摘要:

3iecueefons of tbe $arabap 5ocietp No. 5, 1949 CRYSTAL GROWTH GURNEY AND JACKSON LONDON: 98 GREAT RUSSELL STREET EDINBURGH : TWEEDDALE COURTTits Faraday Society reserves the Copyright qf aLt Communications published in the '' Discussions '* P R I N T E D I N E N G L A N D B Y H A Z E L L , W A T S O N & V I N E Y , L T D L O N D O N A N D A Y L E S B U R YA GENERAL DISCUSSION ON CRYSTAL GROWTH 12th - 14th April, 1949 A GENERAL DISCUSSION on Crystal Growth was held in the Department of Physics, Bristol University (by kind permission of the Vice-Chancellor) on the rzth, 13th and 14th April, 1949. The President, Prof. Sir John Lennard-Jones, K.B.E., F.R.S., was in the Chair and over 300 members and visitors were present. Among the distinguished overseas members and guests welcomed by the President were the following :- Prof.R. Becker (Gottingen), Dr. G. Berkhoff (Geleen, Netherlands), Dr. H. de Bruijn (Geleen, Netherlands), Prof. C. Correns (Gottingen), Dr. P. H. Egli (Washington, D.C.), Dr. P. Franzen (Delft), Dr. W. Gaade (Amsterdam), Mr. I. J. Haven (Eindhoven), Prof. R. Hocart (Strasbourg), Ir. Th. J. J. Hoek (Geleen, Netherlands), Dr. A. N. Holden (Murray Hill, N.J.), Prof. A. Juliard (Brussels), Dr. D. W. van Krevelen (Geleen, Netherlands), Dr. W. C. McCrone (Chicago), Ir. W. May (Delft), Mr. W. M. Mazee (Overleen, Netherlands), hflle. M. Michel-L&vy (Paris), Dr. S. 0. Morgan (Murray Hill, N.J.), Dr. M. H. R. Plusj6 (Geleen, Netherlands), Dr. and Mrs. A. H. Spong (Cape Town), Dr. E. W. R. Steacie (Ottawa), Prof. D.C. Stockbarger (Cambridge, Mass.), Prof. I. N. Stranski (Berlin), Dr. C. E. Sunderlin (U.S. Navy, London), Ir. E. Sweep (Amsterdam), Miss M. G. Ter Horst (Leeuwarden, Netherlands), Mr. R. S. Titchen (Paris), Mr. H. P. J. Wijn (Eindhoven), Dr. J. Willems (Krefeld) and Dr. S. Zerfoss (Washington, D .C .) . 3Cottten ti3 PAGE General Introduction . General Introduction on Crystal Growth. By W. E. Garner . I. Theory of Crystal Growth Introductory Paper.-By N. F. Mott Forms of Equilibrium of Crystals. By I. N. Stranski . Surface Structures of Ionic Crystals. Crystal Growth and Surface Structure- . By K. MoliGre, W. Rathje and I. N. Stranski . Part I. Part 11. By W. K. Burton and N. Cabrera . By N. Cabrera and W. K. Burton . The Influence of Dislocations on Crystal Growth.. Kinetics of the Formation of Nuclei and Statistical Theory of General Principles of Crystal Growth. By Paul H. Egli and S. Zerfoss GENERAL Drscuss1oN.-Dr. F. C. Frank, Dr. S. P. F. Humphreys- Owen, Prof. I. N. Stranski, Dr. H. K. Hardy, Mr. W. K. Burton and Dr. N. Cabrera, Mr. R. S. Bradley, Dr. J. L. Amords, Dr. C. W. Bunn, Dr. U. R. Evans . By F. C. Frank Condensation. By R. Becker . 11. Nucleation and Normal Growth Introdzcctory Paper.-By W. J. Dunning Kinetics of Crystallization in Solution- Kinetics of Crystallization- The Precipitation of Silver Chloride from Aqueous Solutions- By C. W. Davies and A. L. Jones Nucleation and Growth in Sucrose Solutions. By Andrew Van . The Relative Rates of Growth of Strained and Unstrained Ammonium Crystal Growth from Solution- Layer Formation on Crystal Faces.H. Emmett . Crystals. By C. W. Bunn . . Part I. Part 11. Part I. Hook and Arthur J. Bruno Nitrate Crystals. By s. H. Bransom, W. J. Dunning and B. Millard. . By s. H. Bransom and W. J. Dunning. . . By S. Fordham . I. By C. W. Bunn and 11. Concentration Gradients and the Rates of Growth of 4 3 7 11 13 21 33 40 48 55 61 66 79 83 96 103 112 117 119 132CONTENTS 5 PAGE The Growth of Individual Faces of Cubic Sodium Chlorate Crystals The Linear Velocity of Polymorphic Transformations. By N. H. Boundary Migration and Grain Growth. By Walter C. McCrone . 158 Crystal Growth at High Temperatures. By S. Zerfoss, L. R. Johnson and P. H. Egli. . . 166 Some Aspects of the Growth of Quartz Crystals. By A. C. Swinnerton, G.E. Owen and J. F. Corwin . . 172 The Role of Diffusion Potentials in the Growth of Ionic Crystals. By A. R. Ubbelohde . . 180 from Aqueous Solution. By S. P. F. Humphreys-Owen . ' I44 Hartshorne . I49 GENERAL DIscussIoN.-Dr. M. H. R. J. Plusje, Dr. W. J. Dunning, Mr. W. K. Burton and Dr. N. Cabrera, Dr. R. F. Strickland- Constable, Dr. S. Fordham, Dr. F. C. Frank, Prof. W. E. Garner, Dr. K. G. Denbigh, Dr. W. A. Wooster, Dr. D. R. Hale, Prof. A. Juliard, Mr. A. E. Robinson, Mr. L. J. Griffin, Prof. I. N. Stranski, Dr. C. W. Bunn, Mr. Y. Haven, Mr. E. 0. Hall, Dr. H. K. Hardy, Mr. H. E. E. Powers, Dr. A. F. Wells, Prof. A. R. Ubbelohde, Sir John Lennard- Jones. . . 183 111. Abnormal and Modified Crystal Growth Introductory Paper.-By A. F. Wells . I97 Misfitting Monolayers and Oriented Overgrowth.By J. H. van der Oriented Arrangements of Thin Aluminium Films Formed on Ionic The Influence of Foreign Ions on Crystal Growth from Solution. By Oriented Overgrowths and Stabilization at Ordinary Temperatures of the Cubic (I), Tetragonal (11) and Orthorhombic (111) Phases of Ammonium Nitrate. By Raymond Hocart and Mlle. Agn& Habit Modification in Crystals as a Result of the Introduction of The Effect of Dyes on the Crystal Habits of Some Oxy-Salts. By The Effect of Crystal Habit Modification on the Setting of Inorganic Growth and Dissolution of Crystals Under Linear Pressure. By On the Disordering of Solids by Action of Fast Massive Particles. Merwe . . 201 Substrates. By T. N. Rhodin, Jr. . 215 B. Raistrick . - 234 Mat hieu-Sicaud .. 237 Impurities During Growth. By H. E. Buckley . 243 A. Butchart and J. Whetstone. . * 254 Oxy-Salts. By J. Whetstone . . 261 Carl W. Correns . . 267 By Frederick Seitz . . 2716 CONTENTS GENERAL DIscussIoN.-Dr. J. Willems, Mr. J. H. van der Merwe, Mr. P. Woodward, Mr. P. R. Rowland, Mr. H. E. E. Powers, Prof. A. Juliard, Dr. A. F. Wells, Dr. B. Raistrick, Dr. F. C. Frank, Dr. C. W. Bunn, Dr. S. Fordham, Dr. J. Whetstone, Prof. R. J. Hocart and Dr. J. C . Monier, Dr. F. M. Lea, Prof. C. W. Correns . IV. Mineral Synthesis and Technical Aspects PAGE Intyoductoyy Paper. By Dr. F. A. Bannister . The Production of Large Artificial Fluorite Crystals. Improved Crystallization of Lithium Fluoride of Optical Quality. The Growing of Crystals.. Growing Single Crystals from Solution. The Growth of Large Crystals of Ammonium Dihydrogen Phosphate Controlled Growth Inhibition in Large-Scale Crystal Growth. By Hydrothermal Synthesis of Minerals. By Jean Wyart . The Synthesis of Crystals Produced in Gaseous Media by Detonation of Explosive Mixtures. By A. Michel-Lkvy . Factors Governing the Growth of Crystalline Silicates. By R. M. Barrer . The Hydrothermal Crystallization of Vitreosil at Constant Temperature. By G. Van Praagh . The Hydrothermal Synthesis of Quartz. By L. A. Thomas, Nora Wooster and W. A. Wooster . Problems of Crystal Growth in Building Materials. By F. M. Lea and R. W. Nurse . The Growth of Periclase Crystals and Its Importance in Basic Refractories. By E. B. Colegrave, H. M. Richardson and G. R. Rigby . By Donald C. By St ockbarger . Donald C. Stockbarger . By A. C. Menzies and J. Skinner. By A. N. Holden . and Lithium Sulphate. By A. E. Robinson . Hans Jaffe and Bengt R. F. Kjellgren . GENERAL DIscussIoN.-Mr. Y. Haven, Dr. W. Ehrenberg and Mr. J. A. Franks, Mr. P. R. Rowland, Mr. T. A. Kletz, Mr. A. E. Robinson, Dr. E. W. Fell, Dr. B. Raistrick, Prof. R. M. Barrer, Prof. W. E. Garner, Dr. G. R. Rigby, Mr. R. W. Nurse, Dr. D. R. Hale . V. Concluding Remarks Dr. C. W. Bunn, Mr. P. R. Rowland . 283 291 294 299 306 312 31.5 319 323 325 326 338 341 345 352 358 364 Author Index . . 366

ISSN:0366-9033    

DOI:10.1039/DF9490500001

出版商:RSC    

年代:1949

数据来源: RSC

ISSN:0366-9033    

DOI:10.1039/DF94905BX003

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

GENERAL INTRODUCTION ON CRYSTAL GROWTH BY W. E. GARNER Received 3rst January, 1949 In the earlier investigations the two aspects of the growth of crystals, the initiation of crystallization and the rate of growth, were developed independently. It is now realized that each plane of atoms or molecules added to the crystal may involve a fresh initiation of crystallization, and that the rate of crystallization is dependent on the rate of nucleation on the crystal surface. This means that in the fundamental treatment of crystal growth, the two sections are inseparable, and this has been recognized in the grouping of papers for this Discussion. In this introduction, which is mainly historical, the gradual evolution of the present outlook is indicated. Interest in this field has been accentuated by important applications in industry and a brief survey of these applications is included.Initiation of Crystallization. Throughout the nineteenth century there n7as much interest in the crystallization of supersaturated solutions, for example, of solutions of Glauber’s salt, magnesium sulphate, vitriols, etc. Boisbaudron found that spontaneous crystallization took place only in strongly supersaturated solutions and de Coppet, by cooling solutions, determined the limits of solubility at which spontaneous crystallization begins. Ostwald developed the idea of a metastable zone on the solubility diagram showing the limits within which no crystal nuclei could form spontaneously. This theory proved to be of considerable practical importance at the time in explaining some of the phenomena of precipitation and of Liesegang rings. Much attention was paid to the limiting size of particle needed to start crystallization in the metastable zone, and rough estimates gave a minimum size of 10-~-10-~~ g.The thermodynamic criteria developed by Willard Gibbs in 1878 which were applicable to this problem were not very early appreciated, with the result that for a long period the approach to the subject was empirical in character. Tammann’s work on the initiation of crystallization in undercooled organic liquids and inorganic glasses was of the greatest significance and settled many doubtful points. By making counts of nuclei under controlled condi- tions, he showed that the formation of nuclei obeyed the laws of probability and that the maximum probability occurred a t temperatures 4oo-12o0 below the melting point, where the liquids begin to lose their mobility and show marked changes in viscosity.There was a zone of about zoo, below the melting point, where nuclei formation was very slow, which corresponded to the metastable zone found with supersaturated solutions. Tammann showed that nuclei could be formed in this zone if the observer would wait long enough for them. He thought, however, that there might be a metastable region a few tenths of a degree below the melting point, due to an increase in solubility resulting from a decrease of particle size. He also found that the rate of nuclei formation became very slow in the glassy state of under- cooled liquids, where the viscosity was very high.Tammann considered that since the formation of a nucleus was a very rare event, a large number of molecules must meet under limiting condi tions o 78 GENERAL INTRODUCTION ON CRYSTAL GROWTH velocities, orientation and direction of movement, before a nucleus can be formed. The process was so complicated that any simple relations between the probabilities and the stabilities of the forms produced were not to be expected. He concluded that Ostwald’s Law of Stages was not universally applicable. Willard Gibbs showed that a spherical particle of phase 11, p = p”, was in The equilibrium with a continuous phase I, $ = p’, when r = II equilibrium is, however, unstable, for if r is slightly reduced, the particle will decrease in size and finally disappear, and if it be slightly extended it will grow until phase I completely disappears.The work done in the creation of a partide of phase I1 in phase I is always positive up to the value of y = pfl - - p” so that phase I is stable with respect to nuclei formation so long as r is of such magnitude for the surface tension equation to apply. It will break down as r approaches molecular dimensions and 9’’ > p’. It would be expected, therefore, that for an undercooled liquid there would be a metastable region for phase I , where spontaneous nuclear formation could not occur, and a metastable limit below which the system became labile owing to r approaching molecular dimensions. Haber employed the Thomson equation , 2 0 P -$‘* 2cr T, - TI 2aM ~ -- Ts - r Q 4 in a theoretical examination of the crystallization of supercooled liquids.T, is the melting point, T, the melting point of a nucleus of radius r, G the interfacial energy, Qs the heat of crystallization, p the density of the solid phase, and M the molecular weight. He postulated a Spurenschmelxpunkt as the melting point of the smallest ordered aggregate, which determined the temperature of the metastable limit. These considerations of Gibbs and Haber will, however, be modified if there be taken into account the local fluctuations of energy which occur in any fluid and which have been demonstrated in the phenomena of critical opalescence. These local fluctuations will facilitate the formation of nuclei and render the metastable limit less sharp, although the conception of a metastable zone is still of some practical value.Rate of Growth. Tammann’s researches on the crystallization of super- cooled liquids show that the rate of crystallization is very slow down to about 30” below the melting point, increasing to a maximum which is often flat, and falling off as the viscosity increases to that of a glass. The maximum for the rate of crystallization lies at higher temperatures than for nucleation. The low values just below the melting point are due to the slow removal of heat of Crystallization. Tammann concludes that the rate is at its maximum when the temperature of the melt is where To is the melting point, qo the heat of crystallization, and cm the mean specific heat. Surface Flow. Studies of the growth of crystals from the gaseous phase indicate that the flow of molecules over the surfaces of the crystals plays an important role in the rate of crystallization.Volmer and Estermann showed that mercury crystals formed from the vapour consist of very thin flat plates, and that the rate of extension of the main faces can only be accounted for if the molecules colliding over the whole surface of the crystal are available for the growth of the very small areas at right-angles to the basic planes. This requires that the surface flow of a molecule during its = ‘0 - qo/c?nJW. E. GARNER 9 tifetime on the surface is of considerable magnitude. The work of Becker and of Taylor and Langmuir on adsorbed czsium on tungsten, and of Bosworth on potassium on tungsten, at temperatures where the evaporation of the adsorbed atoms is low, shows that the atoms undergo activated diffusion along the surface.For czsium the number of sites covered during the lifetime is at least xo8. Also, Newman has demonstrated that activated diffusion occurs on the surface of heated sodium chloride crystals. The experiments of Volmer and Adikari on the surface flow of benzophenone on glass and of Xowarski on $-toluidine over a crystal of the same substance illustrate the same principle. The extension of this principle to crystallization from supersaturated solutions and from undercooled melts is unavoidable, since in general the work required to move a molecule or ion along the surface is less than that to transfer it to the liquid phase. The Repeatable Step. The energies required to remove ions or mole- cules of sodium chloride from the surface of a crystal into the gaseous phase have been calculated by Kossel and Stranski for the corner, edge and various surface positions.Homopolar lattices have been dealt with similarly by the same authors and by Becker and Doring. The difference between the energies for the various sites is sufficiently great to have an important bearing on the kinetics of crystal growth. In building up a plane of atoms on the surface of a crystal, the greatest energy is liberated at the repeatable step of an uncompleted edge of a covered area. The energy evolved on adsorption on such sites is approx- imately the same as that resulting from embedding the atom half-way in the crystal. The process of crystallization on surfaces large compared with the atomic diameter consists mainly in the repetition of the ' repeatable step.' The adsorption of atoms singly on the plane surface is much less strong than at the repeatable step. Over part of the range of temperatures for which atoms are firmly held at the repeatable step, those on the main surface are readily desorbed.The surface molecules, however, travel by surface flow considerable distances before they evaporate, and therefore it is to be expected that in favourable circumstances the whole surface of the crystal will act as a collecting ground for the repeatable step. Two-dimensional Nuclei. The rate of evaporation is greatest if the adsorbed molecules are held singly on the surface and least when held at a repeatable step on a two-dimensional nucleus, the size of which is above a critical value.In the building-up of new crystal planes, the average time taken to complete a two-dimensional nucleus of this critical size may be considerably greater than that required to complete the plane of molecules by a succession of repeatable steps. Volmer, for iodine crystals growing from vapour, concludes that the formation of the two-dimensional nucleus is such a rare event that the probability of its occurrence determines the velocity of crystallization. Crystals grow the more regularly the lower the supersaturation. At high supersaturations polymolecular sheets are built up, giving a series of steps on the faces of crystals which can be detected by interference colours (Marcellin, Perrin, Kowarski) .These phenomena are of frequent occurrence and are of special interest. Stranski, studying the growth of polished spherical surfaces, shows that the planes with high indices of even simple lattices give uneven surfaces during growth, built up of steps of various heights. It should, however, be borne in mind that some of these phenomena may be due to the discontinuities caused by polishing. It is clear, however, that the mechanism of crystal growth, with complex molecules from strongly super- saturated solutions, can become an involved problem. Phenomena makeI0 GENERAL INTRODUCTION ON CRYSTAL GROWTH their appearance which have not been unambiguously elucidated. It is possible that some of these may be due to Smekal, Zwicky or other types of discontinuity, as suggested by Frank.However, under the simplest condi- tions, with low supersaturation, the conception of the formation of two- dimensional nuclei aided by surface flow may prove to be adequate for the calculation of rates of growth. Crystal-Crystal Interface. The nuclei formation in solid phases obeys similar temperature relationships to supercooled melts, giving maxima at temperatures considerably below the melting point. Volume changes on crystallization, producing cracks, are, however, an added complication. Nuclei formation in processes which are accompanied by gas evolution are one step more complicated, but the phenomena obey the same general rules. In a number of cases in which gas evolution occurs, the activation energy is approximately the same as the thermodynamic heat for the process, which implies a close fit between the lattices of the two phases and a very close coupling between the disappearance of the old and the building-up of the new lattice.This may well be the case, in favourable circumstances, for the growth of one crystal phase out of another. The need for large crystals free from flaws for spectroscopy, piezoelectric measurements and the various purposes of the electrical industry cannot be met from the diminishing natural resources, nor do these give a sufficient variety. This has led to researches on the methods of accurate control of crystallization from the vapour phase, the melt, from supersaturated solutions and by hydrothermal processes at high pressures simulating those in nature.In the natural processes whereby crystals are formed in the earth’s crust, an infinitude of time is available for the manufacture, but on the industrial scale the time available makes it necessary to work at higher supersaturations, where irregularities are the more likely to occur in t h e crystallization processes. The control of crystal shape and size by the addition of surface active substances is a requirement in many industries. In the explosives industry particles with as nearly spherical shape as practicable are advantageous from the point of view of flow properties, bulk density, pelleting properties, etc. It is also possible in cases where two solid modifications are produced to prevent the formation of the unstable modification by the use of suitable additaments. The control of particle size distribution is also important in the manufacture of materials used as the basis of products with good plasticity.The tendency of hygroscopic substances to cake can often be reduced by paying attention to crystal shape, choosing that shape which gives a minimum of contacts between the grains. The surface agents may operate by adsorption on one set of faces, either reducing or preventing growth, as is found by the use of certain dyestuffs. These agents may operate by retarding all growth except in one direction, thereby giving spherulitic growths. The detailed mechanism by which they act is not yet elucidated, although it can readily be seen from current ideas on crystal growth that the effects of adsorption at the repeatable step would have important consequences. There are many processes in which crystallization is the final stage, giving the product its essential properties. Such are the manufacture of cements, bricks, ceramics, etc. Although in these cases the crystallization process is often accompanied by chemical change, the mechanism involves the nucleation by crystals and the growth of crystals such as occurs for the simpler processes, and their study will benefit by the development of the fundamental theory of cryst a1 growth. Practical Applications. The University, Bristol.

