摘要:

Furuduy Discuss., 1993, 95, 1-2 Introduction J. N. Sherwood Department of Pure and Applied Chemistry, University of Strathclyde, Thomas Graham Building, 295 Cathedral Street, Glasgow, UK GI 1XL 1949 saw the publication of the seminal Faraday Discussion on Crystal Growth. At that time this scientific area, which can be regarded as one of the most ancient of scientific crafts, had seen some major theoretical developments and the subject served well as one of the opening series of the Faraday Discussions. Crystal growth was emerging from being ‘still in the alchemical stage’ (a quotation made in the closing remarks to the Discussion). It was beginning to see the initiation of that most essential blend of theoretical development and practical experience which leads to major scientific advances.Discussion No. 5 served to bring together the theorists and experimentalists to initiate the interaction between the two activities. Looking back at the published volume, one finds all the basic elements which are recognised as having underpinned the development of the subject. Stranski’s papers on equilibrium form show the relationship between the crystal structure and the morphology and surface structure of the material. The series of papers by Burton, Cabrera and Frank relate this well to the growth process and to the influence of sub-structural defects (notably screw dislocations) on the growth process. These contributions laid the essential foundation for the development of the modelling of the growth process and the expansion of all aspects of the theory of crystal growth which have developed to the present day.Additionally, we also see, possibly for the first time, the confrontation of theory and experiments on pure systems. The consideration of the growth process is important not only for the scientific development of crystal growth but also for its technological applications. Crystallisation is probably the most common unit process in the chemical industry. Its understanding and control to produce a well defined product have many ramifications in downstream processing. Industrial crystallisation processes take place in the less favourable environ- ment of synthetically impure solutions. The impurities influence both growth process and crystal habit.Experimental studies of these phenomena and their potential use in controlling habit were also represented in the original Discussion bringing the industrial community into contact with the fundamental scientists. The studies reported were devoted principally to inorganic systems and it was freely admitted that the processes are ‘imperfectly understood’. The need for their better understanding was, however, signalled. Finally, the problems of mineral synthesis and the associated art of producing large single crystals for device production were addressed. The latter papers are devoted to what one might regard today as simple materials for optical and piezoelectric applications grown by relatively simple procedures. As we all know, industrial developmment of the growth of semiconductor and optical materials from this simple base was to become rapid and to have a phenomenal impact on both science and society.Thus the 1949 Discussion signalled the beginnings of what were to be major changes in the development of the understanding of the process ofcrystal growth, which were to have a major influence across the whole range of science and technology. The past forty years have seen many developments in the better understanding of the theoretical background to the crystal growth process both in a static and a dynamic sense. The development of the theories of intermolecular and interionic forces have enabled the better prediction of a whole range of phenomena associated with the equilibrium nature of 1 Introduction surfaces and surface processes.Based principally on the development of the Hartman- Perdok theory, these allow not only predictions of equilibrium morphology but also its modification in the presence of solvent and impurities. It has proved possible to extend this to consider the effect of kinetic factors and the development of important processes such as thermodynamic and kinetic surface roughening under extremes of operational conditions. Such variations become extremely important in the operation of industrial crystallisers where they have a marked influence on the nature and purity of the final crystalline product. The development and application of molecular dynamics calculations to the crystallisation process has also played a major role in theoretical development.A major contribution to these developments has been the full acceptance of the role of structural effects on the crystal growth process. This has led to a number of important contributions of which two are particularly worthy of note: first, the absolute identification of sub-structural defects such as screw dislocations and the definition of their role in the growth process. This gave considerable support to the early theoretical approaches and gave confidence as to their viability. Secondly, and more recently, the seminal studies of the group from the Weizmann Institute group on crystal habit modification and in which the influences noted by the earlier workers were explained and defined.These and other developments are of course aided and underpinned by the definition of new and improved experimental techniques which can be used to access data of increased detail and reliability and to examine the nature and development of the crystal surface under operational conditions. Foremost amongst these is of course the computer in all its manifestations. Unavailable to our colleagues of 1949 this instrument has played the greatest part in the development of the theory of crystal growth. In parallel, however, developments in interferometry and microscopy in all its forms from optical to electron, and now atomic force microscopy, allow us an in situ view of surfaces of size from 1 cm2 to 1 nm2. The development of synchrotron X-ray sources and imaging detectors have also opened the way for the detailed structural examination of interfaces. Using the combination of these techniques considerable advances have been made in the understanding of the crystal growth process.There remains, however, much to be done before we could claim to be at the stage where crystal growth could be said to be a predictable process, although we are in almost that position in some areas. Consequently, it seemed appropriate to take stock of the present situation and once more to bring together the principal exponents of the science and technology of crystal growth to discuss the present state of the art. In considering what had been achieved the organizing committee decided that the major outstanding problems requiring solution lay in the more detailed consideration of the ‘interface’ including particularly, the solution phase, which has probably been the least addressed part of the whole process over the years. It was decided therefore to focus the present Discussion on ‘Crystal Growth-Equilibrium Structure, Interface Kinetics, Lattice Defects and their Interrelationships’ and as in 1949 to bring together today’s theoretical and experimental crystal growers to summarize the state of the art and to define future developments.

ISSN:1359-6640    

DOI:10.1039/FD9939500001

出版商:RSC    

年代:1993

数据来源: RSC

摘要:

