous configuration of the N2 molecules during the course of the MD simulation: ( a ) p = 0.0307 k2, ( b ) p = 0.0545 A-2 and (c) p = 0.0636 A-2.104 MOLECULAR DYNAMICS OF ADSORBED N2 (C) Fig.8. (continued). CONCLUSIONS We have presented a number of simulations of fluid N2 adsorbed on graphite. The simulations were performed in the microcanonical ensemble, and the final temperature in each simulation was close to 75 K. The molecule-molecule component of the configurational energy becomes more negative with increasing coverage, but the molecule-surface component becomes more positive as molecules tilt away from the surface. The isosteric enthalpy calculated in the simulation is in good agreement with the experimental measure- ments of Piper et all1 for an isotherm at 79.3 K. The translational structure of the fluid with respect to the underlying surface lattice is monitored with the function p ( F ) , which is sensitive to changes in coverage in the fluid phase and shows a preference for adsorption sites over carbon atoms.The density fluctuation in the adsorbed fluid becomes more pronounced at tem- peratures closer to the freezing point (ca. 49 K). The site-site distribution functions of the adsorbed fluid show a feature at r = I + (T which is a remnant of the cusp in a corresponding hard-core ‘two-dimensional’ fluid. The circular harmonic expansion of the total pair distribution of adsorbed N2 is a rapidly convergent series. The harmonic coefficients are weak functions of density in the fluid and, with the exception of goo(r), they can be predicted with a low-density approximation. The quadrupole moment has a significant but small effect on the in-plane orientational ordering of the model fluid.The average height of the adsorbed layer increases slightly with temperature and there is a marked increase in the thickness of the layer. The order parameter OP,, which describes out-of-plane ordering, increases with increasing coverage, indicating that configurations where the molecule is perpendicular to the surface become more important on approaching the solid-fluid boundary. The change in slope of OP,J. TALBOT, D. J. TILDESLEY AND W. A. STEELE 105 with p at low coverages may result from the onset of significant short-range lateral interactions. We thank NATO for grant no. 216/80 (W.A.S. and D.J.T.), the Division of Materials Research of the N.S.F. for grant no. DMR-8113262 (W.A.S and D.J.T.) and the S.E.R.C. for a studentship (J.T.). J. K. Kjems, L. Passell, H. Taub, J. G. Dash and A. D. Novaco, Phys. Rev. B, 1976, 13, 1446. * J. Eckert, W. D. Ellenson, J. B. Hastings and L. Passell, Phys. Rev. Lett., 1979, 43, 1329. R. D. Diehl and S. C. Fain, Surf: Sci., 1983, 125, 116. M. H. W. Chan, A. D. Migone, K. D. Miner and Z . R. Li, Phys. Rev. B, 1984, 30, 2681. N. S. Sullivan and J. M. Vassiere, 1983, 51, 658. R. D. Diehl and S. C. Fain Jr, J. Chem. Phys., 1983, 77, 5065. S. Ostlund and A. N. Berker, Phys. Rev. Lett., 1979, 42, 843. J. Talbot, D. J. Tildesley and W. A. Steele, Mol. Phys., 1984, 51, 1331. J. Talbot, D. J. Tildesley and W. A. Steele, Mol. Phis., submitted for publication. lo R. M. Lynden-Bell, J. Talbot, D. J. Tildesley and W. A. Steele, Mol. Phys., 1985, 54, 183. l 1 J. Piper, J. A. Morrison, C. Peters and Y. Ozaki, J. Chem. Soc., Faraday Trans. 1, 1983,79,2863. l 2 R. Moreh and 0. Shahal, PAYS. Rev. Lett., 1979, 43, 1947. l 3 C. S. Murthy, K. Singer, M. L. Klein and I. R. McDonald, Mol. Phys., 1980, 41, 1387. A. D. Buckingham, R. L. Disch and D. A. Dunmar, J. Am. Chem. Soc., 1968,90, 3104. l 5 C. S. Murthy, S. F. O’Shea and I. R. McDonald, Mol. Phys., 1983,47, 381. l6 R. D. Amos, Mol. Phys., 1980, 39, 1. l 7 P. A. Monson, W. A. Steele and W. B. Streett, J. Chem. Phys., 1983, 78, 4126. l8 W. A. Steele, Surf: Sci., 1973, 36, 317. l9 W. A. Steele, J. Phys. (Paris), 1977, C4, 61. 2o J. S. Whitehouse, D. Nicholson and N. G. Parsonage, Mol. Phys., 1983,49, 829. J. S. Whitehouse, Ph.D. 7lesis (Imperial College, London University, 1984). 22 D. Chandler, in Studies in Statistical Mechanics, Fluids Volume, ed. E. W. Montroll and J. L. Lebowitz (North-Holland, Amsterdam, 1981). 23 T. Melnyk and W. R. Smith, Mol. Phys., 1980, 40, 317. 14 21

ISSN:0301-7249    

DOI:10.1039/DC9858000091