ISSN:0366-9033    

DOI:10.1039/DF9490500007

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

I. THEORY OF CRYSTAL GROWTH Introductory Paper BY N. F. MOTT Received 7th March, 1949 The theory of crystal growth can, it seems to me, conveniently be divided into three parts. These are : (a) The theory of the rate of growth of a surface in contact with a vapour or solution with a given degree of supersaturation. Or, in the case of a crystal growing from the melt, the theory of the rate of growth for a given degree of supercooling. This will include a discussion of the rates of growth of different crystal faces, and the effect on growth rates of impurities which may be adsorbed on the surface, and of imperfections in the crystals them- selves. The solution of the problems under this heading depends, of course, on a knowledge of interatomic forces. (b) The use of results obtained under the heading (a) to determine crystal forms in as far as they depend in the case of growth from solution, or diffusion of the ions or atoms to be deposited, or in the case of growth from the melt on conduction through the material of the heat liberated.Much of the theory of dendrite formation is included in this category. I t forms a part of classical rather than atomic physics, depending as it does on the equations of diffusion and heat flow. (c) Discussions of the crystal form of the deposit. This will include such problems as the formation during growth of screw or edge dislocations in the crystal; a solution of these problems is very important for the theory of mechanical strength. Then there is the question of the possible pseudo- morphic forms of crystalline films grown on a substrate of different composi- tion ; a contribution to this subject is made by van der Menve in a paper to be presented to this conference.And, finally, there is the question of the state of strain and possible cracking of the surface layer treated by MoliGre, Rathje and Stranski. ( a ) Atomic Theory of Growth. The elements of a theory of crystal growth have been laid down by Volmer, Stranski, Becker and Doring, and new contributions made by Frank, Burton and Cabrera (for references, see the contribution of F. C. Frank to this Discussion). This theory applies explicitly to growth from the vapour; but can probably be applied in principle to growth from solution. The problem of growth from the melt remains an open question. The elements of the theory of growth are as follows: consider a flat crystalline surface of low indices (say, (roo) for a simple cubic or (111) for a close-packed structure) in contact with a vapour.Suppose this surface is partly covered by another layer. Then if the pressure of the vapour is raised by a small amount Ap above the equilibrium vapour pressure, theory indicates that the layer will grow, with a speed proportional to A$, until it covers the surface. But in order to start a new layer, a two-dimensional nucleus must be formed, and, like other nucleation phenomena, the rate of nucleation varies with Ap as e-A/*p, where A is a constant at given temperature. It follows that when A@ is below some critical value the rate is negligibly small. It seems likely that the growth rate depends in general on the rate of nucleation, at any rate for surfaces of low indices; for surfaces of high indices, having a step-like formation anyhow, nucleation is much easier.11I 2 THEORY OF CRYSTAL GROWTH But such surfaces of high indices will, of course, by growing quickly tend to disappear, leaving a crystal surrounded by planes of low index only. It should be emphasized that a flat surface in contact with vapour will have a number of atoms adsorbed on it. Two-dimensional nucleation can occur whether or not these are mobile over the surface ; it is not at present quite certain whether their mobility affects the rate of nucleation. Among the papers presented to this Discussion, Becker gives a valuable account of the relation of his theory to Mayer's theory of condensation.Burton and Cabrera, in a paper to be published elsewhere, have made some refinements to the present theory by calculating the shape of the two- dimensional nucleus when it has reached the size beyond which it will normally spread. This puts the theory on a firmer footing, and does not alter the numerical values very much. Frank points out that the theory suggests a growth rate which is negligibly small unless the supersaturation of the vapour is of the order 1-5, and that this is contrary to experiment, in particular to the results of Volmer and Schultze on the growth of iodine crystals; the degree of supersaturation required is of the order 1-01. He suggests that the presence of dislocations is essential for growth at these concentra- tions, and that the growth rate depends essentially on the density of dis- locations in the material. Theory has at present made little contribution to our knowledge of habit modification.It does, however, follow that, if dislocations are essential for crystal growth, very small concentrations of impurity, which could be adsorbed preferentially at the " ledge '' where the dislocation meets the surface, could profoundly affect growth rates and thus lead to habit modifi- cat ion. ( b ) Phenomena Depending on Heat Flow and Diffusion. It is believed that dendrite formation in the solidification of liquid metals is due to the fact that a thin needle, growing into a supercooled solution, will need to get rid of less heat by conduction than a thicker one and so will grow faster.In the same way, in the formation of crystals from solution, a thin needle will grow more quickly than a thick one into supersaturated solution. Probably the clue to the step formation observed by Bunn will be found along these lines. (c) Physical State of the Crystal as a Consequence of the Mechanism of Growth. Frank, in his paper, gives some reasons for believing that, at finite growth rates, dislocations will be formed in the crystal. They are in no sense present in thermodynamic equilibrium and ideally a long enough anneal would get rid of them ; but, in practice, there appear always to remain a certain number. Stranski and his colleagues reopen the very interesting question of the state of strain of the surface layer. The origin of the " Griffith cracks," responsible for the low stress for fracture of brittle materials, has never been explained, and it is possible that this work will provide a clue.In a later section of the Discussion, van der Menve discusses the crystal structure of thin films deposited on a substrate of differing crystal structure. He shows that the question, whether or not the deposit has a pseudomorphic form, depends on whether the first monolayer conforms to the structure of the substrate or not; and that this in turn depends on the degree of misfit . Equilibrium Crystal Forms The study of the shape of a crystal in equilibrium with a vapour forms an interesting field rather apart from the theory of crystal growth. Burton and Cabrera have found that the equilibrium forrn of the two-dimensional crystalline nucleus on a flat substrate is a rounded polygon, if only oneN. F. MOTT I3 atomic or molecular unit is involved. For ionic forces, on the other hand, it appears that the two-dimensional nucleus may have sharp corners. In the case of three dimensions Stranski has shown that the corners of a crystal are rounded off through the presence of afinite number of planes of higher index, and so are not truly rounded. The microstructure of the surface in equilibrium with vapour or solution is also of interest. As already stated, a flat surface will always contain some adsorbed atoms, and there will always be some vacant lattice points. Burton and Cabrera have made an investigation of the concentration of " Frenkel terraces " on a surface in equilibrium. For faces of low index, there will be practically none for a perfect crystal; any which exist depend on the presence of dislocations. A crystal temperature exists, however, at which they form, but this will in general be above the melting point. H. H . Wills Physical Laboratories, Royal Fort, Bristol 8.