Faraday Discuss., 1993, 95, 3-10 Explanation for the Occurrence of {hklm}Faces on Modulated Crystals P. Bennema,* M. Kremers, H. Meekes, K. Balzuweit and M. A. Verheijen RIM Laboratory of Solid State Chemistry, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands The concept of a morphological theory extended to include a superspace approach is presented. The aim of this theory is to be able to treat the case of incommensurate modulated crystals and consequently to understand the existence of satellite faces. The case of a one-dimensional modulated crystal is discussed. The science of crystallography started with the discovery that crystallographic faces fulfil the law of rational This law implies that each crystal face is characterised by a set of integral indices (hkl),the so-called Miller indices, which determine a reciprocal vector k perpendicular to the face.k = ha* + kb* + lc* (1) where a*, b*, c* are the basis vectors of the reciprocal lattice. The set of symmetry- equivalent faces to which (hkl)belongs is called a crystal form and is denoted by {hkl).The relative morphological importance (MI) of crystallographic forms {hkl) on growth or equilibrium forms (habits) of crystals can be derived from the law of Bra~ais,~ Friede14 and Donnay and Harker5 (BFDH), which can be formulated as follows: Ik,I 'Ik2l +MI1 <MI2 (2) that is, the smaller the norm Jkl,the higher the MI. The corresponding interplanar distance dhklis equal to the inverse of k. Allowed values of k obey the space-group selection rules.The BFDH law has been quite successful in predicting the morphology of numerous crystals."* Recently, the law of rational indices and to a certain extent also the BFDH law has been generalised to describe crystal faces that occur on incommensurate displacively modulated crystals with an average p-K,S04 structure such as Rb2ZnBr4,9 ((CH3)4N)2ZnC1410>1 and the mineral calaverite (AuTe2).12-15 For these incommensurately modulated crystals, the vector k perpendicular to a crystal face does not belong to a three-dimensional reciprocal lattice, but to a so-called Fourier module, which consists of all integral linear combinations of the three basic vectors a*, b*, c* and the modulation wave vectors ql,..,qd.Thus the rank of this module is equal to 3 + d and its dimension is three.Accordingly, the corresponding crystal face can be defined by 3 + d integral indices (hklml...md)such that: k = ha* + kb* + lc* + m,q, + ... + mdqd (3) The examples mentioned above are one-dimensionally modulated, so that in these cases d = 1. The modulation wavevector is indicated by q, defined as: q = aa* + pb* + yc* (4) where, at least one of the coefficients a,p, y is an irrational number. Note that allowed values of k again obey the selection rules imposed by the symmetry of the crystals. In the case of 3 (hklm) Faces on Modulated Crystals incommensurate modulated crystals the symmetry is described by a 3 + d dimensional superspace group.I6 This can be seen as an extension of the BFDH law.The morphological importance of a crystal face, however, seems to be determined not only by the interplanar distance because of the new types of faces that occur on modulated crystals which are called satellite faces and are characterised by having some non-zero index m. Indeed, one then expects that the orientation and magnitude of the modulation wave also represents relevant parameters for the MI of those faces. In any case, the morphology reflects the modulation properties and it has been possible to determine, from the orientation of satellite faces, the modulation wavevectors q with high precisi~n.~-~~J~.~~The morphologically determined values of q are consistent with values determined from X-ray19,20 and neutron diffraction experiments.21 The selection rules following from the relevant four-dimensional superspace group, leaving the incommensur- ate crystals given above invariant, could be shown to be compatible with the available morphological data.22 In some cases, experimental information on the MI is even sufficient to fix the superspace group itself.One can therefore claim that at the present level of knowledge the superspace group approach is supported not only by X-ray structure determination but also by crystal morphology. The occurrence of (hkl)faces on classical crystals fulfilling the symmetry of one of the 230 three-dimensional spacegroups can be explained with the integrated roughening and Hartman-Perdok theorie~.~~-'~In order to predict the morphology of these crystals the following procedure is carried out.(1) For the crystal structure under consideration the overall bond energies between (first-nearest) neighbour growth units (atoms, molecules, complexes) is determined. (2) The growth units are then reduced to centres of gravity or charge and the crystal graph is determined. This crystal graph consists of the centres of gravity or charge and the bonds between these centres. (3) The crystal graph is partitioned in slices with an overall thickness dhkl,the interplanar distance. It is checked whether within these slices the so-called connected nets can be identified. By definition a connected net consists of a net with an average thickness dhkl,where all points of the crystal graph are connected to each other by bonds.Connected nets show in principle a roughening transition. This implies that an absolute roughening temperature TRcan be identified. By definition a connected net consists of a net with an average thickness dhkl,where all points of the crystal graph are connected to each other by bonds. Connected nets show in principle a roughening transition. This implies that an absolute roughening temperature TRcan be identified. If a crystal face is growing below this roughening temperature it will grow with a layer mechanism (spiral growth or a 2D nucleation mechanism) with an orientation (hkl).If a certain orientation (hkl)is growing at a temperature above the roughening temperature, this face will grow as a roughened, rounded face with no fixed orientation (hkl).(4) It is possible to define a hierarchy of faces in the sense that faces having, in principle, the highest dhkl,which means the highest energy content per growth unit, will have the lowest growth rates and hence will have the highest MI. Using such hierarchies growth forms with a high heuristic value can be constructed. It is interesting to note that connected nets correspond to cusps in a raspberry Wulff plot according to Herring,23 26 It has also to be observed that the faces that dominate the 8rowth or equilibrium forms are those having the highest dhkl,the highest energy content (Eir),the lowest surface energy or attachment energy and the deepest cusps in the Wulff- Herring plot.So far only a mathematical recipe for describing the (hklm)orientation of satellite faces that occur on modulated crystals is known. No physical explanation for the occurrence of satellite faces is developed so far. Dam2' found from in situ observations that a satellite face that occurs on a modulated ((CH3)4N)2ZnC14 crystal growing from an aqueous solution P.Bennema et al. showed rotating spiral-like patterns. This suggests that this satellite face behaves as a pseudo-classical face. It is the aim of this paper to show, starting from simple physical principles and using the higher-dimensional crystallography of de Wolff, Janner and Janssen, 16,28 that the occurrence of (hklm)faces on modulated crystals can be explained.It will be shown that in a sense the integrated Hartman-Perdok roughening transition theory can be generalised to include the case of modulated crystals. Modulated One-dimensional Crystal Embedded in a 2D Superspace Fig. 1 shows the simplest possible modulated one-dimensional crystal embedded in 2D superspace. The 2D translational invariant superstructure is spanned on the basis of a, and d, according to the superspace- description. The Fourier module of the 2D space corresponding to a ‘standing-wave’ crystal consisting of transversal waves can be recognised. The d, axis is chosen to be perpendicular to the real a axis. The points of the modulated crystal are generated by the intersections of the real one-dimensional space (the horizontal axis) with the transversal ‘world lines’.The translational symmetry of the 2D superspace corresponds to translations na combined with a phase shift nq*aalong the d, axis. The 2D vectors a, and d, are characterised by their 1D components in the following way: a, = (a, -ad) (5) ds = (07 d) (6) where d is a vector in the d, direction. The reciprocal 2D vectors are then defined as: a: = (a*,0) d,* = (4,d*) Fig. 1 Two-dimensional superspace representation of a modulated one-dimensional crystal. The hatched area corresponds to the bonds cut by the faces (01). The symbols are referred to in the text. (hklm)Faces on Modulated Crystals Thus, qrepresents the projection of d$ on the real a axis. Pis chosen such that the following conditions are fulfilled: a*a*= I &d*= 1 (9)a-d = 0 a*9d = 0 Consequently: a,-a: = 1 d;d: = 1 as-d,*= 0 a$*d,= 0 Eqn.(10) represents the usual relations between the direct and reciprocal lattice. It follows from the conventions used that Iql = b*l = laial (1 1) We assume that for a real crystal a is fixed. If we choose the angle between the real axis and a, to be p = 45", then the phase along the d, axis is measured with the same unit of length as that of the real axis a. The only free parameter left is then the amplitude A of the world lines. Thus, the two periodicities present in the crystal, namely the average lattice parameter (la/) and the modulation wavelength (l/lql) are represented by a and ds, respectively. Concept of Chemical Bond in a Modulated Crystal: Selective Cuts Returning to Fig.1, which represents a modulated real one-dimensional crystal embedded in a two-dimensional superspace, the points of the crystal are given as: X-n, X-,+l,...,XO, XI, x2,...x, ,... with x, = na + A sin(q*na+ t) (13) where t represents a phase shift of the modulation with respect to the origin. The distance a(n) between the adjacent points x,+ and x, is given by a(n) = a + A sin[q-(n + l)a + t] -A sin(q*na+ t) (14) Using eqn. (1 1) we find a(n) = n + A sin[a(n + 1) + t]-A sin(an + t) (15) a(n) is a measure of the bond strength @[a(n)] between the neighbouring points x, + and x,. We implicitly restrict the bonds to the direction of the real space, i.e. in Fig.1 only the horizontal bonds are relevant. In order to obtain a survey of all possible bonds the information of the 2D space will be used. If the phase factor t is varied (01t12~) by travelling along the d, axis the whole continuum of horizontal bond distances is engaged within the period d, of one elementary cell of superspace. Travelling an infinite distance along the real space of the modulated crystal, all distances, a(n), of the continuum of distances 'stored' in the elementary cell of superspace will be met. Nevertheless, the order in the continuum of distances which is realised along the real axis is, though completely determined, mathematically complicated. The points of the modulated crystal are completely determined by a, d, and A. P.Bennema et al.In order to illustrate the identification of distances, equidistant netlines characterised by the Miller indices (Ol), (li)and (10) are drawn in Fig. 1. Since the 2D space is translationally invariant, concepts of classical mathematical crystallography like net lines can be used. We generalise the notion of mesh area hfhk[, for which it holds that the volume v of the unit cell is equal to hfhk&&, in the case of our 2D superspace. Then hfhm stands for a length and V is the area of the unit cell in superspace. Looking at Fig. 1, one can see that a grid of equidistant parallel (01)lines cuts the real modulated 1Dcrystal in equidistant pieces. These observations lead to the following results, which yield the essential ingredient of this paper. The grid (01) cuts only a fraction of the whole continuum of distances (bonds). This is because, as can be seen from Fig.1, only those distances which are cut by the mesh area of (01) occur in the cuts of the grid with the modulated crystal. Thus, it can immediately be seen which fraction of the continuum of distances stored in superspace will be cut by the grid (hatched areas). Changing the phase t will implicitly change this fraction. The total energy involved for a grid (hm)is therefore a function of the phase t. We will refer to those grids (hm)as faces (hm). In order to calculate bond energies the interaction potential between adjacent points @[a(n)]of the modulated crystal is used. With this potential the specific energy Eh,,,(t), i.e.the sum of the energies of all bonds cut in a mesh area divided by the area, can be determined for each face (h,m)as a function of the reference phase t.In this way a minimum specific energy E:,~ is found for a certain phase The modulated one-dimensional crystal is cut in pieces, having on the average a lowest cut (surface) energy and a highest energy within the pieces. Here a kind of generalisation of the Hartman-Perdok theory occurs for a modulated ID crystal embedded in a 2D superspace. This generalisation implies that the modulated 1 D crystal is cut in pieces having on the average a lowest Eattand a highest Eslice. We note that looking at other grids determined by equidistant lines (hm) like for example the (17) grid in Fig.1,it can be seen that the mesh area becomes larger than that of (01). The relevance of the phase t and the mesh area becomes clear in this figure. The findings of this section concerning the selectivity of cuts for a face (hm)can be generalised to a modulated 2D crystal embedded in 3D space and a modulated 3D crystal embedded in 4D space. In the latter case it can be checked whether connected nets are present within a thickness dhkrrn.l4 wulff Plot 2D Plot for Non-modulated Crystals Consider a one-dimensional unmodulated and translationally invariant crystal. This crystal can be obtained from Fig. 1 by making the amplitude A zero. The continuum of bonds of Fig. 1 now becomes a set of equal bonds. In order to calculate specific energies expressed by bond energies divided by some unit of length, we proceed by using the construction as suggested by Herring.26 Thus we write Y($hrn) = cos 4hm (16) where y is the surface energy unit of length and $ is the angle between the normal of the face (Am)and the horizontal line giving the orientation of the face in reference to the continuum of horizontal bonds (h,m are considered as continuous variables) and D is a proportionality constant.Making a polar plot of y yields a plot consisting of two circles. Note that for the face (10) 4 is zero, corresponding to the maximum value of 7. The face (hm)with h = -ma corresponds to the only very steep cusp. The reason is that for the latter face no bonds are cut. The only function of the extra dimension (dsaxis) is to store data (bonds) with their proper phase in 2D space.Using the principles of a Wulff plot construction it can be seen that all orientations (hm) cut the real axis in the same point. This means that all orientations (hm)have the same (hklm)Faces on Modulated Crystals contribution to the equilibrium form, which is a piece of a line with a surface energy zero corresponding to the orientation (10) and an edge energy @(a). As could be expected, it does not make any sense to embed the one-dimensional truly translationally invariant crystal in a 2D space, because no new faces (hm)show up. This situation becomes radically different if a modulation is switched on. Principles of 2D Wulff Plot for a Modulated Crystal In order to proceed, we have to calculate the specific energies for the different faces of a modulated crystal.Instead of labelling the different bonds with n, that is in real space, we use the phase t for this purpose. Implicitly we use that all relevant bonds are engaged, through in a different order, in the correct number of occurrence along the d,axis. Therefore we write a(t)= a + A sin[a(n + 1) + t] -A sin(an + t) (17) The integration runs over a complete mesh area Mhmof the face (hm)in formula Now the reference phase has to be chosen in such a way that the energy becomes minimal. In the exceptional case of (hO) the energy has been minimised by varying the position of the face along the real axis. Results of the Calculation In order to perform the calculations mentioned in the previous section, we chose the example of a 1 + 1 dimensional superspace, describing a modulated crystal with q = 0.3a* and an amplitude of ]AI = 0.21al.For the interaction we used a Coulomb-like potential.In Fig. 2 the results are summarised in a Wulff plot. The spread in energy as a function of the reference phase is indicated by the radial lines. The minimum values correspond to the cusps of a Wulff plot. For these minima, the habit-determining lines are drawn. The actual value of the amplitude determines the size of the spreading interval. When the amplitude approaches zero, this interval shrinks to the value on the two-circle unmodulated case. For Fig. 2 Wulff plot for the crystal in Fig. 1, Faces with 3 as maximal index have been indicated.Note the steep cusp along the d, axis and the spread in energies for the different faces. P. Bennema et al. 9 our choice of the modulation wave vector the face (01) corresponds, apart from the face (am,a),to the deepest cusp. The order of the minimal energies obviously depends on the modulation wavevector. Conclusion In this paper we have demonstrated that the attachment energy of a face on a modulated one-dimensional crystal does not involve the whole spectrum of bonds present in such a structure. Depending on the face (hm)and on the phase t of the modulation only a well determined subset of this spectrum turns out to be relevant. The phase t acts as a minimising factor for this energy. These results are used to make a generalised Wulff-Herring construction in order to obtain a prediction of the equilibrium form of the crystal.In a forthcoming paper the effect of the modulation wave vector on the specific energy will be treated. Furthermore, it will be shown how our two-dimensional model which in principle explains the occurrence of ‘faces’ (hm) can be generalised to three and four dimensions in order to explain the occurrence of faces (hklm).Analogously, we then assume that a four-dimensional grid of hyperplanes (hklm) is partitioning a three-dimensional crystal graph in slices with thickness dhkbn. Assume that a 3D crystal graph is cut in slices with a thickness djklmand that the cuts have the lowest surface energy. It is then reasonable to assume that if the corresponding nets are connected by the bonds of the 3D crystal graph, the corresponding faces (hklm) will grow as flat faces, if the actual growth temperature is below the roughening transition temperature. Since, as a rule the value of djklmof a large number of faces is quite high, many (hklm) faces will be connected and will have relatively high roughening temperatures.This explains to some extent the very large variety of {hklm)forms that occur on calaverite crystals. The relatively high amplitude of the modulation in calaverite probably enhances this effect. In case the amplitude of the modulation is small, cusps in a 2D (and 4D) Wulff plot will be less steep. Consequently, the chance for the occurrence of satellite faces is smaller.This may explain the much lower number of (hklm) faces that occur in the structures having an average potassium sulfate structure, growing from aqueous solution. References 1 J. G. Burke, Origins of the Science of Crystals, University of California Press, Berkeley, 1966. 2 G. J. Schneer, Crystal Forms and Structures, Dowden, Hutchinson and Ross, 1977. 3 A. Bravais, J. Ecol. Polytech., 1850, 19, 1. 4 G. Friedel, Bull. Soc. Fr. Miner. Le Couscle Crystallographie, Herman, Paris, 1911. 5 J. D. H. Donnay and D. Harker, Am. Mineral., 1937,22,446. 6 F. C. Philips, An Introduction to Crystallography, Longman, London, reprinted 4th edn., 1978. 7 P. Hartman, Crystal Growth: An Introduction, ed. P. Hartman, North Holland, Amsterdam, 1973, pp.367402. 8 P. Hartman, Morphology of Crystals, Part A, ed. I. Sunagawa, Terra Scientific Publishing and D. Reidel, Dordrecht, 1987, pp. 271-319. 9 A. Janner, T. Rasing, P. Bennema and W. H. van der Linden, Phys. Rev. Lett., 1980,45, 1700. 10 B. Dam and A. Janner, 2. Krist., 1986, 165, 274. 11 B. Dam and A. Janner, Acta Crystallogr., Sect. B, 1985, 42, 69. 12 B. Dam, A. Janner and J. D. H. Donnay, Phys. Rev. Lett., 1985,551, 123. 13 B. Dam and A. Janner, Acta Crystallogr., 1989, 45, 115. 14 L. J. P. Vogels, K. Balzuweit, H. Meekes and P. Bennema, J. Crystal Growth, 1992, 116, 397. 15 K. Balzuweit, A. Hovestadt, H. Meekes and J. L. de Boer, J. Crystal Growth, submitted. 16 T. Janssen and A. Janner, Adv. Phys., 1987,36, 519. 17 L.J. P. Vogels, M. A. Verheijen, H. Meekes and P. Bennema, J. Crystal Growth, 1992, 121, 697. 18 B. Dam and P. Bennema, Acta Crystallogr., Sect. B, 1987, 43, 64. 19 E. Colla, P. Muralt, H. Arend, R. Perret, G. Godefroy and C.Dumas, Solid State Commun., 1984,52,1033. 20 W. Schutte and J. L. de Boer, Acta Crystallogr., Sect. B., 1988, 44, 486. (hklm)Faces on Modulated Crystals 21 K. Gezi and M. Izumi, J. Phys. SOC. Jpn., 1978,45, 1777. 22 P. Bennema, K. Balzuweit, B. Dam, H. Meekes, M. A. Verheijen and L. J. P. Vogels, J. Phys. D, 1991,24, 186. 23 P. Bennema and J. P. van der Eerden, Morphology of Crystals, Part A, ed. I. Sunagawa, Terra Scientific Publishing, Tokyo and D. Reidel, Dordrecht, 1987, p. 1. 24 P. Bennema, Sir Charles Frank, An Eightieth Birthday Tribute, ed. R. G. Chambers, J. E. Enderby, A. Keller, A. R. Lang and J. W. Steeds, Adam Hilger, Bristol, 1991, pp. 47-78. 25 P. Bennema, Handbook ofCrystal Growth, ed. D. J. T. Hurle, Elsevier, Amsterdam, 1993, vol. I, ch. 7. 26 C. Herring, Phys. Rev., 1951, 82, 87. 27 B. Dam, Phys. Rev. Lett., 1985, 55, 2806. 28 P. M. de Wolff, T. Janssen and A. Janner, Acta Crystallogr., Sect. A, 1981,37, 625. Paper 3/00139C; Received 8th January, 1993