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr J. A. Barker (IBM, Sun .lost!, U.S.A.) said: In considering critical temperatures whether in two or three dimensions one has to consider the shape of the two-body potential and three-body and substrate-mediated potentials as discussed in Prof. Cole's paper. Perhaps one should also consider long-ranged density fluctuations which are certainly vital in determining critical exponents and which may also affect the critical temperature. These effects are presumably not described by techniques such as perturbation theory or the Monte Carlo method. However, I can give some evidence that this is not too important, at least in determining critical temperature and pressure, since for three-dimensional argon Monte Carlo calculations with 108 atoms using the BFW pair potential and long-range many-body interactions gave excellent agreement with experiment on the critical isotherm ( T = 150.87 K) as shown in the following results from ref.(1). v/cm3 mol-' p ( calc.)/atm p ( exptl)/atm 48.39 53 57.46 47 70.73 49 91.94 49 61 50 49 49 J. A. Barker, R. A. Fisher and R. 0. Watts, Mof. Phys., 1971, 21, 657. Dr S. F. O'Shea (Unioersity of Lethbridge, Alberta) said: M. Rami Reddy and I have recently fitted an empirical equation of state to the p V T data available from simulation for the Lennard-Jones fluid phases in two dimensions. Our experience suggests that the critical properties in two dimensions are even more sensitive to perturbations than are those in three dimensions. Relatively modest changes in the data, particularly for points in the coexistence region, lead to significant changes in the critical constants.Because of this sensitivity, we feel that a perturbation approach of this kind is unlikely to be successful quantitatively, and possibly even semiquantitativel y . Prof. W. J. Meath (University of Western Ontario, London, Ontario) said: I have questions concerning the perturbation theory used in Prof. Cole's paper. The first-order corrections for the pressure, relative to a Lennard-Jones potential as a zeroth-order two-body problem, are evaluated for two effects, namely those due to (1) the change in the two-body potential and (2) the addition of three-body interac- tions as represented by the triple-dipole energy. The problem I have is most easily seen from fig. 1 of the paper by Klein and Cole, where it is clear that both first-order correction terms for the three-dimensional reduced pressure p* become much too large as the reduced density n* increases, and indeed their sum becomes larger than the zeroth-order results for n"20.4.Similar comments apply to the treatment of the two-dimensional reduced pressure. 107108 GENERAL DISCUSSION My questions are, first, how does the divergent nature of the theory used here affect the evaluation of the reduced critical temperature and, secondly, do the authors have plans to improve their perturbation theory and to obtain more convergent results? Prof. M. W. Cole (Pennsylvania State University, U.S.A.) replied: Indeed, one should be concerned when the perturbation is large. However, here the relevant domain of application (reduced density n*=0.2-0.3) is such that n z is shifted by only ca.10%. One can reduce the magnitude of the perturbation by judiciously choosing a reference state. Such an approach is currently being pursued. Dr M. L. Klein (National Research Council of Canada, Ottawa) said: I would like to draw attention to certain similarities between the behaviour of ethylene and ethane physisorbed on graphite. Prof. Bienfait remarked that at low coverage solid ethane may exist in a herringbone structure with the molecules lying down on the graphite substrate, while at higher coverage the molecules are standing erect. Analogous phases have also been reported for overlayers of ethylene.',* We have carried out some molecular-dynamics calculations for both systems but extensive results are so far only available for the case of ethylene on a planar (uncorrugated) substrate.It is already apparent that even for this simplified model, the comportment of the ethylene molecules is a very complicated function of both temperature and surface coverage. For example, in the low-density solid the mean canting angle of the C2H4 molecular plane has been found to be a particularly sensitive function of surface coverage. The effect of heating produces structural changes and even generates a rotator (plastic crystal) phase.3 The high-coverage solid phase, in the case of ethylene, seems also to be a herringbone