ISSN:0366-9033    

DOI:10.1039/DF9490500011

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

N. F. MOTT FORMS OF EQUILIBRIUM OF CRYSTALS BY I. N. STRANSKI Received 17th February, 1949 A knowledge of the forms of equilibrium of crystals is important for an understanding of the processes on crystal surfaces, independent of whether the crystal is immediately concerned in these, or merely functions cata- lytically. Furthermore, a comparison between theoretically and experi- mentally deduced forms of equilibrium makes it possible to check the assumptions used in the former, and supplies valuable clues to alteration in structure and changes within the individual lattice surfaces, The following observation should first be made. The theoretical treatment falls into two parts. First the underlying ideas must be developed, then the mathematical work can be started. This is directed by the knowledge at the time of the force functions, and must of necessity bring new problems in its train.In the following I will confine myself to the part dealing with the underlying ideas. The treatment of the forms of equilibrium of crystals has been developed on the basis of two fundamentally different ideas. The older one, historically, made use of an analogy to liquid surfaces. The surface tension here was replaced by the idea of the specific surface energy 0. The values of cr for crystals are dependent upon direction, so that in general the form of equilibrium is a polyhedron which must satisfy Gibbs’ condition : h i . Fi = minimum, at constant volume. If one ignores the edges and corners, it is known that here, also, one arrives at the same relation as for vapour pressure, which is completely analogous t o Thomson’s equation and has the following form : 1 The following recent papers on the Thomson-Gibbs relation are mentioned : Volmer, Kinetik der Phasenbildung (Dresden and Leipzig, 1g3g), p.87 et seq. v. Laue, 2. Krist., 1943, 105, 124. Stranski, 2. Krist., 1943, 105, 91. Honigmann, Moliere and Stranski, Ann. Physik, 1947, I, 181.I4 FORMS OF EQUILIBRIUM OF CRYSTALS vo represents the volume of a crystal unit and ra the centre distance, ci the specific surface energy of the i-th face. $, and $, are the sublimation pressures of the finite- and infmite-sized crystals respectively. Wulff’s method for the construction of equilibrium forms of crystals follows directly from eqn. (I). An exact relation, capable of general application, cannot be derived in this way.For if we wish to take into account the fact that the crystal also possesses edges and corners, and that the specific surface energy and the specific energy of the edges and comers which must further be introduced, ( 5 ) 1 possibly depend on the size of the crystal as 7 well, a relation can only be derived at first FIG. div division of a crystal by for simplified models. This is to be shown in three planes* Two Of these the following for the case where the form are shown as lines where they cut the plane of the of equilibrium of the crystal is represented diagram, the third lies in the by a simple crystalline form, i.e., it is sur- diagram plane itself. The rounded by only one kind of face. For figures in brackets denote this purpose let us refer to the definition of the sections below the plane of the diagram, those with- the specific surface energy, and give the out brackets the sections definition of the specific edge and corner above it.energies in reference to Born and Stern2 The specific edge energy x is defined as the work which must be done in order to separate the crystal sections I and 3, z and 4 respectively (see Fig. I), divided by twice the length of the edge, and given a negative sign. Correspondingly, the specific corner energy E is half the work required to separate two crystal sections situated diagonally in space, with their corners touching, e g . , I from 7, or 3 from 5 (see Fig. I). Assuming that these values are independent of the dimensions of the crystal, one obtains in place of eqn.(I) : Thus, as a result of the existence of edges, an additional term appears as correction. The corners are without influence. In order to be able to discuss the dependence of the values 0, x and E upon the size of the crystal at all, the definitions of these values for finite crystals had first to be found. The definitions given by me at that time will be explained for a simple case with the aid of Fig. 2. If the form of equilibrium is represented by a cube, cU is equal t o the work of separating such a small crystal from a cube face of the infinite crystal, divided by twice the area of one cube face of the small crystal. xu is correspondingly equal to the work of separating such a cube from the infinite crystal quadrant lying diagonally opposite divided by twice the length of a single edge and FIG.2.-T0 define the values cro, xa and for a finite crystal cube with an edge-length a. 2 Born and Stern, Ber. Berlin Akad., 1919, 48, 91 ; Stranski, 2. Krist., 1943, 105, 287. Stranski, Ber. Wien. Akad.. math.-natzrrwi. Kl., 1936, IIb, 145, 840 ; Mh. Chem., 1936, 69, 234.I. N. STRANSKI with a negative sign. Lastly the corner energy E, is equal to half the work of separating a small crystal from the infinite crystal octant lying diagonally opposite in space. Thus the total surface energy of a small crystal with edges of length a is @, = 6a20, + 12 a xa + Sc,. - (3) In this case it is possible to obtain the form of equilibrium of a small crystal simply by taking an infinite crystal to pieces, and CD, can also be defined as the work of separating a small crystal from a crystalline half- crystal position (see Fig.3 ) . However, it should be mentioned that is generally given in the following relation : Nu @u = Na - ~ p 1 / ~ - t ~ p ” . * (4) v = 1 pin is the work of separating a crystal unit from 3.-crysta11ine half- crystal position. @, the half-crystal position (see below). The second is equal to the work term is the work obtained in building up the small of separation of the crystal from its Nu individual crystal units. crystal cube with The example dealt with in the last section is edge-length a, in the especially simple. The important thing is, that as shown positions in the this case already shows that it is not possible to diagram.specify the exact sublimation pressure of a small crystal from the forms of equilibrium, with the aid of the values o,, xu and E ~ , now assumed to be variable. For this purpose, the differentiation of the eqn. (3) is necessary : P a da do, da dx, dE, = 12 c, a - + 6n2 ~ + IZ xu-+ 12 a -+ 8 - P m dN dN dN dN dN dN (5) kTln - = The values mu, o,, Xa and E, would thus have to occur as continuous functions of the number of crystal units N . That is not the case, however, for they present themselves as a series of isolated points. The following possibilities can be discussed. (I) Curves are drawn through these points and differentiated. The result could give the sublimation pressure with sufficient exactitude. (2) The dependence of the values o,, xu and &a upon N can be found to be so small that it can be neglected.Neither possibility, however, can be proved for no standard of comparison exists at present, which gives us the correct pressure values. We will return to these questions below. The advantages of the method using the values G, x and E are not to be denied, for by means of it, all considerations which had been made on liquid systems could be applied in a comparatively simple way, and with little alteration, to crystal systems. Special attention is here drawn to the fact that, on the whole, Volmer’s theory on the frequency of nucleus formation also reproduces the conditions correctly for crystal systems. By continuing the nucleus idea, introducing, namely, the idea of a two-dimensional nucleus, the growth of a crystal could be submitted for the first time to a mathematical method.Many different questions could be answered comparatively simply. The interpretation of Ostwald’s step-rule may be mentioned as an example.5 The values o, x and E do not refer at all to elementary stages of growth and reduction, and the relations which are obtained with their aid can only be applied under certain conditions to kinetic considerations on crystals, and remain difficult to visualize. As is known, the application of Thomson-Gibbs’ But the disadvantage of this method must also be enumerated. Volmer and Weber, 2. physik. Chem., 1926, 1x9, 277 ; Volmer, loc. cit. Stranski and Totomanow, 2. physik. Cham. A , 1933, 163, 399.16 FORMS OF EQUILIBRIUM OF CRYSTALS equation has led to numerous, and often crass, misunderstandings.This method, by simulation of completion, has also prevented many equilibrium questions from being asked and answered a t the right time. The second treatment began to take form as a result of work by Kossel on the one hand, and myself on the other.' The work of separating individual crystal units from the crystal surface was estimated, NaCl being taken as the first example, and with the help of this it was possible to draw a picture of the molecular processes connected with growth and solution. The logical starting point for these considerations is the determination of the length of time a crystal unit remains in the so-called half-crystal position. The crystal unit in the half-crystal position possesses a work of separation which amounts to half of that of a crystal unit in the inside of the crystal.It is thus equal to the negative value of the lattice energy per crystal unit, and determines the vapour pressure of the infinite-sized crystal. Elementary reasons can be given for this conclusion if the position of growth of a repeatable growing crystal face is chosen as model of the half-crystal position. For, in this case, the position as such is retained after any number of separa- tions or addition of crystal units. Thus the crystal would only be in equili- brium with its surroundings, if the probability of a separation of a crystal unit from the half-crystal position is found to be equal to the probability of an addition on this. With the help of the different works of separation, it has already been possible to draw a series of conclusions which, at that time, were justifiably regarded as completely new-fangled.Only as a consequence of these was it asked whether certain faces in equilibrium can be retained as such, or whether their surface structures would have to undergo alterations of a coarsening nature. It should be emphasized here that these questions could have been asked earlier, as a result of the determination of the values, or merely the signs, of the specific peripheral energies of the lattice surfaces concerned. That they have not been asked up to this time is to be explained exclusively by the fact that the older theories were difficult to visualize. Because of its importance the criterion might be given here by reason of which one can decide whether a certain face appears in the equilibrium form of the infinite-sized crystal or can remain as crystal face.If the specific peripheral energy of this lattice face shows the value zero or a negative value, in one direction only, this face cannot appear as a form of equilibrium. Should this condition be fulfilled for one direction only, the face concerned will grow over one-dimensional nuclei and show a typical chain formation. A chain formation alone, on the other hand, is not sufficient argument against the face belonging to the equilibrium form. If this condition is fulfilled for two directions, the one-dimensional nucleus formation is also eliminated. An example of the first case is (011) on the NaCl crystal and of the second, (111) on the same crystal.Another question could also be answered with the help of the work of separation, namely, with what kind of face must the infinite-sized crystal be surrounded ? For it is evident that the only possible form of equilibrium is one in which all comer crystal units are bound a t least as f i r m l j 7 as in the half-crystal position. So that by starting with a simple form, and systema- tically removing all crystal units which are less firmly bound, one could arrive a t forms which no longer exhibit such crystal units, and which then mirror the equilibrium form, in that they possess all the faces of same. In order to arrive at an expression which represents the sublimation Kossel, NacR. Ges. Wiss. Giittingen, 1927, 135 ; Leipziger Vortrlige, 1928, I.Stranski, 2. physik. Chem., 1928, 136, 259.17 I. N. STRANSKI pressure of finite crystals, those crystal units will be taken into consideration which, on evaporating, produce a deviating value for the work of separation. In the case of a single crystal face, that is a very simple matter.8 The mean value Fa appears here in place of = p/, (" work " of separation of the crystal unit in the half-crystal position), where the mean is taken so as to include all crystal units of the uppermost lattice face, and for a process also, in which the crystal units of the lattice face are removed. The logarithm of the relation between this vapour pressure and that of an infinitely extended lattice face is then simply This simple result can be explained as follows : for the faces concerned in equilibrium, the probability that the uppermost lattice face is removed by solution must be equal to the probability that, after removal, it is re- formed by means of a condensation process.But this stipulation is connected with the fact that the work of formation of a lattice face nucleus (two- dimensional nucleus) by condensation is exactly equal to that by superficial solution of an uppermost lattice face. Let us imagine a position where a crystal unit is bound in such a way that the work of separation has the exact value (F;.) required for the position to be occupied by a crystal unit for not more than exactly one-half of a very long observation period. We could then undertake the formation of a lattice face nucleus by condensation, by allowing the crystal units to attach them- selves first at this point from the vapour phase, and forming the nucleus by bringing them each time from there on to the face.If the nucleus contains m crystal units and the whole lattice face n, the following work is necessary for the production of the nucleus : m t (Fa - Cpi) . I By superficial solution of an existing lattice face, on the other hand, the work of formation of the nucleus amounts to n mtr E ((pi - Cpu) * By ba1ancing:the two work equations, one obtains directly The following should also be taken into account. The conditions of equilibrium deduced quite generally apply to both lattice faces and single lattice rows. In the case of the lattice face nuclei, the peripheral rows must be in equilibrium with the surroundings, i.e., the mean work of separation per peripheral row of the lattice face nucleus must show the same value pa on all sides.If we now consider a three-dimensional crystal which is in equilibrium with its surroundings, this implies that the same conditions must be fulfilled for each of its faces. From which it further follows that the surroundings are supersaturated as regards all rows on the edges of the crystal (for the uppermost lattice face of an equilibrium form is greater than the corre- sponding lattice nucleus) ; in the same way, the surroundings are also super- saturated as regards each single point on the surface of the crystal, andr8 FORMS OF EQUILIBRIUM OF CRYSTALS therefore a150 as regards the corner crystal units.This conclusion is instruc- tive. But it is also fundamental for the consideration of the equilibriom of a crystal. It leads directly to an easy method of construction of the form of equilibrium of crystals. In order to obtain the form which corre- sponds to a certain pressure, in the vapour phase, one proceeds as follows : the value of (pa corresponding to the pressure PI is calculated. Then starting from any simple form of the crystal, all crystal units which show a work of separation smaller than (9. are eliminated, one after the other, from its surface. Lastly, the areas of all faces are varied until each single mean work of separation reaches the value Cp,. Another conclusion from the thermodynamic deduction of the sublimation pressure of a small crystal is made especially clear.That is theconclusion which can be drawn directly from eqn. (2) : the vapour pressure is simply a function of the relation between c and the centre distance of any face. Provided the latter remains the same this quotient must remain unchanged independent of whether the face concerned occurs in a simple form or in a combination. When drawn from the thermodynamic deduction, this conclusion is not clear, as the deduction includes only the form of equilibrium itself, and is tied to the assumption that for small evaporation and growth processes the form remains similar. The following explanation, based on the mean work of separation, can be given for this conclusion. To this end, let us begin with a simple form and study a definite face. This form is now allowed to develop into a combination, the centre distance of the face under consideration remaining unchanged.The area of the face decreases but the deviations of the individual works of separation also decrease to the same degree, for the rows on the edges of the Combination border upon more lattice neighbours than the rows on the edges of the simple form. Lastly, eqn. (6) provides the possibility of deciding the question which cropped up on a previous page. It supplies the vapour pressure in a manner which is quite independent of that in eqn. (I) or (5). It is also possible, in this manner, to carry out the calculation for a definite example, namely, for a sim#,?ified NaCl crysta1.l I t showed, though only for this case, that the second possibility is realized, namely, that it is not necessary to include the dependence of the specific energy values 0, x and E on the number of crystal units, in the calculation, for all crystal sizes which actually come into question.It is comparatively easy to obtain the form of equilibrium theoretically for typical ionic crystals, if simplifying assumptions are made. In all cases dealt with up to now, it has been found to be a simple form. It is a cube for NaCl ', a rhombic dodecahedron for CsCl 9, an octahedron for CaF, 10, a rhombohedron for CaCO, or NaNO,. It is also independent of the size of the small crystal. Thus, form of equili- brium and form of growth are here identical (for low supersaturations). In this case the greater the range of the forces between the crystal units, and the nearer Cp.approaches CpW, the greater the number of faces appearing in the form of equilibrium. Table I gives a list of (infinitely great) forms of equilibrium for a few simple lattices as functions of the said range and calculated under the assump- tion that the work of separating one crystal unit from another is always The conditions in the case of non-polar crystals are different. 9 Kleber, Zbl. Miner., Geol., Paliiont. A , 1938, 363. lo Bradistilov and Stranski, 2. Krist., 1940, 103. I.FIG. 4.-Cd mono-crystal, formed in the fused liquid and allowed to grow further in the vapour. The small circular face a t the bottom left-hand corner is { I IZO), the irregular coarsened face above it {I 12 I}. (Eisenloeffel.)l5 To jute page 191I.N. STRANSKI positive and only dependent upon the distance between them.ll The majority of crystals with simple lattices would seem to represent transition stages between the polar and non-polar type. The metals constitute a special class. I t is worthy of note that the experimental data for metals l2 l3 also agree well, on the whole, with the results in the Table, inasmuch as they give the correct order of the faces. It was possible to make a more accurate experimental investigation especially in the case of Zn l4 and recently also for Cd1415, and these results were confirmed. In both cases the experi- ments on growth, carried out accurately on spherical rudimentary forms consisting of one crystal, which grow from supersaturated vapour without TABLE I FORMS OF EQUILIBRIUM The range of the forces between the lattice crystal units embraces Simple cube .. B o d y - c e n t r e d , cubic Face-centred cubic Diamond lattice. . Hexagonal closely packed spheres Adjacent crystals units only 00 I 0 1 I 111, 0 0 1 111, 001 OOOI, roir I oio Next crystal unit but one, also 001, 011, I 1 1 011. 001 111, 001, 011 111, 001, 011 0005, 1oi1 Ioio, II%O roiz Next crystal unit but two, also 001, 011, I11 I I2 011, 001, I12 I11 111, 001, 0 1 1 113, 0122 I35 111, 001, 011 113 0001, IoiI Ioio, IIVO r o i z Investi- gated on W, urotro- Al, Ag, Pt ~~ pine __ diamond Be, Mg, Zn, Cd __ The most far-reaching effect was found in every case to embrace the next crystal unit but two. The underlined examples have been investigated more thoroughly.any signs of coarsening, gave the faces : (OOOI), (IO~I), (IOTO) ; (IIZO), (10E2). It is also very significant that the faces (1120) and ( I O ~ Z ) ~ which are to be traced back to the influence of nearest neighbours but one, exhibit a con- siderably smaller area in the case of Cd, than in the case of Zn (see Fig. 4). This is probably connected with the greater screening capacity of the Cd atoms in the crystal lattice. It is also very noteworthy that W l6 l3 and urotropine l7 l8 which both have the same lattice (cubic body-centred) but belong otherwise to quite different valency types, exhibit exactly the same equilibrium form faces : { 0111 ; (001) ; { 112). In both cases, of the two faces which are to be traced back to the effect of nearest neighbours but two ((IIZ) and (III}), only { IIZ } appears.Concerning further properties of the urotropine crystal, 11 Stranski, 2. fihysik. Chem. B, 1931, 11, 342 ; Ber., 1939, 72, 141 ; Stranski and l2 Straumanis, 2. Physih. Cltem. B, 1931, 13, 317 ; 1932, 19, 64 ; 1934, 26, 246. 1s Stranski and Suhrmann, Ann. Physik, 1947, I, 153. 1 4 Kaischew, Keremidtschiew and Stranski, 2. Metullkunde, 1942, 34, 201. l6 Eisenloeff el, Dissertatim (Techn. Universitat Berlin-Charlottenburg, 1948). l6 Muller, 2. Physik, 1937. 106, 541 ; 1938, 108,668 ; 1943, 120, 270. 17 Kaischew, Jahyb. Univ. Sojia, fihys. math. Fak., XLIII, 1946147, 2, gg. 1 8 Stranski and Honigmann, Nuturwiss., 1948, 35, 156. Kaischew, 2. Krist., 1931,78, 373 ; Ann. Physik, 1935, 23, 330.20 FORMS OF EQUILIBRIUM OF CRYSTALS whose lattice can be considered approximately as homopolar with snperim- posed dipolar forces, see later.A brief study of the relation between form of equilibrium and form of growth (more exactly, final growth form) will be inserted here. The crystals which are investigated are, almost without exception, the product of a growth process. If the form of equilibrium is not a simple crystalline form, the resulting growth form contains only the slow-growing faces large enough to be visible ; the quick-growing faces remain the same size as the same faces in the form of equilibrium, which in general is sub- microscopic. The form of equilibrium here is to be ascribed to the pressure prevailing during the process of growth. Intermediate stages of growth of rounded single crystal forms provide the possibility of making all equilibrium faces visible l1 l4 l5 I t is, unfor- tunately, always possible that in the course of this faces also appear which do not belong to the equilibrium form.The appearance of { 012 ) and { 111 ) in Spangenberg’s l9 and Neuhaus’s 2o experiments on growth with spherical, polished NaCl crystals from aqueous solution may be recalled, for example, although the only form of equilibrium here is { OOI ). It is therefore of great importance to develop a thoroughly reliable method for the experimental production of equilibrium forms. This was achieved for the first time for urotropine, following on observations by Kaischew,l’ Honigmann l* and myself. At low temperatures, at which the transfer of matter takes place almost entirely via the adsorption layer, the growth form, which in the case of urotropine is a rhombododecahedron, re-forms the faces (001) and (112) (see Fig.5). Specially accurate investigations were carried out a t 0°C. (If one subjects the crystal to small fluctuations in temperature the same form appears much more quickly.) In answer to the question why, up to now, one had neither observed a spontaneous appearance of equilibrium forms of visible size (microscopic) ~ nor considered this possible theoretically, one can say that the relative differences in the vapour pressures of faces of an already visible crystal, which are not in proper ratio to one another, are in fact very small. In spite of this, it is not so much the smallness of the differences of the relative vapour pressure which is responsible for retarding the course of the reaction as Volmer’s work of formation of the two-dimensional nucleus connected with the supersaturation.This must appear in the formation of new lattice faces, and as the supersaturation disappears, converges towards infinity. If therefore one succeeds in removing the energy threshold of the work of formation of the two-dimensional nucleus, the process of alteration leading to the equilibrium form on a crystal of the growth form should be possible. It is possible to remove this energy threshold, or to lower it considerably, by the construction of hollow edges starting from which single lattice faces can develop. Only the few crystals whose crystal units show a comparatively high mobility within the adsorption layer at low temperatures will qualify for this.The discrepancy between theory and experiment, already mentioned, evinces itself with urotropine, by the appearance of (112) of the faces referred to the nearest neighbour but two, but not of {XXI). As is to be set forth elsewhere by Honigmann and myself, the experimental result can be explained by the fact that a profound alteration in lattice takes place in the uppermost lattice face of { 112). This is probably a Iattice alteration which is also stable a t a low temperature. With urotropine another phenomenon can be studied. Certain alterations 1s Spangenberg, N . Jahrb. Miner., Miigge-Festbd. A , 1928, 57, I 197. 20Neuhaus, 2. Krist., 1928, 68, 15.( a ) Growth form ( 0 1 1 ) .( 3 ) The form which is formed on tempering (011, 001, 112). (Honigmann.) FIG. 5.-Urotropine crystals. [To jure Page 20I. N. STRANSKI 21 i n the surface lattice do not appear until the temperature is high, i.e., altera- tions also exist which show the character of two- and three-dimensional changes. Above 170' C the mechanism of growth and evaporation, as well as the form of equilibrium of the urotropine crystal, changes fundamenta1ly.l8 The growth and reduction take place now in multimolecular (visible) layers, whose border is completely rounded ; the form of equilibrium is a rhombic dodecahedron whose corners and edges are also rounded. This phenomenon is obviously connected with the fact that new degrees of freedom (rotations) of crystal units of certain lattice surfaces, edges or peripheral rows are aroused by temperatures considerably lower than those in the inside of the crystal. In closing, the question may further be asked, how the equilibrium form of a crystal changes when it is surrounded by a liquid instead of its own diluted vapour. The simplest case would be to suspend the small crystals in their own fused liquid. The specific interface energy of a certain face ~ h k l would be given here by the following relation : where 1c and 2chkJ are the corresponding values for the liquid and the crystal relative to vacuum, and 1 p h k i the work which would be obtained by the contact of unit areas of crystal and liquid. It is seen that dhkl is not only very small if p h k l is very small, but also when 12Ghkl is especially large. The latter is all the more likely to be true, the more continuous the transition from crystal via the interface to liquid. The growth form of Cd which is produced from the fused liquid1415 may be quoted here as an example. It is seen that the face (IIZI} appears here, which as a rule is coarsened on continuing to grow in vapour, as it does not belong to the equilibrium form of the crystal surrounded by vapour phase (see Fig. 4). The general case of an equilibrium form surrounded by a phase of any desired composition has not yet been accurately treated, either experi- mentally or theoretically. Up to the present a certain amount of attention has only been paid to the occasional growth forms showing great deviations, which precipitate from various solutions. Gh/d = lcT + zghkl - 1 2 ~ h k l 1 - (8) Institut fur Physikalische Chemie wad Elektrochemie, Berlin-Charlottenbuurg 2, Hardenbergstrasse 34, Germany.