ISSN:1359-6640    

DOI:10.1039/FD9939500003

出版商:RSC    

年代:1993

数据来源: RSC

摘要:

Faraday Discuss., 1993,95, 11-25 Theoretical Analysis of the Polar Morphology and Absolute Polarity of Crystalline Urea R. Docherty ZENECA Specialities, Blackley, Manchester, UK A49 3DA K. J. Roberts Department of Pure and Applied Chemistry, University of Strathclyde, Glasgow, UK GI 1XL and SERC Daresbury Laboratory, Warrington, UK WA4 4AD V. Saunders SERC Daresbury Laboratory, Warrington, UK WA4 4AD S. Black and R. J. Davey Crystal Chemistry Team, ZENECA, R & T Department, The Heath, Runcorn, UK WA7 4QE Changes to the molecular polarisation of urea associated with its crystallis- ation are considered through ab initio quantum-mechanical calculations starting with the isolated molecule with increasing complexity up to a full 3D calculation using periodic boundary conditions.The calculations accurately reveal the electrostatic nature of the intermolecular forces associated with the hydrogen-bonding network in the solid state. The resulting charge densities are used together with the force field of Lifson, Hagler and Dauber (J. Am.Chem. Soc., 1979,101,511I) in an attachment energy calculation to predict the polar morphology of the material. The resultant simulations are in good agreement with the morphology of crystals prepared from the vapour phase and are used to absolutely assign the polar forms as (1 I 1). Habit modification effects associated with re-crystallisation from polar solvents are also discussed from a structural perspective. The prediction, control and optimisation of crystal morphology has been one of the long- term objectives of crystal growers, crystallographers and mineralogists.Such an aim is also of particular relevance in the industrial sphere where undesirable crystal shape may have detrimental consequences in terms of dissolution kinetics, particle flow and rheology, solute filterability and product purity. Recently the combination of crystal structure data, intermolecular force calculations, quantum-mechanical methods, and the overall frame- work of the Hartman-Perdok’ (see also for example ref. 2) approach to crystal morphology have enabled significant advances to be made, particularly for molecular crystal^.^,^ Hartman and Perdok were among the first to attempt to quantify crystal morphology in terms of the interaction energy between crystallising units.They used the assumption by Born5 that surface energy is directly related to the chemical bond energies and identified uninterrupted chains of ‘strong bonds’ called periodic bond chains (PBCs). A PBC can contain a number of different bonds and the influence of a PBC on the overall crystal shape is governed by the weakest link in that chain as this is the rate-determining step for the speed of growth along the direction of the chain. Polar Morphology of Urea One of the most useful concepts that emerged from PBC theory was that of slice and attachment energies. The slice energy (Esllce)is defined as the energy released on the formation of a stoichiometric growth sli~e.~.~,~ The thickness, dhkl,of such a growth slice is defined through the usual symmetry rules, analogous to those governing X-ray extinction, evolved by Bravais,8 Freide19 and Donnay and HarkerlO usually (see also ref.11 and 12) referred to as the BFDH law which basically states that: ‘taking into account submultiples of the interplanar spacing dhk,due to space group symmetry the most important crystallographic forms will have the greatest interplanar spacings.’ The overall effect of the BFDH rules is effectively to reduce the importance of crystal forms which have a translational symmetry element acting along their growth normals. The attachment energy (EJ is defined as the fraction of the total lattice energy (ECJreleased on the attachment of this slice to a growing crystal surface, i.e.Ecr = Esl1ce + Eatt (1) The attachment energy was proposed as a measure of the relative growth rate of a crystal face. The faces with the lowest attachment energies will be the slowest growing and therefore have the most morphological importance. This assumption that the attachment energy could be used as a measure of the relative growth rate normal to a face was analysed by Hartman and Bennema,6 who showed it to be particularly valid at low supersaturations when the birth-and-spread and BCF13 growth mechanisms dominate crystal growth. By making a classical Wulff plot14 of Eat,as a function of the crystal face (hko normal, the relative growth rates of the various crystal surfaces can be predicted and the crystal morphology simulated. By using a representative intermolecular potential (U) such as that provided by the combined Lennard-Jones15/coulombic (6-12-1) function, i.e.: U = -A/r6+ B/r“ + 332.0g1q,/r (2) one can calculate the strength of the intermolecular bonds in the solid state which can, in turn, be related to the bonds involved in the surface attachment process.It is thus possible through the use of computer programs such as MORANG16 and HABIT” to calculate the slice and attachment energies based on the known bulk crystal structures on a fairly routine ba~is.4~12~18-22 In the area of additive and solvent effects on crystal growth it is now possible, for molecular crystals, to predict and calculate the influence of such external influences on crystal morphology. Thus, for example, calculated binding energies23 for benzamide crystals growing in the presence of benzoic acid as an impurity are now in excellent agreement with ob~ervation.~ Similarly the habit modification of the n-alkanes by iso- derivative^^^ can now be understood and quantified.Other environmental factors such as solvent effects can also be understood; for example such predictions on succinic acid25 are in excellent agreement with measured growth kinetics.26 The one area that has so far eluded theoretical progress concerns crystals which exist in non-centrosymmetric space groups which possess (a) polar __-direction(s). This structural polarity can result in the unequal development of (hk I) and (h k I) faces, leading to a polar morphology. One of the best-studied systems of this type is resorcinol in which the c-axis is the polar axis and the (0 1 I) and (0 Ti) surfaces, although symmetry related, grow at very different rates.27 It is often assumed that these kinetic differences arise from unequal solvent binding to the polar faces and indeed in the case of resorcinol growing from aqueous solution this does seem to be the case.28 Such an assumption is, however, by no means rigorous and indeed structural factors leading to polar morphologies have never been explored. This is primarily because the predictive models which are used (Hartman- Perdok) are based, as can be seen from Fig.1, on the calculation of attachment energies between stoichiometric slices of crystalline materials.These calculations by definition implicitly assume a centre of symmetry in the calculation and thus EJh k I) always equals R. Docherty et al. \I \\XII (hkI)surface Fig. 1 Schematic diagram showing how the classical attachment energy model, based on the bulk solid-state structure, does not allow the prediction of polar effects; note the intermzkc_ular bonds formed on attachment to the (hkl) face (X) are the same as those involved on the (h k l) face (Y) E,,,(h kI) because the coulombic part of the intermolecular potential function (the van der Waals part is, by definition, isotropic) in the bulk solid state contains no surface-specific interaction parameters. In order to examine this problem in more detail a simple organic material, urea O=C(NH,),, has been chosen.By calculating differences in the molecular charge distributions between a single molecule, molecules in the solid-state structure and more specifically those on the surfaces of the { 1 1 1 ]polar faces with the aid of ab initio quantum chemical calculations it is possible not only to predict a polar growth morphology without recourse to solvent effects but also to assign precisely those polar faces which will appear and hence determine the absolute polarity of the ‘as-grown’ crystals. Crystal Structure of Urea and the Nature of its Intermolecular Forces in the Solid State The crystal structure of urea has been of interest for a number of years due to its comparatively simple structure, its highly anisotropic thermal vibrations as well as it being the only example of a structure in which a carbonyl 0 atom accepts four N-H***O hydrogen bonds.Urea crystallises in the highly symmetrical tetra onal space group P42p with two molecules in a cell ofdimensions a = 5.576 A, c = 4.684 1.A full 3D structure was first provided by Vaughan and Donohue and other worker^.^^-^^ In this study we have used the atomic coordinates provided by the most recent structural as summarised in Table 1. The dominant solid-state intermolecular bonds in urea were calculated from the crystallographic data using the computer program HABIT” with the potential function given in eqn. (2) using force-field parameters (see Table 2) described by Lifson et al., 35 which are optimal for amides; the electronic charges used for this were those provided with this potential set.The calculated lattice energy obtained was -22.6 kcal molt’, which is in good agreement with the experimental sublimation enthalpy of 21 .O kcal mol-1,36 thus demonstrating the applicability of this force field to urea. The calculated intermolecular bonds together with their van der Waals and coulombic components are given in Table 3. It can be seen that the intermolecular forces are dominated by both interchain (two type a bonds) and intrachain (four type b bonds) hydrogen bonds (see Table 3). Thus for each urea molecule there are six dominant intermolecular bonds which when added together account for over 85% of the lattice energy.Fig. 2 shows a projection onto the (0 0 1) and (1 1 0) planes illustrating the strong type a and b bonding network in urea after which the next Polar Morphology of Urea Table 1 Atomic fractional coordinates for urea (after Swaminathan et ~21.