摘要:

GENERAL DISCUSSIONS OF THE FARADAY 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 973 974 974 975 975 976 Subject Inelastic Collisions of Atoms and Simple Molecules High Resolution Nuclear 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-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Oxidation For current availability of Discussion volumes, see back cover.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 61GENERAL DISCUSSIONS OF THE FARADAY 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 973 974 974 975 975 976 Subject Inelastic Collisions of Atoms and Simple Molecules High Resolution Nuclear 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-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Oxidation For current availability of Discussion volumes, see back cover.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

ISSN:0301-7249    

DOI:10.1039/DC97661FX001

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Date Subject 1907 1907 1910 1911 1912 1913 1913 1913 1914 1914 1915 1916 1916 1917 1917 1917 1918 1918 1918 1918 1919 1919 1920 1920 1920 1920 1921 1921 1921 1921 1922 1922 1923 1923 1923 1923 1923 1924 1924 1924 1924 1924 1925 1925 1926 1926 1927 1927 1927 1928 1929 1929 1929 1930 1930 Osmotic Pressure Hydrates in Solution The Constitution of Water High Temperature Work Magnetic Properties of Alloys Colloids and their Viscosity The Corrosion of Iron and Steel The Passivity of Metals Optical Rotatory Power The Hardening of Metals The Transformation of Pure Iron Methods and Appliances for the Attainment of High Temperatures in a Laboratory Refractory Materials Training and Work of the Chemical Engineer Osmotic Pressure Pyrometers and Pyrometry The Setting of Cements and Masters Electrical Furnaces Co-ordination of Scientific Publication The Occlusion of Gases by Metals The Present Position of the Theory of Ionization The Examination of Materials by X-Rays The Microscope: Its Design, Construction and Applications Basic Slags : Their Production and Utilization in Agriculture Physics and Chemistry of Colloids Electrodeposition and Electroplating Capillarity The Failure of Metals under Internal and Prolonged Stress Physico-Chemical Problems Relating to the Soil Catalysis with special reference to Newer Theories of Chemical Action Some Properties of Powders with special reference to Grading by The Generation and Utilization of Cold Alloys Resistant to Corrosion The Physical Chemistry of the Photographic Process The Electronic Theory of Valency Electrode Reactions and Equilibria Atmospheric Corrosion.First Report Investigation on Oppau Ammonium Sulphate-Nitrate Fluxes and Slags in Metal Melting and Working Physical and Physico-Chemical Problems relating to Textile Fibres The Physical Chemistry of Igneous Rock Formation Base Exchange in Soils The Physical Chemistry of Steel-Making Processes Photochemical Reactions in Liquids and Gases Explosive Reactions in Gaseous Media Physical Phenomena at Interfaces, with special reference to Molecular Atmospheric Corrosion. Second Report The Theory of Strong Electrolytes Cohesion and Related Problems Homogeneous Catalysis Crystal Structure and Chemical Constitution Atmospheric Corrosion of Metals. Third Report Molecular Spectra and Molecular Structure Optical Rotatory Power Colloid Science Applied to Biology Elutriation Orientation Volume Trans.3 3 6 7 8 9 9 9 10 10 11 12 12 13 13 13 14 14 14 14 15 15 16 16 16 16 17 17 17 17 18 18 19 19 19 19 19 20 20 20 20 20 21 21 22 22 23 23 24 24 25 25 25 26 26GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Date 1931 1932 1932 1933 1933 1934 1934 1935 1935 1936 1936 1937 1937 1938 1938 1939 1939 1940 1941 1941 1942 1943 1944 1945 1945 1946 1946 1947 1947 1947 1947 1948 1948 1949 1949 1949 1950 1950 1950 1950 1951 1951 1952 1952 1952 1953 1953 1954 1954 1955 1955 1956 1956 1957 1958 1957 1958 1959 1959 1960 1960 1961 1961 Subject Photochemical Processes The Adsorption of Gases by Solids The Colloid Aspect of Textile Materials Liquid Crystals and Anisotropic Melts Free Radicals Dipole Moments Colloidal Electrolytes The Structure of Metallic Coatings, Films and Surfaces The Phenomena of Polymerization and Condensation Disperse Systems in Gases: Dust, Smoke and Fog Structure and Molecular Forces in (a) Pure Liquids, and (b) Solutions The Properties and Functions of Membranes, Natural and Artificial Reaction Kinetics Chemical Reactions Involving Solids Luminescence Hydrocarbon Chemistry The Electrical Double Layer (owing to the outbreak of war the meeting was abzndoned, but the papers were printed in the Trunssactions) The Hydrogen Bond The Oil-Water Interface The Mechanism and Chemical Kinetics of Organic Reactions in Liquid The Structure and Reactions of Rubber Modes of Drug Action Molecular Weight and Molecular Weight Distribution in High Polymers.(Joint Meeting with the Plastics Group, Society of Chemical Industry) The Application of Infra-red Spectra to Chemical Problems Oxidation Dielectrics Swelling and Shrinking Electrode Processes The Labile Molecule Surface Chemistry. (Jointly with the Societ6 de Chimie Physique at Colloidal Electrolytes and Solutions The Interaction of Water and Porous Materials The Physical Chemistry of Process Metallurgy Crystal Growth Eipo-Proteins Chromatographic Analysis 1 leterogeneous Catalysis Physico-chemical Properties and Behaviour of Nuclear Acids Spectroscopy and Molecular Structure and Optical Methods of Inves- tigating Cell Structure Electrical Double Layer Hydrocarbons The Size and Shape Factor in Colloidal Systems Radiation Chemistry The Physical Chemistry of Proteins The Reactivity of Free Radicals The Equilibrium Properties of Solutions on Non-Electrolytes The Physical Chemistry of Dyeing and Tanning The Study of Fast Reactions Coagulation and Flocculation Microwave and Radio-Frequency Spectroscopy Physical Chemistry of Enzymes Membrane Phenomena Physical Chemistry of Processes at High Pressures Molecular Mechanism of Rate Processes in Solids Interactions in Ionic Solutions Configurations and Interactions of Macromolecules and Liquid Crystals Ions of the Transition Elements Energy Transfer with special reference to Biological Systems Crystal Imperfections and the Chemical Reactivity of Solids Oxidation-Reduction Reactions in Ionizing Solvents The Physical Chemistry of Aerosols Radiation Effects in Inorganic Solids The Structure and Properties of Ionic Melts Systems Bordeaux.) Published by Butterworths Scientific Publications, Ltd.Volume 27 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 35 36 37 37 38 39 40 41 42 42 A 42 B Disc. 1 2 Trans. 43 Disc. 3 4 5 6 7 8 Trans. 46 Disc. 9 Trans. 47 Disc. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32GENERAL DISCUSSIONS OF THE FARADAY 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 973 974 974 975 975 976 Subject Inelastic Collisions of Atoms and Simple Molecules High Resolution Nuclear 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-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Oxidation For current availability of Discussion volumes, see back cover.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

ISSN:0301-7249    

DOI:10.1039/DC976610X001

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

GENERAL DISCUSSIONS OF THE FARADAY 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 973 974 974 975 975 976 Subject Inelastic Collisions of Atoms and Simple Molecules High Resolution Nuclear 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-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Oxidation For current availability of Discussion volumes, see back cover.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 61GENERAL DISCUSSIONS OF THE FARADAY 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 973 974 974 975 975 976 Subject Inelastic Collisions of Atoms and Simple Molecules High Resolution Nuclear 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-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces Precipitation Oxidation For current availability of Discussion volumes, see back cover.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

ISSN:0301-7249    

DOI:10.1039/DC97661BX003

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

General Introduction BY E. ROY BUCKLE Department of Metallurgy, The University, Sheffield Sl 3JD Received 14th April, 1976 NUCLEATION AND GROWTH OF PRECIPITATES A precipitate is a particulate phase which separates from a continuous medium. The particles may, as the word implies, appear suddenly and fall away from the point of origin but, although it may be important observationally, the behaviour under gravity is not essential to the occurrence of a precipitate in the ordinary course of events. The production of particles by phase change at discrete centres, and the suddenness with which this occurs, is associated with a delay in the response of the homogeneous system to a change in external conditions. These must be altered to a set of values appropriate to a condition of the system in which a new phase accom- panies or replaces the old one.The freedom of the initial phase from catalytic impurities, particularly heterogeneous bodies such as dust particles, is essential to the deep penetration of the metastable region. On the other hand, the delay may be avoided altogether by the addition of a sufficient number of seed particles of the new phase, however fine. The theory of nucleation, founded on these facts, attributes the difficulty of initiating change in a clean system to the low probability of a local configuration of atoms of the appropriate structure and density, and large enough to be stable, arising naturally out of the chaos of molecular motion. A temporary arrangement of a group of molecular neighbours into the pattern of the new phase was termed by Frenkel a heterophase fluctuation,l and the group becomes a permanent focus of growth only if it acquires a sufficient number of molecules to form what Volmer called the germ, or nucleus.2 Volmer’s formulation of the theory of nucleation provided a basis for the calcula- tion of precipitation rates from the physical properties of the phases involved.The nucleus is predicted to be a very small entity, under typical conditions the radius extending over a few molecular diameters. Its concentration is an extremely small fraction of the total molecular concentration and it is only as a result of the very rapid growth of the nucleus that its formation is detectable. Growth of the nuclei remains undescribed by nucleation theory, and the growth stage has never been distinguished experimentally other than by the phenomenon of ageing.The problem of observing a nucleus is necessarily connected with its small- ness and transience, although if it is merely a large heterophase fluctuation it also lacks a property by which it could be distinguished from a smaller or a larger fluctuation. Steady flow experiments are in principle capable of yielding the size distribution of growing clusters. In the flow stream above the threshold of phase change, and where the homogeneous phase is metastable, the cluster concentration should fall to a minimum at the critical size corresponding to the local conditions. The presence of8 GENERAL INTRODUCTION atomic clusters in molecular beams sampled from vapours has been demonstrated mass-spectrometrically in recent years ; observation of the predicted curve would provide valuable support for the theory.In view of the difficulty in obtaining direct information about the nucleus from experiment, an important new development is the study of models by computer calculation. In the three papers on this theme the complexity of the problem is explored at different depths. Hoare and McInnes’s clusters are isolated systems of identical atoms with total energy restricted so as to confine displacements to solid-like vibrational amplitudes. The computer searches among the frozen configurations for examples of high relative stability. The relationship of the stability to the symmetry properties is of fundamental interest. In Briant’s molecular dynamics method cold assemblies of chosen configurations are fed kinetic energy to examine their reaction to temperature changes (there is still no exchange of material with the surroundings).For certain starting configurations a heating and cooling cycle increases the total energy and the structure acquires additional disorder, suggestive of a glass transition. In the Monte Carlo study of Abraham and co-workers gas-phase association of rigid water molecules with ions is modelled. Free energies of reaction can be tested against equilibrium data from molecular beam work. The detection of the metastable limit is at the heart of the majority of experimental studies. Assuming the nuclei to grow without restriction the position of the limit may be predicted from nucleation theory, and a comparison with experiment used to check the values of the physical constants.However, the mechanism by which nuclei enlarge to occupy substantial volumes in the system is of practical importance in crystallization, and competition between the growing centres may lead to the actual disappearance of nuclei if the original concentration is sufficiently high. Errors in the counting of nuclei to obtain the gross nucleation rate could result from the failure to recognize this possibility. In treating this communal ageing process theoretically, Kahlweit chooses to ignore latent heat and uses a macroscopic transport equation to describe the growth of individual particles under the control of isothermal mass transfer in the matrix. This gives a first-order dependence on the supersaturation c-c,.It is frequently assumed that latent heat is unimportant in studies on precipi- tation from aqueous solutions, but there is a tendency for a salt crystal to grow at a rate proportional to ( c - c , ) ~ for some of the time.3s As in the paper of Davies and Lewis, an isotropic, short-range diffusion model, analogous to collisions at a smooth wall, is commonly used to describe the growth of a nucleus in a simple melt. It was pointed out long ago by Volmer and Weber5 that growth might be sufficiently sensitive to crystal structure to require a nucleation step of its own. Jackson and Gilmer apply various lattice models to investigate the con- ditions under which the crystal surface roughens for intrinsic reasons, making such surface nucleation unnecessary. Even when this is the case the efficiency with which arriving atoms are assimilated is still sensitive to the stepped surface profile below the critical roughening temperature.BEHAVIOUR I N THE VICINITY OF CRITICAL POINTS During the past decade a certain amount of common ground has been established between nucleation and the critical phenomena of co-existing phases.”’* At the critical point of a first-order phase transition the density jump and the interfacial tension 0 vanish and the correlation length of fluctuations (a measure of the interfacial thicknessll) diverges. The nucleation event can therefore no longer exist, and it is possible to pass from gas to liquid (but not to solid) without producing an interfaceE .ROY BUCKLE 9 by avoiding the co-existence line12 (fig. 1). The interesting question now arises as to FIG. 1 .-One-component p , V,T diagram (schematic). ab bc bde dd ff b d LS, GL G'L' sublimation line melting line dewpoint line critical isochore critical isotherm triple point critical point paths of phase change requiring nucleation condensation path not requiring nucleation how far down the co-existence line one must make the crossing in order to induce a recognizable precipitation effect. The problem also occurs of the values to be assigned to terms in nucleation theory to make it applicable in this region. According to the theory of exponents at the gas-liquid critical point,13 CTLG K (T,- T)p and pjd-pG cc (Tc- T)P on the co-existence line (fig. 1, bde) and 1 /C, cc I T,- TIa along the critical isochore (fig. 1, dd').Metastability in C 0 2 has been detected calorimetrically by slow heating and cooling across the co-existence line at super- critical densities close to the critical point.14 The gas phase should nucleate on this side of the critical point (to the left of d in fig. 1). A metastable condition corre- sponding to about 0.5 K of supercooling was achieved and C, remained a smooth function of T. This suggests that the value of the exponent dL is unchanged when the liquid supercools. The use of exponents to re-write the nucleation rate equation has been proposed for such a situation,15 and scaling laws relating various exponents have been adopted in the comparison of nucleation models with condensation data.16 The increase in CT below T , and the decrease in interfacial thickness, expressed in terms of y and the correlation length exponent v, respectively, are linked by a scaling hypothesis l1 p+v = 2-a.A similar set of exponents may be applicable in the vicinity of the liquid-liquid critical point of a binary system.17 In addition to the behaviour of a system close to a critical point, the critical exponent formalism may be of assistance in the prediction of physical properties under conditions far removed from it. Various empirical rules for the temperature dependence exist18 for quantities like r~ and the latent heat L under ordinary con- ditions which are useful in approximate calculations using nucleation theory. These may be compared with exponent theory in appropriate cases and interpolation formulae constructed.10 GENERAL INTRODUCTION Searching tests of nucleation theory are made possible by experiments of the kind described by Wegener and Wu and by Katz and Virkler.An area of uncertainty frequently blamed for the failure of theory is the effect of interfacial curvature (particle size) on the value of c. In wind tunnel work, particularly, conditions can be made to change so rapidly along the flow that one can explore the region of onset when condensation depends on growth of clusters through a very small critical size. The Volmer theory is unsuitable for the prediction of such detail, and there is a pressing need for a more realistic step-by-step description of the growing particle containing fewer than a hundred molecules.SYMMETRY I N PHASE CHANGES INVOLVING CRYSTALS The change in density and the heat evolved in a first-order transition become gradually less marked as the crossing point approaches the critical point. Transitions which involve a pure crystalline solid, however, are not processes which can be brought about continuously (for example, lines ab or bc in fig. 1 must be crossed), and this is connected with the fact that the symmetry elements of the crystal lattice always appear or disappear abruptly, however gradually the atomic positions are varied.lg This is exemplified by the martensitic transformations graphically described by Thomas and co-workers. The nucleus of a crystal formed by heterogeneous fluctuation will adopt and maintain from the outset the lattice symmetry of the crystal form belonging to the phase field into which the matrix was cooled.But what happens when the critical size is comparable to the lattice parameter ? The question assumes practical as well as conceptual significance when applied to condensation if the co-existence line is crossed rapidly enough at the triple point or just above it. Aspects of the problem arise in connection with the work reported by Davies and Lewis, where the change from melt to crystal was indefinitely postponed by the speed of quenching, and in that of Buckle and Pointon, in which it appears that one condensation path crossed bd to be followed by freezing (bc) while at the same time another crossed ab direct to the solid (fig. 1). COMPLEX PHENOMENA ACCOMPANYING PRECIPITATION From the purely theoretical standpoint there are advantages and disadvantages in the study of complex systems.Motion of particles may result from general con- vection in the matrix or by direct interaction with external fields. Mazeau and Zarzycki’s ultrasonic waves accelerated nucleation in the glass transition region, apparently through their effect on T and p? and the number of growing particles was multiplied by disintegration. Another mechanical effect is collision breeding,20 observed here by Nielsen. From the influence of sterically significant additives Nielsen concludes that it could be the growth rate rather than the nucleation rate that controlled the onset of the heat effect in his experiments. Experiments with long-chain compounds are particularly suited to the investigation of the effect of increasing molecular complexity on a phase change, as described by Urry.In systems forming liquid mesophases, nucleation, which may be the vital step in the change in properties of the whole phase, should be carefully distinguished from chemically specific processes of limited aggregation, such as micellization, which may precede it. Jamieson and co-workers identify as a nucleation-controlled process the spontaneous separation of the liquid coacervate phase from tropoelastin solutions on heating. The critical size in their scheme is that of the ordered micelle, but the heterophase fluctuation would seem to involve randomization of the molecular chainE. ROY BUCKLE 11 configurations inside it.This is an intriguing extension, if not a complication, of the conventional theoretical ideas we are here to discuss. J. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955). M. Volmer, Kinetik der Phasenbildung (Th. Steinkopff, Dresden and Leipzig, 1939). A. E. Nielsen, Kinetics of PrecQitation (Pergamon, Oxford, 1964). M. Kahlweit, Adv. Colloid. Interface Sci., 1975, 5, 1. M. Volmer and A. Weber, 2. phys. Chem., 1926, 119,277. J. W. Cahn, Trans. Met. Soc. AIME, 1968, 242, 166. F. H. Stillinger, Jr., J. Chem. Phys., 1967, 47, 2513. D. Stauffer, C . S. Kiang, A. Eggington, E. M. Patterson, 0. P. Puri, G. H. Walker and J. D Wise, Jr., Phys. Rev. By 1972, 6, 2780. ’ M. E. Fisher, Physics, 1967, 3, 255. lo J. S. Huang, W. I. Goldburg and M. R. Moldover, Phys. Rev. Letters, 1975, 34, 639. l1 B. Widom, Phase Transitions and Critical Phenomena, vol. 2, ed. C . Domb and M. S . Green i2 J. S. Rowlinson, Liquids and Liquid Mixtures (Butterworths, London, 1969), pp. 2-4. l3 P. A. Egelstaff and J. W. Ring, Physics of Simple Liquids, ed. H. V. N. Temperley, J. S. Rowlin- l4 J. Straub, 3rd. Internat. Con$ Chem. Thermodyn. and Symp. Phys.-Chem. Techniques at High l5 J. S. Langer and L. A. Turski, Phys. Rev. A , 1973, 8, 3230. l6 C. S. Kiang, D. Stauffer, G. H. Walker, 0. P. Puri, J. D. Wise, Jr. and E. M. Patterson, l7 Ref. (12), p. 155. l8 J. R. Partington, An Advanced Treatise on Physical Chemistry, vol. 2 (Longmans, London, 1962). l9 L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, London, 1958). 2o R. F. Strickland-Constable, Kinetics and Mechanism of Crystallization, (Academic Press, (Academic Press, London, 1972), p. 79. son and G. S. Rushbrooke (North-Holland, Amsterdam, 1967), p. 253. Temp., Baden, Sept. 1973, p. 40. J. Atmos. Sci., 1971, 28, 1222. London and New York, 1968), p. 112.