ISSN:0366-9033    

DOI:10.1039/DF9490500013

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

I. N. STRANSKI 21 SURFACE STRUCTURES OF IONIC CRYSTALS BY K. MOLI~RE, W. RATHJE AND I. N. STRANSKI Received 16th February, 1949 There is no doubt that the structural peculiarities of crystal surfaces play a role in many physical and chemical processes in which the surfaces of solid bodies participate. Up to the present no experimental method exists which is dependable enough for the determination of surface structures. The only way to decide how the positions of equilibrium of the atoms in the surfaces can differ from those of the infinitely extended space lattice is to employ theoretical considerations with the use of simple models.22 SURFACE STRUCTURES OF IONIC CRYSTALS In the following report is an account of calculations on the model of the rocksalt lattice, the aim of which was to determine the surface structures caused by the effect of the ionic polarizability.Our work* follows on an investigation by Lennard- Jones and Dent 1 published twenty years ago, but goes further in that not only the displacement of the ions in the direction of the surface normals are taken into consideration, but also the tangential distortion of the surface lattice. As in the case of Lennard-Jones, besides the electrostatic forces, only forces of repulsion of small range according to Born’s power function are taken into account; here also a simplified surface structure is assumed where only the ions of the uppermost lattice face undergo deformation and displacement, while the rest of the lattice below it remains undeformed. Only average ionic properties of the two components are taken into account (except for the signs of the charges), i.e., the potentials of repulsion and the polarizabilities for both kinds of ions are regarded as equal.13 c FIG. I. The idea directing our calculation was the consideration that the polariza- bility of the ions can have the effect of lowering the symmetry. This idea was applied by Born and Heisenberg2 and by Hund3 to ionic molecules, and by the latter to three-dimensional ionic lattices. If one imagines the polarizability of these to be continuously increased, the co-ordination lattices, which at first are stable, change at a certain polarizability into less sym- metrical structures, namely, to layer- or molecule-lattices. These polariza- tion sub-structures” are distinguished by the fact that the ions prefer those positions where the electrical field has the highest possible intensity, in other words, where the gain in polarization energy is as great as possible.From the fact that an isolated (001)-lattice face in equilibrium must have a lower lattice constant compared with a compact space lattice (2.68 A for NaC1, calculated using the function of repulsion which is obtained from the lattice constant 2-81 A and the compressibility of the space lattice) Lennard- Jones and Dent l came to the conclusion in their above-mentioned investi- gation that a tendency to contract exists in the surface, which they treated as analogous to the surface tension of a liquid. The question of how such a contraction could be brought about was not followed up by Lennard- Jones and Dent ; above all, the part played by the ionic polarizability in a tangential deformation of the surface was not considered.We will show that no tendency towards tangential contraction exists in the case of low polariza- bility in the surface. Only a decrease in the distance between the two uppermost lattice faces is to be expected, similar to that already calculated by Lennard-Jones and Dent. With more highly polarizable ions, on the 1 Lennard-Jones and Dent, Proc. Roy. Soc. A , 1928, 121, 247 ; see also Madelung, Physik. Z., 1919, 20, 494 ; Zwicky, Helv. physic. Acta, 1930, 3, 269 ; Stranski, J b . Univ. Sofia, 1927-28, 24, 297 ; Z . physik. Chem., 1928, 136, 259. Hund, 2. Physik, 1925, 34, 833. 2 Born and Heisenberg, Z.Physik, 1924, 23, 388. * An account of our calculations which have been supplemented in the meantime by further results has already been published in 2. Physik, 1948, 124, 421 and 429.K. MOLIERE, w. RATHJE AND I. N. STRANSKI 23 other hand, tangentially deformed surface structures are favoured from the standpoint of energy. These are also adapted to the periodicity of the space lattice situated below, but in such a way that adjacent ions collect together to form small isolated molecular complexes.* That is to be seen best from a consideration of one- and two-dimensional structures. One- and Two-dimensional Lattices.-The investigations on stability carried out by Hund3 in three dimensions, but only approximately, can easily be calculated exactly for one- or two-dimensional lattices.How will a chain consisting of alternate positive and negative ions and an isolated lattice face of the rocksalt (001)-type behave respectively by a continual rise in the ionic polarizability ? Certain limiting assumptions must first be made concerning the form of the sub-structures of low symmetry which result from the co-ordination structures, and which one can imagine as being produced from the first by a homogeneous deformation. The chain or lattice face will break up into single insular complexes, and it seems plausible to expect that similar crystal units in the structure thus formed will assume equivalent positions as regards energy and structure and that the ionic complexes formed are electrically neutral, Thus, for the alternating chain only a division into double ion molecules comes into question, whilst for the lattice face four-ion insular complexes are con- ceivable too.Furthermore, " chain lattices " can also be formed here (analogous to the layer lattices in three dimensions), i.e., the complexes formed can extend over the whole surface in simple co-ordinative relationship. If one introduces the additional assumption that the complexes formed by a division of the original co-ordination lattice in the manner suggested are deformed by the effect of the polarization forces with no loss in their own symmetry, one is bound to arrive, in the case of the (001)-lattice face, a t the types shown in Fig. 2 . These structures may be characterized, as indicated in the figures, by means of a relative parameter y (the relative approach between immediate neighbours, referred to the distance between the ions 6 in the undeformed structure).The lattice energy of the chain or lattice face must now be formulated as a function of y. I n addition to the coulomb ionic effect and the energy of repulsion we introduce the polarization energy for one ion in the form ++ + - p . + + 2 / 2 C C -+ + where is the dipole moment produced in the single ion ; E is the " self-field intensity " arising from all other charges and dipoles ; a is the mean polar- izability. The components of p can be eliminated by putting the partial differential quotients equal to zero as was done by Born and Heisenberg.2 The following equation t is then obtained for the lattice energy for an ion Dair : + e is here the elementary charge, A .r-fi the repulsion potential of two ions (we use n = 9) ; VB) is Born's repulsion potential, V P ) , F(p), Fjd) self- potentials or self-field intensities for corresponding forms with the lattice constant 6 = I, originating from the poles ($) and dipoles (d). These * A similar hypothesis was already put forward by one of us in 1928. t The easily proved rule must be taken into account that for all configurations of the type under consideration the mutual effect between one dipole and all other ionic charges is equal in value to that between one ionic charge and all other dipole moments.24 SURFACE STRUCTURES OF IONIC CRYSTALS FIG. 2 b. FIG. 2 c. FIG. 2 d . "1K. MOLIERE, W. RATHJE AND I. N. STRANSKI 25 values are functions of y, and can be calculated according to the well-known methods of Madelung and Ewald.5 From the condition s u / a Y 2 = 0, one can find the " critical polarizability " above which the tangentially deformed structures must be more stable than the co-ordination structures.If, for the lattice constants of the latter, one inserts the equilibrium values of 6 = 2-49 A for the chain and 6 = 2.68 A for the lattice face, one obtains : for the chain . . .. CG$\ = 1-82 A3 ; for the lattice face . . Fag:: = 1-95 As. But if one stretches the chain or lattice face under force, so that their lattice constant adjusts itself to the value 6 = 2-81 A of the compact space lattice, acdt. becomes much smaller, namely : for the chain . . . . c ~ $ \ = 0.58 A 3 ; The mean value of the polarizabilities of Na+ and C1- amounts to CC = 1-61 A.The function U(y) for this value of a and for 6 = 2-81 A was plotted graphi- cally. for the lattice face . . Fazi8'; = 1-50 A'. It shows an energy minimum : for the chain at . . .. y = z o y o ; for the lattice face at .. ~ " 6 % . For the (001)-lattice face the structure type (a), the chain lattice parallel to the edge, proves to be the most stable. The figures quoted refer to this type. If one puts the lattice face or edge as the surface in a space lattice, the deformation of the surface is diminished through the effect of the undeformed remaining part of the lattice, as is shown below. The (001) -Surface.-The expression for the lattice energy of the lattice face must now be completed by the terms which express the potential energy of the ions of the uppermost lattice face in the field of the half-lattice lying below, which is undistorted and infinitely extended.* The field intensity where the surface ions are situated now has components perpendicular to the surface (z-direction).The lattice energy depends therefore upon a further parameter 5, for which we choose the relative distance (referred to 8) between the two uppermost lattice faces (see Fig. 3). The lattice energy per pair of ions - U(,,, is now The values FV, FF represent the self-potentials and self-field intensities of the surface already introduced above, V , HF the potentials and field inten- sities induced by the ions of the half-lattice at the points where the surface ions are situated; these are defined, as previously, for lattices with a distance between the ions of I.They can be calculated best according to Madelung's method. 4 Madelung, Physik. Z., 1918, 19, 524. Ewald, Ann. Physik, 1921, 64, 253- * These terms, of course, must be substituted in the expression for the lattice energy of the uppermost lattice face in their full amounts (or if one refers to a pair of ions, with the factor 2), and must not be halved like the self-mutual effect of the surface ions. This fact was overlooked by Lennard-Jones and Dent.'26 SURFACE STRUCTURES OF IONIC CRYSTALS U 0'1 I b 0.08 C 0.05 d 0'01 As the direct application of the conditions of equilibrium 3U/3y = o and 3U/3< = o is too complicated, we determine the minimum potential energy (maximum lattice energy) by a graphical method.If (- U ) for certain values of the tangential parameter y is plotted against the vertical distortion co-ordinate <, curves are obtained of the type in Fig. 4 a. The figure refers to the type (a) (Fig. 2 ) , the chain lattice parallel to the edge, and the polarizability (a = 1-61 A3), all the maxima lie at OSg7 0.97 y = 0, i.e., the lattice face retains its 0.96 complete symmetry. At the most, the 0.96 f23 f#-f3 1 jZ6 1 1 i f2.5 f25 'IK. MOLIERE, W. RATHJE AND I. N. STRANSKI 27 predominating polarization sub-structure found for the (001)-face. Then follow the types (b) and (c) and with a greater difference the diagonal chain lattice (a). From the result one can conclude that for NaCl probably no decrease in the symmetry of the face is to be expected on the cube faces.This should only set in with higher polarizability. The [OOlI-Edge.-It is in line with the approximate method used up t o now to deform independently only the ion chain which forms the edge of the crystal, whilst all inner atoms of the crystal quadrant remain unpola- rized in their normal positions. Those structures in the surface lattice faces adjacent to the edge are to be fixed which, according to the calculation above, represent the configurations of minimum energy for the infinitely extended surface. In this case also we allow the whole row on the edge t o move its position relative to the rest of the crystal, confining ourselves to a displacement in the plan which divides the angle between the adjacent cube faces.0.00 0.05 0.10 055 aa0 - r - r FIG. 5 a . FIG. 5 b. According to the previous section, when the polarizability is low the faces remain undistorted tangentially, their ions are polarized perpendicularly to the face and the distance between the two uppermost lattice faces is some- what decreased. In this case the critical polarizability for a conversion of the edge to a linear molecular lattice amounts to = 1-4 pi3. If one substitutes the polarizability 1-61 A3 (corresponding to NaC1) the energy minimum then lies at y = 0.04, which indicates a 4 yo approach between neighbouring ions. The edge is displaced about 5 yo in the direction of the remaining crystal quadrant. If we now use the higher polarizability of 2-1 A3, we must assume the chain lattice structure type (a) (see Fig. 2) in the cube surfaces.As the chain structure can take up different positions relative to the edge, different types of combinations must be considered, from which we pick out the three types in Fig. 6 as those most favoured from the standpoint of energy. According to the calculation carried out, type (a), in which the chains run vertically to the edge on both sides, is found to be the most stable. A molecular structure is obtained in the edge with an 18 yo approach between the nearest neighbours (compared with 11 yo in the adjacent28 SURFACE STRUCTURES OF IONIC CRYSTALS faces). In equilibrium, the whole edge alters its position relative to the rest of the crystal by 4 yo (compared with 2.5 yo €or the face). The Surface Structures for Crystals of the Rocksalt Type.-The formulz for the lattice energy of faces and edges contain the polarizability in the combination a/S3 only.If one makes the assumption (somewhat rough, of course) that the repulsion potentials for all lattices of the rocksalt type can be represented by means of a power function with the same repulsion exponent (n = g), Alp-' is a common constant for all lattices. (A is calculated from the equilibrium lattice constant of the space lattice which is obtained experimentally.) Thus the value 6.U is a pure function of the quotient afS3. It is thus now possible to specify critical values for this quotient, which are determining factors in a sub-structure formation in the faces and edges. These are (a/S3)crit. = 0.064 for the [OOI]-edge ; (a/83)crit-= 0.073 for the (001)-face. No decrease in symmetry in the faces and edges is to be expected in the alkaline halides for : Salt KF NaF RbF CsF KC1 LiF a/S3 0.047 0.047 0.056 0.061 0.062 0.063 The cube faces retain their full symmetry, whilst the edges show molecular structure for the salts : ---I- RbCl KBr RbBr NaCl 3 l L I a/ 63 0.065 0.069 0.070 0.073 The edges have strongly defined molecular structures, the faces exhibit chain lattice structures in the following : Salt RbI KI NaBr LiCl NaI LiBr LiI a/63 0.079 0.080 0.082 0.091 0.096 0.101 0.117 The way in which the lattice energy behaves at the polarizability a = 2.1 Hi3 shown above in Fig.4 and 5 might fit the case NaI. The anion is the chief determining factor for the values of the mean polarizability.On the other hand, the denominator of the quotient a/S3 is the lattice constant 6, which has especially low values for salts with small (strongly polarizing) cations. Therefore, the most strongly defined polarization sub-structures in the surfaces are to be expected in salts with large anions and small cations. ---I Decrease in Symmetry in Individual Faces.-It must be assumed that the decrease in symmetry in individual faces will show itself externally in some way in the physical properties of the faces. No direct method for the determination of surface structures exists, however, as yet; it is known that when electron diffraction is brought about, space lattice regions of considerable thickness are always involved. Clues to a decrease in sym- metry in the surface are given by numerous experiments using the etching method,6 in which a lower symmetry actually was found than that corre- sponding to the space lattice.But up to the present, no clear connection Brauns, N. Jahrb. Miner., 1886, I, 224 ; 1889, I, IZI. Rosicky, N . Jahrb. Miner., 1916, z, 15.K. MOLIERE, w. RATHJE AND I. N. STRANSKI 29 with our theory of polarization sub-structures could be established, as in the etching method apparently there are too many unpredictable conditions playing a part. FIG. 6 b . FIG. 6 4 b . Reorientation Processes ; Decrease of Symmetry of the Habitus.- It can be seen from Fig. 5 that the different sub-structure types of the (001)-face are in fairly keen competition with one another from the stand- point of energy (ordinate scale 10-l~ erg/ion pair), One must further note that each type of structure possesses four possible forms (e.g., orientation of the chain lattice in the x- and y-direction).I t must be assumed that spontaneous reorientations take place between the different structures.30 SURFACE STRUCTURES OF IONIC CRYSTALS It is possible that, a s co-operative processes, such changes would require considerable energy of nucleus formation or activation. As a result the frequency of such reorientation processes would be dependent upon temperature. As the molecular structures are especially well defined on the edges, as was shown above, there should be the fewest reorientation processes taking place there. Thus the edges function as nucleus-forming centres, i.e., they determine the structures of the adjacent parts of the faces, and the most favoured are chain lattice structures of the type (a) (Fig.6 a) with the chain running perpendicular to the edge. i coofl FIG. 7. rhombic dodecahedron. In the long needles was observed. If one considers the complete cubic. crystal-body, it is conceivable that parallel sub-structures will be formed on the cube faces of a zone through mutual influence across the common edge. Such a case is shown diagrammatically for the zone (001) in Fig. 7. According to the tem- perature, a more or less frequent change in structure will set in on the two faces which do not belong to the zone, and this can also lead to occasional reorientation of the complete zone structure beyond the edge, but on the whole the parallel surface structures of the faces of the zone in question would be especially stable.The difference in the degree of orientation of the two faces could perhaps show itself in their growing and adsorption properties, by which a lower symmetry of the whole crystal body (crystal habitus), in this case a tetragonal form, could be simulated. Nothing like this is known up to the present for the rocksalt lattice, but an analogous case might be found in the cubic space-centred lattice of crystallized ammonia. The growth form here is the presence of pectic acid, crystallization in Influence of External Forces ; Tensile Strength.*-There is no doubt that surface structures are sensitive to the effect of external forces, in which, incidentally, one may include the forces which proceed from an adsorption layer or a neighbouring phase, mentioned in the previous example.Tangential electric fields, for example, would probably favour the structure types (c) and (d) of Fig. 2 which possess a tangential electric moment, from the standpoint of energy. This might express itself in dielectric or optical anomalies, unless the effects are too small to be observed. If one submits the crystal to a tensile force in the direction of a cube edge, the tendency to form molecular lattice structures in the edges and faces running parallel to the direction of the force will increase greatly with only a small stretch. Molecular cracks, perpendicular to the direction of the force, are thus produced in the surface. It is conceivable that at a certain tensile force, which lies far below the theoretical tensile strength 7 Ehrlich, 2.anorg. Chem., 1932, 203, 26. 8 Stranski, Ber., 1942, 75, 1667.K. MOLIERE, W. RATHJE AND I. N. STRANSKI 3= of the infinitely extended crystal, these surface cracks will extend further into the inside of the crystal. The cracks produced would coincide with the breaking-faces of the crystal. Thus the presence of atomic sharp edges would be a decisive factor for the lowering of the tensile strength, compared with the theoretical value, as is actually confirmed experimentally. The experimental data of the Joff6 effect are in agreement with this, namely, FIG. 8 a. FIG. 8 b. if the edges are removed by superficial solution of the crystal, the tensile strength (that is, its upper limit in a great number of experiments) increases until almost at the theoretical value. With a renewed growing process (new formation of the atomic sharp edges) the tensile strength again greatly decreases.9 Joffg, Kirpitschewa and Lewitzky, 2. Physik, 1924, 22, 286.32 SURFACE STRUCTURES OF IONIC CRYSTALS By carrying out the necessary calculations, we have convinced ourselves how the surface structure of a superficially dissolved crystal might be expected to behave. This is known to consist of numerous atomic cube face steps. It was found that the ion chain which forms the steps does not exercise a directing influence upon the chain structures of the two adjacent parts of the face, as does the edge. The crack structures produced in the face are isolated to a certain extent by the steps.This result could be supported by means of calculations on the equilibrium structure of the (011)-face. This might be taken as representative for the structure of the superficially dissolved surface, as it is made up exclusively of atomic cube steps. The only sub-structure types of the (oII)-face which come into question are shown in Fig. 8 a, b. The calculation of the lattice energy shows that the type A is z x 10-l~ erg/ion pair more stable than type B. The distortion parameter (defined according to Fig. 8) amounts in equilibrium to 8 yo for NaCI, 20 yo for NaI. Thus for the (oII)-face it would seem that surface structures will be chiefly formed in which the molecular surface cracks do not lie in the track of possible breaking surfaces, as is the case in type B.In the more stable type A, on the other hand, the surface cracks follow [ I ~ I ] , i.e., in the track of rhombic dodecahedron faces; these are possible sliding-faces of the crystal. This could explain the increase in plasticity of superficially dissolved crystals. From our considerations, however, we are not able to produce a mathe- matical theory for the cracking. Among other things, the influence of statistically distributed lattice disturbances would have to be included. But it seems certain to us that structural irregularities in the surface will have to be taken into consideration in any exact theory of the future. Considerations Concerning the Justification of the Assumptions Made.-One objection which could be made to our way of calculating refers to the use of the linear polarization expression, + + $ = a E . This is known to be valid for homogeneous fields and small field intensities only. I t is certain that neither of these conditions are fulfilled in crystal surfaces. Estimates as regards energy, which refer to alkaline halide mo1eculesJ10 give cause for the assumption that, in reality, the share of the polarization energy in the total bond energy is considerably greater than the share calculated from the polarizability in the homogeneous field. If one subtracts the energy of repulsion, calculated from crystal lattice data, from the energy of dissociation known from spectroscopic data, and makes a correction for the effect of van der Waals’s forces, the remainder is more than twice as great as the classic polarization energy. The term remaining contains the quantum-mechanical mutual effect of the electron-clouds, which is difficult to estimate, but it can scarcely be assumed that this is very great. It can therefore be assumed that in the mutual effect of the ions in a crystal surface also the polarization share is still greater than that calculated by us according to the classical method. The data which we give for the surface distortions probably represent, therefore, a lower limit of the structural deviations realized in nature. Institut f u r Physikalische Claemie und Elektrochemie, Berlin-Charlottenburg 2, Hardenbergstrasse 34, Germany. Hellmann and Pschejetzkij, Acta Physicochiwz., 1937, 7, 621.