~~) atom X Y Z + 0.0 + 0.5 + 0.3260 + 0.0 + 0.5 + 0.5953 + 0.0145 + 0.6459 + 0.1766 + 0.2575 + 0.7575 + 0.2827 + 0.1441 + 0.6441 + 0.0380 -0.1459 + 0.3541 + 0.1766 -0.2575 + 0.2425 + 0.2827 -0.1441 + 0.3559 -0.0380 Table 2 Details of the van der Waals intermolecular potentials used in the calculations (after Lifson et interaction atoms Alkcal k6mol-' Blkcal A-6 mol-' c-0 3428.0 3.81 x 10 C-N 5366.0 1.10 x 10 C-H 0.0 0 0-0 2098.0 1.15 x 10 0-N 3285.0 3.3 x 10 0-H 0.0 0.0 N-N 5141.0 9.49 x 10 N-H 0.0 0.0 H-H 0.0 0.0 Table 3 Calculated dominant intermolecular bonds in urea (units kcal mol-I) interaction energy fraction of position of bond total lattice second molecule attractive repulsive coulombic total type energy (YO) 00 11 -2.02 + 2.10 -3.73 -3.64 a 16.1 00-11 -2.02 + 2.10 -3.73 -3.64 a 16.1 10 02 -1.74 + 1.26 -2.53 -3.00 b 13.2 -11 02 -1.74 + 1.26 -2.53 -3.00 b 13.2 00 02 -1.74 + 1.26 -2.53 -3.00 b 13.2 01 02 -1.74 + 1.26 -2.53 -3.00 b 13.2 -10-12 -1.35 + 1.11 -0.41 -0.65 C 2.9 01 -12 -1.35 + 1.11 -0.41 -0.65 C 2.9 -11-12 -1.35 + 1.11 -0.41 -0.65 C 2.9 00-12 -1.35 + 1.11 -0.41 -0.65 C 2.9 ~~ All bonds are calculated from an origin molecule at U V WZ of 0 0 0 1, where U V Ware the basic unit- cell lattice translations and Z defines which of the two independant molecules is involved in the interaction.R. Docherty et al. Fig. 2 Crystal packing projections of the urea crystal structure showing the strong intermolecular bonds together with the structure of the { 1 1 01, (0 0 11 and { 1 1 l} slices; (a)projection onto (0 0 l} and (b)projection onto { 1 1 O} strongest intermolecular interaction, the type c bond, is nearly an order of magnitude weaker. In fact it is only the strong hydrogen-bonding network that enables this eight-atom molecule to crystallise at room temperature; note the fully methylated derivative (tetramethylurea) is a liquid at room temperature. The morphology of vapour-grown urea crystals, shown in the scanning electron micrograph (Fig.3), clearly illustrates that in the absence of a solvent, this material exhibits polar { 1 1 1) capping forms in a overall prismatic morphology which is bounded (see inset) by { 1 1 O] sides and (00 l} end facets. Urea crystals re-crystallised from polar solvents such as water and methanol, as shown in Fig. 4, show the same forms as those grown from the vapour, although their habit is more needle-like, being elongated along the crystallographic c axis. The surface chemistry of the polar { 1 1 l} surfaces is shown in the molecular packing diagram illustrating the projection of the bulk structure onto the (1 0) plane given in Plate 1. In the urea structure the two molecules in the unit cell are symmetry-related by the four- Polar Morphology of Urea Fig.3 Scanning electron micrograph of urea showing the morphology of urea crystals that results after crystallisation from the vapour phase at a temperature of ca. 203 K using a simple cold finger system (horizontal scale 0.36 mm). The figure illustrates the well-defined prismatic polar morphology bounded (see inset) by {1 1 0) sides with (0 0 1)end and polar (1 1 1) cap facets. fold inversion axis of the space group. Of these two molecules, one (labelled M 1) has its plane parallel to the projection whilst the other (labelled M2) is oriented perpendicular to the plane of projection. The net polarity for this growth slice can be understood by considering dipolar orientation of these two molecules in turn: molecule MI has the carbonyl group and one amino hydrogen protruding onto the {1 1 1} surface whilst the {I 1 1) face has two amino hydrogens at its surface; molecule M2, in contrast, has only amino hydrogens on the {1 1 l} surface whilst the {1 1 1 }face has the carbonyl group at its surface.Thus, summing these two molecular contributions, ___it becomes clear that the difference in the surface chemistry between the {1 1 1) and {1 1 1) faces responsible for the polar effect is very small, being due only to an additional amino hydrogen on the {1 I 1) face. Morphological Prediction based on the Classical Attachment Energy Approach The crystal morphology was predicted using the computer program HABIT.” This approach (see Fig.5) involves the calculation of all the interactions between a central molecule and the other molecules in 3D based on the crystal structure to a defined radius R.Docherty et al. Fig. 4 Scanning electron micrographs of urea showing the elongated crystal morphology that results after re-crystallisation from polar solvents (a) crystals grown from aqueous solutions (horizontal scale 2.4 mm); (b)crystals grown from methanol solutions (horizontal scale 5.8 mm) (typically ca. 30 A). The slice energy is calculated by summing the interactions between a central molecule and all the molecules within a slice of thickness dhkl.The attachment energy is calculated similarly by summing all the interactions between a central molecule and all the molecules outwith the slice.The centre of the slice can be defined as the centre of gravity of a molecule or an atom in the molecule or indeed at a lattice point of each of the symmetrically independent sites in the unit cell. The slice and attachment energies are then averaged over all these sites. The calculated attachment energies are given in Table 4 (column 2) and the predicted growth morphology, assuming Eat,is proportional to the centre-to-face distance used in the Wulff plot (drawn with aid of the computer program SHAPE3’), is given in Fig. 6(a). The results show that, as expected, the { 1 1 l} and (777)faces have the same relative growth rates Polar Morphology of Urea N* V I / I / adjust slice boundaries h kl)to maximise Eslice .limiting raaius +-for summationN Y Fig. 5 Basic approach for calculation of intermolecular interactions using atom-atom method showing how the lattice energy is partitioned between the slice and attachment energies within a limiting sphere. A is the central molecule, B is a molecule outside the slice, D is a molecule inside the slice, N+ is the growth normal to the planes (h k 0, N-is the growth normal to the planes (%Eq,AB and AD are bonding vectors, dhk, is the interplanar spacing, 8is the angle between the growth normal and the bonding vector and AC is the component of the vector AB parallel to N, the growth normal. Note that the slice-boundaries defined by dhk, may be shifted along the growth normal to obtain the energetically most stable slice.and that the polar morphology is not reproduced. Apart from this the overall crystal habit is in good agreement with that observed following growth by sublimation (see Fig. 3). Molecular Charge Distribution Calculations Given that crystal morphological changes can be effected through only subtle changes to the attachment energy the calculation of the polar effect and hence the absolute polarity of the crystal require accurate calculation of the partial electronic charges. Caution is also needed in using the charges provided with this force field which has been optimised by empirical fitting to crystallographic and thermodynamic data. Against this perspective our Table 4 Surface attachment energies (units kcal molt') for the important forms in urea calculated using the classical model and its refinement for predicting polar morphology attachment energy face classical bulk and isolated surface and isolated (hkl) model charge model charge model (1 1 0) -4.5 -4.2 -4.5 (0 0 1) -5.2 -4.5 -5.2 (1 1 1) -5.7 -4.5 -6.0 (TTT) -5.7 -4.9 -7.0 R.Docherty et al.1001t ti iit (001t l111tI 111 I11 ot 1ot Fig. 6 Simulated crystal morphologies for urea using the computer program HABIT with different charges for host and attaching molecules: (a)classical model with the host and attaching molecules have bulk atom charges, (b)polar simulation with host atoms having bulk charges and attagktg atoms isolated molecuIe charges and (c) polar simulation with host atoms having {1 1 11 and (1 1 I> surface charges and the attaching molecule having isolated molecule charges approach has been to calculate charge distribution over the urea molecule using ab initio quantum-chemical methods.To obtain self-consistent results we have used a series of calculations, increasing in structure complexity and ending in a calculation of the solid- state properties using full periodic boundary conditions. For the latter we used the program CRYSTAL38 running on a CRAY XMP48. For calculations on isolated molecules and molecular clusters the GAMESS code39 running on an FPS164 was used. For both approaches the STO-3G basis set was employed. The development, in recent years, of ab initio quantum-chemical calculations through the CRYSTAL38 quantum-chemical package has meant that we are now able to understand the influence of crystal-field effects on molecular p~larisation.~.~~ In terms of crystal growth such capabilities offer a potential route towards the prediction of surface charge densities so that additive and solvent effects can be modelled.In this work we use such calculations to predict and understand the nature of changes to the molecular polarisability of the urea molecule associated with crystallisation. These calculations reveal that the crystal environment favours a more ionic charge distribution than that predicted for the isolated molecule. This effect is illustrated in Fig. 7 which shows the electron interaction density, i.e. the difference between the charge density in the isolated molecule and that in the crystal.The data show the more ionic nature of the urea molecule in the condensed state and most strikingly demonstrate that all the peripheral zones of the molecule in the molecular plane have negative deformation densities. This confirms that in the crystalline state part of the intramolecular bonding electrons are transferred to the non-bonded intermolecular bonds associated with hydrogen-bond formation. Such fine structure in the electron density is usually obtained from careful structure factor measurements but even these fail to reveal the finer details afforded by the ab initio calculations. The resultant charge distributions are shown in Table 5, columns 2 and 9.Differences in charge are found for all atoms, the most significant differences being found on hydrogens. The differences result from the close hydrogen-bond interactions in the crystal which are not considered in an isolated molecule model. Some feeling for the changes to the electrostatic interaction due to intermolecular packing in the solid state can be obtained by using simple dimers and trimers as models.2s Comparison of the charge densities on a dimer pair (Table 5, columns 3 and 4) bonded by two type a hydrogen bonds (i.e. along the c-axis) with those on the isolated molecule (Table 5, column 2) reveal subtle changes in the molecular polarisation reflecting the more ionic nature of urea when hydrogen bonded. This effect becomes stronger for the trimer (see Table 5, columns 5-7) where the hydrogens on the outer molecule of the trimer have a charge that is very similar to the isolated molecule (0.196 isolated, 0.206 in trimer).However, for the same hydrogens in the molecule at the centre of the trimer the charge is Polar Morphology of Urea Fig. 7 Interaction electron density map of urea projected onto the { 1 1 0) plane showing the difference between the bulk and isolated molecule molecular charge distributions which clearly illustrate hydrogen-bond formation in the solid state associated with the transfer of charge density from the H atom sites to the N and 0atoms. Continuous, dashed and dot/dashed lines refer to positive, negative and zero values respectively. The contour interval is ca.0.003 e k3. 