ISSN:0301-7249    

DOI:10.1039/DC9766100007

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Statistical Mechanics and Morphology of very small Atomic Clusters BY M. R. HOARE AND J. MCINNESJ~ Department of Physics, Bedford College, Regent’s Park, London NWl 4NS Received 19th January, 1976 Mechanical and thermodynamic systems of a very few atoms (N < 100) interacting through central-forces are considered in the light of their role in an ideal model for “ precipitation ”. We discuss the stability of such systems at low energies and present detailed numerical investigations of the potential-energy surfaces of “ clusters ” of up to thirteen atoms interacting through Lennard-Jones and Morse potentials. Our main results comprise what we believe to be an almost exhaustive survey of the distinct minima available to systems of this size for the potentials used. Each such stable structure obtained is vibrationally and rotationally analysed and its symmetry is examined.The most striking feature of these results is the extreme sensitivity of the number of possible stable configurations to the range and softness of the pair potential. Thus, of no fewer than 988 minima for 13 Lennard-Jones atoms, only some 36 are supported by the (a = 3) Morse potential. The minima available are also classified geometrically and it is shown that non-crystallographic con- figurations predominate in structures of both greatest and least binding energy. A preliminary account of the statistical mechanics of cluster systems based on the rigid-rotor/har- monic oscillator approximation is given. 1. INTRODUCTION In recent years it has been possible for the first time to generate and examine very small clusters of atoms (N < 100) under effectively “ free-space ” conditions and to obtain information on their structural, mechanical and electronic properties to an accuracy sufficient to invite detailed theoretical investigation at a level hitherto reserved for stable, chemically-bonded molecules.The greatest single advance in the physics of “ microclusters ”, “ Van der Waals molecules ” or whatever else we may wish to call such aggregates, has undoubtedly been due to the development of the supersonic nozzle-beam in combination with various size and structure-discrirninat- ing detectors. Its use in conjunction with mass-spectrometry,1-6 electron diffrac- tion 7-9 and optical methods has made possible not only the determination of geometrical characteristics but also, in favourable cases, properties such as ionization potential or work f~nction.~ These experiments, which are unfortunately not represented at this Discussion, could be said to provide the most clear-cut instances of “ precipitation ” so far known and, as such, offer the most attractive possibilities for the comparison of theory and experiment. At the same time, advances in electron diffraction and microscopy have brought down the threshold for observation of small- particle deposits to the region of N < 100 atoms and supplemented the results of beam experiments under considerably more varied, if less controllable, conditions.The importance of these experiments, which we can only allude to here, ranges from catalysis to astrophysics and crosses the boundaries of subjects such as nucleation, crystal growth and surface physics in all its variety.For further background and bibliography we refer to earlier papers.10-12 Present address: Department of Physics, University of Warwick, Coventry CV4 7AL.M. R . HOARE AND J . MCINNES 13 In this Discussion paper space limitations prevent us taking up more than the first stages in an attempt to develop a mechanics of few-atom systems in what, in the absence of well-defined phase transitions, we can only call the “ solid-like ” region. By this we refer to the indefinite condition in which a cluster can be said to be at Sufficiently low energy for it to possess a definite geometrical identity on an experi- mental time-scale yet sufficiently high energy to undergo structural equilibration and perhaps sublimation.Most of the results we shall present here will be those of various explorations of the potential-energy surfaces of clusters in few-atom con- figuration space, designed to discover first the positions and multiplicity of the stable minima available and the dependence of these on the nature of the inter-atomic potential, then to construct an approximate statistical mechanics based on their local geometry. The result is an as yet primitive “ statistical morphology ” of very small particles which may provide the basis for more special treatments of electronic properties, catalytic activity, nucleation kinetics etc. 2. THE POTENTIAL-ENERGY SURFACE Consider the mechanical system of a few atoms (say for present purposes N 10) interacting in free space with a specified Hamiltonian.Let the system be classical and governed entirely by central two-body forces derivable from a realistic pair- potential ~ ( r ) . Under these conditions the total potential energy of the system can be expressed as JvN) = $2 c NI~L-rJl) (2.1) i j # i where ri is a position vector for the i’th particle and rN = (rl,r2, . . . rN} specifies a point in configuration space of 3N-6 dimensions. (Although not evident in the notation, the six degrees of freedom for rotation and centre of mass motion are assumed removed, e.g., by requiring: rl = (O,O,O), r2 = (x2,0,), r3 = (x3,y3,0).) Under these conditions V(rN) defines a potential energy surface in a space of 3N-5 dimensions inheriting certain characteristics (boundedness, differentiability, behaviour at infinity) from the pair potential ~ ( r ) , but otherwise embodying geometrical informa- tion of extraordinary complexity.We shall attempt to analyse certain aspects of this under the simplifications introduced above. Needless to say, further complica- tions would arise in a more realistic treatment involving non-central forces, many- body potentials and quantum-mechanical effects. STATICS Certain conditions are required on the pair potential function in order that it support definite stable configurations of the N atoms with finite separation. In- tuitively we expect the usual well-potentials of Lennard-Jones or Morse type to provide for this; somewhat more generally we may assert that a sufficient condition for the existence of at least one stable configuration of minimum potential energy is that ~ ( r ) should be differentiable, bounded below, convex at infinity and positive infinite at r = 0.Under these conditions, V(rN) will have a greatest lower bound at some absolute minimum configuration rN(o). The centres of the atoms will occupy points in a definite geometrical figure, the external vertices of which will form a poly- hedron in 3-space. * Physically we can refer to this object as a cluster of N atoms or, * Evidently the physical cluster at its minimum will strictly correspond to a set of N! points in configuration space differing only in the numberings of the set of atoms. We refer to these collec- tively as a minimal cluster or absolutely minimal cluster as appropriate.A further problem arises14 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS more briefly, as an N-mer. The positive quantity V( GO)- V(rN(o,) with V( GO) the poten- tial energy when all atoms are infinitely separated, can be identified with the binding energy of the absolute minimum and this is conveniently measured in units of the well depth in the individual pair-potential. As is easily imagined, the set of configurations absolutely minimal in potential energy will be only one of a multitude of such sets with lesser binding-energy (higher potential-energy). The precise number of such local minima is a mysterious quantity, though it can be expected to be sensitive to the nature of the pair-potential v(r).Each minimum will be characterized by the vanishing of all partial derivatives (aV/ax,) and the positive-definiteness of the Hessian matrix (a2 V/ax,8x,) in any internal co- ordinate system (principle of virtual work). The eigenvalues of the Hessian matrix correspond geometrically to the principal radii of curvature at the minimum in the potential energy surface and physically to the normal-mode frequencies of vibration governing small motions. A number of invariants can be defined to test whether two given minima are in fact geometrically distinct. A sufficient condition for the geometrical distinctness of two configurations is that the inertia tensors of each corresponding cluster should differ. A less easily computed sufficient condition for their identity is that the inter- atomic distance matrices D i j = lri-vjl should be identical for some choice of number- ing.This does not, however, distinguish between enantiomorphic pairs. To test whether a structure possesses an enantiomorph we may use the fact that, if a plane of symmetry is present, two of the principal axes of the inertia ellipsoid will lie in it. To establish the absence of a plane of symmetry it therefore suffices to test the sym- metry of the three planes determined by the principal axes. 3. MINIMUM SURVEYS How many geometrically-distinct minimal structures exist for, say, ten Lennard- Jones atoms interacting through the pair-potential V(r) = r-l2--2rm6 and, of these which is absolutely minimal? If we replace this by the Morse potential V(r) = (1 -exp(3(1 -r))}2- 1 does the number change and, if so, can subsets of the minima for each potential be put into one-to-one correspondence with each other? Can the sets of minima in either case be related to the alternative question of enumerating the number of distinct packings of hard spheres in contact? Only in the cases N = 3 and N = 4 can we give unequivocal answers to these questions. (By writing the potential energy in terms of " bond-lengths " between neighbours it follows that the unique minima possible are the equilateral triangle and regular tetrahedron respectively.) Already at N = 5 the problem is highly non- trivial and the futility of trying to discover minima by systematic searching of con- figuration space becomes clear.(N.b., *N(N- 1) > 3N-6 for N > 4.Any attempt to write out the partial derivatives and solve for their stationary values is equally futile.) where two minimal configurations referred to the same co-ordinate system have identical energy by virtue of being equivalent under the operations of translation, rotation or reflection (improper rota- tion). We shall say that such configurations are geometrically equivalent and refer to all possibilities collectively as a single isomer. In the last case, however, we may wish to distinguish between enantio- morphic pairs sharing the same energy. These can occur with N 2 6.M. R. HOARE AND J . MCINNES 15 Nevertheless, the possibility exists of finding an algorithm which will discover at least a large proportion of the existing minima for N > 5 if not all of them.The key to such an algorithm is to be found in observing the way hard spheres can be composed in rigid contact. Just as is suggested by sphere packings, it is reasonable to expect that a large proportion of the minima available to N+1 atoms consists of those derivable by addition of the (N+l)’st atom to a favourable position at the surface of a previously established N-atom cluster. Although in certain cases the suggested minimum may fail to exist, and the cluster collapses to one of more or less unrelated geometry, computer experiments confirm that potentials of the general range and hardness of the Lennard-Jones function are capable of supporting the growth of successive minima by the stepwise process just described. It is too much to hope that we could find all existing minima in this way because we cannot rule out the possibility that the addition of several atoms simultaneously and their positioning in some co-operative fashion may lead to a stable structure of a different kind.As we shall see, however, it is almost inconceivable that the absolutely minimal structure for N = 10 would go undiscovered by this method. Descriptions have been given elsewhere of the methods available for the computer “ optimization ” of many variable functions, i.e., for the precise determination of local minima in the neighbourhood of a given starting configuration and with func- tions of the complexity of eqn (2.1).13 These methods are sufficiently fast and convenient, even with functions of some hundreds of variables, that we can afford to speak in physical terms of “ relaxing ” a structure to a minimum or carrying out an experiment in “soft-sphere packings ” without going into details of the numerical analysis.* To convert the above ideas into a definite algorithm we need to identify all the smallest “ seed-structures ” that seem likely to occur and “ grow ” minima upon these by making all possible positionings of additional atoms at their surface, comparing every N-atom minimum so obtained and deleting all geometrically indistinguishable structures from the list before continuing the process to size N+ 1. The choice of small seed-structures does not seem to be a serious problem. The two most important are clearly the N = 4 tetrahedron and the N = 6 octahedron. To these we may add a small number of others in the size range of N < 10 from Bernal’s list of canonical polyhedra. (The Archimedean antiprism, the capped trigonal biprism etc.15) In outline, the required algorithm goes as follows.(1) Pick a seed-structure of N atoms and relax it to its minimum in the required potential. Keeping this configuration fixed, test the structures obtained on adding an additional atom to each stable surface position in turn. Store all configurations which are geometrically distinct. Confirm that they remain distinct and delete any which collapse to another structure in the list. (3) Take each of the (N+1) atom structures discovered in this way and repeat the procedures under (1) and (2) above using it in place of the seed-structure. (4) Terminate the sequence when computing time becomes unreasonable.Pick a new seed-structure and begin again, checking each new minimum for dis- tinctness against the full list from all previous stages. We carried out this programme using the London University CDC7600 computer and investing most of our computing time in a study of the minima for the Lennard- Jones potential. Our main object was to reach the size N = 13 which we knew * Our recent work on the “relaxation” of the Bernal random close-packed structure is an example of a calculation carried out relatively easily in n. configuration space of some 3000 variables.“ (2) Relax these configurations to their true (N+l) atom minima.16 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS would include the very important minimum configuration in which twelve atoms are grouped in an icosahedral shell about a thirteenth.This appears to be the smallest possible structure in which one atom is completely enclosed in a co-ordination shell. We were able to enumerate all the N = 13 Lennard-Jones minima generated by the algorithm, analyse the statistics of their distribution in potential energy and later carry out a vibrational analysis of each one to determine its normal-mode frequencies. The principal moments of inertia of each were also determined for application in statistical mechanics. For present purposes, however, we shall confine ourselves mainly to the first stages, which might be put under the somewhat quaint heading of ‘‘ statistical statics ”. 4. RESULTS FOR THE LENNARD-JONES POTENTIAL As we indicated earlier, one can make a reasonable guess at the outcome of the first stages of the computed growth sequence by experimenting with the packing of real spheres.Thus it is clear that, starting from the N = 4 tetrahedral seed, the first isomeric minima (3) occur at N = 7, giving rise to five distinct at N = 8 and thereafter rising extremely rapidly. * The first stages in the tree-like interrelationship of minima of the tetrahedral family are identified in fig. 1 which, though very limited, nevertheless FIG. 1 .-Interrelationships for tetrahedral minima of up to eight Lennard-Jones atoms. The letters in each circle identify the structures, the small numbers identify numbers of equivalent surface facets for the particular growth step.The following have recognizable geometry: 6A,7A,8A (Boerdijk spiral) ; 7B (Pentagonal bipyamid) ; 8E (Stellated tetrahedron). The other structures have various planes and axes of symmetry some of which may be recognized in the pictorial representations in fig. 2. The Boerdijk spiral configurations alone have no plane of symmetry and thus possess enantiomorphs. illustrates several of the features rspeated in far greater complexity for the larger isomers. When the computation was stopped at N = 13 it was found that no less than 988 distinct minima had been verified and listed, of which 131 possessed enantiomorphs. MULTIPLICITY OF ISOMERS The final tally of isomers for N = 6 to N = 13 is set out in table 1 with sub-totals of the three main types: tetrahedral, octahedral and others.It is relatively easy to * Enumeration of structures by hand model-building is somewhat unreliable. In an earlier paper we listed only four of the five N = 8 isomers, a mistake later corrected by Bonnissent and Mutafschiev.16M. R. HOARE AND J . MCINNES 17 make computer-generated drawings of the actual minimum configurations in sphere- packing, ball-and-spoke and other conventional renderings. We illustrate a few of these in fig. 2 and 3. Fig. 2 shows the tetrahedral subset of minima for N = 7 to N = 10. In fig. 3 we have listed separately parts of the lists of minima for N = 13 TABLE 1 .