ISSN:0366-9033    

DOI:10.1039/DF9490500021

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

CRYSTAL GROWTH AND SURFACE STRUCTURE Part I BY w. K. BURTON* AND N. CABRERA Received 15th February, 1949 Introduction. As a preliminary to the study of the rate of growth of crystals, we consider in Part I of this paper the influence of surface structure on the rate of advance of the growing surface. If, for the time being, we confine our attention to crystals with perfect lattices, it is found that crystal surfaces can be divided into two classes, (a) close-packed 7 surfaces and (b) non-close-packed or " stepped " surfaces, which possess essentially different properties. A surface is close-packed if, when it is as flat as possible, all the surface molecules are at the same distance from a plane parallel to it ; in all other cases the surface will present a stepped appearance, as in Fig.I, the height of each step being of molecular dimensions. By way of illustration, in the simple cubic system (100) surfaces, (111) surfaces and (110) surfaces are close-packed, all other surfaces are stepped. In a stepped surface FIG. I . the terraces in Fig. I are pieces of close-packed surface. In the case of a (11zo) surface of a simple cubic crystal, the terraces will be (100) surfaces. This point of view is fruitful because it can be shown that for a stepped surface the portions between the steps play almost no part in growth phenomena at low supersaturations. In fact, deposition on a close-packed surface can only take place by surface nucleation : small " islands " of molecules collect on the surface and grow so as to produce a new layer, a process which is very slow at low supersaturations.On the other hand, deposition on the edge of a step A (Fig. I) can take place without there being a linear nucleation process. Hence the growth problem for a stepped surface is essentially solved, once the corresponding problem for steps has been solved. If a crystal grows at all, some kind of steps must exist at some time in the surface. These steps may be of the kind already mentioned, or they may be boundaries of two-dimensional nuclei. Growth essentially depends on the existence of " kinks " in these steps. Easy growth is guaranteed if these * Seconded from I.C.I., Ltd., Butterwick Research Laboratories, The Frythe, Welwyn, t Note that our definition of " close-packed " differs somewhat from current usage.B 33 Herts.34 CRYSTAL GROWTH AND SURFACE STRUCTURE kinks are always present, and this criterion can be reduced to the question of the existence of kinks, when the external concentration is the equilibrium value. For if kinks are present at equilibrium, then when the external concentration is raised there are already suitable deposition points available. If there are no kinks at equilibrium, then these must be created, and a large hindrance to growth appears. It can be shown that the concentration of kinks in a step in equilibrium is high, and that the concentration of kinks in a close-packed surface at equilibrium is negligible. This again speaks in favour of our classification of surfaces into stepped and close-packed surfaces. In this paper we are concerned with the equilibrium structure and rate of growth of an infinite surface.It is, of course, clear that an infinite surface is in equilibrium with the same external concentration (e.g., vapour concentra- tion) whatever the surface. A finite surface will not be in equilibrium in the same sense as in the case of an infinite surface, and consequently, in general, some change will tend to take place. But changes of orientation can take place only by means of processes which occur at the boundary of the surface, and hence for a surface of observable size the change will occur at an unobservably slow rate, the associated relaxation time tending to infinity with the size of the crystal. Therefore, if we confine ourselves to a region on a finite surface which is almost flat, then its structure will be the same as that in an infinite surface having the same orientation. The two basic equilibrium problems are now (a) the equilibrium structure of an infinite step, and (b) the equilibrium structure of an infinite close- packed surface.e3 A - + FIG. 2. The Infinite Step. Consider then, a semi-infinite layer of molecules on an infinite close-packed plane crystal surface bounded by a connected line. This we call a step : it can have any mean direction. At T = oo K, the step will be perfectly straight, but as the temperature is increased it will consist of a number of “ kinks,” separated by certain distances as in Fig. 2, a certain number of adsorbed molecules (A) and a certain number of holes (B). We need to know only the concentration of kinks to form a picture of the mean structure of the step.This idea, introduced by Frenkel l simplifies the treatment of the problem very much: we call the kinks “ Frenkel kinks ” (F.k.). I t is clear that the concentration of kinks in a step will depend on its orientation, and that there will be orientations for which the number of kinks is a minimum. For instance, for the (001) face of a simple cubic crystal, the (10) steps will have the smallest number of kinks. This minimum number will tend to zero with T. Accordingly, if we can show that a (10) step contains a large number of kinks at T>o under equilibrium conditions we know that steps of all orientations also contain a large number of kinks. If we use a simple cubic model with nearest neighbour interactions (Kossel crystal) it is easy to find the equilibrium concentration of adsorbed molecules, Frenkel, J .Physics, U.S.S.R., 1945, 9, 392.W. K. BURTON AND N. CABRERA 35 holes and kinks in a (10) step. Let the energy of interaction between neigh- bouring molecules be 9. Then the energy necessary to fonn an adsorbed atom in the step (Fig. 3) will be q. The energy to fonn a hole is also 9, since an energy 2 9 is required to form a hole and an adsorbed molecule. The energy to form a kink is, however, only Qq, since from Fig. 4 and Fig. 5 only an energy 2 9 is required to form four kinks. There is no change in energy in going from Fig. 4 to Fig. 5. The numbers of positive and negative kinks (Fig. 2) are, of course, equal. We conclude that the probability for having a hole or an adsorbed molecule at a given place on the step are both given by and that the probability for having a kink at a given place on the step is given by n = exp (- cp/kT) .(1) n+ = n- =exp (- cp,kkT) . (2) If T - 600' K and we take a typical value of 'p as 0.2 eV, we find that there is a kink for every ten molecules in the step, and an adsorbed molecule or hole for every hundred molecules. We have, of course, simplified the problem very much : there is a con- siderable probability for the existence of kinks of multiple height, particularly for steps which deviate a great deal from the [IO] direction (in the simple cubic case). The complete theory has been developed elsewhere.2 FIG. 3. FIG. 4. FIG. 5. In view of the fact that steps of different orientations have different concentrations of kinks in them, the edge-free energy per unit length of a step varies with the orientation and is a minimum for the (10) step.It might be thought that steps other than (10) steps are not in real equilibrium and that there must be a tendency for these steps to change into (10) steps. If we are considering infinite steps, this conclusion would be erroneous, for it can be shown that steps of all orientations are in equilibrium with the same external concentration of molecules, despite the fact that the concen- tration of Frenkel kinks varies with the orientation. Frenkel has treated the kinetical problem of the transformation of any step into a (10) step, assuming on energy grounds that the others are not in equilibrium.He obtains in this way a time of relaxation independent of the length of the step. This result is incorrect because the " torque " which applies to the steps not in a [IO] direction is evaluated by Frenkel by taking the derivative of the potential energy with respect to the orienta- tion angle. But his formula contains only points corresponding to the equilibrium positions of the steps, and not the intermediate non-equilibrium positions through which the step would have to pass in order to change at all. In fact for an infinite step, each equilibrium position is surrounded by infinitely high potential barriers which cannot be surmounted. Such a step can only be in equilibrium with a supersaturated or undersaturated external phase, and then the equilibrium is unstable and subject to stringent restrictions as regards shape.The sharper the corners of a finite step, the greater the For a finite step, the situation is different. Burton and Cabrera (to be published elsewhere).36 CRYSTAL GROWTH AND SURFACE STRUCTURE rate of evaporation, and an arbitrarily oriented step tends to become a (10) step during the evaporation, The time required for this process to take place increases with the length of the step, because the processes which permit the transformation occur only at the corners. It must not be thought, however, that all the considerations which apply to kinks in a step apply to steps in a surface. It is still true that surfaces of all orientations are in equilibrium with the same vapour concentration in the same sense as for steps, but the fact that for some surfaces the concen- tration of steps is large does not imply that' double steps, treble steps, etc., will be frequent.* The difference between the energy of two single steps and one double step is proportional to the step length, and is very large if the interactions are not of the nearest neighbour type.Similarly, there is no question of steps being formed by thermal fluctuations,* as kinks are formed in a step, since the energy of formation of a step is proportional to its length and is enormous for long steps. Thus a stepped surface tends to become as flat as possible, and at equilibrium, only single steps will appear. It follows that the macroscopic steps which have been observed, e.g., on metals by Graf,3 on growing crystals have nothing to do with equilibrium problems, but are essentially kinetic in origin.If a surface is produced with macroscopic steps in it, it is obvious that the rate of approach to macroscopic equilibrium is negligibly small and the structure is essentially frozen in. Close-packed Surfaces. The circumstance which makes stepped surfaces so easy to treat is that the steps themselves present a one-dimensional problem. In each position on the step we have a variety of possible states : occupation by a kink of positive, negative or zero height. Each of these possible states can occur independently at each point, and hence the proba- bility for the occurrence of a compound state affecting more than one position is the product of the probabilities for the individual states at each of the individual positions.We have assumed so far that those parts of the crystal surface between steps can be ignored, and this assumption is shown to be reasonable in the following discussion. However, if there are no steps in the surface, which is the normal case in a close-packed perfect crystal surface, then we are presented with an essentially different two-dimensional problem. We assume that in the close-packed surface of a crystal there can be differences of level, i.e., that " jumps " can occur in the surface. The presence of jumps provide suitable places for evaporation and condensation, provided that the jumps are not due merely to the presence of adsorbedmolecules and holes. The problem is to estimate the number of jumps at equilibrium as a function of temperature.In this case the jumps themselves cannot be assigned independently, since it is possible to have twice as many jumps in a surface as there are molecules. Fig. 6 shows a picture of part of a surface ; the small squares represent molecules seen from above. The heights of these molecules above some arbitrary plane can, of course, be assigned independently, but the distribution of jumps across the full lines in the figure cannot. For suppose we start at the molecule A and follow any closed path such as ABCDEF, then although we can have any jump we choose between neighbouring molecules on this path, providing we do not close it, the necessity for finishing at A at the same level at which we started implies that the magnitude of any jump on a closed path must be fixed by the magnitude of the others.3 Graf, 2. Elektrochem., 1942, 48, 181. * At least, if the interactions are all attractive.W. K. BURTON AND N. CABRERA 37 So there are innumerable sets of relations, corresponding to all the closed paths on the crystal surface, between the jump probabilities. In fact, to specify the probability for the existence of a jump at a given point involves the knowledge of the state of the surface at every other point. So we are faced with a so-called co-operative phenomenon. This makes the Frenkel kink picture employed previously almost unworkable. We must therefore look for some other method. We have made preliminary calculations on the basis of a model which is somewhat oversimplified : we suppose the levels in the crystal surface to be capable of two values only.The method employed is that due to M~ntroll,~ Kramers and Wannier,5 Onsager,* Onsager and Kaufman and Wannier, originally devised for the treatment of ferromagnetism, using the two-dimensional Ising model. Just as there is a transition or critical tem- perature associated with an infinite specific heat in the case of the two- dimensional ferromagnet, so there is in the case of this crystal surface model. FIG. 6. The problem is to find the increase in potential energy of the surface due to the presence of jumps in it. The mean number of jumps can then be found. If the surface of the crystal were perfectly flat we should say that the surface potential energy, for example, of the (100) surface of the Kossel crystal per molecule was (pl/z in the nearest neighbour model, corresponding to one unused " bond " per molecule, which we can imagine as sticking out perpendicular to the crystal surface.If, however, the surface is not flat, then there will be additional unused bonds sticking out parallel to the surface, and each of these bonds will contribute (pJ2 to the potential energy 4 Montroll, J. Chem. Physics, 1941, 9, 706. 5 Kramers and Wannier, Physic. Rev., 1941, 60, 252, 263. 6 Onsager, Physic. Rev., 1944, 65, 117. 7 Onsager and Kaufman, Report Int. Conf. on Fund. Particles and Low Temperatures (Cambridge, July, 1946)~ Vol. 11. : Low Temperatures, Physical Society (1947). Wannier, Rev. Mod. Physics, 1945, 17, 50.38 CRYSTAL GROWTH AND SURFACE STRUCTURE of the surface.If we take our zero of energy to correspond to a flat surface, then if we evaluate the potential energy per molecule of the surface at equilibrium and divide it by pl/z we get a figure for the number of unused bonds in the surface which are parallel to it. This figure, s, represents the " roughness" of the surface. This is the quantity we aim to evaluate as a function of temperature. We expect it to go from o to I as T goes from 0 to co. The crystal surface model we are considering is equivalent to a square lattice of units which we call atoms capable of two states which we designate by + I and - I. If two neighbouring atoms have the same state their interaction energy is zero, otherwise it is ql/z. The first possibility describes the two molecules in the crystal surface when they are at the same level, the second when their levels are different. For the sake of generality we assume, following Onsager, that the interactions can be different in the two directions [ro] and [OI] : in the case of the (100) surface we shall equate them. - 0-8 FIG. 7. The problem is solved by studying the effect on the partition function for the surface by adding an extra row of molecules. The final result is, for the (100) surface where S = L - - . 1 . (I + 2k2 K , / x ) coth H , (3) H = q#kT ; k , = 2 tanh2 H - I ; - dw (I - liI2sin2Cr))'/e ; kI2 + k,2 = I . . (4) K , is the complete elliptic integral of the first kind. A graph of s against T is shown in Fig. 7 (a). The curve possesses a vertical tangent at T = T, given by k , = I or k , = 0, i.e., by sinh H , = I , or If we had assumed the jumps to be independent, the result would have been 7-c H, = <~~,/zkT, = In cot - - 0.9 . (5) 8 (6) 2 exp (- CPpl/zkT) 1 + exp (- cpll2kT) ' S = which gives rise to curve b in Fig. 7.W. K. BURTON AND N. CABRERA 39 separating the rbgime where the Lattice Surface jump concentration is negligible (100) . .{ (110) and surface nucleation is re- (111) quired for growth, T< Tc, from that rkgime T>T,, where the jump concentration is high, and Face-centred cubic . . (1x1) no nucleation is required for Simp1e cubic. * T c I 0 O O 0 C 400' C -3oOC 1700' C