0.230 which is much closer to the bulk value of 0.233. The trimer is a simplified model of the structure since it contains molecules pointing in only one direction and only one type of hydrogen bonding (i.e. type a). By constructing a molecular cluster one can build up the environment to mimic the solid state. Fig. 8 shows the resulting molecular geometry based on a seven-molecule cluster which contains all the six important surrounding molecules which are hydrogen bonded to the central molecule with type a and b bonds. The central molecule in the cluster should interact in a manner closely resembling that of the molecule in the bulk crystal environment. The resultant charges, shown in Table 5, column 8, closely mimic and bridge the changes to the electron density noticed when comparing the isolated molecule and 3D periodic calculations. Finally, in order to mimic the surface effect more explicitly, we constructed a molecular cluster with a thickness equal to the slice thickness dhklso that the surface charges on the polar { 1 1 1} faces could be calculated. The charge distributions over the two symmetry- related molecules calculated using this approach is given in Table 5, columns 10 and 1 1.Morphological Prediction based on a Polar Attachment Energy Model Polar attachment energies were calculated using an early version of the computer program HABIT94,42 which uses a computational procedure similar to that adopted for the classical attachment energy calculation.With this program different charge sets can be used for the host and adsorbing species when calculating the intermolecular bonds involved in the attachment energy. In this way effects due to surface polarity may be quantified. Note that for the intermolecular force field calculations it is not the absolute values of the electronic charges that are important but the differences between surface and bulk charges. The published force field of Lifson et al.35has its own charge distribution. In order to be consistent with this force field the differences in the STO-3G isolated/bulk/surface charges were used to scale these values. The published charges were assumed to be that of the bulk for two reasons: first, the charges have been derived for a force-field describing bulk properties; secondly, the published charges were in better agreement with the calculated bulk charges than with the calculations as the isolated gas-phase molecule.Thus the published charge (see Table 2, column 6) were assumed to represent the bulk solid-state Table 5 Comparison between the atomic charges for an isolated molecule; molecular dimer, trimer and cluster and 3D crystal models as calculated using ab initio quantum-chemistry calculations with programs GAMESS and CRYSTAL using the STO-3G basis set dimer trimer surface isolated molecule molecule molecule molecule molecule cluster 3D M1 M2 atom molecule 1 2 1 2 3 (central) crystal (1 1 I) (1 1 1) -0.347 -0.376 -0.376 -0.370 -0.385 -0.372 -0.437 -0.425 -0.372 -0.438 + 0.409 + 0.395 + 0.420 + 0.406 + 0.423 + 0.428 + 0.438 + 0.426 + 0.378 + 0.427 -0.446 -0.442 -0.417 -0.461 -0.456 -0.441 -0.461 -0.467 -0.443 -0.427 + 0.219 + 0.195 + 0.225 + 0.201 + 0.208 + 0.224 + 0.231 + 0.234 + 0.200 + 0.239 + 0.196 + 0.215 + 0.194 + 0.225 + 0.230 + 0.206 + 0.233 + 0.233 + 0.216 + 0.234 -0.446 -0.442 -0.417 -0.461 -0.456 -0.441 -0.461 -0.467 -0.442 -0.427 + 0.219 + 0.195 + 0.225 + 0.201 + 0.208 + 0.224 + 0.231 + 0.234 + 0.185 + 0.191 + 0.196 + 0.215 + 0.194 + 0.225 + 0.230 + 0.206 + 0.233 + 0.233 + 0.222 + 0.246 Polar Morphology of Urea 6 Fig.8 Molecular cluster used for the ab initio charge calculations showing the first coordination sphere of six urea molecules hydrogen bonded to a central atom.The two primary strong bonds (types a and b) are identified. structure and then scaled using the benchmarks provided by the surface and isolated molecule calculations to produce scaled values consistent with the published force field. For the polar calculation the charges on the oncoming molecules are taken to be different from that in the bulk of the crystal. These charges were then used to scale the published force field charges to produce isolated molecule charges (see Table 6, column 3). The resultant predicted attachment energies are shown in Table 4, column 3 together with the predicted morphology in Fig. 6(b).Here the data, for the first time, reveal a polar morphological prediction with the growth rate along { 1 1 1) predicted to be less than that along { 1 1 1).However, the use of the bulk/isolated charge set pair is not sufficient to remove the faster growing { 1 1 1) face from the overall predicted crystal morphology. The surface charges for molecules M 1 and M2 are shown in Table 5, columns 10 and 1 1, and were used to scale Hagler’s published charges to produce surface charges consistent with Hagler’s parameters sets. The silrface charges were assigned to the { 1 1 1 ) slice with the isolated distribution to the oncoming molecules. The resultant calculated polar attachment energies are shown in Table ---4, column 4. Using the surface charges/isolated charges the growth rates of { 1 1 1) and { 1 1 1) forms are different. The computed morphology reflects this with the polar morphology shown in Fig.6(c). The growth rates of { 1 1 0) and (00 1) are unaltered in this plot. Thus through this polar simulation we can determine the absolute sense of the polar capping faces to be the { 1 1 1) form rather than its Freidal opposite the {iii). Confrontation of the Theoretically Predicted Morphology with Experimental Data A comparison of the polar morphological simulation with the crystals prepared by sublimation (see Fig. 4) reveals excellent agreement; the three forms are all predicted, the aspect ratio is correct and the polar effect is reproduced. This is a clear demonstration of the validity of the approaches in this study and shows that the simulation of polar morphological effects can be directly computed, thus providing an alternative to the use of R.Docherry et al. Table 6 Isolated molecule, bulk and surface charges used in the polar simulations together based on the force field of Lifson et al.35against which they have been scaled to remain consis tent surface ~~ atom bulk isolated M1 M2 -0.380 -0.310 -0.333 -0.392 + 0.380 +0.365 + 0.337 + 0.381 -0.830 -0.793 -0.787 -0.759 + 0.415 +0.388 +0.355 +0.424 +0.415 +0.349 +0.385 + 0.416 -0.830 -0.793 -0.785 -0.759 + 0.415 +0.388 +0.328 + 0.339 +0.415 +0.349 + 0.395 + 0.438 Ref. 34. tailor-made additives (e.g. ref. 3) to assign the absolute configuration of chiral structures and the absolute polarity of polar crystals. In the case of urea our data reveals the observed polar form to be { 1 1 1) rather than its Freidel pair { 1 1 I ), thus assigning the observed faces to have the more electropositive crystal surface of the two forms.At this point it is informative to re-examine the crystal morphology of urea when recrystallised from polar solvents (see Fig. 5) such as water and methanol. In this case the observed crystal morphology, although still polar, is more needle-like (aspect ratio of ca. S), reflecting a lower growth rate for the (1 1 0) face when compared to the vapour-grown crystals (aspect ratio of ca. 1.5). Examination of the packing diagram, Plate 1, provides the likely origin for this effect and shows the growth retarded { 1 1 0)forms to have the carbonyl (electronegative) groups for both molecules M1 and M2 located at the centre of the slice with the surface regions rich in H atoms and exhibiting a cavity-like surface topography.These factors are conducive to the binding of the hydroxy groups from the solvent molecules at this surface which presumably have the effect of blocking the movement of surface terraces on this face, thus retarding the growth. The other (1 1 1) and (0 0 1) forms exhibit a more complex surface structure with both amino hydrogens and carbonyl groups at the surface; the resulting mixed charge environment being likely to deter substantial solvent absorption. Conclusions The development of quantum chemistry codes with full periodic boundary conditions enable molecular charge distributions in the solid state to be evaluated and the applicability of this approach to studies of urea have been verified through the use of self-consistent calculations on isolated molecules and molecular clusters.Such methods when used in combination with attachment energy calculations have enabled the polar effects associated with crystal morphology to be predicted and understood. From this work the polar morphology of urea has been successfully predicted and found in good agreement with the observed morphologies of crystals prepared from the vapour phase. The calculations have also enabled the direct assignment of the absolute polarity of the crystal ___which reveals the observed form to be the (1 1 11 form rather than its Freidel opposite (1 1 1). The habit modification of urea when re-crystallised from polar solvents can also be understood in terms of structural considerations as the hydroxy groups from the solvents Plate 1 Molecular packing projection of urea (colour code: 0,red; C,grey; N, blue;H, white), drawn using the CERIUS molecular graphics package.The projection is made onto the (1 1 0) plane showing the surface chemistry involved in the (1 1 1) and (TTT) polar crystal surfaces; note the orientation of the two symmetry-related urea molecules one of which (M 1) has its plane parallel to the projection whilst the other (M2) is oriented perpendicular to the plane of projection. The c-axis is vertical with the a-axis (coming out of plane) and b-axis (going into plane) pointing in the positive sense horizontally left to right.cfacingp. 23) 24 Polar Morphology of Urea such as water and methanol appear to easily bind to the hydrogen-rich (1 1 0) surfaces rather than the { 1 1 l} and (0 0 1) end faces where the surface chemistry involves a mixed anionic and cationic environment. Research on understanding the role played by molecular structure in being able to predict, control and optimise the morphology of crystals has been supported at Strathclyde for a number of years through research grants from the SERC, Exxon Chemical and ICI Chemicals and Polymers to whom we are most grateful. We also gratefully acknowledge SERC Daresbury Laboratory and SERC Rutherford Appleton Laboratory for computer time on, respectively, the FPS164 and CRAY XMP supercomputers as well as L.MacCalman who re-crystallised and photographed the solution grown urea crystals and P. 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Ramanarayanan, W. Kern, M. Larson and S. Sikdar, Am. Inst. Chem. Eng. Symp. Ser. 284, 1991, vol. 87, p. 130. 21 P. Bennema, Xiang Yang Liu, K. Lewtas, R. D. Tack, J. J. M. Rijpkema and K.J. Roberts, J. Crystal Growth, 1992, 121, 679. 22 L. MacCalman, K. J. Roberts and B. A. Hendriksen, J. Crq'stal Growth, 1993, 128, 1218. 23 G. Clydesdale, K. J. Roberts and R. Docherty, J. Crystal Growth, 1993, submitted. 24 G. Clydesdale, K. J. Roberts and K. Lewtas, Mol. Cryst. Liq. Cryst., 1993, submitted. 25 R. Docherty, Modelling the Morphology of Molecular Crystals, Ph.D. Thesis, University of Strathclyde, Glasgow, 1989. 26 R. J. Davey, J. W. Mullin and M. J. L. Whiting, J. Crystal Growth, 1982, 58, 304. 27 R. J. Davey, B. Milisavljevic and J. R. Bourne, J. Phys. Chem., 1988, 92, 2032. 28 F. C. Wireko, L. J. Shimon, F. Frolow, Z. Berkovitch-Yellin, M. Lehav and L. Leiserowitch, J. Phys. Chem., 1987,91,472. 29 P. Vaughan and J.Donohue, Acta Crystallogr., 1952, 5, 530. 30 A. Carron and J. Donohue, Acta Crystallogr., 1964, 7, 544. 31 A. W. Pryor and P. L. Sangster, Acta Crystallogr., Sect. A, 1970, 36, 543. 32 H. Guth, G. Heger, S. Klein, W. Treutmann and C.Scheringer, Z. Kristaflogr., 1980, 153, 237. 33 S. Swaminathan, B. M. Craven and R. K. McMullan, Acta Crystallogr., Sect. B, 1984,40, 300. 34 S. Swaminathan, B. M. Craven, M. A. Spackman and R. F. Stewart, Acta Crystallogr., Sect. B, 1984,40, 398. 35 S. Lifson, A. T. Hagler and P. J. Dauber, J. Am. Chem. Soc., 1979, 101, 51 11. R.Docherty et al. 25 36 J. D. Cox and G. Pilcher, Thermochemistry of Organic and Organometallic Materials, Academic Press, New York, 1970. 37 E. Dowty, Am. Mineral., 1980, 65, 465. 38 S. Dovesi, C. Pisani, C. Roetti, M. Causa and V. R. Saunders, CRYSTAL 88, QCPE program no. 577, Bloomington, Indiana, 1989. 39 M. F. Guest and J. Kendrick, GAMESS General Atomic and Molecular Electronic Structure, University of Manchester Regional Computing Centre, June, 1986. 40 S. Dovesi, M. Causa, R. Orlando, C. Roetti and V. R. Saunders, J. Chem. Phys., 1990, 92, 7402. 41 V. R. Saunders, C. Freyria-Fava, R. Dovesi, L. Salasco and C. Roetti, Mol. Phys., 1992, 77, 629. 42 G. Clydesdale, K. J. Roberts and R. Docherty, Comput. Phys. Commun., 1993, to be submitted. Paper 3/00537B; Received 26th January, 1993