-MULTIPLICITY OF L-J ISOMERS size 6 7 8 9 10 11 12 13 tetrahedral 1 3 5 11 25 69 171 483 octahedral 1 1 3 6 29 60 143 338 others 0 0 0 1 3 16 52 167 total 2 (1) 4 (1) 8 (1) 18 (3) 57 (11) 145 (19) 366 (47) 988 (131) with tetrahedral and octahedral types respectively.In each case the most stable structure appears top left with isomers in descending order of binding energy along the rows. Thus the 13-atom icosahedron-beyond reasonable doubt the absolute minimum-appears first in fig. 3a and one notices immediately the trend towards more extended structures as the binding energy decreases. With the single exception of the N = 6 octahedron (a " seed-structure ") the N = 7 (3) N = 8 (51 N = 9 (11 I I N = 10 (251 FIG. 2.-Tetrahedral isomers for N = 7 to N = 10. Automatically printed projections of the computed minimum configurations for the Lennard-Jones potential. Binding energies decrease from left to right and in successive rows.The Boerdijk spiral configurations are marked with a cross.18 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS FIG. 3a.-Tetrahedral isomers for thirteen Lennard-Jones atoms. The figure shows only the first and last 43 configurations of the total of 483 arranged in order of decreasing binding energy. The structure of greatest binding energy, the 13-atom icosahedron, is seen top left. The Boerdijk spiral is again marked with a cross. minima of greatest binding energy for the L-J potential are all of " polytetrahedral " type and follow the Werfelmeier sequence-the progression of structures obtained on adding atoms around the five-fold axis of a 7-atom pentagonal bipyramid to give the icosahedron.ll9 l7 A few of the configurations seen have been mentioned in sphere- packing studies, for example the Boerdijk spiral configuration resembling a chain of tetrahedra face to face.18 These minima (marked with a cross in the figures) are, some- what surprisingly, not the least stable available but are followed in the lists by numer- ous others of less systematic extended structure.It is interesting that in the course of generating the sequence of tetrahedral minima, no major structural rearrangements were found to occur on carrying out stage (2) of the algorithm with the Lennard-Jones potential. An example of mechanical collapse was, however, found in the octahedral sequence. The eight-atom structure consisting of an octahedron with two extra atoms in a skewed configuration (one of four possi- bilities) proves unstable and moves to a structure of tetrahedral type already accounted for.Another of the four N = 8 octahedral structures is also anomalous-it distortsM. R . HOARE AND J . MCINNES 19 FIG. 36.-Octahedral isomers for thirteen Lennard-Jones atoms. The first and last 38 structures of the total of 338 are shown in order of decreasing binding energy. Note how in many structures, though not invariably, the 6-atom octahedral sub-unit has distorted to part of an 8-atom " dodeca- deltahedron " at the minimum. spontaneously with a fundamental change of symmetry to form the " dodecadelta- hedron " (" DD "), a minimum envisaged by Werfelmeier in his 1937 study of alpha-particle structures and later listed by Bernal as a canonical polyhedron.15* l9 These special features of octahedral minima not only affect the number of larger isomers which can form, but may point to a mechanical disability shared by very small nuclei containing vestiges of cubic close-packed structure.This could be of importance in crystal growth-theory. (See also discussion in ref. (1 l).) Numerous " DD " structures can be distinguished on close examination of fig. 3b. STATISTICAL DISTRIBUTION OF POTENTIAL ENERGY OVER MINIMA The N = 13 isomers are sufficiently numerous to provide an adequate sample for statistical analysis of the distribution of binding energies over the whole set. Fig. 4 shows a histogram of the full set of N = 13 minima without inclusion of enantio- morphs. The curve obtained is noticeable asymmetric with a pronounced " foot " at high energies leading to the detached single point for the icosahedral structure.The latter is well separated from its closest competitor in energy (E = 44.33 and E = 41.47 respectively in pair units). The width of the distribution is narrower than20 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS might be expected considering the variations of shape and compactness which occur. This, however, reflects the fact that, so long as clusters are too small to contain an appreciable number of " interior " atoms, changes of shape can only alter potential energy appreciably through variation in the distribution of second nearest neighbour distances. The asymmetry of the peak indicates that it is easier to form alternative structures by variation of the less stable, more elongated structures than the more compact units tending to the icosahedral configuration.35.0 45.0 binding energy FIG. 4.Distribution of number of isomers by binding energy for thirteen Lennard-Jones atoms. Solid line: tetrahedral isomers. Dashed line: octahedral isomers. Energy is given in units of the L-J pair-energy and the isolated point is the icosahedral structure. The points shown make up 521 of the 988 minima found. ANALYSIS BY TYPES Although we have previously studied the relative stability of crystalline and non- crystalline units by means of isolated examples l1 it was not previously possible to make a systematic survey of the symmetry properties of minima over a list as exhaus- tive as that generated here. The main trend of the results is that octahedral structures in the size-range N 5 13 are comparable in numbers to the tetrahedral types but possess binding energies more narrowly peaked about the most probable.For N = 13 the first fourteen minima in order of binding energy are tetrahedral and thus seem likely, in the absence of special entropic effects, to dominate the equilibrium population of isomers in a condensing system. Fig. 5 is an attempt to present the statistics of structural types graphically. Each of the three symbols marks the structural type of a particular minimum in the complete N = 13 Lennard-Jones set when these are arranged in decreasing order of binding energy as in the pictorial plots of fig. 2 and 3. The predominance of tetrahedral minima at both extremes of binding energy and the relatively uniform distribution of other types nearer the peak of the distribution is clearly distinguishable. 5.RESULTS FOR THE MORSE POTENTIAL We earlier posed the question of whether the numbers of distinguishable minima for a few atoms under central two-body forces would be markedly sensitive to the shape of the well in the pair-potential. Previous studies on particular structures suchM. R. HOARE AND J . MCINNES 21 FIG. 5.-Distribution of structural types for 988 Lennard-Jones minima of 13 Lennard-Jones atoms. Each structure is coded by a symbol according to type and these are printed as in fig. 3. in order of decreasing binding energy, illustrating the " interleaving " of different structural types not apparent in fig. 4. The lower figure shows the tetrahedral minima alone and illustrates the bunching of these at lower and higher energy ranges.Q = tetrahedral, 0 = octahedral, El = other. as the icosahedron have indicated that the range and softness of the potential can have considerable effect on the compactness of the minimum found 2o and in certain cases be a deciding factor in whether a particular minimum has perfect or broken symmetry.21 It did not seem justifiable to repeat the extensive computing of the growth algorithm described in Section 3 with a change of potential. Instead we used a short cut which seemed likely to generate a comparable set of Morse minima from the Lennard-Jones22 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS set already obtained without the costly repetitiveness of the original scheme.We accordingly took each distinct N-atom Lennard-Jones minimum and applied a Morse potential to the same configuration. The resulting points on the new potential-energy surface were no longer minimal, but from this starting point a new minimum could be sought by a relaxation programme as before. While there could be no certainty that the programme would not " miss " a very shallow minimum in this neighbourhood and lead the configuration point over a saddle to an alternative one, it seemed reason- able to assume that the method would discover those isomeric minima which, while altering somewhat their position in configuration space, nevertheless retained their nearest neighbour topology and thus could be said to be " supported " by the Morse potential.In the event a surprisingly small fraction of the Lennard-Jones minima survived in the Morse potential energy surface-no more than 36 of the original 988! (table 2). This is less surprising at second sight when we consider the tree-like interrelationship of the minima on potential energy surfaces of progressively higher dimension. Thus, if the minima corresponding to some particular geometrical motif is " squeezed out " on softening or lengthening the pair-potential, this can only eliminate progressively more minima from each generation. We are still investigating the geometrical characteristics of the minima which are interconverted on replacing the Lennard-Jones potential with the Morse. It should not be difficult to identify precisely which structural features lose their stability on softening the potential, and to verify that the disappearance of only one or two types of sub-structure can suffice to cut the number of distinct minima as drastically as found.Our preliminary finding is that certain tetrahedral configurations (including the single N = 6 structure) '' open-up '' to give octahedral-type minima on softening the potential, the longer-range potential thus showing a greater tendency for form crystallographic structures. Whether this type of rearrangement is alone responsible for the very considerable smoothing-out of the topography of the P-E surface remains to be seen. A particularly striking result is the reduction in the number of stable Morse isomers on going from N = 11 to N = 12 (24 reducing to 22).This result need not be paradoxical if we visualize what is involved. With N = 11 the apparently absolute minimum is the icosahedron with two adjacent gaps. It may well be that, for the Morse potential a number of variants of this structure have a precarious existence close to low saddle-points. Addition of the twelfth atom may have sufficient shrink- ing effect to tip the balance and destroy a number of arrangements with stable 11- atom sub-units. The role of the form of the pair potential in determining the stability of alternative structures is evidently a complex one, though not altogether impossible to imagine- TABLE 2.-MULTlPLICITY OF MORSE, L-J AND HARD-SPHERE ISOMERS size 6 7 8 9 10 1 1 12 13 morse 1 3 5 8 16 24 22 36 L-J 2 4 8 18 57 145 366 988 hard-sphere 2 4 >10 >32 >113 >473 * * * The numbers quoted for hard-sphere isomers refer only to tetrahedral-type configurations.M.R. HOARE AND J . MCINNES 23 in somewhat picturesque terms we might say that the packing of Morse spheres is like packing frog-spawn; packing Lennard-Jones spheres is nearer to the packing of caviar. These peculiarities are entirely a function of the extreme smallness of the clusters considered here and can only be weakly reflected in the properties of the bulk phase or the infinite surface. Nevertheless, they could prove to be one of the more important on the properties of special systems such as nucleating films or ultra-finely dispersed metal catalysts. 6. HARD-SPHERE ISOMERS We also commented on the relationship between the multiplicity of minima for realistic potentials and the problem of enumerating configurations of a few hard spheres in contact.As a number of authors have pointed out,22 this problem is not well-defined inasmuch as the tetrahedral space-filling deficit leads to small gaps or " clearance regions " which, in clusters of more than seven atoms, can themselves be permuted around a structure, leading to a considerable increase in stable configura- tions only marginally different from each other. On imposing a realistic potential and " relaxing " the structure, the clearance regions predictably " heal-up ", usually to give almost perfectly symmetrical pentagonal motifs. * The numbers of sphere- packing isomers for a few atoms are thus of limited interest.For the sake of com- parison, however, we determined the multiplicities up to N = 11 and have included the figures in table 1. We have also investigated the possibility of enumerating packing isomers by graph theoretic methods with only limited success.23 The analogous, though simpler, problem of finding the numbers of " animals " of N adjacent points on a lattice is well known to be extraordinarily diffi~ult.'~ 7. CLUSTER EQUILIBRIA The extension of our results on the statics of cluster isomers to a statistical mech- anics of equilibria and growth-kinetics is possible, though subject to a number of drastic assumptions. If we take it that all the configurational states available to an " N-mer " at a given temperature are solid-like and characterized by small motions in the neighbourhood of stable minima, then the problem of describing the equilibrium populations of each is analogous to that of describing the state of a molecule with an unusually large number of tautomers interconvertible via energy-barriers.Provided we accept the harmonic-oscillator/rigid-rotor approximation, a partition function can be written for each distinct isomer and these in turn can be summed to give the partition function for all states, both configurational and internal, of the N-atom cluster. By further combination of these a grand partition function may also be constructed as a function of N. To implement this calculation the normal vibration frequencies and principal moments of inertia of every single isomer are required- these we have obtained for the full set listed in table 1 up to N = 13.It is less easy to obtain rotational symmetry factors in every case and we have so far been unable to apply these systematically. (N.b., a-factors can be as high as 60 for the N = 13 icosahedron and have a pronounced effect on the free-energy and equilibrium con- centrations.) It will be noted that a tacit assumption in the above is that, for tem- * Much larger soft-packed clusters may show a breaking of their symmetry at certain minima if the well in the pair-potential is sufficiently narrowed. This effect, analogous to the formation of clearance regions in hard packings, seems only to occur for the Lennard-Jones potential in the size- range N > 40, too large to be studied by the present mefhod.'O* *'24 MECHANICS AND MORPHOLOGY OF VERY SMALL ATOMIC CLUSTERS peratures of interest, the whole range of accessible isomers remain solid-like and harmonic even though the lowest in potential energy may tend to melt before the highest isomers can be “ excited ”.The results of these calculations, which we have only in preliminary form, may be applied in various ways. It is relatively easy to compute the “ mix ” of isomers of particular size as a function of temperature as the higher-energy structures become “excited”, and to study the effect of the potential on this. By computing free- energies of formation it is also possible to obtain equilibrium constants for growth and decay of clusters and to evaluate the “ single-configuration ” approximation used by previous authors 16- 25 who assumed that the structure of lowest potential energy is minimal in free-energy also and effectively determines equilibrium concentrations and growth rates.In all, the surprising features of these results lie more in the static properties dis- cussed earlier than in the statistical calculations. Within the approximations used, there appears to be a general tendency for vibrational frequency patterns to be some- what insensitive to the energy and character of the respective minima and thus for equilibria to be determined largely by energetic and configurational factors rather than vibrational and rotational entropy. We hope to expand on these remarks in the Discussion and subsequent publications. This work was carried out during the tenure of an S.R.C. research studentship by one of the authors (J. McI.), and substantially aided by additional funds from the same source. We are indebted to Dr. J. Andrew Barker for programming assistance. E. W. Becker, K. Bier and W. Henkes, Z. Phys., 1956, 146, 333. E. W. Becker, R. Klingelhofer and P. Lohse, 2. Natrrrforsch., 1962, 17a, 342. F. T. Greene and T. A. Milne, J. Chem. Phys., 1967, 47,4095. P. J. Foster, R. E. Leckenby and E. J. Robbins, J. Phys. B, 1969, 2, 478. E. W. Becker, J. Gspann and G. Krieg, Entropie, 1969, 30, 59. 0. F. Hagena and W. Obert, J. Chem. Phys., 1972, 56, 1793. P. Audit, J. Physique, 1969, 30, 192. J. Farges, B. Raoult and G. Torchet, J. Chem. Phys., 1973, 59, 3454. B. Raoult and J. Farges, Rev. Sci, Instr., 1973, 44, 430. lo M. R. Hoare and P. Pal, Adv. Phys., 1971,20, 161. l1 M. R. Hoare and P. Pal, J. Cryst. Growth, 1972, 17, 77. If M. R. Hoare and P. Pal, Adv. Phys., 1975, 24, 675. l3 R. Fletcher and C. M. Reeves, Comp. J., 1964, 7, 149. l4 J. Andrew Barker, J. Finney and M. R. Hoare, Nature, 1975, 257, 120. l5 J. D. Bernal, Nature, 1959, 183, 141; 1960, 185, 68. l6 A. Bonnissent and B. Mutafschiev, J. Chem. Phys., 1973, 58, 3727. l7 W. Werfelmeier, 2. Phys., 1937, 107, 332. l9 J. D. Bernal, Proc. Roy. SOC. A, 1964, 280, 299. 2o M. R. Hoare and P. Pal, Nature, 1972, 236, 35. 21 M. R. Hoare, Ann New York Acad. Sci., 1976, (in press) 22 Y. Fukano and C. M. Wayman, J. Appl. Phys., 1969,40, 1656. 23 J. McInnes, Thesis (University of London. To be submitted). 24 F. Harary, Graph Theory and Theoretical Physics (Academic Press, New York, 1967), p. 33. 25 D. J. McGinty, J. Chem. Phys., 1972, 55, 580. A. H. Boerdijk, Phillips Res. Repts., 1952, 7, 303.