structure (at least in the molecular- dynamics simulation). This transforms on heating to a uniaxial rotator phase before melting.' We have yet to investigate in detail the liquid phase or the effect of including a corrugated substrate.Since it is evident from the observations of Coulomb and Bienfait that the substrate corrugation plays an important role, even in the liquid, it is imperative that we extend our model calculations to allow for such effects. Fortunately, well established methods exist for doing this.4 ' M. Sutton, S. G. J. Mochrie and R.. J. Birgenean, Phys. Rev. Lett., 1983, 51, 407. ' S. K. Satija, L. Passell, J. Eckert, W. Ellenson and H. Patterson, Phys. Rev. Lett., 1983, 51, 411. S. Nos6 and M. L. Klein, Phys. Rev. Lett., 1984, 53, 818. J. Talbot, D. J. Tildesley and W. A. Steele, Faruday Discuss. Chem. SOC., 1985, 80, 91. Prof. S. C. Fain (University of Washington, U.S.A.) said: In the phase diagram shown in the talk by Prof.Bienfait [taken from ref. (2)], what is the difference between I1 and L? I propose that there is just a continuous change in the radial and azimuthal widths of the LEED spots and of the diffusion times as the temperature is changed with constant coverage. The existence of LEED spots which have a greater azimuthal width than radial width has also been observed in unpublished work done in my laboratory on argon, diatomic oxygen and normal hydrogen phases that are inferred to be fluids from thermodynamic measurements. In addition, X-ray measurements have observed a similar effect for the adsorbed xenon fluid.' Ethane seems to be a beautiful system to study the decreasing order in the fluid phase as a function of increasing temperature [see ref.(2) of the paper]. Theories of the fluid state need to be revised to include the bond-orientational correlations imposed by the substrate field. A first attempt is given in papers on xenon.'.2GENERAL DISCUSSION 109 ’ S. E. Nagler, P. M. Horn, T. F. Rosenbaum, R. J. Birgeneau, M. Sutton, S. G. J. Machrie, D. E. ’ E. D. Specht, R. J. Birgeneau, K. L. d’Amico, D. E. Moncton, S. E. Nagler and P. M. Horn, J. Phys. Moncton and R. Clarke, Phys. Rev. B, 1985, 32, 7373. (Paris), 1985, 46, L-561. Prof. M. Bienfait (University of Marseille, France) replied: The fluid phases of ethane films adsorbed on graphite in the submonolayer range (coverage <0.7) have been termed I, and L from the early neutron diffraction works published by the Marseille and Missouri groups.’.’ From the first recorded diffraction patterns, it was clear that the melting of the S1 herringbone structure exhibited unpredicted features.Just above melting, a fluid-like phase with a large unexpected correlation length (30-5OA) was observed. It was termed I1 because it was ‘intermediate’ between the solid S, and the usual two-dimensional liquid (L) phase observed at higher temperature (correlation length ca. lOA). The index 1 in I, permitted differentiation of this fluid-like phase from a more compressed one, Iz, occurring at higher coverage. At this time, the limited number of recorded diffraction patterns did not allow one to draw definite conclusions about the properties of the I, and L phases, although one could observe that the 11-L transition was continuous.This transition has been revisited in ref. (2) and it has been shown that the lattice liquid I continuously loses its positional and bond orientational order with temperature and becomes a surface liquid L with a slight residual positional order. All of these observations as well as those reported in our paper raise interesting questions about the influence of the surface periodic potential on the properties of the adsorbed liquids. ’ J. P. Coulomb, J. P. Biberian, J. Suzanne, A. Thorny, G. J. Trott, H. Taub, H. R. Danner and F. Y. Hansen, Phys. Rev. Lett., 1979, 43, 1878. * H. Taub, G. J. Trott, F. Y. Hansen, H. R. Danner, J. P. Coulomb, J. P. Biberian, J. Suzanne and A. Thorny, in Ordering in Two Dimensions, ed.S . K. Sinha (North-Holland, New York, 1980), p. 91. Dr R. K. Thomas (University of Oxford) said: The paper distinguishes a lattice liquid at 95 K and below from a two-dimensional liquid at 122 K. What would make the distinction quite clear and would also give valuable information about the nature of the diffusion in the lattice liquid would be an Arrhenius plot. The activation energies would presumably be different for the two types