ISSN:0366-9033    

DOI:10.1039/DF9490500033

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

CRYSTAL GROWTH AND SURFACE STRUCTURE Part I1 BY N. CABRERA AND W. K. BURTON* 1. Introduction. In Part I of this paper we studied the microscopic equilibrium structure of crystal surfaces and distinguished two kinds, stepped surfaces and close-packed or saturated surfaces. This second type is essen- tially the same as the first one, when the temperature is above a certain critical value. Below this temperature the close-packed surfaces behave in quite a different way. In Part I1 we treat the kinetic problem of the growth of these surfaces from the vapour ; in all cases it is necessary to consider the diffusion of adsorbed atoms on the surface of the crystal. First, we treat the growth of stepped surfaces; quantitative formulae are given as a function of the inclination. The growth of close-packed surfaces below their critical tem- perature by the mechanism of two-dimensional nucleation is also considered.Treatments of two-dimensional nucleation have been given by several authors, especially by Becker and Doring,l neglecting the effect of the diffusion of adsorbed atoms. We discuss in this paper the influence of this diffusion and we show that in spite of the fact that the predicted rate of growth is perhaps different from that given by Becker and Doring, it will certainly not account for the experimental fact that in the small number of cases where a critical supersaturation for growth has been observed, it is of the order 0.01 at most. Finally, a treatment of the initiation of growth of an imperfect crystal containing a random distribution of dislocations is given, and is shown to be in agreement with experiment. The results of this work will be given here in a preliminary way ; the complete treatment will be published elsewhere.2. Growth of Stepped Surfaces. In Part I we showed that each step in the surface contains in equilibrium a very high concentration of Frenkel kinks, much higher than the concentration of adsorbed atoms in the edge of the step. The equilibrium concentration of adsorbed atoms on the surface is I - e-WJkT, . * (1) a2 where a is the interatomic distance and Ws the evaporation energy from the step into the surface. When the vapour pressure is increased by a factor cc (saturation ratio = a ; supersaturation = a - I), the concentration of adsorbed atoms remains practically equal to that in equilibrium near the steps, because of the high concentration of Frenkel kinks in them.The saturation ratio ccs on the surface increases from the value I near the steps to a maximum value between the steps. The current of atoms condensing into the steps will be controlled essentially by the diffusion of adsorbed atoms on the surf ace. 1 Becker and Doring, Ann. PhysiFz, 1935, 24, 719. * Seconded from I.C.I., Ltd., Butterwick Research Laboratories, The Frythe, Welwyn, 40 Herts.N. CABRERA AND W. K. BURTON 41 The actual variation of as can be obtained solving the corresponding diffusion problem, the result being cosh px a - GL(X) = (a - I) ~- coshpx,’ * (4 where xa is the distance counted from half-way between the steps, 2x,a is the mean distance between them, and W , being the energy necessary to take an adsorbed atom from the surface into the vapour (W = W, + W , is the total evaporation energy) and U the activation energy for diffusion on the surface.Therefore p2 is the ratio between the probabilities for an adsorbed atom to evaporate into the vapour and to diffuse on the surface. Usually W , > U ; if U is very small, p will also be very small, and the concentration of adsorbed atoms on the surface will be practically uniform and equal to that in equilibrium, unless x, is very big. As U increases the non-uniformity on the surface becomes more important. If W , < U , the diffusion on the surface does not play any role, the condensation into the steps takes place directly from the vapour.We do not think that this is likely to occur. In order to illustrate the values that p can take, let us consider a face- centred cubic crystal with a stepped surface consisting of terraces (I, I , I ) and steps in any direction, and let us call ‘p the interaction energy between nearest neighbours. Then W , - 3q1 and U - cp, therefore p = exp { - cp/kT} -0.05, if, eg., cpjkT - 3 . The current j of atoms condensing per sec. per cm. into each step will be equal to the current of atoms condensing from the vapour on the surface between two steps; therefore, from (z), p 2 = exp { - (W, - U ) / k T } , . (3) ’ (4) = 2 xe-W/kT - I tanh Px, , . a P where v is the frequency of vibration of the adsorbed atoms. The velocity of advance of the steps, will be a function of the distance between steps.For x,~<I the velocity of the steps is proportional to the distance between them; as x, increases above P-l, 11 tends to a maximum value v, given by which represents the velocity with which a single step would move. For the example considered above putting v - 1o12 sec.-l, a - I O - ~ cm., cp/kT - 3 and a - I - IO-~, we get v z - I O - ~ cm./sec. v = a?, v, = z(a - 1)va exp( - (W + Ws + U ) / z k T ) , . exp{ - (W + ws + U ) / k T } = exp{ - 5cp/kT} ; (5) The total current J of atoms adsorbed per cm.2 per sec. is J = N j , where N = 1/2x,a is the number of steps per cm., therefore Na < @; J = zNaJolp . - (7) For values of Na - p, the current J will depend on the distribution of steps in the surface. The curve, Fig. I , has been calculated assuming a random B*42 CRYSTAL GROWTH AND SURFACE STRUCTURE distribution.For Na N o and Na - I the stepped surface transforms into close-packed surfaces of quite different character (see 9 3). We see that the rate of growth of stepped surfaces is practically given by formula (6), down to small values of Nu. FIG. I.-Rate of growth of stepped surfaces as a function of the number Ar of steps per cm. Recent experimental work by Graf,2 and Mahl and Stranski,3 suggests that the stepped surfaces present a striated structure during growth from the vapour, the distance between the striations being of the order of 10-* cm, We showed in Part I that this striated structure cannot be in equilibrium with the vapour, the surfaces always having a tendency to be as flat as possible.We believe that the formation of these striations has an essentially kinetical character and is probably related to the fact that the velocity of displacement of the steps is a function of the distance between its nearest neighbours. A correct theory has not yet been obtained. 3. Growth of Close-packed Surfaces. Let us consider now the surfaces without steps. The surfaces with small indices will correspond to this type. We considered in Part I the equilibrium structure of these surfaces, and we have shown that they will remain practically flat if the temperature is below a certain critical temperature, but they will contain a considerable number of Frenkel kinks if the temperature is above it. It is easy to show that for close-packed surfaces above their critical temperature the rate of growth is again given by (6).This type of surface can occur, for instance, as the limiting case of a stepped surface when Nu N I. SURFACE NUCLEATION PRocEss.-Let us now consider the growth of habit surfaces, where Frenkel kinks do not occur. A long time ago Gibbs, and later on Volmer, suggested that the growth of these surfaces requires a two-dimensional nucleation process. The theory of nucleation, especially in the case of formation of three- dimensional liquid or solid nuclei from the vapour, has been developed by V ~ l m e r , ~ Stranski and Becker and D0ring.l In that case the diffusion in the vapour plays a small role and the supersaturation is practically the same all over the volume. The tendency for small nuclei to evaporate is very big and the supersaturation required for them to grow has to be high 2 Graf, 2.Elektrochem., 1942, 48, 181. a Mahl and Stranski, 2. Metallkunde, 1943, 35, 147. Volmer, 2. physik. Chern., 1926, 1x9, 277. Kaishew and Stranski, 2. physik. Chern. B, 1934, 26, 317.43 N. CABRERA AND W. K. BURTON (a - I of the order of 4). The number of nuclei of critical size formed per sec. per ~ m . ~ turns out to be of the form, * (8) I = B e-AIkT, where B - 1o20 c111.-~ sec.-l and A is the increase in free energy necessary for the formation of a nucleus of critical size ; A is tremendously high for all values of a smaller than 4. applied the same ideas to cases in which there is a diffusion of the atoms condensing into the nuclei, such as occurs in precipi- tation in a supersaturated alloy.They assumed that the only change in formula (8) to be made in this case is to multiply by a factor exp( - U/kT), where U is the activation energy for diffusion. Actually this assumption is not entirely correct ; it implies that the supersaturation is the same all over the volume, which is not true if diffusion exists. The surface nucleation required for the growth of habit surfaces can be treated in a similar way, and the diffusion of atoms on the surface has also to be taken into account. Before studying two-dimensional nucleation, let us consider the growth of a single nucleus on the top of a habit face. We assume a circular shape of radius pa ; if p is the number of atoms contained in it, If the total energy of a nucleus of p = np2 atoms is Frenkel and Becker p = z/*..np2ws - 2qy, where y is the edge energy per atom, the mean evaporation energy from the nucleus is Let a be the supersaturation ratio in the vapour ; assuming the nucleus to be big enough, it will not change appreciably before a steady distribution of adsorbed atoms around it has been formed. Under these conditions the diffusion problem can be solved, and the supersaturation ratio a,(r) on the surface around the nucleus is given by W P > = ws - (Y/P) * (9) 1, and KO are the Bessel functions of first and second kind with imaginary argument. a&) is the supersaturation ratio near the edge, which by definition is where is the supersaturation ratio which would be in equilibrium with the nucleus, and j ( p ) is the current of atoms condensing per sec.into the nucleus. The current is then calculated as the number of atoms condensing from the vapour per sec. all over the surface. The result is e-W/kT j ( p ) = 4rrpv[a - aSe(p)] ___--~~___ __ __- - (11) P2[2P Io(PP) Ko(PP) + I1 6 Frenkel, Sowjet Physik, 1932, I, 498. 7 Becker, Ann. Physik, 1938, 32, 128; Proc. Physic. SOC., 1940, 52, 71.44 CRYSTAL GROWTH AND SURFACE STRUCTURE The radial velocity is 4 P > = m/w* pc = y/kTln a . * (12) Current and velocity change sign when a = ase(p), which defines the critical nucleus to have a radius p. given by For p < p. the nucleus evaporates, for p > pc the nucleus grows. The maximum velocity of growth of the nucleus, for p 3 pc and p p > I, reduces of course to formula (5).If pc is big, such that @pc > I, the velocity is given by v ( p ) = ', (.I - $) J ' (13) valid for p > pc. For ppc < I, the velocity curve becomes steeper near the critical size. This is illustrated in Fig. 2. The concentration a ( ) near the edge of the nucleus is seen to be practically the same as ase(p) ; is a little bigger when p < pc and smaller when p > pc. The correction increases as p decreases but remains small. C FIG. 2.-Radial velocity of a nucleus bigger than the critical size. Now let us consider the current J of atoms condensing per cm.2 per sec. on a close-packed surface. In a steady state, when a constant supersatura- tion a - I is maintained in the vapour, there will be on the surface a stationary distribution n, of nuclei of different sizes. They grow until they collide with other nuclei in the same atomic layer and disappear ; we can take this into account by assuming that there is a certain maximum value M for p.Given the actual distribution n,, there will be a surface super- saturation ratio ccs ( x , y), a function of position. The total current J will be equal to where Zs is the mean value of as all over the surface. When a is very near I, the number of nuclei on the surface is small, and therefore most of the surface has a supersaturation ratio as - a ; consequently CC, - a and the current J will be very small. As a increases the number of nuclei increases and the mean value as decreases, becoming a, - I when the proportion of big nuclei is high and the distances between them small.We expect therefore that J as a function of a will be represented by a curve such as is illustrated in Fig. 3, where the straight line corresponds to the formula (6). The calculation of CC, as a function of a is a very difficult problem ; it requires, of course, the knowledge of the distribution n, of nuclei of different size as a function of a. Nevertheless for small values of a - I for which the current J remains small (region OA in Fig. 3), it can be estimated using a method proposed by Becker and Doring which we cannot develop here.N. CABRERA AND W. K. BURTON 45 We assume that the growth of any nucleus is due fundamentally to the condensation of single adsorbed atoms ; this assumption is correct onlyzfor a small density of nuclei, and therefore when a - I is small.Under these conditions, the number I of nuclei of any size p formed per sec. per cm.$ from nuclei p - I can be calculated if the ratios q(p)/a(p) of the mean probability of evaporation q(p) to that of growth a(p) for every nucleus are known. The current J of atoms J condensing per sec. per cm.2 is then calculated from J = I M , where M is the maximum value of p. In general, the ratio of the supersaturation ratio which would be in equilibrium with the nucleus to that actually existing near it. unimportant, and therefore as(p) = cc for all nuclei. tions one gets for the current J the expression, 4(p)la(E”) = ase(d/as (P), 0’’ FIG. 3.-R,zte of growth of a - I. as a function Let us first assume that the influence of the diffusion on the surface is Under these condi- where A , = - xpC2kT In a + 2xpcy = y2x/kT In a, is the increase in free energy necessary for the formation of a critical nucleus.The factor 4 multiplying A , comes from the calculation of M , which happens to contain a factor exp{ $A,/kT} ; this of course is assuming that the surface itself is much bigger than M . It is easy to see that expression (15) will give a negligible rate of growth, unless a is of the order of 2 . Actually, according to Volmer and Schultze * a linear rate of growth is observed above a - I - I O - ~ . For this supersaturation, taking the exponent in (15) becomes A/3kT N 103, and therefore no growth should occur at all. Let us now consider the influence of diffusion. This is a very difficult problem, for which only a qualitative answer has been found.There are two conflicting effects. First of all, the nuclei bigger than the critical size, which are therefore on the average growing, decrease the con- centration of adsorbed atoms in the neighbourhood of their edge, with the result that the current of condensation in these nuclei is now much lower than before. On the other hand, the nuclei smaller than the critical size, which are in the average evaporating, may tend to increase the concentration of adsorbed atoms around their edge, and consequently the probability for them to grow further is higher than it was before. From the study of the evapora- tion of a single nucleus we showed that there was an increase of concentration near its edge. Nevertheless, we do not think that the same considerations apply to the assembly of nuclei in the nucleation process.In that case owing to the fact that the distribution of nuclei n, is a decreasing function of p more nuclei are coming to the size p per sec. by growth from p - I than nuclei coming by evaporation from p + I, and this difference increases when p decreases ; therefore we think that the supersaturation ratio a&) near the nuclei p must tend to the value a for values of p not very small compared with the critical size pc. y - ‘p, and cp/kT - 3, Volmer and Schultze, 2. physik. Chsm. A , 1931, 156, I .46 CRYSTAL GROWTH AND SURFACE STRUCTURE As an illustration, let us suppose that a&) = a a 4 4 = ase(P.> for nuclei p smaller than a certain size b, and for bigger nuclei.Then the total current J can be shown to be where is the increase in free energy necessary for the formation of a nucleus of size EL.. = 7cpo2. This formula will give a bigger current than (15) if A , < 4 3 , therefore if This is reasonable if we consider that for supersaturations of the order a - I - I O - ~ , assuming always pc is of the order 105. On the other hand, it can be shown that formula (16) will account for the observed rate of growth at supersaturations of the order IO-~, only if p, < TO which is certainly too small. We conclude from these considerations that, in spite of the fact that the diffusion perhaps changes the current given by the simple nucleation theory, it does not agree with the current experiment ally observed. ROLE OF DISLOCATIONS IN CRYSTAL GRoWTH.-According to Frank, the surface of any real crystal must contain a certain number of dislocations, with a screw component, terminating in the surface and producing steps which do not disappear during growth.Under these conditions the two- dimensional nucleation is no longer necessary. The current of condensation in the special case of a random distribution of these dislocations can be estimated in the following way. Let us suppose that we have N dislocations per cm.2 distributed at random. We assume also that there is practically the same num- ber of dislocations of both signs. Steps will then exist between pairs of dislocations of different sign. During growth the coupling be- tween dislocations will change, but length x between neighbour dislocations always between neighbouring dis- of different sign.locations of different sign. The proportion f(x)dx of steps of length between x and x + dx (times the interatomic distance a) will be represented by a function illustrated in Fig. 4. If we assume that the steps are always formed between two nearest neighbour dislocations of different sign, then < 0.04 x t~... y/kT - rp/kT - 3, f w x , %= FIG. 4.-Distributions of steps of different the steps be Let a - I be the supersaturation in the vapour. Then all the steps of length x bigger than the diameter x, of the critical nucleus, given by Frank, This Discussion.N, CABRERA AND W. K. BURTON 47 will grow freely, until they collide with other steps. The length of the steps remains always of the order of x.The steps of length x < xc will have a very small probability for growth ; we shall neglect their contribution to the total current. The current of atoms condensing per sec. into the steps of length x , j(x) will also be a function of the distance to the nearest neighbour steps ; never- theless, provided the condition Pxo > I is satisfied (see $I), we can use formula (5), that is to say, 2v j ( x ) = - e-W/kT (a - I)X. The total current J of atoms condensing per sec. per cm.2 in surface will now be given by J = y J M.W&- Fig. 5 illustrates the current obtained from (19) as a function of ( a - I) and for a given value of N . For x, > x, (a - I very small), J is given by the expression 13 ' (19) N " xc J = - Na2v e-WIkT exp { - k)2} , Pa2 where xo and xc are given by (17) and (18). For x c < xo the current tends to a linear law of the form The critical supersaturation (point C in Fig. 5 ) for which the current becomes practically linear is given by xo/xc -2, J FIG. 5.-Rate of growth of a close-packed surface containing N dislocations per cm.2 as a function of the supersaturation o! - I. The general shape of the curve represented in Fig. 5 agrees with the experimental results of Volmer and Schultze,8 for the growth of iodine crystals at oo C. The critical supersaturation is in their case a - I - I O - ~ , which agrees with the value given by (zI), taking y / k T - 3 and assuming N = ro8 cm.-2 which agrees with the value generally assumed to explain the mechanical properties of crystals. Volmer and Schultze (loc. cit.) observe also a linear law of growth, as a function of a - I, for a - I > IO-~. The experimental value of the rate of growth for a - I = I O - ~ is of the order of 102 atomic layers per sec.48 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH Formula (20) is strictly speaking only applicable to simple monoatomic substances; in order to apply it to complicated structures as iodine, we have just to calculate W and v in such a way to account for the saturation vapour pressure of iodine. Using the experimental values of Gillespie and Fraser,lo one obtains W = 0.7 eV, v = 0.4 x 1017 sec.-l. Putting these values, and N - ro8 cm.-2, p -IO-~, into formula (20) one obtains a rate of growth of the order of the experimental value. Departmefit of Physics, University of Bristol. 10 Gillespie and Fraser, J . Amer. Chem. SOC., 1936, 58, 2260.