ISSN:1359-6640    

DOI:10.1039/FD9939500011

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年代:1993

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Furuduy Discuss., 1993, 95, 27-36 Thermal Roughening investigated by Scanning Tunnelling Microscopy Joost W. M. Frenken, Laurens Kuipers and Jaap B. Sanders FOM-Institutefor Atomic and Molecular Physics, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands The scanning tunnelling microscope (STM) can be used to study surfaces of crystals at elevated temperatures. In this paper it is shown how the roughening transition on a metal surface can be observed in STM images. From such observations an accurate value of the transition temperature is obtained. In addition, two technical steps are described that are necessary for the analysis of short-range correlation functions in the images in terms of the kink formation energy and the step-step interaction energy.The first step concerns an algorithm for the pattern recognition of kinks and step edges, which is used to acquire automatically the statistical kink distributions from a large number of STM images. The second step deals with the theory of short-range correlation functions, and is employed to derive the two energy parameters of interest from the experimental statistical distributions. Most surfaces of metals, semiconductors and molecular crystals develop a more or less severe degree of disorder at high temperatures. The mildest form of disordering, usually taking place at the lowest temperatures, is the loss of a surface superstructure, such as the (7 x 7) reconstruction of Si(111)’ and the (2 x 1) reconstruction of Au( 1 The other extreme is the almost complete loss of surface lattice order, known as ‘surface melting’,3 which typically occurs at temperatures close to the bulk melting point.At intermediate temperatures, the roughening transition converts the initially flat surface into an atomically rough configuration, without affecting the lattice periodicity at the ~urface.~ Each of these three types of disorder may influence the behaviour of the material on a mesoscopic or even macroscopic length scale. In particular, the roughening transition has been recognized as a potential cause for changes in crystal growth beha~iour.~ The scanning tunnelling microscope is an ideal tool to study local arrangements of surface defects, such as thermally generated steps and kinks on an otherwise flat surface.Employing a sufficiently symmetrical STM design, crystal surfaces can be imaged at elevated temperatures. In this contribution we demonstrate that the roughening tempera- ture, TR,of a metal surface, in this case Ag( 119, can be determined straightforwardly from a qualitative comparison of temperature-dependent STM images.6 For a quantitative analysis of the densities of, and correlations between, steps or kinks, each measured STM topograph has to be reduced to its underlying pattern of steps and kinks, from which the required densities and correlations can be counted directly. This reduction can be performed automatically by a pattern recognition algorithm. In the interpretation of the resulting statistical information we make use of the theory developed by Villain et al.for the roughening of vicinal surface^,^ adapted to yield Zocal probability distributions.8 From the comparison between the statistical mechanical theory and the STM observations both the stepstep interaction energy and the kink formation energy should be obtained. The required consistency between these energies and the observed roughening temperature should serve as a test of the employed theory.’ 27 Roughening investigated by STM AgW5) The structure of the Ag(l15) surface is illustrated in Fig. 1. It consists of narrow (001) terraces of 7.5 A (2.5 atom rows) in width, separated by monoatomic steps of 1.6 A in height. The roughening transition of such a vicinal surface is the result of the thermal generation of kinks in the pre-existing step edges.Each kink changes the local terrace width by + 1 or -1 atomic row, which is accompanied by a change in the local height of its step edge by -0.8 A or + 0.8 A. Lines or loops of kinks form the boundaries between areas on the surface that differ by 0.8 A in height. In the following we refer to such a boundary as a 'STEP', in order to distinguish it from the ordinary 'step' on the ideal (1 15) surface. It is important to realize that, on a sufficiently large length scale, real surfaces are never perfectly oriented and completely flat. The local deviations from the (1 15) orientation must be accommodated by STEPs. At temperatures well below TRthese are the only STEPs present, and they minimize their length by running as straight as possible.At TRthe free energy for STEP formation vani~hes.~ The resulting strong meandering of the pre-existing STEPs and the additional high concentration of thermally generated STEPs wash out the original STEP structure. It is this feature that appears most prominently in the STM images and that is used to identify the roughening temperature. Experimental Observations The STM results presented here have been obtained with a microscope that had been modified to image warm ( < 300 "C) surfaces.6 The Ag( 1 15) specimen was prepared by standard polishing, etching and in situ sputtering and annealing treatments.6 Fig. 2 shows STM images of the Ag( 1 15)surface, at 20,58,98 and 260 "C.Before each of [oii] Fig.1 Side view (a) and top view (b)of the Ag(ll5) surface. A contour followed by the tunnelling microscope is indicated schematically. A kink in a step edge leads to the narrowing or broadening of the adjacent terraces, thus causing height differences. A line of kinks forms a STEP (see text), separating areas that differ one level in height. J. W.M. Frenken et al. (b) Fig. 2 STM topographs from the Ag( 1 15) surface.6(a)20 "C,585 x 220 &;(b)58 "C,1 180 x 375 A2; (c) 58 "C,enlarged perspective view of a small section of (b);(d)98 "C,450 x 120 A'; (e) 260 "C, enlarged perspective view. these measurements, the surface had been annealed at 600°C and stabilized for several hours at the measurement temperature. At all temperatures, the images clearly show the steps of the (1 15) surface, although the experimental corrugation, of typically 0.25 A, is much smaller than the true step height of 1.6 A.The atomic structure on the (001) terraces has not been resolved. As explained, it is in the meandering of the STEP edges that one most easily recognizes whether the surface is below or above TR.The STEPs at room temperature [Fig. 2(a)]are relatively straight, which indicates that they involve a non-zero free energy. At 58 "C the STEPs meander more strongly than at room temperature, from which we infer that this temperature is much closer to, but still below, the roughening temperature [Fig. 2(b)].Fig. 2(c)shows a perspective representation of a small section of Fig.2(b).A loop of kinks in this section encloses a portion of the surface that is one level (0.8 A) below its surroundings. Also visible in this enlarged view is the effect of surface diffusion. Several scan lines seem to 'protrude' one atom row from individual step edges. This corresponds to temporary displacements of one of the kinks on the edge, displacing the location of the edge by one atom row during one scan line (ca. 1 s). Kink dynamics have been proposed as the Roughening investigated by STM mechanism behind the so-called ‘frizziness’ of individual step edges on Ag( 1 1 1) and cu(001).9~~ At temperatures of 98 “Cand higher [Fig. 2(+(e)] the STEP edges are ‘delocalized’. The surface morphology is dominated by a high density of kinks in the step edges, and the resulting STEPS easily bridge the non-thermal surface height variations.Based on this observation, we conclude that the roughening temperature of Ag( 115) is between 58 and 98 “C, i.e. TR= 78 f20 “C. The density of kinks at the higher temperatures (in particular, above 200°C) seems much higher than one would expect theoretically (see below). Probably, this is caused by the strong surface diffusion, which changes the surface during the scan, thereby introducing apparent kinks. Possibly this could also affect our identification of the rough state, in which case it would render our value for TRa lower estimate of the true roughening temperature. Pattern Recognition We will show below that the kink formation energy and the stepstep interaction energy can be inferred from the statistical distribution of kinks.For the evaluation of the kink distribution in an extensive set of images, it is essential that this statistical information is extracted from the measurements in an automatic manner. Especially at high temperatures, where the kink density is relatively large, an unambiguous identification by hand of the kinks in a large image is almost impossible. Although the signature of the (1 15) steps seems rather strong in Fig. 2, the experimental (1 15) corrugation in most images is below 0.3 A, i.e. much smaller than the geometrical step height of 1.6 A. This easily leads to ambiguities in the interpretation of the images which usually results in inconsistencies in the patterns of kinks obtained from an image by visual inspection. We have developed a simple pattern recognition procedure that avoids inconsistency and finds for each ambiguous situation the best possible ‘guess’ that is consistent with the rest of the information in the image.It is rather insensitive to statistical noise in the image and is extremely robust with respect to bad resolution of the tunnelling tip, and even ‘survives’ tip-resolution changes within one image. The two main difficulties that the method has to overcome are illustrated in Fig. 3. First of all, the data are superimposed on a heavily tilted background. Because of the elevated crystal temperature the thermal drift in the vertical direction has been extremely strong in these measurements.The drift depended on temperature and was not equal for different images measured at the same temperature. The slope of the background is almost entirely due to this drift, so that the precise orientation of the (1 15) plane with respect to the scanning plane has to be determined for each image separately. Secondly, as already mentioned, in most images the corrugation am litude related to the (1 15) steps is much smaller than the geometrical corrugation of 1.6 8:. Additional sources of possible error are noise in the images, temperature-dependent variations in the piezoelectric coefficient of the translation element of the microscope and hysteresis of the piezo element, which is reflected in non-linearities in the distance scale along the scan lines and artificial curving of the steps.Our pattern recognition approach makes use of the fact that the scan lines have been recorded in a direction approximately perpendicular to the steps. It employs the crystallographic information of the step height and stepstep distance on Ag( 119, and combines this with the restriction (see next section) that the distance between two neighbouring steps may differ from the regular distance by no more than plus or minus one atom. Each image is processed line by line, starting from the middle scan line of the image. We follow a prediction-correction procedure, commencing at the step closest to the centre of the line. This step is found as the local maximum nearest to the middle of the line.The next maximum along the line, for example to the right, is expected at or close to one of three locations, either at the regular distance Ax, at one atom too near, Ax -dx (STEPup by one level), or at one atom too far, Ax + dx (STEP down by one level). Corresponding to these three possible step locations are three different heights. Rather than just examining the J. W. M. Frenken et al. +START-* measured STM contour I I I II I I I I identified surface levels -1 -1 -1 -1 0 0 0 -1 -1 Fig. 3 Schematic illustration of the pattern recognition in an individual STM scan line. Starting from the central local maximum, the algorithm searches in both directions for the best match with a distance to the next step edge of Ax (regular distance), Ax -dx (STEP up) or Ax + dx (STEP down).The procedure is iterated to the end points of the line, and repeated for all lines in the image. measured heights at the three lateral distances, the algorithm sums all measured heights between the first maximum and each of the three possible locations for the next maximum, in order to minimize the effect of noise. The three resulting average heights are compared with the average heights expected for the three possible step locations. These expected heights are calculated for a non-corrugated surface, i.e. a surface for which the area between neighbouring steps is interpolated linearly. This makes the method virtually independent of the actual corrugation amplitude in the image. The step location that works best is selected and the procedure is then applied to the three possible stepstep distances measured from this location.This process is iterated up to the image edges. The result is a set of step