of liquid. Does Prof. Bienfait have enough information to do such a plot? Fig. 3 of the paper shows that the jump translation of the molecules gives rise to a quasielastic broadening comparable with the resolution function. Under these conditions the parameters extracted from the data may be unduly sensitive to the model assumed for the molecular rotation.Since the model used for the rotation is rather an arbitrary one, can Prof. Bienfait indicate how reliable are his conclusions about translation in the lattice liquid? Prof. M. Bienfait (University of Marseille, France) replied to Dr Thomas: ( 1 ) Both quasi-elastic neutron scattering [this paper and ref. (3)] and LEED experi- ments [ref. (2)] show that the fluid L is much more isotropic than I,. However, there is no clear-cut transition between them; the orientational and positional ordering changes continuously with T when going from I t to L. This result is also supported by an Arrhenius plot of the translational diffusion coefficient [ref. (3)] that shows that I, and L have the same activation energy for diffusion within the experimental uncertainty.110 GENERAL DISCUSSION (2) (i) The model used for the rotation is not an arbitrary one, because the rotation at 66.4,71.4,76.2, 84.1 and 87 K can be interpreted with an isotropic model only and not at all with an uniaxial model [see fig.3, ref. (3)]. (ii) The molecular translational motion does give rise to a quasielastic broaden- ing comparable with the instrumental resolution (i.e. experimental width ca. 45 peV at Q = 0.83 &' vs a 35 peV resolution). Still, the interpretation is reliable because the instrumental function is well known and has a triangular shape. The translational quasielastic broadening has a Lorentzian shape whose wings extend much further than the foot of the instrumental function. We also tried to interpret fig.3 with a rotational model only (without translation), but we never obtained a good fit to the data. Finally, fig. 4 in ref. (3) shows that translation in the lattice liquid is an activated process whose activation energy is quite reasonable (ca. 1.4 kcal mol-'). Prof. G. Coma (KFA Jiilich, West Germany) said: What is the influence of the multiple scattering (in particular with the carbon atoms of the graphite substrate lattice) on the LEED patterns which Prof. Bienfait showed in order to demonstrate the various stages between isotropic liquid and lattice liquid? To be more specific: why is the LEED pattern of a submonolayer two-dimensional isotropic liquid on the graphite lattice at 122 K a simple non-modulated ring without any hexagonal pattern originating from the electron scattering at the substrate lattice; and how important is the contribution of this scattering on the LEED patterns (modulated ring, six elongated spots etc.) observed at lower temperatures? Prof.M. Bienfait (University of Marseille, France) replied: The highest tem- perature for which a LEED pattern of the C2H6 submonolayer has been recorded is 102 K [ref. (2)]. It still exhibits a slight modulation of the liquid ring. At this temperature, the LEED pattern does not show any ring centred around the (01) graphite spot, which seems to indicate that multidiffraction effects are negligible in that case. I do not believe that this conclusion can be modified at lower T because dynamical effects are not very temperature dependent. Prof.W. A. Steele (Pennsyluania State University, U.S.A.) said: It seems quite possible that the assumption of isotropic reorientation made for ethane in the monolayer is incorrect. This molecule is sufficiently anisotropic and the molecule- solid forces are sufficiently strong that one might better assume that the molecule exhibits reorientation such that the C-C bond remains essentially coplanar with the surface but rotates around an axis perpendicular to the surface. Theoretical correlation functions for such motion are known, both for jump and for small step (diffusional) reorientation.' The question is: are the experiments capable of distin- guishing between the limiting cases of the single axis and the random axis model? ' For a review see W. A. Steele, Adv. Chem.Phys., 1976, 34, I, section 111. Prof. M. Bienfait ( University of Marseille, France) answered: The theoretical EISF curve for a model where C2H6 molecules perform a rotation about an axis perpendicular to the C-C axis, is located between those