ISSN:0301-7249    

DOI:10.1039/DC9766100012

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

On a Structural Model for Amorphous Metals: Implications from Microcluster Studies BY CLYDE L. BRIANT Department of Metallurgy and Materials Science, University of Pennsylvania, Philadelphia, Pennsylvania Received 8th December, 1975 The structure of amorphous metals may be based on a 13 atom icosahedron. The argument in 1. The similarity of the interference functions of amorphous metals and the 1 3 atom icosahedron. 2. The observation of a 13 atom icosahedron in a larger microcluster which was quenched from the liquid phase to an amorphous state in a molecular dynamics computer experiment. 3. The readiness with which a 1 3 atom icosahedron is expected to form during atom-by-atom crystal growth. We picture the structure of amorphous metals as one in which these 1 3 atoms icosahedra exist as structural units or nuclei throughout the bulk but are surrounded by large areas of disordered material.favour of this model rests on the following facts: 1. INTRODUCTION The study of amorphous metals is currently a very active area of research.l One main field of investigation is the study of the structure of these material^.^-^^ In recent years several structural models have been proposed. At one extreme one can picture a complete absence of any order; another view could be that the entire material is composed completely of tiny microcrystals or micrograins. Between these two limiting models one could suggest that structural nuclei do exist in the bulk, although much of the material remains disordered. The standard experimental techniques for studying the structure of amorphous metals are X-ray, neutron, and electron diffraction. These yield interference func- tions, and from these one can attempt to develop models.One method is to pack large numbers of hard spheres together and calculate the interference function of the m0de1.I~ Another approach is to calculate interference functions for small model structural nuclei4 If any of these reproduces the major features of the experimental curves, the results may be held to suggest that very similar units exist in the bulk material. Recently, many studies have been made on the structure and thermodynamic properties of free microcl~sters.~~-~~ Hoare and Pal have made a detailed study of the structure and crystal growth sequence of solid, rare gas microclusters; and molecular the computer solution to Newton’s equations, has allowed a very good description of these microclusters in both the solid and liquid phase.22 This type of work is of great value for studying the structure of amorphous metals. The crystal growth and structural studies provide model microcluster units which could easily form during a rapid quench.One can calculate interference functions for these clusters and compare them with experimental curves. Molecular dynamics can be used to perform heating and cooling experiments on microclusters. Thus,26 ON A STRUCTURAL MODEL FOR AMORPHOUS METALS liquid clusters can be quenched to amorphous solids and the structure of the quenched solid completely determined. In this paper we use both of these approaches to study the structure of amorphous metals.First we give a brief description of molecular dynamics, since it was used to relax the solid model microclusters near 0 K and also to perform the quenching experiments. We then report our results : the comparison of interference functions for model microclusters with experimental curves and the structure and interference functions of the quenched clusters. We then use these results and the crystal growth FIG. 1.- The 13 atom icosahedron. sequence developed by Hoare and Pal to suggest a structural model for amorphous metals. We will argue that structural nuclei do exist, although much of the material may remain disordered. In particular, we suggest that many of these nuclei are 13 atom icosahedra, fig. 1. Our view is based on three facts: 1.The similarity of the interference functions of many amorphous metals with that of the 13 atom icosahedron. 2. The observation of a 13 atom icosahedron in a larger microcluster which was quenched from the liquid phase to an amorphous state in a molecular dynamics computer experiment. 3. The readiness with which a 13 atom icosahedron is expected to form during atom-by-atom crystal growth. The importance of the 13 atom icosahedron has been noted previously in theoreti- cal studies of micro~lusters.~~-~~ It is thought to be the minimum energy packing structure for 13 atoms.18 2. MOLECULAR DYNAMICS CALCULATIONS Molecular dynamics is the computer solution to Newton’s equations. Alder and Wainwright 23 developed this technique to study liquids.It has been used extensively for this purpose 23-25 and more recently for the study of microc1usters.22* 26 Molecular dynamics allows the structure and thermodynamic properties of the system to evolve under the assumed potential. Thus, one makes no assumptions about the structure or phase of the system. To develop model microclusters which might be structural nuclei in bulk amor- phous metals, we built up clusters based on the icosahedral growth sequence l8 and relaxed them near 0 K under a Lennard-Jones potential.27 We found that this was the lowest energy configuration for all clusters containing seven or more atoms. Clusters of 13 or 55 atoms which were initially given face-centred-cubic coordinates sheared spontaneously to form icosahedra. This result also holds for various Morse p0tentia1s.l~CLYDE L .BRIANT 27 To prepare liquid microclusters we gradually heated the solid clusters by adjusting the atomic velocities. Once the cluster was completely liquid, we allowed it to equilibrate at a fixed total energy. Quenching was performed by rapidly decreasing the kinetic energy by adjusting the velocities. Details of these heating and cooling experiments have been given elsewhere.22 3. RESULTS First we consider the interference functions of model microclusters. We then examine the quenching experiments. A. INTERFERENCE FUNCTIONS The interference function of a model cluster is given by 28 : 2 sin ( 2 ~ k . r ~ ~ ) Z(k) = 1t-C N i, j 271k.rij where N is the number of atoms in the microcluster, k is a reciprocal lattice vector, rij is the distance between atoms i and j in the microcluster, and the sum is over all atom pairs in the microcluster. In fig.2 we plot the interference functions for micro- I (&) I I 147 I00 5 5 33 13 7 6 .o 2 4 6 8 ka, FIG. 2.4nterference functions for model microcrystals. The sizes of the microcrystals are given on the right of each curve. clusters containing 3, 4, 5, 6, 7, 13, 33, 55, 100 and 147 atoms. The structures of those containing 7 or more atoms were based on the icosahedral growth sequence. Below this size, the icosahedral and face-centred-cubic (fcc) growth sequences are indistinguishable. Fig. 3 shows several experimentally determined interference functions for amorphous metals. We note one characteristic feature for all of these28 ON A STRUCTURAL MODEL FOR AMORPHOUS METALS I i__ _I__-- 0 2 4 6 8 koo FIG.-3.Experimental interference functions for amorphous materiakg experimental curves, the shoulder on the large angle side of the second peak. The calculated interference functions for the 13 and 33 atom microclusters also have shoulders on the second peak similar to those found experimentally. The interference function for the 13 atom microcluster, especially, resembles the experimental curves. The 13 atom microcluster is an icosahedron; the 33 atom microcluster is the 13 atom icosahedron plus an atom in each of its 20 triangular faces. As the interference function is quite sensitive to the assumed structure of the microcluster unit, the similarity between the experimental curves and that of the 13 atom icosahedron suggests that these icosahedral units exist in amorphous metals.In fig. 4 we compare the interference function of the 13 atom icosahedron with that of a 13 atom fcc unit. 1 - I I I I I3 ATOMS FA7 E - C EWERE D - , CLIBIC, U\A IL- 0 '/c 2 4 6 8 kao FIG. 4.-Calculated interference functions for microclusters. The top two curves are for 13 and 33 atom microcrystals whose structure is based on the icosahedron. The lowest curve is the interference function for a 13 atom face centred cubic microcrystal.FIG. 5.-Photograph of a 55 atom cluster. The right hand photograph is a solid cluster and the left hand photograph is a liquid cluster. (Photographs prepared at the Graphics Facility for Interactive Display, Department of Biological Sciences, Columbia Univer- sity, New York, Nen York.) The cluster was projected on to a plane.[To face page 29CLYDE L . BRIANT 29 Note that the fcc unit does not have the shoulder on the high angle side of the second peak. Recently, neutron diffraction studies of amorphous metals have extended the interference function beyond the fourth peak.29 An asymmetric fifth peak was observed. The interference function for the 13 atom icosahedron, fig. 2 and 4, also shows an asymmetric fifth peak. The shape is quite similar to that observed in amorphous metals.29 B. QUENCHED MICROCLUSTERS To perform the computer quenching experiments, we first heated the solid clusters into the liquid phase. We detected melting by calculating the diffusion coefficient and by visual inspection.In the solid phase the diffusion coefficient was less than cm2 s-' and in the liquid phase it was approximately cm2 s-l. We visually inspected the clusters by projecting the coordinates of the atoms on a plane. The solid phase was ordered and the liquid disordered, fig. 5. Fig. 6 shows (energy, temperature) curves for clusters containing 7, 13, and 55 atoms prepared by both heating and cooling experiments. In molecular dynamics, 60 40 TI K 2 0 0 I ' 55 ' -5 -3 - I -9 - 7 er?ergy / 10-A erg atom-' FIG. 6.-Heating and cooling curves for microclusters containing 7, 13 and 55 atoms. The smooth curves are for heating experiments and the individual points are cooling curves. the temperature is calculated by time-averaging the kinetic energy.25 The heating curves were prepared by starting with the low energy solid and slowly increasing the kinetic energy of the clusters.The cooling curves were rapid quenches. Note that the heating and cooling curves superimpose for the clusters containing 7 and 13 atoms, indicating that the quenched solid is the low energy structure. However, the quenched 55 atom solid has a higher energy than its low energy structure. This suggests that we have formed an amorphous cluster. We now examine the interference functions of these quenched clusters. Those for clusters containing 7 and 13 atoms are identical to their respective curves in fig. 2. Fig. 7 shows the interference functions for a quenched 55 atom cluster as well as a quenched 19 atom cluster. Note that both of these clusters exhibit a shoulder on the second peak similar to the experimental curves in fig.3 and the 13 atom icosahed- ron, fig. 2 and 4.30 ON A STRUCTURAL MODEL FOR AMORPHOUS METALS 55 ATOMS QUENCHED 19 ATOMS QUENCHED I ' I I I 0 , 2 4 G 8 kao FIG. 7.-Calculated interference functions for microclusters quenched from the liquid phase. The interference function for the quenched 19 atom microcluster is easily explained. In the lowest energy configuration the 19 atom microcluster consists of the 13 atom icosahedron plus a cap of 6 atoms.18 The resulting structure resembles two inter- locking icosahedra, fig. 8. When the 19 atom liquid is quenched, there is little hindrance to the initial formation of a 13 atom icosahedron, the minimum energy structure for 13 atoms which is formed when 13 atoms are quenched from the liquid phase.The remaining five atoms could either form the cap or sit in the triangular faces of the icosahedron. l a 1 l b ) /C/ FIG. 8.-Structures of model microclusters. a-The 13 atom icosahedron. b-Minimum energy structure for 19 atom cluster. c-55 atom icosahedron. The result for the quenched 55 atom microcluster is particularly interesting. In its low energy structure this cluster consists of two complete icosahedral shells around a central atom, fig. 8; the 13 atom icosahedron contains only the first shell. The interference function for the 55 atom icosahedron does not have a shoulder on the second peak, fig. 2, whereas the interference function of the quenched 55 atom cluster does, fig. 7. This suggests that the quenched 55 atom cluster contains a 13 atom icosahedron.In examining the 55 atom quenched cluster we found that a 13 atom icosahedron had formed. Interestingly, it did not form at the centre of the cluster but included atoms near the surface. This off-centre icosahedron would make it much more difficult for the entire cluster to form the large 55 atom icosahedron, fig. 8. It also suggests that if one quenched larger clusters containing 200 to 300 atoms, several 13 atom icosahedra might form throughout the cluster. Finally, if one quenched the bulk phase, many icosahedra might form. However, much of the material may remain disordered.CLYDE L. BRIANT 31 4. FORMATION OF THE 13 ATOM ICOSAHEDRON BY Experimentally, amorphous materials are produced in one of two ways.Either atoms are deposited very rapidly on to a substrate kept cold enough to prevent long range diffusion, or a bulk liquid sample is cooled so rapidly that no long range dif- fusion can occur during solidification. In either case clusters of solid phase material condense atom by atom. In the vapour deposition experiment, isolated clusters form on the substrate and are otherwise surrounded by vacuum; when the bulk liquid is solidified rapidly, initially solid clusters form surrounded by still liquid material. In both methods, the structures of the initially formed solid clusters will be determined by the minimum energy structure of small clusters. We examine here the probable growth sequence for a microcluster or structural nucleus starting with one atom and steadily adding one atom at a time.This follows the pioneering work of Hoare and who have shown that the growth sequence to be described here applies to materials interacting by simple pair potentials. This theoretical work is based primarily on growth in a vacuum. Work by Mader3' indicated that the substrate will not alter this growth sequence if the atoms of the growing crystal do not interact strongly with the substrate. Two atoms come together to form a dimer, fig. 9a. When a third atom is added to form a trimer, the equilateral triangle, fig. 9b, is the stable structure. Addition of a fourth atom produces a tetrahedron, fig. 9c. At this point the growth deviates from normal fcc packing, which would require the addition of an extra atom as shown in fig.9d. This configuration is unstable and relaxes into the trigonal bipyramid, fig. 9e. Addition of another atom to the 5 atom trigonal bipyramid produces the octa- hedron, fig. 9f. Addition of another atom to the 6 atom octahedron produces a CRYSTAL GROWTH f (6) J g (7) FIG. 9.-Growth sequence to form a 13 atom icosahedron, h. The numbers of atoms in the clusters are indicated in the figure. The 5 fcc structure, d, is unstable, and deforms to structure e. The 13 atom fcc structure, i, is unstable and deforms spontaneously into the icosahedral packing, h.32 ON A STRUCTURAL MODEL FOR AMORPHOUS METALS pentagonal bipyramid, fig. 9g. Addition of 5 atoms on the five upper faces of the pentagonal bipyramid pictured in fig. 9g plus one further atom on the fivefold sym- metry axis produces the 13 atom icosahedron, fig.9h. Note that the occurrence of the 13 atom icosahedron is a consequence of the attractive part of the potential.l’g l6 It is stable because the 13 atom icosahedron has six more nearest neighbour bonds than do 13 atoms in a face-centred-cubic packing.31 Thus, hard sphere packing studies would not necessarily show icosahedral packing. 5. DISCUSSION All three sets of results indicate that the 13 atom icosahedron might exist as a structural unit in amorphous metals. Since it is the result of atom-by-atom crystal growth it is expected that such clusters should be formed during either vapour phase growth or rapid quenching. If the metal is allowed slowly to crystallize, the bulk symmetry must eventually be nucleated.However, at the high supersaturations and rapid cooling rates used during the formation of amorphous metals, this early growth sequence should be the most important in determining any structural nuclei which might form. Based on these results we suggest that amorphous metals do contain structural units and that a predominate unit or nucleus would be the 13 atom icosahedron. We do not imply that the entire material has ordered itself into these units and is thus completely microcrystalline ; rather there could be large areas of disordered material. Note that while theoretically the 55 atom cluster could have quenched into four ico- sahedra, only one was found. By quenching larger clusters of perhaps 200 to 300 atoms, one could then begin to understand how many nuclei would exist in the bulk material and also begin to examine the regions between these nuclei.6. CONCLUSIONS Three independent types of evidence indicate that the 13 atom icosahedron, fig. 1 , may exist as structural nuclei in amorphous metals. These are : 1. The similarities of the interference functions. 2. Observation of the 13 atom icosahedron in quenched liquid microclusters in computer experiments. 3. The expectation that the natural atom-by-atom growth sequence should lead to this packing unit. Based on these results we conclude that it is highly likely that amorphous metals contain structural nuclei, and that these nuclei are 13 atom icosahedra. This does not provide a complete model for amorphous metals. For instance, we can say nothing about the regions between icosahedra.Such information should result from quenching larger clusters. The author is particularly grateful to Dr. James J. Burton of Exxon Research and Engineering Company for many helpful discussions. J. J. Gilman, Physics Today, 1975, 28,46. C. N. J. Wagner, J. Vac. Sci. Tech., 1969, 6, 650. A. Bienenstock and B. G. Bagley, J. Appl. Phys., 1966,37,4840. G. S . Cargill, 111, J. Appl. Phys., 1970, 41, 12. G. S. Cargill, 111, J. Appl. Phys., 1970, 41, 2248. A. K. Sinha and P. Duwez, J. Phys. Chem. Solids, 1971,32, 267. C. N. J. Wagner, T. B. Light, N. C. Halder and W. E. Lukens, J. Appl. Phys., 1968,39, 3690.CLYDE L . BRIANT 33 B. G. Bagley and D. Turnbull, J. Appl. Phys., 1968,39, 5681. P. L. Maitrepierre, J. Appl. Phys., 1669, 40, 4826. lo D. E. Polk, Acta. Met., 1972, 20, 485. l1 G. S. Cargill, 111, Structure of Metallic Alloy Glasses, to be published in Solid State Physics, l2 J. G. Wright, J. Phys. C., 1974, 4, L187. l3 P. K. Leung and J. G. Wright, Phil. Mag., 1974, 30, 185. l4 J. F. Sadoc, J. Dixmier and A. Guinier, J. Noncryst. Solids, 1973, 12, 46. l5 J. J. Burton, J. Chem. Phys., 1970, 52, 345. l6 J. J. Burton, Nature, 1971, 229, 335. l7 M. R. Hoare and P. Pal, Nature (Phys. Sci.), 1971, 230, 5. M. R. Hoare and P. Pal, J. Cryst. Growth, 1972, 17, 77. l9 D. J. McGinty, J. Chem. Phys., 1971, 55, 580. ’O J. J. Burton, Cat. Rev.-Sci. Eng., 1974, 9, 209. ’l J. K. Lee, J. A. Baker and F. F. Abraham, J. Chem. Phys., 1973, 58, 3166. ” C. L. Briant and J. J. Burton, J. Chem. Phys., 1975, 63, 2045. 23 B. J. Alder and T. E. Wainwright, J. Chem. Phys., 1959, 31,459. 24 B. J. Burne and G. D. Harp, A h . Chem. Phys., 1970,17,63. 25 A. Rahman, Phys. Rev., 1964,136, A405. 26 C. L. Briant and J. J. Burton, J. Chem. Phys., 1975, 63, 3327. 27 T. Kihara, J . Phys. SOC. Japan, 1948,3, 265. 29 J. Dixmier, Table Ronde on the Structure of Disordered Solids, Orsay, France, 1974. 30 S. Mader, J. Yac. Sci. Technol., 1971, 8, 247. 31 F. C. Frank, Proc. Roy. SOC. A, 1952,215, 43. ed. F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1976). C. W. B. Grigson and E. Barton, Brit. J . Appl. Phys., 1967,18, 175.