ISSN:0366-9033    

DOI:10.1039/DF9490500040

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

48 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH THE INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH BY F. C. FRANK Received 16th February, 1949 The kinetic theory of the nucleation of new phases, developed especially by Volmer,l by Farkas,2 by Kaischew and Stranski3 and by Becker and DOring4 indicates that under typical conditions * the self-nucleation from vapour of new crystals, new liquid drops and fresh two-dimensional mono- layers of molecules on a “ saturated ” crystal face require respectively supersaturations of the vapour by factors of typically 10, 5 and 1.5 respec- tively in order to proceed at appreciable rates. Experimentally, the first two of these figures are apparently correct : but the third is much larger than the actual supersaturation required to cause further growth of a crystal already formed.In fact, the existence of a critical finite supersaturation for further growth has only been established for a few materials, and then for individual faces of individual crystals, being different from case to case ; at the most it is about I yo. Volmer and Sch~ltze,~ who found a critical supersaturation of 0.8 yo for the growth of an iodine crystal from the vapour, interpreted this as the critical supersaturation for two-dimensional nuclea- tion : but the quantitative discrepancy is far too great (for details of the growth rate formulzJ see the contributions of Burton and Cabrera to this Discussion). However, this discrepancy is not in the least surprising. One ought not to expect that any visible crystal will exhibit a completed perfect face needing fresh two-dimensional nucleation in order to grow.Investigation of the mechanical properties of solids shows that no macroscopic specimen ever exhibits the theoretical strength of the perfect crystal; and this enormous discrepancy (a factor of 100, say, and more for “good” 1 Volmer and Weber, 2. physik. Chem., 1926, 1x9, 277. Volmer, Kinetik dev Phasen- 2 Farkas, 2. physik. Chem., 1927, 125, 236. 3 Kaischew and Stranski, 2. physik. Chem. B, 1934, 26, 317 ; Physik. Z . , 193 j, 36, 393. 4 Becker and Doring, Ann. Physik, 1935, 24, 719. bildung (Leipzig, 1939). Volmer and Schultze, 2. physik. Chem. A, 1931, 156, I. * Typical conditions may be taken as such that the vapour pressure lies within a few powers of 10 of I mm. Hg : or the temperature between about 0.5 and 0.S times Th, the boiling point in “K.F.C. FRANK 49 crystals) is attributed to the presence of dislocations. In the early stages of the development of dislocation theory by Polanyi,6 Orowan and Taylor,8 only one aspect of the dislocation was recognized, in which the displacement direction was normal to the dislocation line. In 1939 Burgers drew attention to the ‘ I screw ’’ form assumed by the dislocation when the displacement is parallel to the dislocation line, and the developments of dislocation theory by Mott and Nabarro,lo Frank l1 and others emphasize the fact that disloca- tion lines can be curved and exhibit any orientation. Fig. I shows (a) as a continuous defonnation of a plane, (b) in a block model, the form of a simple cubic crystal when a screw dislocation emerges normally at the FIG.r.-The end of a screw dislocation. cube-face. I t is clear that when dislocations of this type are present, the crystal face always has exposed molecular terraces on which growth can continue, and the need for fresh two-dimensional nucleation never arises. If just one dislocation of this type emerges at the centre of the face, that crystal face can grow perpetually “ up a spiral staircase.” If there are two, respectively right- and left-handed, we can show that the terrace connecting them will grow indefinitely if the supersaturation is raised to such a value that the diameter, Zo, of the critical two-dimensional nucleus is less than the distance between them. More precisely, since the critical nucleus is not 6 Polanyi, 2.Physik, 1934, 89, 660. 7 Orowan, 2. Physik, 1934, 8g, 634. 8 Taylor, Proc. Roy. SOC. A , 1934, 145, 362. Burgers, Proc. Kon. Ned. Akad. Wet., 1939, 42, 293. 10 Mott and Nabarro, The Strength of SoZids (Physical Society, London), 1948, p. I , 11 Frank, ibid., p. 46.50 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH circular, we should say that growth occurs when the critical nucleus, correctly oriented, will pass between two points in the positions of the two dislocations. Likewise when a single dislocation is close to the boundary of the face, growth occurs if the critical nucleus will pass between the dislocation and the boundary. If it will not pass, the terrace rests in a position corresponding to a portion of the boundary of the critical nucleus (Fig.2). For rough calculations we may approximate the boundary as a circle of radius iZo, and we have a close analogy to the equilibrium (or non-equilibrium) of a bubble at an orifice. Measuring in molecular spacings, we have lo = cp/kT In a, where 0: is the saturation ratio : since the supersaturation is small, (a - I) may be written in place of In a. (p is the neighbour-neighbour binding energy of the crystal. For a rule of thumb we may use Trouton’s rule and take y / k T as 3-5 Tb/T. Thus at an absolute temperature of 0.6 of the boiling point of the material, and a supersaturation of I %, I, is about 600 molecular spacings : or 6000 at 0.1 74. In the theory of mechanical defonna- tion, it is commonly estimated that in an annealed metal crystal there are about 1oS dislocation lines intersecting each square centimetre.This appears to be the limiting perfection attainable in metals, and would be classed, in- deed, as high perfection from the point of view of X-ray diffraction. (This limit may well be connected with the handed screw dislocations ending in order-of-magnitude estimate, and does a crystal face. not necessarily apply to other mate- rials than metals: but it serves as a guide, and suggests that critical supersaturations of the order of a fraction of I yo are reasonably to be expected. We are perforce limited for quantitative discussion to the case of growth from the vapour of homopolar crystals, since the classical nucleation rate theory has not been quantitatively developed for any other case.But experimental indications, and such theoretical guesswork as we can make, suggest that the conditions governing the growth from solution of crystals, including ionic crystals, are substantially similar. It is possible, but less certain, that the same is true of growth from the melt. Perhaps the most distinctively different case is that of the growth, either from vapour or solution, of crystals of highly non-equiaxed organic molecules. There are indications that in such cases growth proceeds through the formation of adsorption films, dense, but differing in molecular orientation from the bulk crystal (e.g., liquid, or liquid-crystalline) within which subsequent rearrange- ment occurs. The general importance of dislocations for crystal growth accounts immediately for many observations, such as the individual behaviour of each crystal face, particularly on the microscopic scale, leading sometimes to such unexpected results as the formation of lath-shaped crystals for a lattice of cubic symmetry.Under steady uniform supersaturation the terrace attached to an isolated growth-promoting dislocation in a crystal otherwise perfect in its neighbour- hood will grow outwards in a spiral of which the spacing between turns, and the rate of their advance, will be uniform at a considerable distance from FIG. 2.-One right-handed and two left- impurity ‘Ontent-) This is Only anF. C. FRANK 51 the centre. Near the centre the rate of advance must be less, since the radius of curvature of the spiral terrace-line must remain less than +Zo.At given supersaturation and distance from other terrace-lines, the rate of advance, o, is an increasing function of the radius of curvature, being o when it is +?, and, say, v, when the terrace becomes straight. Supposing it increased very steeply to voo for a very small increase of the radius of curvature above +Zc, the inner portion of the spiral would be an arc of a circle of this radius, and the spiral would make vm/xZ, turns per second. The spacing of turns in the outer region of the spiral would then be nZ,. Actually v will not increase infinitely steeply to vm, so that the number of turns per second will be less, and the spacing greater, but still of the same order of magnitude : let us guess zxZo. Macroscopically this spirally terraced hill would appear to be a flat cone, with its sides inclined at an angle of $do radians to the true lattice surface-say, I min.for a supersaturation of I yo, and in other cases pro- FIG. 3.-The influence of random dislocations on crystal growth. portional to the supersaturation. We have so far neglected the dependence of the rate of advance of the terrace on its direction. On this account, instead of a flat cone, we shall have a more or less sharply defined flat pyramid having the symmetry appropriate to the crystal face. When growth spreads from an isolated pair of dislocations, respectively right- and left-handed, the observable result will be practically the same provided that the supersaturation suffices to make Z, less than their separa- tion. If it is not, no growth occurs at all.If it is, the terrace-line connecting them repeatedly spreads out on one side, wraps round, and meets itself on the other side : thus forming a short connecting terrace-line again, and an outward-growing closed loop. The rate of formation of such loops will be the same in order of magnitude as the rate of formation of turns of the spiral from a single dislocation. The result will be a flat cone or pyramid indis- tinguishable from the previous case. In the case that there is a random distribution of dislocations a variety of phenomena occur, exhibited pictorially in Fig. 3 ( I to 12). In these pictures there is supposed to be positive supersaturation, so that each terrace makes a curve convex on its cliff side, running from one dislocation t o another of opposite sense, or to the boundary of the crystal face.The supersaturation is supposed to be low, and fluctuating, so that in general52 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH only one terrace-line-normally the longest-moves at a time. In each picture the terrace due to move next is specially marked. Shading indicates the area covered by fresh growth after picture 1. Among points to be specially noticed are : (i) the holding up of one terrace-line behind another (pictures 2, 8, 9, 10) or at a dislocation connected to another terrace (9, 10, 11, 12) ; (ii) the way in which a close pair of dislocations connecting a terrace facing the advancing terrace break it and so impede its passage (many examples, especially pictures 4, 5, 6 ) . In these two ways, a dislocation pair holding a terrace facing either way is an obstruction to the passage of an advancing terrace ; (iii) the way in which the obstructions of type (i) are broken down (10, 11, a) ; .(iv) the relatively impregnable region in the bottom left-hand corner. In 6 growth into this region has ceased by obstruction of type (ii) : a new advancing terrace is held up again (at 9) by obstruction of type (i) : but this obstruction is more easily broken down, and by 12 all but a small portion of the crystal face has increased in thickness by at least one mono- layer. A third advancing terrace, now commencing, will finally overrun this strong-point : however, a continuous line of closely spaced dislocations could be totally impenetrable below a critical supersaturation. Obstruction of type (i) requires a little more consideration.One way in which type (i) obstruction can break down is the following. Each " pinned " terrace-line holds its equilibrium form by small statistical fluctuations back and forth. When two terraces lie together, the lower cannot fluctuate back under, nor the upper fluctuate forward over the other. Thus they exert a small effective repulsion on each other, and the lower one may be pushed beyond the critical curvature when the upper one arrives. Secondly, there is only a small region of seriously deformed crystal structure and hence of seriously reduced binding energy for new molecules, in the immediate neigh- bourhood of the dislocation. When the two terraces at such a point of obstruction face each other at an acute angle, as in pictures 9, 10, 11, only a small number of molecules need condense in these unfavourable positions to enable the terrace to link across and pass on.If the obstruction arises from a straight row of dislocations spaced I , apart, this angle becomes smaller as I, decreases, diminishing rapidly from 120' when I, becomes less than 21,. The obstruction will be ineffective when E, is significantly smaller than this value. Even when it is greater, a comparatively small number of mole- cules (say, 6 to 10) in unfavourable positions suffice to overcome the obstruc- tion, so that statistical fluctuations (negligible with regard to obstruction of type (ii)) can be effective. Thus obstruction of type (i) probably only imposes a delay rather than a total prohibition of growth.In view of this, one might suppose it a particularly important question whether the numbers of right-handed and left-handed dislocation-ends in a face are equal or not : the surplus of one kind might have very long terrace- lines linking them to the boundary. This is not the case for reasonably uniformly or randomly distributed dislocations, unless the relative numbers are very different. If, in an area A , n dislocations of one kind and m of another are formed randomly, like head and tail throws of a coin, the probable excess of one kind, n - m, is 1-35 (n + m)'/z 1.9 nl/a : but if it does not exceed m z ' l ~ (where n > m) the dislocations linked to the boundary can all be close to it, and the longest necessary terrace inappreciably longer than in the case n = m, i.e., still of the order of magnitude (A/n)*!a.In a systematically deformed crystal it is possible to have n > m, and then there is always a terrace-line at least as long as the (m + r)th furthest dislocation of the n from the boundary.F. C . FRANK 53 The most important aspect of the type (i) obstruction by a fence of dis- locations all of the same kind is that every boundary between two crystal blocks inclined at a small angle to each other constitutes such a fence (see Fig. 4, and Fig. IZ in ref. 9). The distance between the dislocations in the fence, measured in molecular spacings, is equal to the reciprocal angle of rotation between the two blocks, whether for " tilt," " twist " or more general boundaries, so long as the angle is small.A possible cause of the visible growth terraces sometimes seen advancing over a crystal face is that a number of molecular terraces have accumulated behind such a fence, and then been set free by a rise in supersaturation: but it is possible to think of alternative causes which can bring about the same " bunching " of molecular growth terraces into visible ones. I FIG. 4.-'' Twist " boundary- crossed grid of screw dislocations. I t remains to elaborate the concept of dislocations somewhat, to free it from the simplifications introduced by considering only simple cubic crystals, with cube faces. In the general case the important property of a dislocation is its displacement vector (or Burgers vector). If this vector has a component nonnal to the crystal face on which the dislocation line ends, there will be an associated molecular terrace in the face, promoting crystal growth.But there is also an important distinction between perfect dislocations, whose displacement vectors are lattice vectors, and imperfect dislocations whose displacement vectors lead in general from a lattice position to a twin-lattice I I 1: position. Such is the dislocation with displacement vector (-3, -3, -j), produced in the close-packed cubic crystal by omission of part of a (111) close-packed plane of atoms. This dislocation cannot glide,l2 but must lie in its (111) plane : ending in a (II~) or similar surface, it attaches a mole- cular growth terrace in the usual way-but every time this growth terrace reaches the trace in the surface of the missing plane of atoms, the terrace must pause unless there is a definite supersaturation.At this boundary the lattice is not continuous but has a translation-twin relationship. Instead of the usual 6 contacts per added atom, characteristic of cubic close-packing, one row of atoms added at this boundary make 7 contacts each, and the next row only 5. The latter loosely bound row will only be formed at a definite supersaturation, or by statistical fluctuation after delay. la Frank, Proc. Physic. Soc. A , 1949, 62, 202.54 INFLUENCE OF DISLOCATIONS ON CRYSTAL GROWTH We must give a brief account of the origin of the dislocations which, it is suggested, dominate crystal growth. The chief origins which have been thought of so far are : (i) Surface nucleation of layers in improper (e.g., twin) positions and proper positions simultaneously on the same face.Where these meet there is a dislocation. It must be remembered that the initial nucleation of the crystal always takes place at high supersaturation, much more than adequate for the Becker-Doring condition for surface nucleation on a perfect face. (ii) Formation of one-dimensional dislocations in the edge row of the growing terrace (cf., van der Menve’s contribution to this Discussion). Such one-dimensional dislocations have an energy similar to the latent heat of evaporation of a molecule, and consequently exist in thermal equilibrium. During rapid growth at high supersaturation they can be trapped in an edge row, developing into two-dimensional and thence into three-dimen- sional dislocations.(iii) The development of curvature in the growing crystal owing to the presence of impurities (a subject to be treated at length elsewhere). This ultimately leads to stress in the surface which demands a certain super- saturation for further growth, which can then continue if, and only if, dislocations are formed. (iv) When systems of dislocations are present (particularly su b-grain boundaries) it is probable that the stress they would cause in perfect crystal compels the formation of further members of the system in growth : i.e., sub-grain boundaries are propagated in lineage structure. (v) Aggregation of molecular vacancies into flat collapsed cavities (the edges of which are dislocation loops) whenever the temperature of a crystal is lowered.This is very likely the process responsible for the intensification of X-ray reflection (the so-called establishment of mosaic structure) when an organic crystal is plunged in liquid air. (vi) Plastic yield under mechanical stress : this is believed not to create dislocations ab initio but to multiply those already present.ll These various considerations indicate that the initial dislocations necessary for growth are formed inevitably in the conditions needed for nucleation : and that to secure the best attainable perfection thereafter we must ensure small supersaturation (this involves good stirring, or there will be large supersaturation at the corners when it is small at the centre of a face), high purity of materials, steady temperature and absence of mechanical stresses. The effect which dislocations have upon crystal growth produces a rather odd natural selection both of imperfection and perfection in crystals. The nucleation stage with high supersaturation makes a population of initial nuclei of varied, mostly rather great, imperfection. If growth is now carried out at low supersaturation, only a few of these seeds, in which dislocations are relatively far apart, will grow. The lower the supersaturation at this stage, the fewer and the more perfect the seeds which will actually grow. But the completely perfect crystal will not grow in any circumstances : the conditions which could cause it to grow would also soon make it imperfect. H . H. Wills Physical Laboratory, Royal Fort, Bristol 8.