describing the isotropic (random axis) and uniaxial (about the C-C axis) rotation models [see fig. 3 of our ref. (16)]. Our experimental data are closer to the 'isotropic' curve. However, one cannot rule out for I, the existence of more complicated motions where the ethane molecules perform librations in addition to rotations around an axis perpen-GENERAL DISCUSSION 111 dicular to C-C. This model and the isotropic model can only be distinguished at large Q (Q> 1.5 where our measurements are too inaccurate to draw any conclusion.However, our experiments are capable of distinguishing between the limiting cases of the single-axis and the random-axis model. They favour the latter. Dr D. A. Young (Imperial College, London) said: It is a little surprising that after so much work there has not developed a generally valid, accepted description of the orientation of physisorbed homonuclear diatomics and non-polar dumb-bells as the adsorption coverage on (0001) of graphite increases from 8 = 0 towards 8 = 1, with the temperature not far from the two-dimensional T, siwh that long-range crystalline order does not determine matters. Certainly one understands that for 8 < 0.005, say, adsorption occurs preferentially at defects, so orientation is then not of the essence. However, as the coverage increases, the adsorbed molecules will first diffuse across atomically smooth substrate as an ideal two-dimensional gas, the molecular polarisability tensor determining orientation: presumably the molecule will prefer to lay flat - always, whatever the temperature? At still higher coverages the gas will manifest imperfection and the molecules will form islets of a condensed phase, which itself shows only short-range, liquid-like positional disorder, ultimately covering the whole surface.But here, so far as orientation is concerned, the molecules have a choice. They can lie parallel to each other and to the substrate, or they can lie parallel to each other but not to the surface, or parallel to the basal plane but not to each other (herringbone?).Libration and free rotation at the highest tem- peratures will require a statistical description. Some recent work on the special case of (N,), by van der Avoird' in another context indicates the delicacy of the choice that has to be made. The reason for my interest comes from a need to interpret some preliminary data on the spectroscopic ellipsometry of bromine adsorbed on, and at higher pressures intercalated into, well oriented pyrolytic graphite. Pace Professors Scoles and Fain this is not a chemisorption system in any way comparable with CO on nickel. Rather should it be treated with respect as a senior, if unruly, member of the physisorption family. Thus up to 8<0.3 a wide variety of circumstantial evidence, suggests that the adsorbed species is Br,, possibly librating about the c-axis direction of the graphite ~ubstrate,~ probably adopting a hexagonal structure on account of intralayer dipole-dipole r e p ~ l s i o n .~ As the coverage increases beyond 8 =: 0.3 these intralayer repulsions ultimately nullify the otherwise dominant image forces, and ionic adsorption ceases. Neutral Br, is then added to complete the monolayer. There is little doubt that once the intercalation threshold has been exceeded the bromine molecules (ions) located within the interplanar spaces adopt some sort of flat herringbone structure. In this connection it is important to note that using basal-plane spectroscopic ellipsometry on the well oriented compound C ,6Br2 at room temperature with p ( Br,) = 170 Torr, one can detect the 211g,3/2,1/2 +- '2; elec- tronic transition on the Br, molecule ion at ca.1.24 eV in the extraordinary reflected ray (electric vector and optic axis of the substrate both in the plane of incidence). However, no such detection was possible below the intercalation threshold when bromine was adsorbed only on the external (0001) surface. Lack of sensitivity is not the explanation, for weaker transitions on neutral Br, were observed. However, if the Br, were oriented perpendicular to the basal plane (perhaps as the axle in a pinwheel structure) the II +X transition would not be observed in the reflected p-ray for reasons of selection rules, while the s-ray would be quite impotent to induce the transition because the in-plane conductivity of graphite is so good.112 GENERAL DISCUSSION So my comment to Drs Bienfait, Fain, Thomas and Tildesley asks for their views on a code of conduct for homonuclear diatomic molecules, coupled with an educated guess on the role of charge transfer in determining orientation.’ A. van der Avoird, Faraday Discuss. Chem. SOC., 1982, 73, 33. ’ J. D. Hibbs and D. A. Young, Chem. Phys. Lert., 1978, 53, 361; A. S. Bender and D. A. Young, J. Phys. C, 1972, 5, 2163. E. A. Stern, J. Vuc. Sci. Technol., 1977, 14, 461. J. J. Lander and J. Morrison, Surf. Sci., 1967, 6, 1. Prof. M. Bienfait (University of Marseille, France) (communicated). As far as the liquid monolayer of ethane adsorbed on graphite is concerned, we have shown by quasielastic neutron scattering that above the two-dimensional triple point, the C2H6 molecules perform rotational motions that depend on coverage [this paper ref.(3)]. Between 0.4 and 0.63 layer, the hydrogen atoms perform an isotropic motion around the C2H6 centre of mass. At and above the monolayer completion, the molecules lie parallel to each other and are perpendicular to the graphite surface; they perform a uniaxial rotational motion about their C-C axis, in addition to their diffusive translational motion. However, it would be risky to claim that the results represent the typical behaviour of non-polar dumb-bells adsorbed in fluid submonolayer films. Much work must be done in that direction to draw a comprehen- sive picture of the molecular orientation in two-dimensional fluid phases. Prof. S. C. Fain (University of Washington, U.S.A.) (communicated).In the low-temperature limit, I expect that a single diatomic molecule prefers to lie down as eggs on a table do. This allows the atoms in the molecule to take advantage of the attractive van der Waals forces with minimal cost in overlap energy. The calculations of crystal field in our paper give an example of where lateral forces at higher coverage want the molecules to stand up, just as for closely packed eggs. (I owe this analogy to John Berlinsky, who must have handled a lot of eggs.) In the high-coverage solid phases of monolayer oxygen on graphite, the molecules are all standing up almost perpendicular to the substrate.* This also occurs in the large positive crystal field limit of Harris and Berlinsky. As temperature is increased, there will certainly be a tendency for a single molecule to tilt out of the surface due to thermal excitation.I was surprised to see how little the nitrogen molecules tilt out of plane for the high-temperature fluid phase simulated by Talbot, Tildesley and Steele for nitrogen. I have not thought about cases involving charge transfer and can provide no insight at this time into your interesting data on bromine on graphite. The system of bromine intercalated graphite has provided some interesting physical realizations of two-dimensional models, as discussed in part by Erbil et a1.2 Some work on caesium adsorbed on graphite has been done by Hu et aL3 M. F. Toney and S. C. Fain, Phys. Rev., 1984, 30, 11 15-1 118 and references therein. Hu, Wu and Ignatiev, Bull. Am. Phys.SOC., 1985, 30, 331. * A. Erbil, A. R. Kortan, R. J. Birgeneau and M. S. Dresselhaus, Phvs. Reu., 1983, 28, 6329. Dr R. K. Thomas (University of Oxford) said: I do not wish to attempt a general answer to Dr Young’s questions. We have, however, been studying Br2 adsorbed on graphite using X-ray diffraction. Although it is too early to decide exactly what is going on, we find that its behaviour on the surface is quite different from any other physisorbed system that we have so far studied. We find evidence for fourGENERAL DISCUSSION 113 phases, in agreement with Lander and Morrison (LM).' The two lowest-coverage phases are unusual in that the first gives no discrete features in the pattern (possibly a lattice gas of LM) and the second gives a liquid-like pattern and must presumably be an amorphous structure.Of the two solid structures formed at coverages around the monolayer, the high-temperature one ( b 200 K) has a structure similar to though not exactly the same as the intercalate structure: i.e. the molecules form zig-zag chains and are lying nearly flat on the surface. More recent EXAFS data3 also indicate that the molecules lie flat from a coverage of 0.2 to 0.9 monolayers. Our structure is not at all consistent with either of those proposed in ref. ( 1 ) . However, it should be emphasized that we expect to repeat our measurements before publishing