ISSN:0301-7249    

DOI:10.1039/DC9766100025

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

The Thermodynamics and Structure of Hydrated Halide and Alkali Ions BY FARID F. ABRAHAM, MICHAEL R. MRUZIK IBM Research Laboratory, San Jose, California 95193 AND G. MARSHALL POUND Materials Science Department, Stanford University, Stanford, California 94303, U.S.A. Received 12th December, 1975 Gibbs free energies were calculated for the gas phase reaction: ion(H20)~- 1 +H&apour) = ion(H2Oh for the Li+, Na', K+, C1-, and F- ions and for N = 1 to 6. The Monte Carlo method "as used to evaluate the appropriate classical expressions of statistical mechanics by employing the inter- molecular potential functions recently developed from ab initia Hartree-Fock calculations. Enthal- pies and structural information were also calculated. Agreement with experiment is sufficiently good to demonstrate the feasibility of this approach.1. INTRODUCTION Adapting the intermolecular water-water and water-ion potential functions developed from ab initio Hartree-Fock calculations, we demonstrate a successful application of the Monte Carlo method of classical statistical mechanics to evaluate the thermodynamic and structural properties of hydrated halide and alkali ions. This study deviates from heuristic theories of gas phase hydration of ions where the physical cluster is pictured as a " liquid droplet," a " microcrystallite," or some other model construct. While these models provide an insight into the origin of certain features of a physical cluster, they cannot serve to elucidate the cluster's true molecular structure, since this is assumed in one way or another.With the advent of large scale computers, pursuing a first-principles approach for many-body systems has begun where only a form of the intermolecular potential function between two molecules is a s s ~ m e d . ~ Recent examples of this approach have been the molecular dynamics and Monte Carlo simulations of Lennard-Jones atomic clusters, and Monte Carlo simulation of water clusters.8 In Section 2, the Hartree-Fock potential functions for water-water and water-ion interactions are presented. The Monte Carlo method is outlined in Section 3, and expressions for various thermodynamic state variables are derived in terms of quanti- tives that can be measured using the Monte Carlo method. A brief description of recent experiments 9* lo on gas phase hydration of alkali and halide ions is given in Section 4.In Section 5, we present recent theoretical free energies of formation of ion-water clusters for comparison with experiment." In particular, Gibbs free energies have been calculated for the gas phase reaction: i0n(H,0),,~- I-/- H20~vapour) ion(H,O), (1)F A R I D F . ABRAHAM, MICHAEL R . MRUZIK A N D G . MARSHALL P O U N D 35 for the Li+, Na+, K f , C1-, and F- ions and for N = 1 to 6 . Enthalpies, and structural information are also presented. Agreement with experiment is sufficiently good to demonstrate the feasibility of this theoretical approach. 2. WATER-WATER AND WATER-ION POTENTIAL FUNCTIONS Water-water potential functions have been produced which are based on first principles ca1culations.l These functions obtain their form from various point charge models and are adjusted to agree with the ab ivlitio calculations of the Hartree-Fock potential energy surface. The C-XI1 function is the most recent in a series of suc- +Q *H2 FIG.1.-The Beriial and Fowler charge model l2 is employed for the C-XI1 water-water potential function.' The symbol H represents hydrogen atoms; 0 represents oxygen; and M is a fictitious point along the line of symmetry of the water molecule. The OM distance is 0.225 954 A and the OH distance is 0.957 A. The HOH angle is 105". cessively improved formulas of this type. This function uses the Bernal and Fowler charge model l2 illustrated in fig. 1 and is given by: UC-XII = UHartree-Fock+ Udispersion UHartree-Fock = Q2(1/r13+1/r14+ l/r23$.1/r24)+4Q2/r78 -2Q2(1/r18+ 1/r28+ 1/r37+ 1/r47) +a, exp(--b,r,,) (2) + a2[exp(- b2r13) + exP(--b2r14) +exp(- b2r23) + exP(- b2324)I +a3 [exp(-b3r16) + exp(-b,r26) +exp(--b3r35) + exp(- b3r45)1 Udispersion = cl/r566-c2/r586+c3/r4~ (3) where the constants are given by, Q2 = 139.272 kcal A/mol a, = 71533.4 kcal/mol b, = 3.96994/A c1 = 922.781 kcal A6/mol a2 = 779.885 kcal/mol b2 = 3.12544/A c2 = 17283.5 kcal A8/mol a, = 4084.02 kcal/mol b3 = 3.91443/A c3 = 24119.7 kcal bilo/mol The Hartree-Fock energy surface has been modified to include correlation effects which decrease, by making more negative, the total water-water binding energy by 15% to 30%. The dispersion terms were developed from the data of the Quantum Chemistry Group of the University of Warsaw, directed by Prof.W. K0l0s.l~ A rigid geometry is assumed for the water molecule and multibody effects are ignored. Errors in the water-water potential function are not highly significant in this study, since the dominant force in ion-water clusters is the ion-water interaction. For example, the empirical ST2 water-water potential function,14 which has been " effec- tively " adjusted so as to include the average multibody and other effects present in bulk water, gives essentially the same results in this study as does the C-XI1 function." As in the case of the water dimer, water-ion potential functions have been obtained36 THERMODYNAMICS AND STRUCTURE OF HALIDE AND ALKALI IONS from ab initio Hartree-Fock calculations.2 The functions which approximate this Hartree-Fock potential energy surface were based on two arbitrary point charge models: the " simple " ion-water model, which is a modification of the Bernal and Fowler version, and the slightly more complicated " accurate " model, as illustrated in fig.2. The corresponding functions are : / a ) Simple Model Ion PI '10 Ql ( b l Accurate Model FIG. 2.--Charge models employed for the water-ion potential functions.' Point charges are placed on the hydrogen atoms (HI and H2), oxygen atoms (0), along the water molecule line of symmetry (M), and along the bisectors of the OM distance (MI and M2). In (a) all lines are in the plane of the paper, and in (6) all dotted lines are parallel to their respective three dimensional axes. where the numerical values of the constants are listed in tables V and VI of ref, (2). The correlation terms are negligible.We have used the " simple " ion-water poten- tial function for the Li+ and Na+ ions and the " accurate " potential function for F-, Cl-, and K+. For a given ion-water configuration, both functions give almost identical values of potential energy, but the " accurate " version requires slightly more computing time. Although both ion-water potential energy functions were very close to the Hartree-FARID F . ABRAHAM, MICHAEL R . MRUZIK A N D G . MARSHALL P O U N D 37 Fock limit, errors were introduced by assuming a rigid geometry for the water mole- cule (rigid rotor assumption) and by neglecting correlation and multibody effects. The net result of these assumptions and omissions is that the ion-water potential functions should yield values of potential energies which are too positive by about 5% for clusters containing three or less water molecules and which are too negative by as much as 10% for larger clusters containing up to six water molecu1es.l1 To further investigate the effects of these errors, the enthalpy and free energy of reaction (1) for the Na+ ion and N = 6 were determined for a 10% positive change in ion- water and water-water potential functions.The free energy was reduced by 13% to -5.025 kcal/mol, and the enthalpy was reduced by 16% to -10.71 kcal/mol. These estimates are representative of the expected error in the Monte Carlo calculations. 3. THE MONTE CARL0 METHOD In the Monte Carlo calculations, the ion-water clusters are defined by the criterion that all water molecules of the cluster should lie within a spherical volume, V,, centred on the ion.This constraining volume was chosen large enough so that the strong ion-water binding naturally restricted the water molecules to configurations well away from the constraining boundary. Calculations performed with respect to this frame of reference with the ion fixed at the origin were later corrected for the loss of three degrees of freedom associated with cluster translation. This cluster definition did not otherwise affect the canonical averaging process since those excluded con- figurations represented very high energy states, each with a negligible probability of occurrence. The standard state for all calculations was taken as 298 K and 1 atmos- phere.Since Metropolis et aZ.15 first applied the “ Monte Carlo ’’ method to statistical mechanics, this method has been discussed in detail by various author^.^ We will only present a brief review. The method is based on a stochastic process which generates a Boltzmann-weighted chain of configurations of a given N-particle system. The mean value of any function of the system’s coordinates over all configurations in the chain provides an estimate of the canonical ensemble average of that function. For example, the mean potential energy leads to an estimate of the internal energy. The procedure is as follows (see fig. 3). We consider N water molecules in an initial ice cluster configuration of uniform density equal to the experimental water density at temperature T.The cluster is centred on the ion in a constraining spherical boundary with radius R, and volume V, = SV,, where V,, is the volume for bulk liquid water of N molecules at temperature T. Within the framework of our definition of the ion-water cluster, we generate configurations in the following manner: (i) select a molecule at random, (ii) select displacements Ax, Ay, Az, each uniformly distributed on (-A/2, A/2), (iii) select x , y , or z axis at random, (iv) select an angle 8 uniformly distributed on (-w,w); (v) if these displacements move the centre of mass of any water molecule outside the spherical constraining boundary with origin corresponding to the position of the ion, reject the try and accept the old configuration; otherwise (vi) calculate the change in potential energy SUN on displacing the chosen molecule by (Ax, Ay, Az) and rotating it through 8 about the chosen axis; (vii) if SUN is nega- tive, accept the new configuration; otherwise (viii) select a number h uniformly distri- buted on (O,l), (ix) if exp(--GU,/kT) < h, accept the old Configuration; otherwise, (x) the new configuration and the new potential energy become the “current” properties of the system.These rules ensure that averaging over long chains approaches classical canonical averaging, with weighting of configurations proportional to exp(- U,/kT). The38 THERMODYNAMICS AND STRUCTURE OF HALIDE A N D ALKALI IONS The Monte Carlo Procedure 1. Given (old) Configuration Mo lecu I e Moves Out of v 2. Select 3. Displace and Rotate Water Molecule 4.Calculate Energy Change 5. Metropolis Test M.:h t i - random no I 6. Accepted (New) I FIG. 3.-The Monte Carlo algorithm for calculating canonical averages of state variables for ion- water clusters. angle ly was fixed at 0.2 radians and A was chosen so that approximately half the attempted moves were actually made. The change in Helmholtz free energy AFN-l,N for reaction (1) is given by 1 A F N - 1 . N = {l <u,,o(4> +kT In (N- Iy, PdV, (6) >Vc Y(l atm.) where { UHzo(;l)> is the canonical average of the potential energy of the water molecule (monomer) which is being added to the cluster in the following way : the dimensionless coupling parameter, 0 < 3, < 1, scales the interaction of the monomer with the ion and the other N- 1 water molecules.When the interaction is turned on, A == 1, and the monomer is indistinguishable from the other N - 1 water molecules ; but when the interaction is turned off, 3, = 0, and the monomer is completely non-interacting and behaves as an ideal gas. The work of expanding the constraining volume until the unattached monomer (3, = 0) is at a pressure of one atmosphere is described by the pdV term of eqn (6). To compute the canonical average of the monomer potential energy, ( UH,o(A)), 100000 configurations were generated for a number of discrete values of 3,. A numerical integration of these values as a function of 3, furnished the corresponding free energy term. Standard deviations of the thermodynamic variables were obtained from a series of mean values, each of which represented the average of a block con- taining 10 000 configurations.To compensate for the numerical integration difficulties caused by the rapidF A R I D F . A B R A H A M , MICHAEL K. MRUZIK A N D G . MARSHALL POUND 39 increase of (UHzo) as R goes to zero, as illustrated in fig. 4, the following transforma- tion was employed: The value of Z was fairly insensitive to a choice of rn and varied only about 2% for 0.10 < m < 0.75; m was arbitrarily set equal to 0.25 since the average energy for a particle in contact with a fixed energy source is proportional to %-3/4 when the inter- action is dominated by a repulsive Lennard-Jones term of order 2/r.12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 FIG. 4.--The canonical ensemble average potential energy of a water molecule (UH~O(~)) as a fiinc- tion of the 1 interaction parameter.For the purpose of comparing the theoretical free energies with experiment, the Helmholtz free energy was converted to Gibbs free energy A G N - 1 . N using the relation: A G N - 1 , N = AFN-i,N+A(PV) = AFN-i,N-kT, (8) for the standard state (298 K and 1 atmosphere). The formula used to calculate the changes in internal energy and enthalpy are : A E N - 1 , N = { uN(L=l))-{ u N - 1 ( ] * = I)), (9) AHN-1,N = AEN--,N-kT. (10) 4. EXPERIMENTAL Recent experimental advances 9 9 lo have provided the first empirical thermodynamic data on small ion-water clusters. To make an effective comparison with theoretical calculations, it is appropriate to review the general nature and accuracy of these experiments. Equilibrium constants for the incremental hydration of the alkali metal and halogen ions as per reaction40 THERMODYNAMICS AND STRUCTURE OF HALIDE AND ALKALI IONS (1) were determined as follows (see fig.5 ) : (i) the appropriate gas-phase ions were produced and electrostatically directed to a reaction chamber which contained a known pressure, pH20, (usually about 1 Torr) of water vapour; (ii) a high pressure mass spectrometer recorded the relative concentrations of ion-water clusters as a function of mass number; and (iii) the equilibrium constant, K, and Gibbs free energy, h G N - i , N , were obtained from, Production of Negative Ions Production of Positive Ions (0, c NF3 7 F- i NF:, + 02) 0, Carrier Gas with NF3 I Platinum Grid +- 0 h, Electron Beam or a - particle Source /' __ Anions ~ Focusing Grid Water Molecule To Mass Spectrometer Ion (H201N-, + H,O = Ion (H,O)N FIG.5.-Schematic of experiments for obtaining equilibrium constants for the incremental hydration of the alkali metal and halogen ion as per reaction (l), [ref. (9) and (lo)]. where IN is the concentration of clusters comprised of an ion and N water molecules. Meas- urements of the equilibrium constant as a function of temperature gave the enthalpy of the reaction (l), AHN-i,N, from the van't Hoff relation, Although errors were not estimated in the experimental report, it has been suggested elsewhere that standard deviations of & 1 to i-2 kcal/mol for the enthalpy would not be unreasonable. These estimates presumably include errors such as the dissociation of clusters in vacuum immediately after entering the mass spectrometer.Since the time re- quired for the initial electrostatic acceleration in the mass spectrometer is the same order of magnitude (microseconds) as the period for evaporation of water molecules from the cluster, this effect would decrease the apparent concentration of larger clusters, resulting in more positive free energy measurements. Considerations such as the attainment of equilibrium among the assorted cluster sizes in the reaction chamber and the prevention of anomalous cluster growth, caused by cooling on adiabatic expansion into the mass spectrometer, were examined and probably contributed little to the total error. In addition, different experimental conditions for positive and negative ions complicated a comparison of the error trends of the two groups.Negative ions were produced by ionizing the oxygen to 0; which collided with gaseous NF, or CC14 to create the desired anion. Positive ions were produced by thermionic emission from a platinum filament coated with the appropriate salt. The resulting temperature gradient, which for Li+ began at 900 "CF A R I D F. ABRAHAM, M I C H A E L R . M R U Z I K A N D G. MARSHALL POUND 41 at the filament, was difficult to isolate and would have the effect of making the observed free energies too positive for the positive ions. Cluster sizes larger than six water molecules ( N = 6) could not be studied experimentally due to condensation on the chamber walls which eventually closed the port to the mass spectrometer. A Kf ton (Experimental) A K+ Ion (Monte Carlo) 9 Na' Ion (Experimental) 0 Na' Ion (Monte Carlo) Lit Ion (Experimental) 0 Lit Ion (Monte Carlo) 1 2 3 4 5 6 N, Number of Water Molecules FIG.6.-Enthalpies are given for the reaction M+(H20)N-1+H20 = M+(H~O)N for the cations indicated. All values are for 1 atmosphere and 298 K. L n 5 -24 Q -32 0 CQ- Ion (Experimental) 0 CQ- Ion (Monte Carlo) F' ion (Experimental) F' Ion (Monte Carlo) 1 1 1 1 I 1 1 2 3 4 5 6 #, Number of Water Molecules Fro. 7.-Enthalpies are given for the reaction X - ( H 2 0 ) ~ - ~ + H Z 0 = X-(H20)N for the anions indi- cated. All values are for 1 atmosphere and 298 K.42 THERMODYNAMICS A N D STRUCTURE OF HALIDE A N D ALKALI IONS 5.RESULTS AND DISCUSSION Monte Carlo and experimental lo enthalpies for ion-water clusters containing the F-, Cl-, K+, Na+, Li+ ions and from 1 to 6 water molecules are presented in fig. 6 and 7. As the clusters increase in size, the enthalpy change AHN--l,N for the addition of a water molecule becomes more positive. This is caused by a decrease in ion-water attraction and, in some cases (notably Li+ and Na+), an increase in water-water repulsion with increasing cluster size as illustrated in table 1. The TABLE MONTE CARLO VALUES OF THE AVERAGE ION-WATER POTENTIAL PER WATER MOLE- WATER MOLECULE, ( Uww)/N. ALL VALUES ARE EXPRESSED IN KCAL/MOL CULE, (ulw)/N, IS COMPARED WITH THE AVERAGE WATER-WATER POTENTIAL ENERGY PER 1 -21.89 0 - 10.93 0 -33.44 0 2 - 21.76 0.65 - 10.73 0.08 -33.33 1.25 3 - 21.44 1.39 - 10.58 0.1 1 -33.15 2.74 4 - 21.04 2.08 - 10.08 -0.19 - 32.71 4.15 5 - 19.46 1.71 -9.44 -0.53 -31.94 5.35 6 - 17.98 1.35 -9.18 - 0.74 - 30.76 5.99 1 - 24.32 0 - 16.92 0 2 - 24.24 0.79 - 16.66 0.37 3 - 24.02 1.82 - 16.58 1.10 4 -23.65 2.74 - 16.02 1.39 5 -22.82 3.24 - 14.81 0.98 6 -21.53 3.18 - 13.75 0.57 effect of the high surface-to-volume ratio causes the enthalpy change to become more positive than that for the condensation of a water molecule from the vapour to the bulk liquid (-10.52 kcal/mol) in the cases of K+ and C1-.The Monte Carlo enthalpies are more negative than those from experiment 9* except for the case of C1- where the opposite is true. For the positive ions, discrepancies between the Monte Carlo and experimental data increase in going from K+ to Li+.One notes that the discrepancy between Monte Carlo and experimental enthalpies is generally greater than the estimated error of the Monte Carlo calculations in the cases of Li+ and F-. A comparison of the Monte Carlo free energies with those of experiment 9* lo for the incremental hydration of the different ions is given in fig. 8, 9, and 10. For the positive ions, the Monte Carlo free energies are more negative than experiment, while for the negative ions, the relationship is mixed. The negative deviations of the Monte Carlo from the experimental free energy data for cation clusters increase in passing from Kf to Li+. The experimental free energy for the F- ion displays an anomalous decrease at N = 5. Besides being physically unreasonable, this type of behaviour is not otherwise observed and is most likely an artifact of the experiment.16 For C1-,F A R I D F .ABRAHAM, MICHAEL R . MRUZIK A N D G. MARSHALL POUND 43 which has the best agreement with experiment, the Monte Carlo free energies are slightly more positive than experiment for small clusters (N < 3), and slightly more negative for larger clusters. For the largest clusters studied, N = 6, the experimental free energies approximately converge to the same value (fortuitously close to the value for condensation of vapour to bulk liquid, -2.055 kcal/mol) while the Monte Carlo E -9 CI -12 L 0 u) 0 r ? z u I Ion A KS ton (Experimental) A K+ Ion (Monte Carlo) I I I I 1 1 2 3 4 5 6 ti X , Number of Water Molecules -3 2 .- n -15 0 b) Sodium ton 3 0 Na+ ton (Experimental) 3 Na’ Ion (Monte Carlo) 0 l - 1 I I I I 1 1 2 3 4 5 6 N , Number of Water Molecules FIG.8.-Gibbs free energies are given for the reaction M+(H20)N-l+H20 = Mf(H20), for (a) M+ = K+ and (b) M+ = Na+. All values are for 1 atmosphere and 298 K. free energies are still dependent on the type of ion. One notes that the discrepancy between Monte Carlo and experimental free energies is generally greater than the estimated error of the Monte Carlo calculations in all cases except CI- . In fig. 11 and 12, radial density distributions for oxygen and hydrogen atoms show the formation of a well defined first hydration shell in which the water molecules are constrained to orientations closely resembling the optimum configurations of fig.13. For the anions, there is a mixing of configurations 13(b) and 13(c). Clusters with ions of the same polarity had nearly identical (average) structures differing principally in the distance between the ion and the first hydration shell as determined44 THERMODYNAMICS AND STRUCTURE OF HALIDE AND ALKALI IONS -3 r -30 FIG. 0- I I I I I I - 9.-Gibbs - 21 2 4 1 -27 E Li' Ion (Experimental) Lit Ion (Monte Carlo) Li + ( HzO)~. m I 0 C!2- Ion (Monte Carlo) W F- Ion (Experimental) El F- Ion (Monte Carlo) q- 1 2 c) 5 6 #, Number of Water Molecules FIG. 10.-Gibbs free energies are given for the reaction X-(HZO)N-l+HZO = X-(HzO)N X- = F- and Cl-. All values are for 1 atmosphere and 298 K. All forF A R I D F. A B R A H A M , MICHAEL R . MRUZIK AND G .MARSHALL POUND 45 18 16 for K " ( H ~ o ) , L 4 2 0 0 1 2 3 4 5 6 b) Radial Densities for Na' (H20l6 I,/ Oxwen R, Radial Distance from Icn ( A ) FIG. 1 1 .-Radial densities of oxygen and hydrogen atoms are given for cation clusters of (a) K+(H20)6, (6) Na+(H20)6, and (c) Li+(H20)6.46 THERMODYNAMICS AND STRUCTURE OF HALIDE AND ALKALI IONS 10 9 - 8 - I v1 *% . 7 e 5 - 3 2 6 - L .- .- Q >. 4 - 3 - 2 - 1 - - Ln .- I 1 - b) RadiaCDensities for ce- (H2016 - I a) Radial Densities for F-(H20)6 .(Hydrogen 2 . 3 4 5 6 'R, Radial Distance froin Ion (A) by the first peak in the radial density distribution. Detailed observations of many of the individual configurations generated by the Monte Carlo method indicated no structures reminiscent of the ices or clathrates.In this paper we have performed the first theoretical cluster free energy calculations where a direct verification with experiment is possible and thereby have demonstrated the feasibility of applying the Monte Carlo method in calculating free energies for clusters containing only a few molecules. a) Optimum Cation- C J HB Configuration Water Configuration &!ion l \ l d k r Aiiiuii W3lt.r Aillor1 Wdier - FIG. 13.-Different configurations for ion-water pairs; in (a) and (6) hydrogen atoms are equidistant from the ion and in (c) one hydrogen atom is on a direct line joining the anion and the oxygen atom. All atoms lie in the plane of the paper.FARID F. ABRAHAM, MICHAEL R . MRUZIK A N D G . MARSHALL POUND 47 G . C. Lie and E. Clementi, J . Chettz. Phys., 1975, 62, 2195. W. W. Wood, Physics of Simple Liquids, ed. J. S. Rowlinson, G. S. Rushbrooke and H. N. V. Temperley (North-Holland, Amsterdam, 1968). M. Volmer, Kinetik der Phasenbildung (Theodor Steinlopff Verlag, Dresden, Germany, 1939). E. F. O’Brien and G. W. Robinson, J . Chem. Phys., 1974,61,1050. D. J. McGinty, J . Chem. Phys., 1973, 58, 4733. F. F. Abraham, J. Chem. Phys., 1974, 61, 1221. I. Dzidic and P. Kebarle, J. Phys. Chem., 1970, 74, 1466. M. Mruzik, F. F. Abraham, D. E. Schreiber and G. M. Pound, J. Chem. Phys., to be published. J. D. Bernal and R. H. Fowler, J. Chem. Plzys., 1933, 1, 515. l3 Unpublished data reported in ref. (1). l4 F. H. Stillinger and A. Rahman, J. Chem. Phjs., 1974, 60, 1545. ’ H. Kistenmacher, H. Popkie and E. Clementi, J. Chem. Phys., 1973, 59, 5542. ’ J. K. Lee, J. A. Barker and F. F. Abraham, J. Chern. Phys., 1973, 58, 3166. lo M. Arshadi, R. Yamdagni and P. Kebarle, J. Phys. Chem., 1970,74, 1475. N. Metroplis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem. P/?vs., 1953,21, 1087. l6 P. Kebarle, personal communication.