positions with level numbers, the levels differing by -1,O or 1 between neighbouring steps. Several corrections are necessary to make the procedure perform properly. The tilt angle of the (1 15) plane with respect to the scanning plane has to be estimated in advance. It enters the comparison between measured and predicted height averages. Since a small error in the initial estimate of the tilt angle would lead to the accumulation of errors in the predicted average local heights, we use, at every step in the described iteration, a certain fraction of the difference between measured and predicted heights to correct the subsequently predicted heights.Once a scan line has been processed, the difference between the measured line and the line reconstructed on the basis of the pattern recognition is used to calculate the tilt angle more accurately. This better value for the angle is used in the second and subsequent passes through the same algorithm. The procedure starts from a local maximum near the centre of the scan line. As a result of noise in the image, this local maximum usually is statistically displaced somewhat from its ideal location. In the algorithm the small displacement of the starting location propagates through the line and may lead to errors in the pattern of recognized steps.Therefore, after the first pass of a scan line through the recognition algorithm, the displacement of the starting position is calculated from the comparison between the measured and reconstructed lines, so that the subsequent processing of the line starts from the most suitable location. Finally, after the described processing, the algorithm reinspects each STEP, in order to ensure that a better description of the line cannot be obtained by placing the STEP one step earlier or later along the line. In this way, the recognition is made independent of the order in which the steps are examined. When a line has been analysed by the algorithm, the tilt angle, height offset and starting position are used as first estimates for the analysis of the next line.Once all lines have been converted into step positions and level numbers, the tilt-angle average of the whole image Roughening investigated by STM can be used in subsequent iterations of the entire algorithm, which then again starts near the centre of the image. The final result can be confronted with the original image by reconstructing an image from the pattern of steps and STEPS combined with all non-idealities, such as the finite tip resolution and the piezo creep, recognized by the algorithm. Both a quantitative x2 comparison and a visual comparison show that this type of recognition works extremely well. Each image [e.g.Fig. 2(a)-(e)] contains no more than just a few locations in individual scan lines where the algorithm prefers to place a STEP in a position one step different from that which we would expect visually.However, at those locations the situation in the original image is usually sufficiently unclear that it indeed allows for different local interpretations. Also at high temperatures, where the STEP density is very high, the reconstructed image is virtually identical to the original. It should be emphasized that in that case the algorithm outperforms us, as we find ourselves unable to analyse the high- temperature images by hand, without running into severe decision problems and substantial internal inconsistencies in the result. Theory of Displacement Statistics The model we adopt to predict the low-temperature displacement statistics on the (1,1,2n + 1) surface of an f.c.c.crystal is based on the terrace-ledge-kink model introduced by Villain et al.7 Two energies determine both the low-temperature statistics and the roughening transition at higher temperature, namely the formation energy, Wo,of a kink and the interaction energy, w,, per step atom for two steps that are one unit, dx, closer to each other than the regular distance Ax. Multiple kinks, displacing a step position by more than one unit, are not allowed in the model. The repulsive interaction between steps is assumed to be sufficiently large to neglect safely, and thus forbid, the possibility of two neighbouring steps reducing their distance by more than one unit from their regular distance: The Hamiltonian constructed according to these rules is7 JF = Wo CIUmOi + 1) -umoi)l+ Cg[um+lcV)-umo/>I (1) m,.v m,?-Here, umo/)= x,o/) -rnAx is the deviation of the rnth step from its regular position mdx, and the function g is defined as X>_O g(x) = { $,; x = -dx 00; x<-dx For the interpretation of the STM images, we consider the vicinal surface at a temperature far below TR.At low temperature, kinks in step edges are very rare, so that the model can be simplified drastically to that of a single step with kinks in either direction between two neighbouring step edges that remain fixed in their regular positions (Fig.4). Then, the two terms of the Hamiltonian reduce to single, instead of double, sums. where now g( fdx) = w, and g(0) = 0. The summation over y is a summation over a one- dimensional row of lattice sites.We start with a row of N sites and impose the periodic boundary condition: site N + 1-site 1. One can rewrite the Hamiltonian (3) as the one for a one-dimensional Ising-system, by assigning a ‘spin’, u= + 1, to every site that is on its regular position, and (Z = -1 to every displaced site: N N JF = --W~~(U,,+,-1) + WnC(a, -(4):[/=1 I= I J. W. M. Frenken et a1 wo L 7-Fig. 4 Schematic representation of a part of the (1 15) surface at a low temperature. One of the step edges contains a short excursion, with two kinks (energy W,) and four displaced step edge sites (energy w,). In our one-dimensional low-temperature model we need to consider only the three step edges marked by the heavy lines.The corresponding partition function is where v and L are reduced energy parameters, depending only on W,, w,and T. In the thermodynamic limit, N +a,the one-dimensional Ising system can be proven to be equivalent to a Markov chain. The conditional probability for the kth spin, ok, to have a particular value, given the values of the preceding spins ol.. .ok -depends only on the value of (Tk -1. In other words, the statistics along the chain can be described on the basis of four transition probabilities P, In ref. 8, these probabilities are derived as: with a, A and B depending only on v and L.* The step edge configurations that appear most useful for the experimental determi- nation of W, and w, are displaced step edge sections between two kinks.An excursion of length r can be described as the spin configuration 1(-1)'l. The probability of finding such an excursion, starting at a specific location along the step edge, is where PI= 1/A is the probability for an arbitrary spin to have the value 1. Using the theory of runs and other recurrent patterns, developed by Feller" and Bizley,12 we have derived the average number of excursions of length r per unit of length along the step edge. Simplifying the problem by counting multiple excursions of length r, separated by just one regularly placed site, e.g. I( -1)'1( -l),l, simply as a single excursion, we find for the excursion frequency* This frequency is lower than E, because of the exclusion of multiple excursions.Explicitly counting the pattern 1( -1)'1( -1),1 as a double excursion and also allowing higher-order multiple excursion patterns, we arrive at the more complicated expression for the frequency* Roughening investigated by STM The latter frequency is slightly higher than E,., because the last '1' in each excursion 1( -1)'l forms the proper starting element of a possible following excursion 1( -1)'1, which makes multiple excursions a little bit more probable than uncorrelated sets of single excursions. In the limit of zero temperature, expressions (7)-(9) all reduce to the simple Boltzmann estimate for the probability of Y displaced sites between two kinks, P,: 2W0+rw, The factor 2 accounts for the fact that we consider two equally probable excursion directions. As an example, we show in Fig.S the excursion length frequency at 300 and 500 K, calculated with expression (9) for the typical values of W, and w,of 1500 and 120 K, obtained by Lapujoulade for Cu( 115).13The higher temperature is above the estimate for the roughening temperature of 380 K given by Lapujoulade,13 but it is below the theoretical value of 635 K obtained for this combination of W, and wn, using the VGL the~ry.~ The open circles in Fig. 5 illustrate the result of a two-dimensional Monte Carlo simulation of the VGL model described in eqn. (1) and (2). The analytical expression for a one-dimensional king chain gives an adequate description of the low-temperature statistics of step edge excursions on the two-dimensional surface.At 300 K the Boltzmann factor deviates from the exact result by no more than a few per cent. At higher temperatures, e.g. 500 K, collective displacements of neighbouring step edges increase the excursion frequencies above our one-dimensional estimate. The recipe for determining the energies W,and w,from the semilogarithmic plot of the measured low-temperature excursion length frequency is evident from Fig. 5 and expression (10). The (initial) slope of the curve is equal to w,, while the extrapolated value for Y = 0 corresponds to 2W0. Summary In this paper, all ingredients have been supplied for an experimental test of theories of the roughening transition on crystal surfaces vicinal to a low-index orientation.From STM images measured as a function of the crystal temperature, we directly obtain the transition temperature, TR.The images can be further analysed by use of a pattern recognition algorithm. From the experimental frequency distribution of the length of excursions in individual step edges, at temperatures well below TR,the values for the kink formation energy and the stepstep interaction energy can be determined within the framework of the employed theoretical model for the roughening transition.' The STM images presented here have not been subjected to the described type of statistical analysis, since these measurements have been performed too slowly (ca. 1 s per line) with respect to the typical time between subsequent diffusion events, even at room temperature.Consequently, the surface diffusion would have strongly corrupted the experimental excursion length distributions, which would have led to unacceptably large errors in the derived energy parameters. There are two possible experimental solutions to the diffusion problem, namely faster scanning and measurements on a cooled surface. At present, we are repeating our measurements on Ag( 1 15) with a high-speed high-temperature STM. With this new apparatus, the observation speed has been increased by more than two orders of magnitude (< 10 ms per line), which should be sufficient to measure the true room-temperature kink statistics. One of the authors (J.W.M.F.) would like to express his gratitude to J. E. Demuth and R. J.J. W. M. Frenken et al. 10-2c, I I I I,, , , , $,I,, , I,, , , (b) length Fig. 5 (a)Frequency distribution for excursions of length Y along the step edge at a temperature of 300 K. W,and MI,are set to the experimental values of 1500and 120K for Cu( 115).j3 (-), excursion length frequency F,, eqn. (9),for a one-dimensional Ising chain. (--), simple Boltzmann expression (10). The dashed line almost coincides with the solid line. (0),frequency of excursions of length Y in a two- dimensional Monte Carlo simulation according to the model by Villain et al.’ (b)Same as (a) for a temperature of 500 K. Hamers for their support of the STM experiments at the T. J. Watson Research Centre (IBM). This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM) and was made possible by financial support from the Netherlands Organisation for the Advancement of Research (NWO).References 1 W. Telieps and E. Bauer, Surf. Sci., 1985, 162, 163. 2 J. C. Campuzano, M. S. Foster, G. Jennings, R. F. Willis and W. Unertl, Phys. Rev.Lett., 1985,54,2684. 3 J. F. van der Veen, B. Pluis, A. W. Denier van der Gon, in Chemistry and Physics of Solid Surfaces, ed. R. Vanselow and R. F. Howe, Springer, Berlin, 1988, vol. VII, p. 455. Roughening investigated by STM 4 H. van Beijeren and I. Nolden, in Structure and Dynamics of Surfaces, ed. W. Schommer and P. von Blanckenhagen, Springer, Berlin, 1987, vol. 2, ch. 7. 5 W. K. Burton, N. Cabrera and F. C. Frank, Philos. Trans. R. SOC. London, Ser. A., 1951,243, 299. 6 J. W. M. Frenken, R. J. Hamers and J. E. Demuth, J. Vac. Sci. Technol. A, 1990,8, 293. 7 J. Villain, D. R. Grempel and J. Lapujoulade, J. Phys. F, 1985, 15, 809. 8 J. B. Sanders and J. W. M. Frenken, Surf. Sci., 1992, 275, 142. 9 M. Poensgen, J. F. Wolf, J. Frohn, M. Giesen and H. Ibach, Surf. Sci., 1992, 274,430. 10 S. Rousset, personal communication. 11 W. Feller, An Introduction to Probability Theory and its Applications I, Wiley, Chichester, 3rd edn., 1968. 12 M. T. L. Bizley, J. Inst. Actuaries, 1962, 88, 360. 13 J. Lapujoulade, Surf. Sci., 1986, 178,406. Paper 31002651; Received 14th January, 1993