ISSN:0366-9033    

DOI:10.1039/DF9490500048

出版商:RSC    

年代:1949

数据来源: RSC

摘要:

KINETICS OF THE FORMATION OF NUCLEI AND STATISTICAL THEORY OF CONDENSATION BY R. BECKER Received 25th February, 1949 In the theory of the formation of nuclei it has generally been assumed that in every gas there exist, in addition to single molecules (2, per unit volume), aggregates containing 2 , 3, . . . . n molecules, and if n is not too small these aggregates can be regarded as spherical drops containing n molecules each. I t may now be asked, what is the equilibrium number n of drops containing n molecules ? Provided that the vapour is not supersaturated there is only one solution to this problem. I t is possible to treat the problem from a thermodynamic standpoint by considering the equilibrium, or alternatively, kinetically by an examination of the rate of formation and disappearance of the number of drops containing n molecules.This may occur by means of an aggregate A,+x taking up a molecule, or by an aggregate An+I losing a molecule. A t equilibrium 2, must be a constant. This kinetic method of approach is less exact than the thermodynamic method as it is necessary to make certain definite assumptions concerning the rates of evaporation and condensation. However, it is superior to the thermo- dynamic approach in that it is applicable to systems which are not in equilibrium. For example, it is possible to determine kinetically the rate of change of 2, with time, and furthermore, the method leads to the solution of the problem of the frequency of formation of nuclei as a function of the degree of supersaturation. independently developed a statistical theory of condensation which must be regarded as one of the most important advances in statistical mechanics made in recent times. Mayer attacked the problem in a more general way by assuming that N molecules occupy a volume V and that between any pair of molecules there is a potential energy of interaction vr which depends only upon the distance apart r and which rapidly tends to zero as r increases.In an original discussion of the general integral of state Mayer shows that this may be expressed as a sum of terms represented by the symbols m,, m2 . . . . ml . . . . ; this series indicates that ml clusters containing I molecules are present. Even if the physical meaning of the clusters, which arises in Mayer’s theory from purely mathematical considerations, is not altogether clear, it is important to note that at temperatures which are not too high and for values of I which are not too low, the rnz terms of Mayer are the same as the drop number 2, of the earlier theory.Thus, fundamentally &layer’s theory indicates a method of calculation of 2, and in particular for the derivation of a numerical factor common to all values of Zn, hitherto uncalculated on previous theories. An important confirmation of this concept in Mayer’s theory is obtained if the potential energy associated with the earth’s gravitational field is introduced into the general partition function. It appears from this that individual clusters behave as particles of mass I times the molecular weight in relation to their distribution at various heights; i.e., the large clusters tend to sink to ground level.55 nA, +A,, About twelve years ago Mayer 356 KINETICS OF THE FORMATION OF NUCLEI A particularly brilliant aspect of Mayer's method is in its application to the Einstein treatment of the condensation of helium based on Bose statistics. The condensation is generally described as a difficultly conceivable process in the moment space but Uhlenbeck showed that it could be regarded as a true formation of clusters, analogous to that occurring for ordinary vapours in Mayer's theory. Even if no energy other than kinetic energy is introduced into the partition function the application of the Bose statistics to helium gas shows that an effect exists which amounts to a tendency for mutual attraction between the atoms, comparable to a potential energy of interaction, and which leads ultimately to the formation of clusters distributed in a gravitational field in the normal way.Thus the Einstein condensation may be regarded as a particularly simple case of Mayer's theory which has the special advantage that all the integrals concerned may be completely evaluated. Unfortunately its significance is lessened by the fact that it cannot take into account the van der Waals' attraction between the atoms which is decisive for the observed condensation of helium.56 The contribution of the theory to this particularly important phenomenon, which is so clearly related to the existence of helium 11, is not so great therefore as had been hoped. The Chemical Equilibrium.The methods outlined above may now be considered in somewhat greater detail. We shall consider first the reaction nA, + A n as a chemical equilibrium according to van't Hoff's method. Consider an equilibrium box in which there are 2, single molecules per ~ r n . ~ and 2% drops each containing n molecules ; let this box be connected to a vessel containing n mols of A, at a concentration 2,". We shall consider the work W , gained by a single molecule, when n mols of A, at concentration 2," are reversibly and isothermally mixed with one mol of the drops A,, at a concentration 2%". If this process is carried out in the usual manner using semipermeable membranes, so that the reaction takes place within the equilibrium box, then This reaction may also be carried out by first compressing the n mols from the initial concentration 2," to a concentration Zsat.(that of the saturated vapour over a flat liquid surface), then condensing the vapour to a liquid, and finally by producing one mol of drops from the liquid, each drop having a surface area Fn. If in this last operation the drops are regarded as macroscopic entities, the work done is OF,, where CT = surface tension, and the total work gained a t this stage in the process is therefore 2 " kTn log -1 - OF,. &at. There now arises a difficulty which is characteristic of the whole problem. In the above considerations we have arrived at a stage where the drops, regarded as macroscopic entities, are situated adjacent to each other, whereas in the statement of the problem they were described as gas molecules existing a t a concentration 2,".Actually we have departed from a strict thermodynamic cycle by considering the individual drops as being formed from the liquid one by one. The following solution of the above difficulty may be suggested. If Vn is the volume of a drop, it is possible to consider the drops situated adjacent to each other as a gas of concentration r/Vn. IfR. BECKER 57 this is permissible then the work gained in reaching the concentration 2%" is - kT log Zn"V,, hence Z " &at. n log - 1 - 1OgZn"Vn As W = W' then If F, the pressure vapour pressure, is If we substitute Kn/n for the somewhat uncertain quantity 1'n-l then we have in which the factor Kn depends on n in a manner which is not accurately known.The total number of molecules is N N = XnZ,,. . ' (4 I This series converges only when p < 9,. For p > p,, however, it diverges. Zfi considered as a function of n has a minimum value and from this value the terms increase indefinitely. The minimum may be calculated in the following way. Let In = radius of a drop containing n molecules ; then (vfiq. = the specific n'i; and therefore, From this we have 4r 3 Va = - rn3 = n . vfiq. volume of the liquid) ; hence Ffi is proportional to Here p a is the equilibrium vapour pressure of a drop of radius rn, i.e., Therefore the series of numbers Zn in (I) approaches so closely to zero for p < 9, that ZnZ,, converges ; for p > p,, however, it may be seen from (3) that the series has a minimum value for that value of the drop size for which the vapour pressure f i n is just equal to the given vapour pressure 9.In the latter case therefore equilibrium is impossible and the kinetic theory must be used. The Kinetic Treatment. In the kinetic treatment we consider the growth and disappearance of the drop separately. Let a. = the number of single molecules arriving per sec. per cm.2 at the surface of the drop An, and 4% = the number of molecules which evaporate per sec. per cm.2 from the surface of a drop A,. The ratio a,/qn is equal to $ / f i n , i.e.,58 KINETICS OF THE FORMATION OF NUCLEI The number of processes A,, -+ An+r occurring per second is given on these assumptions as Z,, .F,+Iao, and the number of processes An+x -+An as Z n + I . Fn+I . qn+x. The excess of the latter number over the former is designated by J , where J = adin+= (..- z ~ + ~ ~ p":') For the equilibrium condition, J = 0, and an expression essentially the same as (I) is obtained again for 2,. If, however, p >poo then all 2, values for n > ?Lk may be placed arbitrarily equal to zero (i.e., any nuclei formed are removed). The quasi-stationary state where J is independent of n may then be considered. Thus it follows from eqn. (8) that by eliminating all the terms Z2, Z3 . . . . Znk a value for J is obtained, and this may be called the frequency of formation of nuclei. Thus for example one obtains where A = Q a F k , the work which must be done isothermally and reversibly to produce one critical drop. In this theory N atoms are con- sidered with a potential energy v(r) dependent only on distance and the corresponding partition function c is Mayer's Theory of Condensation.If the integration is carried out with respect to the momentum then A, the de Broglie wavelength at a temperature T , may be written as h = h(znmkT)-''s and the term e - n as I + f ( r ) . Then c = (I +fi2) (I +fi3) . . . (I +fj~) . . . . dr,. . . . d?". . (4) If therefore an The f ( r ) values only differ from zero for small values of r. integral such as J f12f23dr1dr2dr3 has to be evaluated, then the integration may be performed firstly for a given r1 from o to a over r1 and r2. Only the last integration over rl contains the volume factor V . In separating the terms in (4), all those may be taken out in which, for example, the first three particles form a cluster at E = 3.These terms will be the ones containing the factors fiZf23 or fi2fi3 or f12f13f23 and in which the indices I, 2, 3 do not otherwise occur. From the sum of all these terms a factor c may be split out where the term b3 is defined so that it no longer contains the volume. From the sum remaining after the elimination of this factor the terms containing f4, and in which the indices 4, 5 do not otherwise occur may be extracted. Proceeding in this manner may finally be split up into terms cm,, m,. . m l . . , where the individual sums are represented by : (5) m, clusters containing I atom, m2 ,, ,, z atoms, mL ,, S J > )R. BECKER 59 With b defined by { fi2 f23 . . . . fLI, i + . . . . . 1 dr1 . . . drz . * (6) for the contribution of such a series of terms, (Vl ! bl)"l 1 is obtained.I Now in general there are N ! II ~ possibilities of distributing the N The final value obtained for c is 1 l!mlml! particles in the clusters given by (5). therefore m l The Z is taken over all values of the series m,, m2, . . which satisfy the condition Z lml= N . If one of the terms in the sum is relatively so great that for thermodynamic purposes it may replace c, then the indices ml of this term give the most probable values Zl for the numbers ml of the clusters containing I molecules. 1 The appropriate calculation gives = VbiAll, . (8) where the value of the parameter A is given by the condition N = XZml, i e . , by N 1=1 N = VClbLA' . ' (9) Using this approximation log% = - 3NlogA + log N ! - N log A + VZbi A' 3 JV is obtained.From this it follows that the pressure p = - kT __ log c, or Hence the clusters introduced by ( 5 ) affect the pressure as independent particles of an ideal gas. Expression (9) is of particular interest to us because it gives the value of A as a function of the given density N/V of the substance. If A , is the convergence limit of the power series XZbiAz, then the finite sum of (14) assumes enormous values as soon as A > A,. For increasing values of NJV, therefore, A can only increase up to this limiting value. For further increases of N / V , A retains the constant value A , and hence, from (8), the term Zl/V has also a definite value, i.e., blAl,. We are then in the region of condensation, where an isothermal diminution of volume only causes an increase in the amount of condensate but does not give rise to any change in the vapour phase.Also, as may be seen from (IO), the pressure remains independent of the volume in this region. If the convergence limit A , is introduced for the series (9) in (8) and if where 21 is the concentration of the clusters containing I atoms, then60 KINETICS OF THE FORMATION OF NUCLEI From this equation a comparison may be made with the previous equilibrium eqn. (6) for spherical drops, by taking and From (8) it is immediately seen that the parameter A signifies the concen- tration of the single molecules. In this the potential energy is not at first considered. In the calculation of the partition function The Bose-Einstein Condensation. I for N helium atoms in a volume V , E, has the form I - [El2 + ......+ &7. 2m If 'p.(rl . . . . . . r ~ ) is the symmetrical eigenfunction belonging to the energy E, of the total system with 19 I2dY1 ...... drhr = I, J then ?, may be written in the form This sum, taken over all eigenvalues, may be accurately evaluated and gives the result The summation must be carried out over all the N ! pennutations P of the spaces q. In a particular permutation, for example, rl should be taken as rp1, r2 as rp,, etc. Thus, a single partial integral arising in this manner from (12) is similarly to eqn. (4a) of Mayer's theory discussed above. of length I which occur within it. may be easily evaluated. Defining the value of b again, this time by Each single permutation may be denoted by the number mi of the cycles In this case the appropriate integrals With this value of hl, the partition function given by (7) may be formulated, and hence the numbers of the corresponding clusters given by (8) can beR. BECKER 61 obtained. The saturation density may also be accurately calculated. For the series resulting from the combination of (9) with (13)~ i.e., I = > EblA1 = - A3 2 (Ah3)1, 1 1 V there is a convergence limit at Ah3 = I, and in this case it yields the limiting value This corresponds to the well-known fact that the Einstein condensation begins when the single atoms have only the volume A3 available to each of them. In this equation h = h(axmkT)-'/*, i.e., the de Broglie wavelength corresponding to the temperature T. 2-61 Institut fiir theoretische Physik der Universitat Gottingen, Gcttingen, Bunsevtstrasse 9. 1 Volmer, Kinetik der Phasenbildung (Dresden, 1939). 2 Becker and Doring, Ann. Physik, 1935, 24, 719. 3 Mayer, J . Chem. Physics, 1937, 5, 67 ; and subsequent volumes. 4 Mayer and Goeppert Mayer, Statistical Mechanics (New York, 1946). In particular 6 Born and Fuchs, Proc. Roy. SOC. A , 1938, 166, 391. 6 Kahn, Dissertation (Utrecht, 1938). Chap. 13, 14.

ISSN:0366-9033    

DOI:10.1039/DF9490500055

出版商:RSC    

年代:1949

数据来源: RSC