ISSN:0301-7249    

DOI:10.1039/DC9766100034

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

On the Kinetics of Precipitation BY MANFRED KAHLWEIT Max-Planck-Institut fur biophysikalische Chemie (Karl-Friedrich-Bonhoeffer-Institut), D-34 Gottingen, W. Germany Received 1 1 th December, 1975 In a recent paper, it was shown that the predictions of the LSW theory on the ageing rate and the time development of the size distribution of a precipitate are valid, if at all, only for a rather limited time interval during the early stages of a precipitation. The solution of the differential equation for the later stages, however, shows an unphysical behaviour in that it contains poles. We suggest that the reason for this is to be found in a simplification of the original differential equation which does not seem to be justified. In this contribution we report attempts to solve the original equation.Fig. 1 shows a schematic representation of the size distribution N(r,t) of the pre- cipitate during the early and intermediate stages of an isothermal precipitation of crystals from a solution. The N(r = 0 , t ) plane illustrates the course of development of the (mean) concentration c of monomers with time, while the (r,t) plane shows that of the critical radius r,. The diagrams parallel to the N(r,t = 0) plane then represent the size distribution as it changes with increasing time, This paper deals with theMANFRED KAHLWEIT 49 present state of the quantitative treatment of this process considering in particular the precipitation of crystals from a stirred liquid solution, the growth of which follows a first order law i = kV,[c-~S(r)]; k 2 D/6.(1) This problem has already been dealt with by the LSW theory,' which predicts that the size distribution asymptotically approaches a time independent shape, so that N(r,t) may be written as N(r,t) -+ f(t)g(p) for t 4 00 p = r/rc. This result follows from the conclusion that the ageing rate defined in this particu- lar case by d, = d(rt)/dt (3) asymptotically approaches a constant positive value d, --+ b/2 > 0 for t --f 00 b = 2kVA ac,/oRT. (4) As has been shown by us earlier,2 the latter conclusion is incorrect. Instead, 6, being a measure for -dc/dt as may be seen from fig. 1, increases rather rapidly after the nucleation period and then passes through a maximum during the early stages of precipitation to slowly approach zero for t -+ 00. Experiments further indicate that the ageing rate even at its maximum is in general much lower than b/2 [eqn (4)].We now discuss the consequences of these findings for the evaluation of the size distribu- tion. The change of the size distribution with time is determined by the continuity equation aiv/at+aJ/ar = o ( 5 ) J = -6(aN/ar)+iN. (6) where the " flux " J of the particles can be written as Here, 8 denotes the number of monomers incorporated into the surface of a particle in unit time, multiplied by (dr/dQ2, where i is the number of monomers of which the particle consists. When applying eqn (6) to the ageing of a precipitate it was hitherto assumed that, during the later stages of precipitation, the shape of the size distribution of the particles was sufficiently flat so that 6xlaN/arl < i N .(7) With this assumption, eqn (5) simplifies to aN/at+a(iN)/2r = 0. After introducing p, this equation reads a l n N . a l n N 1 a 3 - at +PF r, ap +- - 0. In the case of crystals with a linear growth law [eqn (l)], one has50 O N THE KINETICS OF PRECIPITATION With these expressions it follows from eqn (9) (1 1) F F + ( , - s P ) F + y = @ > l n N p-1 6 > I n N 1 0. Accepting eqn (4) to be valid, this equation was solved in the LSW theory by setting 6 = b/2. Then the coefficient of the second term in eqn (1 1) as well as that of the third term depend only on p, while the coefficient of the first term depends only on t. For this reason, one can separate the two variables by inserting eqn (2) and then evaluate the time independent shape g(p) of the size distribution.However, since eqn (4) holds, if at all, only for a limited time interval during the early stages of precipitation, namely, around the inflection point of curve c(t) in fig. 1, we have tried to find solutions for the later stages of precipitation by setting 6 =pb/2; 0 < p 6 1. (12) We may then again separate the two variables t and p by applying eqn (2), leading to p p-1 d l n g 1 0 4 p2 d l n p p2’ - a = __- -_- 0 d l n f b dt - --- We thus obtain for the time dependence of N(p,t) f(t) - 0 - 2 a l B . (14) The value of a can be obtained from the mass balance condition which now reads Here, yo denotes the number of moles of the precipitating substance in the system divided by its volume. The smaller p becomes, i.e., the slower c changes with time, the weaker will the right hand side of eqn (15) depend on time.Since the integral on the left hand side is time independent, we thus conclude f(t) -+ for p -+ 0. (16) (17) By comparison with eqn (14) it then follows a -+ p for p -+ 0. We thus obtain from eqn (13) as differential equation for the shape g o ) for suf- ficiently small values of p + p = 0. The solution reads g(p) = const. x p x (PQ)-5/2 (Q/P)’I2 E (19) 3 where P = p - fr (1+&) P 2 P E2 = 1 -p. Q = p - - (1-E)MANFRED KAHLWEIT 51 The behaviour of this solution reflects the singularities contained in eqn (18): The slope of the size distribution dg/dp has either a pole or a zero at p = (2//3)(1-c). For 0.817 < p < 1, this point is a zero, for 0.67 <j3 < 0.817, the point is a pole with dg/dp = - a, for p < 0.64 it is a pole with dg/dp = + co.We believe that the reason for this unphysical behaviour is to be found in the relation (7) which is the basis for the simplification of the general eqn (5) and (6). Since we are considering the later stages of precipitation at which the particles have grown rather big and the mean concentration in the parent phase is changing rather slowly with time, we shall neglect the dependence of 8 on r and t and set 8 z 8(r = roc = c,) z const. (20) Instead of eqn (8) we then have and instead of eqn (19) 81d2g p p-1 d l n g 1 bgdp"'(4 p2 ) d l n p p2 + P = O --- --- -_- where in this case 8fb = vRTf8zNAar,2 (23) and rc may be set equal to 7. In view of the fact that we seek solutions mainly for the size distribution around r = rc, we may simplify eqn (22) by expanding b [eqn (lo)] around p = 1. We then obtain = ;[( 1 ) .. . and instead of eqn (22) * To examine the limiting behaviour of eqn (22), we rewrite it as In the neighbourhood of the origin, p -+ 0, the limiting form of the equation is while in the asymptotic region, p + co, it is 6d2g P dg -- +-p---.+ pg = 0. bdp2 4 dp After the nucleation period, the evolution of the size distribution of the larger particles is no longer influenced by the particular shape of the distribution near the * Received 17th May, 1976.52 ON THE KINETICS OF PRECIPITATION origin. We may then simplify the problem by letting g(p) approach zero for p 3 0. Suppose it vanishes according to the s-th power of p, namely, g = ;Ips where A is a constant. Substituting this into eqn (26), we have e b S ( S - 1) + s - 1 = 0 i.e., The physically acceptable solution is s = 1 so that we have g - p f o r p - 0 .(28) This result indicates that independent of the ageing rate p, the size distribution On the other hand, the asymptotic form eqn (27) may be transformed into starts linearly from the origin. - where x = 22/8/bpp. The solution to eqn (29) is the parabolic cylinder function3 (30) Hh,(x) = (-l),-l( -&--l d exp (-x2/2) with n = -4. The explicit form of the asymptotic solution is therefore Hk,(x) = x(x2 - 3) exp (-x2/2) - 2 exp (-x2/2). Transforming back to the original variable p, we have g -p3 exp (---$p2) for p +. co. (31) This result indicates that the right-hand shoulder of the size distribution steepens as time proceeds, i.e., as the parameter p decreases. I am indebted to Dr. S.-M. Clian for valuable assistance. I. M. Lifshitz and V. V. Slezov, J. Phys. Chem. Solids, 1961,19,35; C. Wagner, Ber. Bunsenges. Phys. Chem., 1961,65, 581. M. Kahlweit, Adv. Colloid. Interface Sci., 1975, 5, 1. H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics (Cambridge University Press, London, 1956), Chap. 23.