any firm conclusions. J. J. Lander and J. Morrison, Surf: Sci., 1967, 6, 1. A. Erbil, A. R. Kortan, R. J. Birgeneau and M. S. Dresselhaus, Phys. Rev. B, 1983, 28, 6329.E. A. Stern and S. M. Heald, in Handbook on Synchrotron Radiation, ed. Koch (North-Holland, Amsterdam, 1983), vol. 16. Dr D. J. Tildesley (University of Southampton) said: The recent molecular- dynamics simulations of the adsorption of N2 on graphite have helped to elucidate the behaviour of an adsorbed homonuclear diatomic molecule. In the course of the simulation the distribution function n(cos p ) is routinely calculated by sorting the observed out-of-plane orientations of the molecular axis into a histogram. The angle p is between the bond of the diatomic and the perpendicular to the surface and n(cos p ) d(cos p ) is the normalised probability of finding a molecule with cos p in the appropriate interval. In the simulation we use a histogram with 50 intervals for cos /3 in the range - 1 .O to 1 .O.In the case of a homonuclear diatomic this distribution should be symmetrical around cos p = 0. Simulation studies have been performed over a significant part of the phase diagram for N2 on graphite, and we can comment on the behaviour of n(cos p ) as a function of both temperature and density, although the picture is not yet complete. Fig. 12( a ) of ref. ( 1 ) shows the behaviour of n(cos p ) for the J3 x d3 commensur- ate solid at a coverage of one monolayer (0.064 molecules k') and a temperature of ca. 19 K. The maximum in the distribution is at cos p = 0 and it is sharply peaked. There are no molecules standing perpendicular to the surface. If we increase the temperature to ca. 42 K the molecules remain in the commensurate solid phase but undergo an in-plane orientational transition.At this temperature n(cos p ) is much broader with a slight upturn at cos p = *l [fig. 12(b) of ref. (l)]. The increase in temperature enables the molecules to sample all orientations with respect to the surface, although cos p = 0 is still the most probable orientation. An increase in density to a coverage of 1.05 causes the molecules to form the compressed uniaxial phase, where the compression is towards the lide-line of the low-temperature herringbone structure. Simulations of this phase show n( cos p ) developing a bimodal structure [see fig. 6 of ref. (2)]. The more pronounced peaks at cos p = *1 indicate the formation of transient pinwheels in the in-plane structure. Recent simulation studies of Vernov and Steele3 show that this bimodal structure becomes more pronounced with increased packing of the monolayer and in the buildup of the bilayer.The coverage dependence of n(cos p ) in the fluid phase at ca. 75 K has been presented at this meeting (see fig. 2 of the paper by Talbot et al. in this volume). The results can be represented quite accurately by the functional form B n(cosp)=ao+a, exp(-b,cos2~)+a2exp[-b2(l-cos2~)]. ( 1 )114 GENERAL DISCUSSION It is necessary to go to temperatures in the fluid monolayer above 120 K before n(cos p ) is uniform over the whole range of p. It should be remembered that the fine details of these results may be sensitive to the precise functional form of the potentials used in the simulation. The potential used in these studies gives an adequate fit to the properties of bulk condensed phase NZ, and reproduces some of the structural and thermodynamic properties of the monolayer accurately. However, there is room for improvement, particularly in the inclusion of the image charge interactions, which are not in the present model. A possible weakness of the potential is the failure of the simulation results to produce the incommensurate ‘two-out’ herringbone phase observed by You and Fain and reported in this volume. This structure would require a maximum in n(cos p ) for some /3 between 0 and 1. In the absence of simulation results on significantly longer molecules we can only conjecture that the torques supplied by the neighbouring molecules to rotate a particular molecule out of the surface would need to be larger than in the case of N2 and n(cos p ) would be correspondingly sharper. ’ J. Talbot, D. J. Tildesley and W. A. Steele, Mol. Phys., 1984, 51, 1331. * J. Talbot, D. J. Tildesley and W. A. Steele, Surf: Sci., in press. A. I. Vernov and W. A. Steele, to be published.

ISSN:0301-7249    

DOI:10.1039/DC9858000107

出版商:RSC    

年代:1985

数据来源: RSC