ISSN:0301-7249    

DOI:10.1039/DC9766100048

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Critical Surface Roughening BY K. A. JACKSON AND G. H. GILMER Bell Laboratories, Murray Hill, New Jersey 07974 Received 9th February, 1976 Various models for the structure of crystal surfaces are reviewed. The critical roughening transi- tion which is present in the two dimensional mean field model used to treat adsorbed layers, is not present in the multi level mean field or pair approximation models, but is present in low temperature expansion results and in computer simulation. Experimental data on melt growth indicate that the critical roughening temperature is higher than predicted, whereas vapour growth results indicate that it is lower than predicted. Growth rate curves have been generated for a variety of surface tempera- tures and surface conditions. The structure of the surface of a crystal depends on the co-operative interactions between atoms in the surface layers.It has been known for a long time that such interactions play an important role in crystal growth. For example, they account for the large anisotropies in the growth rate with crystallographic orientation. Con- siderable effort has been expended on the analysis of the crystal surface structure and its motion; most investigators have employed the Kossel-Stranski model, which is equivalent to the Ising2 model used to discuss magnetic transitions. Frequently, however, the co-operative processes involving large clusters of atoms have been ex- cluded as a result of the approximations necessary to obtain analytic or convenient numerical solutions. In the last few years computer simulation by Monte Carlo techniques has provided data which are exact in principle, although the accuracy is limited in practice by the amount of computer time available for the calculation.The Ising model clearly has limitations for describing crystallization processes, but since it is the simplest possible model which incorporates the co-operative processes, it must be understood in detail before proceeding with more complex models. In this paper, an outline of various treatments will be presented. The insights which have been gained from these studies into surface structure and crystal growth processes will be discussed. ONE-LAYER SURFACE MODELS The structure of a surface adlayer was treated by Langrn~ir,~ giving rise to the adsorption isotherm which bears his name.The Langmuir model assumes an adsorption energy V A d , but no interaction energy between admolecules. The adatoms or molecules are assumed to be randomly distributed on the surface, resulting in a free energy of the form n N-n F = n(yA,+p)+nkT In -+(N-n)kT In - N N ’ where n is the number of adatoms, p their chemical potential, N is the number of adsorption sites available, and kT is Boltzmann’s constant times the temperature. The last two terms are the entropy of mixing. This equation has been used exten-54 CRITICAL SURFACE ROUGHENING sively to describe experimental adsorption data. This model, involving no interaction between adatoms, is valid for systems in which this is a reasonable approximation. Again using a monolayer model, Fowler and Guggenheim4 added an interaction pl between the adatoms in the mean field model: n N-n F = n(qAd+p)+2n p,+nkT In-+(N-n)kT In N - IN For equilibrium between the crystal and the phase adjacent to it, p = -pB--2p,, so that eqn (2) becomes n (N-4, F = 2npl(N-n)/N+nkT In R+(N-n)kT In - N (3) For T less than some critical value T, this free energy has two minima of the same value, corresponding to a low density and a high density phase.This is just another way of saying that the surface adatoms will cluster. On one part of the surface the adpopulation will be small, and on the other part, it will be large (close to unity). Deposition occurs by the expansion of the high density regions. The mean field model does not treat clustering properly, but it does indicate that the interaction between atoms will give rise to clustering under appropriate conditions.For T > T,, there is only one minimum in the free energy, corresponding to a random distribution of the adatoms. Fowler and Guggenheim point out that there are two possibilities for an adlayer. The adatoms can be bound to lattice sites, in which case a two dimensional crystal model is appropriate. Alternatively, the adatoms can be bound to the surface, but the lateral potential wells can be too weak to bind the adatoms to specific sites. In this case the adlayer will resemble a two dimensional liquid or gas. These models, derived for adsorption, clearly have relevance to crystal growth. In crystal growth, the adatoms interact with each other with the same magnitude of interaction energy as the binding energy to the layer below.The monolayer (two dimensional model) is not so obviously appropriate, since many layers can be involved in the interface. In their well-known treatment of crystal growth at screw dislocations, Burton, Cabrera and Franks also discussed the equilibrium structure of the surface, using a two dimensional model. The need for defects such as screw dislocations to provide continuous growth steps was occasioned by the application of nucleation theory to the initial formation of the next layer. Assuming that the specific free energy of a step is similar to the specific surface free energy, surface nucleation theory predicts that a crystal should not grow at small undercoolings. But, of course, crystals do grow at small undercoolings.Burton, Cabrera and Frank suggested that the growth at small undercoolings is due to screw dislocations. They realized, however, that the application of classical nucleation theory was not correct. And so they developed a statistical mechanical model for the equilibrium surface. However, they were unable to treat the crystal growth problem properly. Indeed, this is a very difficult problem. The correctness of their concern is indicated by the current commercial-scale growth of large disloca- tion-free silicon crystals. These grow at much smaller undercoolings than those predicted by classical nucleation theory, and yet there are no dislocations to aid the growth. So the screw dislocation model provides only part of the answer. The notion that the classical version of nucleation theory was saved by screw dislocations is incorrect.The classical version of nucleation theory is wrong because it does notK. A . JACKSON AND G , H . GILMER 55 take into account properly the co-operative interaction of the atoms at the crystal surface. This will be demonstrated below when computer simulation results are discussed. The Burton, Cabrera and Frank treatment of these co-operative processes was based on the earlier treatments of adlayers. They used the Onsager6 exact solution for a two dimensional layer, instead of the mean field model used by Fowler and Guggenheim. The critical temperature which occurs in this model TgD is given by is approximately half the bulk (3D) critical temperature, because of the larger number of nearest neighbours in the bulk than in a two dimensional surface layer.Burton, Cabrera and Frank termed this two dimensional critical point the surface melting point, although this is not a particularly appropriate designation. Above this temperature, the surface atoms are not clustered, but they are still on lattice sites as required by the nature of the model. This transition is not to a two dimensional liquid. Presumably, there is a critical point in the Fowler-Guggenheim liquid-gas monolayer which is analogous to this one in the crystalline monolayer. However, the critical point is unlikely to be the same for the two cases. The critical surface temperature of a low index vapour-crystal surface, based on the Ising model, should be above the melting point of most crystals, and so BCF did not expect that it could be observed.However, we will discuss below the recent experi- mental observation of this critical point. Jackson7 applied these same ideas to melt growth in order to explain the experi- mentally observed morphologies of various melt grown crystals. The differences between the bond energies in the melt and the crystal are smaller than the heat of vaporization, so this transition is readily observable in melt growth. However, at atmospheric pressure, the melting point of a crystal cannot be varied, and extreme pressures are necessary to produce significant changes in the melting point, so that only one observation can be made for a particular crystal: its crystal-melt interface is either above or below the critical temperature at its melting point.Thus Jackson was able to classify various melt growth morphologies according to where their melting points lie with respect to the critical surface roughening temperature. The morpholo- gies were classified using L kT5 a = - where L is the heat of transformation, and 5 is a geometrical factor which depends on the geometry of the crystal surface, and determines the order in which the various crystal faces roughen. Surfaces with a < a, (where a, corresponds to Tc) are rough, whereas those with a > a, are below their roughening temperature. MULTI-LAYER MODELS Ternkin* developed a multi-layer mean field model for an interface, again based on a Kossel crystal. This was a solid-on-solid model in which atoms in layer " i " could sit only on filled sites in layer " i-1 ".The extra free energy of the interface as a result of roughening was given by56 CRITICAL SURFACE ROUGHENING under equilibrium conditions with p = -3y. Here Ci = nJN. In this model the interface can be extended over several layers, and the interface width depends on 2y,lkT ( = a for this case). For large a the transition is confined to two layers. For small a, the transition region extends over several layers. This corresponds qualitatively with the two- dimensional models: indeed, the correlation is exact for large a. But in this model there is no critical point. The surface gets rougher continuously as the reduced surface temperature increases. The surface roughness for these models is compared in fig.1. I I 1 0.5 1.0 1.5 kT - € 1 Fig. 1 .-Surface roughness plotted against reduced temperature for the Onsager one-level mode. (......), for the mean field one-level model (------), and for the multi-level mean field model (--). A similar treatment for a multi-level pair approximation model9 leads to a similar result as the mean field model. The surface roughness increases with increasing temperature, and there is no critical point. For both these models, that is both the mean field model and the pair approxima- tion, growth rate kinetics for the motion of the interface can be calculated. This is done by the simultaneous solution of the rate equations for each layer. In the case of the mean field model, and Here P+ is the rate of impingement of atoms on to the crystal surface, and kT', = Ap is the driving force for crystallization.This is a system of non-linear coupled differential equations which can be solved numerically by selecting an initial profileK . A . JACKSON AND G . H. GILMER 57 and following its time evolution. A similar, but more complex set of equations apply in the pair approximation. Examples of calculated growth rates are shown in fig. 2 and 3. In general the growth rate is faster for rougher surfaces (i.e., the growth rates driving force for various rnodels<L/kT = 4-5). - 0 0.5 1.0 1.5 &/kT FIG. 3.-Normalized growth rate plotted against chemical potential driving force for various models (L/kT = 6.0). are faster in fig. 2 than fig. 3). There is a region of zero growth rate for small under- coolings, which is evident in fig.2 for the mean field case, and present, but not evident, in the pair model. The region is evident for the pair model in fig. 3, but off scale for the mean field. The growth rate is calculated to increase rapidly beyond this region of zero growth, in a manner predicted qualitatively by Cahn.lo Beyond this region the58 CRITICAL SURFACE ROUGHENING growth rates agree with each other. The extent of the region of zero growth decreases significantly on going from the mean field to the pair approximation. This region of zero growth would decrease further with higher order approximation, and it is an artifact of these models. It is not present in the computer simulation results, also shown, which will be discussed below.The region of zero growth arises because the model assumes the interface to be infinite in lateral extent and that the whole layer must move from plane to plane simultaneously. In other words, nucleation pro- cesses are excluded. The larger clusters which are important to the formation of new layers are not present in these models. The contribution of these large clusters to the growth rate is particularly important at small undercoolings. Indeed, it is sur- prising that these models do as well as they do at small undercoolings. Weeks, Gilmer and Leamyl' treated the equilibrium structure of the interface using a low temperature expansion method. This model starts with the fact that the equilibrium interface is flat at the absolute zero of temperature. Near absolute zero only configurations which increase the energy by a small amount will occur.In the lattice model, the smallest increase in energy is a single adatom on the surface, or a single vacant site: these have large entropy, since they can sit on any site. The next lowest energy is a pair of adatoms, side by side, or a pair of vacancies side by side: these can also exist on a large number of sites. All the configurations which con- tribute n extra bonds to the flat surface can be determined, where the maximum n depends on the perserverance of the authors. Weeks, Gilmer and Leamy determined 9 terms in the series. The last term had contributions from over three thousand configurations which were determined using a computer. This then gives an ex- pression for the energy of the surface in a series form determined exactly out to nine terms.This series is a very good approximation at low temperatures where the higher order terms are unimportant, but is increasingly poor (due to the truncation) at higher temperatures. Pad6 approximants were then used to determine an analytic expression which would approximate this series. From this analytic expression, several properties of the surface can be determined, such as the second moment of the surface which is a measure of its width, as well as the surface specific energy, which measures the rate of increase in the number of bonds with temperature. The second moment is predicted to have a singularity at a critical point by this method. The critical point lies just above the critical point of a 2D Ising model.Above this critical point, the interface width diverges. This result may seem strange, but it corresponds to the divergent width of a fluid-fluid interface separating two immiscible liquids in a field-free space. If there were no gravity, for example, a fluid-fluid interface would have large-amplitude, very long wavelength displacements. The amplitude diverges (logarithmically) as the area of the interface increases, The interface cannot be located in such a system. A gravitational field removes the divergence. The crystal interface exhibits similar properties above its critical point. In a fluid-fluid system, there is no lattice to localize the interface at one plane and perturbations of the surface are always possible. For the crystal, only discrete, atom- size, perturbations are possible.Therefore a finite temperature is necessary before thermal fluctuations become large enough to produce the long wavelength displace- ments. Below the critical roughening temperature the interface is locked on to the crystal planes ; above this temperature, the interface position fluctuates. Below the critical temperature, the interface moves by the motion of steps, above the critical temperature the interface is free to move under the influence of infinitesimally small thermo- dynamic driving forces. The picture becomes clearer in light of computer simulation results on surfaces,K . A . JACKSON AND G . H . GILMER 59 Fig. 4 shows equilibrium surface configurations generated in the coniputer. These surface configurations were produced by a simulation scheme based on the Ising model.Atoms arrive randomly at surface sites, but the departure (evaporation or melting) of an atom depends on the number of nearest neighbours. These surfaces contain steps which were built into the computer model. At temperatures below the critical roughening temperature, (kT,/u, = 0.63 for this case), the steps are clearly visible. As the temperature increases, the surface becomes rougher, more adatoms are present, and the steps become more irregular. Above the critical roughening temperature, the steps are lost in the general surface roughness. FIG. 4.-Typical equilibrium computer-generated surface configurations for stepped surfaces at various reduced temperatures: kT/q = 0.428, 0.545,0.571,0.060, 0.632, 0.667.Leamy and Gilmer12 have examined this effect quantitatively. In fig. 5, the free energy of a step on the surface as a function of temperature is shown, as determined by computer simulation. It is evident that the step free energy goes to zero (within the accuracy of the data) at the critical roughening temperature. Above the critical temperature, steps do not contribute to the surface free energy, and so they can form spontaneously. The surface is not locked to the atom planes, and long wavelength irregularities in the surface will occur. The surface roughening temperature has a profound effect on growth kinetics. Above the surface roughening temperature, growth can occur by the addition of atoms to a large fraction of the surface sites.Below the surface roughening temperature, the motion of steps is important, and at low temperatures step motion dominates the growth. Also shown is the Wil~on-Frenkell~ growth rate which assumes that all surface sites are active growth sites, and is thus an upper limit to the growth rate. Above the critical Fig. 2 and 3 show growth rates obtained by computer simulation.60 CRITICAL SURFACE ROUGHENING roughening temperature, the growth rate initially depends linearly on the chemical potential difference between the two phases. Below the surface roughening tem- perature, the growth rate is smaller at small undercoolings, and then increases more rapidly. The increase in growth rate at large undercooling is due to dynamic rough- ening of the surface. This arises because, at larger undercoolings, the arrival rate of atoms increases with respect to the evaporation rate, so the number of adatoms 0.c I b FIG.5.-Free energy of a surface step as a function of reduced temperature. etc on the surface increases. This effect makes the growth difficult to calculate by analytical methods: the growth rate depends in detail on both the number of atoms in each layer, as well as on their arrangement in the layers. Fig. 6 shows growth rate determined by computer simulation for surfaces containing screw dis- locations. For L/kT = 6, close to the roughening temperature, screw dislocations have little effect on the growth rate, since the formation of new layers is relatively easy, in contradiction to classical nucleation theory. For L/kT = 12, the screw disloca- tions increase the growth rate significantly at small driving force, but are less important at high driving force.Thus screw dislocations contribute to the growth rate only at low reduced surface temperatures, T< TR, and at small driving force. EXPERIMENTAL RESULTS Melt growth observations have been mentioned above. In addition to observa- tions on a large number of pure materials, there are also observations on eutectic systems. The micro-structures of hundreds of eutectics have been classified according to the entropy of fusion of the phases.14 The results are quite clear cut. For both phases with L/kTEht2, lamellar or rod eutectics occur, because the intrinsic growthK. A . JACKSON AND G . H. GILMER 61 rates are rapid and isotropic. When one or both phases have L/kT>,2, plates, irregular structures, cuneforms, etc, are observed.There are a few apparent dis- crepancies where L/kT,-2, but these are perhaps cases where the crystallographic factors are contributing. For melt growth, the accumulated evidence from growth morphologies places the critical value of a at about 2. This is a factor of almost 2 smaller than the 2D critical temperature, the low temperature expansion singularity and the computer L/ kT= G 0.4 0 1 2 3 Ap/ kT FIG. 6.-Normalized growth rate as a function of chemical potential driving force obtained by computer simulation. The squares and hexagons are data for perfect surfaces. The I symbols are data for surfaces containing screw dislocations. simulation critical temperature.The discrepancy here is probably due to the fact that the Ising model used for the calculations is not a very good model of a crystal- melt interface. The co-operative processes which give rise to the critical point, and which dominate the growth behaviour are properly treated. But other effects, such as the geometry of the liquid adjacent to the crystal are not included. And so the transition temperature is incorrectly predicted, although the growth rates relative to the critical temperature have the proper dependence on a reduced temperature scale. Recent observations l5 on vapour phase growth of C2C16 and NH4C1 have indicated the presence of the surface roughening transition at L/kTE- 16. The transition can be observed readily in the morphology of growth which changes markedly in a narrow temperature interval, indicating a sharp transition in surface structure.Vapour phase growth has the advantage that the crystal and vapour can exist in equilibrium over a range of temperatures, for modest variations in pressure (unlike melt growth). The observed critical temperature L/kTE-16 is smaller by about a factor of 2 than the theoretical critical roughening temperature. If the transition had been at the theoretically predicted temperature, it would have been above the triple point in both materials as Burton, Cabrera and Frank had suggested. The discrepancy for vapour phase growth is opposite to that for melt growth. For melt growth L/kTE is smaller than predicted. For vapour phase growth, it is larger. The reasons for the discrepancy are not properly understood.It is possible that the adatoms are mobile, as suggested by Fowler and Guggenheim. That is, surface adatoms are not located at lattice sites, but rather form a two-dimensional gas. The transition then would62 CRITICAL SURFACE ROUGHENING correspond to the " condensation " of this two dimensional gas. This would occur at a lower temperature than the surface roughening temperature in the Ising model. Further detailed exploration of these discrepancies must await molecular dynamics results where atoms positions and motions are not restricted to lattice sites. CONCLUDING REMARKS The effects of surface roughening are clearly apparent in crystal growth experi- ments, both from the melt and from the vapour phase.A critical surface roughening transition is present in the king model, as demonstrated by low temperature expansion results as well as by computer simulation. The growth kinetics, and morphology, both experimentally and theoretically, depend on whether the growth occurs above or below the critical roughening temperature. Detailed Monte Carlo predictions of growth kinetics based on the Ising model are now available. These treat the co- operative aspects of growth properly, and provide the best available predictions of growth kinetics. In addition, significant insights into surface co-operative processes have been obtained. W. Kossel, Nachr. Ges. Wiss Giittingen, 1927,135; I. N. Stranski, 2. plzys. Chern., 1928,136,259. See for example K. Huang, Statistical Mechanics (John Wiley, N.Y. 1963), p. 329. I. Langmuir, J. Ainer. Chern. SOC., 1918, 40, 1361. R. Fowler and E. A. Guggenheim, Statistical Thermodynamics, (Cambridge Univ. Press, London, 1939), p. 430. W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. A, 1951,243,299. L. Onsager, Phys. Rev., 1944,65, 117. D. E. Temkin in Crystallization Processes, ed. N. N. Sirota et al. (Consultants Bureau, N.Y. 1966), p. 15; Soviet Phys.-Cryst., 1969, 14, 344. H. J. Leamy and K. A. Jackson, J. Appl. Phys., 1971, 42, 2121. J. D. Weeks, G. H. Gilmer and H. J. Leamy, Phys. Rev. Letters, 1973, 31, 549. ' K. A. Jackson in Liquid Metals and Solidification (ASM, 1958), p. 174. lo J. W. Cahn, Acta Met., 1960, 8, 554. l2 H. J. Leamy and G. H. Gilmer, J. CrystaZ Growth, 1974, 24/25, 499. l 3 H. A. Wilson, Phil. Mag., 1900, 50, 238; J. Frenkel, Phys. Soviet Union, 1932, 1, 498. l4 J. D. Hunt and K. A. Jackson, Trans. Met. SOC. AZME, 1966, 236, 843. l5 K. A. Jackson and C. E. Miller, to be published.

ISSN:0301-7249    

DOI:10.1039/DC9766100053

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. M. La1 (Port Sunlight) said: The paper by Hoare and McInnes highlights only the minimum-energy configurations. From the statistical thermodynamic standpoint, however, the relevant configurational states would be those which correspond to the minimumfree energy of the system. It is only at 0 K that the minimum binding energy would coincide with the minimum free energy. At other temperatures entropic contributions, such as those arising from atomic vibrations and the relative flattening of the high-density regions in the configuration phase space, might be significant. How will the consideration of the entropic factors alter the picture as conveyed in the present work? Another important point concerns the presence of many-body interactions in condensed phases, particularly in the solid phase.It has been generally known that in such phases the assumption of the pair-wise additivity for interatomic interactions is inadequate and one niust consider at least three-body interactions in order to establish reliable relationship between the configurational state and the binding energy. Thus it would perhaps be worthwhile studying the influence of non-additive contributions on the configurational stability. The non-additive part for the three body interactions can be conveniently estimated using the Axilrod-Teller formula. Dr. M. R. Hoare (Bedford College) said: La1 is quite right that free-energy consider- ations are paramount in determining the actual occurrence of particular structures at a finite temperature. We do not, however, neglect this, as a reading of Section 7 of the paper will show.Further accounts of the relative contributions of energetic and entropic effects will be found in ref. (1 1) and (12). Although we have not yet pub- lished full details in terms of the complete isomer ensembles, our preliminary conclu- sion is as stated, namely that the relative insensitivity of vibrational frequency patterns to change from one minimum to another tends to minimize the contribution of en- tropic effects to the relative stability of different isomers, at least in the temperature- range for which the harmonic approximation is valid. The main exception to this is in the rotational contribution of clusters with very high symmetry, such as the icosahedron. We agree that the effect of many-body forces should be taken into account, though we have not found a way to do this in the computing time available.We suspect, in fact, that many-body contributions may be amplified in small particles, where surface effects are dominant, and particularly in the case of metals. Dr. J. F. Qgillvie (Newfoundland) said: In regard to the results of Hoare and Mc- Innes,l the great reduction of numbers of stable isomers according to the Morse potential, relative to those obtained with the Lennard-Jones potential, appears very striking. But if one compares the forms of the pair-potential functions, the reasons become evident. The figure contains three potential curves (in reduced form), for the Lennard-Jones function, V(r) = (1 -4-y- 1 M. R. Hoare and J.McInnes, this Discussion.64 GENERAL DISCUSSION where for the usual (6-12) function n = 6, and two Morse curves, V(r) = (1 -e--nr)2- 1 where n = 3 or 6. Examining the two curves with n = 6, we observe that the functions are quite similar, the significant difference being in the region of r - 2 where the Morse curve indicates a much smaller interaction energy than that of the Lennard-Jones function. The relation of these two curves to real molecular systems is that for two argon atoms, for instance, the true potential lies between these two model functions in the region r - 1.8 but rises more steeply than either in the repulsive region r < 0.9. The Morse function with n = 3 provides in contrast a much greater t . V 0 .o -0.5 - 1 .o FIG. 1,-Potential energy functions in reduced form for the Morse (M) or Lennard-Jones (L) models with the indicated exponents. binding energy at r < 2 than either of the n = 6 functions.The consequence is that the central atom can exert a much greater influence, under this potential, on second- nearest neighbours ; therefore, the less compact isomers are relatively much less stable. The Morse function with n = 3 is a good approximation to the true potential function of such strongly bound molecules as carbon monoxide,l and probably to diatomic molecules of metallic elements, but only in the lower region of the potential well. At large separations (r > 2.5), the space dependence tends to a limiting r-6 form. Thus this Morse function seems to have doubtful applicability as a model in the considera- tion of metal-atom clusters, in which long-range interactions are important.Dr. M. R. Hoare (Bedford College) said: While we obviously had metals in mind when studying the stability of the Morse-clusters, we were more concerned to demon- strate qualitatively the sensitivity of the number of minimum locations to the type of potential than to arrive at realistic values of the binding energy and thermodynamic properties. There are various estimates of the Morse exponential parameter in the metallurgical literature, some of which are surprisingly low, (e.g. a from about 1.18 for aluminium to 1.58 for gold).2 J. F. Ogilvie and R. W. Davis, Faraday Disc. Chem SOC., 1973,55, 189. Weaire, Ashby, Logan and Weins, Acta Metallurgica, 1971, 19, 779-88.GENERAL DISCUSSION 65 Dr.F. Abraham (San Jose) said: Until recently, the physical cluster has been pic- tured as a ‘‘ liquid droplet ”, a “ microcrystallite ”, or some other model construct, i.e., the statistical mechanical theories of a physical cluster have been heuristic, de- pending mainly upon the chosen model. The heuristic approach requires a successful choice of a suitable model which (l), represents reasonably well the system of interest, and (2), provides a relatively simple partition function from which numerical quantities can be obtained. While these models provide an insight into the origin of certain features of a physical cluster, they cannot serve to elucidate the cluster’s true molecular structure since this is assumed in one way or another.With the advent of large scale scientific computers, pursuing such an approach for the statistical thermodynamics of many-body systems has begun where only a form of the intermolecular potential function between two molecules is assumed. Two recent examples of this approach have been molecular dynamics and Monte Carlo simulations of clusters of Lennard- Jones atoms. In the Monte Carlo paper, we present a formal physical cluster theory for an im- perfect gas that is valid for an arbitrary definition of a “ physical cluster ”. The role of the deJnition of the physical cluster is stressed. For a particular definition of the physical cluster, which may be appropriate in nucleation theory, the Helmholtz free energy of 13-, 43-, 60-, 70-, 80-, 87- and 100-atom argon clusters is calculated in the classical limit for temperatures ranging from absolute zero to 100 K using Monte Carlo techniques.It is found that a cluster’s free energy is almost independent of its definition provided that the definition is reasonable and the temperature is sufficiently low (T < 75 K for an 87 atom cluster). The results are compared with the predic- tions using the harmonic approximation, deviations occurring for T 15 K. We make special note of the fact that the multiconfigurational states of the cluster arise as a direct consequence of the Monte Carlo computational procedure. Prof. H. Reiss (Los Angeles) said: Since the physical cluster is so prominent in nucleation theory, attempts are frequently made to define it rigorously. Abraham in his comments on physical clusters, described the work of himself and certain of his colleagues, and made the point that under certain conditions there may be no rigorous method of definition, and that in any event the definition should always be tailored to the measurement for which the theory is being developed.I thoroughly agree and would like to illustrate the point by reference to a physical cluster problem involving a one dimensional fluid.3 Here I will try to describe physical clusters which can be used in the development of the equilibrium equation of state of the fluid. Furthermore these clusters have a close connection to those which Abraham has studied and will probably be of value in nucleation theory. The most rudimentary physical cluster theory of the vapour phase assumes the clusters to be “ independent ” molecular species such that they constitute a mixture of ideal gases.Under this condition the configurational partition function for the entire gas may be expressed as , where {m,) indicates that the sum is over all terms such that 2 Iml = N. 1 D. J. McGinty, J. Chem. Phys., 1973, 58,4733. * J. K. Lee, J. A. Barker and F. F. Abraham, J. Chem. Phys., 1973,58,3166. H. P. Gillis, D. C. Marvin and H. Reiss, J . Chenz. Phys., 1977, in press.66 GENERAL DISCUSSION In these equations, N represents the total number molecules in the gas, ZN the total configurational partition function, zl, the configurational partition function for a cluster of molecules, and ml the number of clusters of l molecules. The immediate (rigorous) statistical mechanical consequence of eqn (1) and (2) is that the pressure of the vapour is given by where V is the volume of the vapour and viil the average of (if N + co) the most probable number of clusters of size 1.This is, of course, simply Dalton's law of partial pressures. Now, if the goal is to deJine the cluster, and therefore choose zl in eqn (l), such that the 2, are exact, this is impossible unless ZZ = Vbl (4) where bl is the reducible mathematical cluster integral involving l molecules. (Al- though implicitly obvious, this point has never been explicitly stated in regard to theories of physical clusters. It follows directly from the reversion of eqn (1) which yields 21 = 21 ( 5 ) L L The quantities on the right are just Vbl, Vb2, and Vb3, etc.) Since choosing thephysical cluster, such that its partition function is a mathematical cluster integral, is not very physical, especially since some of the GI will then have to be negative, this choice buys nothing in the way of a model, and so the strict goal of achieving (3) withphysical clusters is impossible.Thus, with a physical cluster approach, Dalton's law is strictly out of the question, and another formulation besides eqn (1) is necessary. The simplest adjustment is to include the so-called excluded volume contributions' in eqn (1). In reality this is a counting restriction associated with the combinatorics of expressions like eqn (l), i.e., it is not associated with exclusion due to physical potentials, but it has the same effect. For a three-dimensional system an approximate inclusion of excluded volume would replace eqn (1) with relations of the following sort for zl, z2, z3, etc.z, = z;e(i, o,o, 0 . . .) = z;v= z1 5 2! = 9 L ( 2 , 0 , 0 . . .)+z;e(o, 1 , o . . .) (6) - 2 3 - - A O ( 3 , @'I3 0, 0 . . .)+z;Z;o(l, 1, 0 . . .)+ZiO(O, 0, 1 . . .) 3! 3! etc. where z ' ~ = zl/Y and where, typically, O(0, i, j , 0 . . .) is an integral, much like a con- figuration integral, for a system containing no monomers, i dimers, j trimers, no tetramers, etc. The integral treats the clusters as each surrounded by a hard shell beyond which there is an arbitrarily defined minimal field of interaction with other F. H. Stillinger, J. Chein. Phys., 1963, 38, 1486.GENERAL DISCUSSION 67 clusters assumed to lie within their own shells.The shells are therefore like " hard " molecules and represent the configuration integrals of assemblies of such hard mole- cules. It can be shown that in one-dimension eqn (6) can be made exact. To illustrate this we choose linear molecules with pair interaction potentials where a, b, and E are constants and x is the distance between the centres of two mole- cules. Thus the molecules have hard-rod cores and attractive, square-well, outer potentials. A cluster of Z molecules is now defined as a group of I, such that every molecule is interactionwise connected. Put in another way, there must be no con- figuration in which the distance between two adjacent molecules exceeds 6. The maximum length of a cluster can be Zb, but, at that maximum length, not every internal configuration meets the cluster criterion.Tn fact, if we define Az as the length of an I cluster, in terms of the distance between its terminal molecules, an application of Markhoff's method, using eqn (7) shows that for In this equation, zl(Al) is the partition function of the physical cluster when its length is Az and L is the length (volume) of the entire system. The quantity r is an integer indexing terms in the sum, while Cf-' is the binomial coefficient. Thus the physical cluster must be defined not only by 2, but by its length &. In three dimensions its shape as well as its extent would have to be involved in its definition. With the definition of the physical cluster given above, it is possible to show that the following alteration of eqn (1) is exact. 5 2! = 0" 2! 6(2, 0)+rz;(A2)LdA2 a etc.Here the 8's have the same meaning as in eqn (6) except the hard-shelled molecules are replaced by hard rods of lengths determined by the various Al. The integers in eqn (9) are really sums over different cluster partition function products ; different values of ;Il corresponding to different clusters, even though I is the same. These integrals, which would be essentially impossible in three dimensions, have one redeeming feature which, fortunately, can be immediately investigated in one dimension, i.e., when Z is large enough the integrals have contributions only from a very small range of This is illustrated by fig. 1, 2, and 3 which are plots of z;(AI), from eqn (S), versus At in dimensionless units, for I = 3, 4 and 12, respectively. Notice how quickly z'(Al) moves toward &function behaviour; at I = 12, the half width is less than 5%68 GENERAL DISCUSSION I ' 00 0.4 0.8 I2 16 2.0 2.4 2.8 3.2 3.6 4.0 A 3 FIG.1. 1 L - L 1.8 2.4 3.0 3.6 4.2 4.8 A 4 FIG. 2. of the entire range of iZl. For larger valuecof 2, ~'(1~): canlclearly be treated as a 8- function in the integrals of eqn (9). Now z; has been defined in term of an origin in some particular moZecu2e in the cluster, together with the overlap criterion enunciated above. The question arises as to whether it can be defined in terms of an origin at the cluster's centre of mass, as in Abraham's cluster, together with the overlap criterion and, if so, can the transformation between the two clusters be easily accomplished.The answer is affirmative, so that the same 8-function-like behaviour should occur.GENERAL DISCUSSION u 69 66 8 8 11.0 132 154 17.6 19.8 2 2 0 i, 12 FIG. 3. Another definition would involve dropping the overlap criterion but insisting that the cluster be confined within a given interval of length, centred on the cluster’s centre of mass, exactly as in Abraham’s cluster. For large enough clusters, configurations in which the overlap criterion is violated should carry very little weight, and the results should be essentially the same as with overlap. The significance of being able to dispense with the overlap criterion in this manner is not major, in one dimension, where it can be accounted for in a straightforward manner. However, in three dimensions a calculation including this criterion would be prohibitive.Thus as with Abraham’s cluster, the cluster in three dimensions can be defined in terms of confinement of the molecules to the interior of a sphere centred on the cluster’s centre of mass, and then even if with neglect of overlap its partition function converges to the same z ’ ~ obtained with both the requirement of overlap and the requirement that the origin be at a given molecule rather than at the centre of mass, the problem of evaluating z’~ becomes tractable. Furthermore eqn (6) becomes accurate under these circumstances. Dr. E. R. Buckle (Shefield) said: Hoare mentioned astrophysics, and I wonder whether small collections of atoms in configurations of absolutely minimal energy can be formed other than in an inter-stellar environment. With the computer one can choose them at will, but Briant’s conclusions pre-suppose the computer’s choice of configurations to be the same as the natural growth sequence.Might not the suc- cession be governed by kinetic mechanism? Is that what is meant by special entropic effects? Dr. M. R. Hoare (Bedford College) said : As you suggest, the mechanism of growth of small particles under realistic conditions might well exclude certain otherwise favoured configurations for essentially kinetic reasons, which may just as well operate in astrophysical conditions as in, say, nozzle-beam condensation. Put slightly differ- ently, one has no right to assume that solid-like clusters, formed under more or less70 GENERAL DISCUSSION rapid cooling, are effectively " annealed ", far less in true thermodynamic equilibrium.We have considered making an assumption of extreme quenching, assuming that particles are formed with equal probability in all accessible minima and computing the effective thermodynamic properties for this mix. However, this is just as likely to misrepresent kinetic and steric factors in collisions and would give weight even to chain-like configurations which are hardly likely to occur. Presumably the truth lies somewhere between the two extremes. So long as we are not trying to determine the morphology of the particles for its own sake, it would be satisfying if the nucleation parameters proved to be quite insensitive to which of the two extreme models we adopted.If' the growth process is kinetically determined, it might well be quite differ- ent according to conditions, presence or absence of carrier-gas, for example. Prof. J. Zarzycki (Montpellier) said: In calculating the interference functions of his model cluster Briant takes into account only the distances rlj between atoms in the same microcluster-and neglects distances between atoms in the adjoining clusters which could be of the same order of magnitude within a dense liquid. The problems discussed are similar to those already encountered in the study of glasses when the choice between the random network and microcrystallite theories appears now as a matter of degree. The difficulty lies in defining the " interstitial matrix " joining more or less locally ordered regions.Attempts at using pentagonal dodecahedra1 models for describing glass lattices go back to Tiltonl and Robinson.2 This model has recently been generalized to larger clusters by Ga~kell.~ Neither X-ray diffraction studies nor high-resolution electron microscopy, however, enable a choice to be made between these different hypotheses. Dr. C. L. Briant (Philadelphia) said: In our calculation of the interference function for the quenched 55 atom structure we include not only the distances within the 13 atom icosahedron but those in the disordered material around the isosahedron as well. This should consider the contribution of the interstitial matrix to the interference function. To completely answer this question, one must quench larger numbers of atoms which might then contain more than one ordered region.This would then allow a much more complete analysis of material interconnecting the ordered regions. Dr. M. La1 (Port Sunlight) said: The model for a microcluster considered in Briant's paper constitutes n(= 13) atoms suspended in vacuum, whereas in the paper by Abraham and Mruzik a microcluster is regarded as an assembly of particles (atoms, molecules or ions) enclosed in an impenetrable " cell ". Will the author please comment on the apparent contrast between the models. Dr. C. L. Briant (Philadelphia) said: The difference in these two models is basically as follows. In the model we used, an atom boiling off the surface of the cluster is allowed to do so and could only return to the cluster if the potential energy drew it back.In Abraham's model an atom passing beyond some constraining radius is directed back at the cluster rather than being allowed to escape. In the solid and low temperature liquid clusters examined in this paper, one never finds any atoms boiling off the surface so the result would be the same for either model. Only if one wanted to study high temperature liquid clusters would the model critically affect the results. L. W. Tilton, J. Res. Nat. Bur. Stand., 1957,59, 139. H. A. Robinson, J. Phys. Chem. SoIids, 1965,26,209. P. H. Gaskell, Phil. Mag., 1973,32,211.G E N ERA L D IS CU S S I 0.N 71 Dr. F. F. Abraham (Sun Jose) said: You noted that in examining the 55 atom quenched cluster you found that a 13 atom icosahecron had formed which included atoms near the surface, in contrast to forming at the centre of the cluster.Don’t you feel that it would be more appropriate rapidly to quench the bulk phase simulated by the use of periodic boundary conditions so as to eliminate any “ surface ” effects in the creation of the amorphous state? (e.g., see A. Rahman, J. J. Mandell and J. P. McTague, Molecular dynamics study of an amorphous Lennard-Jones system at low temperature, J. Chem. Phys., 1976, 64, 1564.) Dr. C. L. Briant (Philadelphia) said: It is true that one would like to quench more bulk-like systems in order to establish a model for amorphous metals. However, the use of periodic boundary conditions presents problems, as they can induce rather artificial structures and also give misleading ideas as to the frequency of various seed structures.I feel the best approach to this problem would be to quench large numbers of atoms using free boundary conditions. One could then examine the structure of the centres of these quenched aggregates where there should be little surface influence. Also such a method should give more information about the inter- connecting tissue between the structural units. Prof. H. Reiss (Los Angeles) said: There is some work,l* employing molecular dynamics, which relates closely to research described by both Hoare and Briant. This work generally confirms the conclusions of both Hoare and Briant. I I I I 0.5 I .o 1.5 2.0 2.5 3.0 l/u FIG. 1. First an extremely supercooled (“ glassy ”) Lennard-Jones fluid. A 500-particle liquid sample was compressed and rapidly cooled to a reduced temperature (kT’/c) of 0.11 and a reduced density (pa3) of 0.95. The radial distribution function, shown in fig.1, is similar to that found by other investigators for random-close-packed systems, The doubled second peak in the radial distribution corresponds to the A. Rahman, M. J. Mandell and J. P. McTague, J. Chein. Phys., 1976, 64, 1564. M. J. Mandell, J. P. McTague and A. Rahman, J. Chein. Phys., 1976, in press.72 GENERAL DISCUSSION shoulder in the second peak of the structure factor, which was described by Briant. The diffusion constant was zero within the limits of their measurement. The dynamic structure factor (left, fig. 2) shows that the amorphous system has propagating longitudinal phonons. The longitudinal sound speed is roughly equal to that for solid argon.The transverse current correlation function (right, fig. 2) 0 10 20 cd.I 0 10 20 r3.z FXG. 2. indicates propagating transverse phonons at long wavelength, with a sound speed about 25% less than that for solid argon. In the course of beginning the study of the " glass transition " these investigators found that by using sufficiently slow cooling rates the Lennard-Jones fluid would spontaneously crystallize. Fig. 3 records the history of a run made with 108 particles at a density of 0.91 with constant energy. The nucleation event is characterized by simultaneous vanishing of diffusion, increase in temperature and decrease in pressure. Similar results have been achieved for 256 and 500 particles. For the larger systems, crystallization requires higher densities and/or lower temperatures.Mandell developed the technique of searching for the absolute maximum (away from the origin) of the structure factor in reciprocal space. The history of this quantity for the nucleation event illustrated in the previous figure is shown in fig. 4. In the super- cooled fluid one infers partially ordered structures which are fairly long-lived. The interpretation that these structures are sub-critical nuclei is tempting. The appearance of the Bragg peaks at crystallization is indicative. Fig. 5 shows the projection of a 256-particle crystal. The sample has been quenched after its formation at high temperature. Close-packed (1 1 1) planes areGENERAL DISCUSSION 73 I R2 1 .o 0.5 (3.0 500 250 0 .80 .70 .60 .50 I .... press u re tempera! ure c vertical and perpendicular to the plane of the figure. Unlike the 108-particle system, the larger samples invariably form imperfect crystals. Nonetheless, these samples are large enough that their crystalline structures can be unambiguously recognized; defects such as vacancies and dislocations can be located. The final figure (fig. 6) is the projection of a spherical region taken from a 256-particle sample shortly before a crystallization event. The region contains 33 particles (28 shown and 5 hidden). The distances are consistent with having vertical close-packed planes as in the previous slide, and the orientation coincides with that of the not-yet-formed crystal. Optimistically, this might be regarded as the first observation of a spontaneously-formed crystal nucleus.74 a 1 GENERAL DISCUSSION I .I I I J 1 o4 c p ,of Y 1 02 0 I 0 */ 0 0 A -(1.2, -1.2, 6.4) o -14.8, 3.7, 2.6) -(1.2, 1.2, 6.4) A -(-3.7.3.7,3.71 FIG. 4. 4 , 1 , - 1 0.1.1 5 r A I I I - 3 - 3 - 2 - 1 - 0 - -1 -2 -3 - - - + + 4 + + + * * 4 b + * + * + + ' * ; 4 : -4 + + -7 -3 FIG. 5.GENERAL DISCUSSION 75 + + + +f + 4- t f ++ + + + + + t + + ti t i- +. + +I + FIG. G. Dr. M. La1 (Port Sunlight) said: Monte Carlo technique based upon Metropolis sampling is most suitable when the phase space contains a single minimum energy " well ',. The method produces a sample in which most of the points (configurations) lie in the " well " in the vicinity of the minimum. If, however, the configurational phase space is composed of multitude of minima separated by energy barriers, realiza- tion of a truly converged sample might be difficult within the present computational limitations.In this context the question concerning the magnitude of the energy barriers separating the various stable configurations of the clusters, as posed by Everett, is most relevant. If such barriers are high, then the suitability of the method to the cluster problem is questionable. Dr. F. F. Abraham (Sun Jose) said: For finite temperatures, the exact structure of the water-ion clusters corresponding to the lowest energy (T = 0 K) is of marginal chemical interest since many conjigurations with nearly equal stabilization energy exist (€3. Kistenmacher, H. Popkie and E. Clementi, J.Clzem. Phys., 1974, 61, 799). For this reason, we use the Monte Carlo method to generate the Boltzmann-weighted configuration space and, hence, to get thermodynamic and structural properties, especially for the high temperature of 298 K. It should be noted that the Monte Carlo method has been proved successful for calculating the equilibrium properties of dense systems (e.g., liquids) where there exists no single minimum energy well. Prof. J. M. Thomas (Aberystwyth) said: My comment refers to the optimum cation-water configuration envisaged for ion-water pairs (fig. 13(a)). There are a number of monovalent cations which possess substantially similar ionic radii (Rb+ = 1.48, NH+ = 1.43, T1+ = 1.40A) but which, on the basis of their differing polarizabilities, give rise to quite widely varying properties in, for example, ease of cation migration, particularly in the solid state. One wonders whether the heuristic approach outlined in Abraham's paper is capable of taking into account these relatively subtle differences between ostensibly similar cations. Dr. F. F. Abraham (Sun Jose) said: If given an accurate potential energy surface for the cation-water system, the variation of physical properties due to differing polarizabilities would be taken in account when calculating the equilibrium properties using the Monte Carlo method. Dr. E. R. Buckle (Shefield) said: In regard to the discrepancies in fig. 8 and 9, the vapours of the alkali-metal salts contain polymeric species which increase in abun- dance in the order K, Na, Li. This could complicate the interpretation of the mass spectra.76 GENERAL DISCUSSION Dr. F. F. Abraham (Sun Jose) said: Such possible errors were examined by Kebarle (ref. (9) and (lo)), and he concluded that they probably contributed little to the total experimental error. Dr. E. R. Buckle (Shefield) said: Turning to the work of Kahlweit, what is being described as ageing is analogous to the course of a homogeneous chemical reaction. It keeps things simple if the constituents of the mixture can be chosen so as to give the correct predictions with the ideal laws of kinetics and thermodynamics. If the description of the growing particle and the way it behaves towards its neighbours were correct the course of reaction would follow from the laws of motion. The trouble, and here the problem is quite general in liquids,l lies in distinguishing the motion of a coherent group of atoms from the motions of the atoms themselves. E. R. Buckle, The Chemical MetaZZwgy ofIron and Steel (Iron and Steel Institute, London, 1973), p. 52.

ISSN:0301-7249    

DOI:10.1039/DC9766100063

出版商:RSC    

年代:1976

数据来源: RSC