ISSN:1359-6640    

DOI:10.1039/FD9939500027

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年代:1993

数据来源: RSC

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Faraday Discuss., 1993, 95, 37-54 GENERAL DISCUSSION Prof. J. P. van der Eerden (Universityof Utrecht, The Netherlands) opened the discussion on Prof. Bennema’s paper: I agree that in your paper, Fig. 1 and 2,in particular, pave the way to understanding the selection rules which decide which of the higher dimensionally indexed faces really appear on the equilibrium form of a modulated crystal. I consider Fig. 2 of your paper to be a convenient way to plot the relevant attachment energies of normal and satellite slices. I do not understand, however, the last step, where you use a Wulff construction to select finally the thermodynamically most favourable inequivalent slices. Another possibility, for example, would be to use directly the distance of the bars from the origin in Fig.2of your paper as a criterion. Do you agree that other possibilities proceeding from Fig. 2might finally turn out to be physically more relevant? Prof. P. Bennema (University of Nijmegen, The Netherlands) responded (in part communicated): I am quite happy with your question. It gives me the opportunity to explain in more detail the basic concepts of our new theory. During our trip home to Nijmegen the problem became more and more clear to me, thanks to our discussions. After discussions with the authors of our paper I can now give the following answer (see also ref. 1). Before you left our group to become Professor in Utrecht somewhat less than two years ago we had already discussed these problems. We have really made progress since then.The text of the presented manuscript is somewhat vague. A much more precise and extensive publication has been submitted and will be followed by two others (see ref. 1). This paper is the first preliminary presentation of (we believe) quite new ideas. You will agree with us that for an unmodulated crystal relation (16) of our paper holds. A corresponding surface energy in real one-dimensional (1 D) space will fulfil the relation Here yLrnis the ‘surface energy’ in a 1Dcrystal. This implies energy divided by the dimension of a dimensionless point or just energy. r(@h,) in 2D space has the dimension of energy per length. After substituting eqn. (16) of our paper into eqn. (1) we obtain YLn = D (2) which means that in real 1D space all 2Dorientations (hm) have the same ID ‘surface energy’, D.This corresponds to the diameter of a circle in Fig.2of our paper. So no cusps due to selectivity of ‘faces’ (hm)having different rLrnvalues occur, as expected. If a modulation is switched on, then in eqn. (16) of our paper, the fixed value D must be replaced by an integration over mesh area Mhrnin 2Dbond space divided by the mesh area Mhm. The cos Qhm plays the same role as in eqn. (1 6) of our paper. This is because the bond energy per unit surface area depends on the cosine of the angle, Qhm, under which the mesh area of orientation hm is ‘seeing’ the horizontal continuum of bonds. We note that if Qhm= 0 the bond energy per unit length is maximal and if @hm = ~/2it is minimal and equal to zero.Note that we use integration over a mesh area in 2Dspace, to obtain an average value of yip,, for a particular orientation (hm) of an infinite number of cuts made from the intersections with real space of the set of net planes (hm).The obtained 1D equilibrium form can be considered as an average equilibrium form, resulting over an infinite number of equilibrium forms, having unique values of rim. For 2D equilibrium forms of modulated crystals embedded in 3D space or 3D 37 General Discussion equilibrium forms embedded in 4D space, the corresponding average values of yhklmor y;lklm can be interpreted as the average edge or surface energy, respectively, over infinitely long extended faces having the lowest surface energy.Note that from a formal mathematical point of view, our 2D Wulff plot does not differ from the usual Wulff-Herring raspberry-like polar plots. From a mathematical point of view the only unusual step is that we introduce the limit limlim y(h + ma) = (h+ma)+O y’(h + ma) = 0 (h+ma)+O (3) Note that h and rn are positive or negative integers. By taking h and rn as integers of opposite sign, h + ma becomes increasingly closer to zero with irrational a. This can be seen in the following way: Writing (h + ma) = m(i + a) and taking h and m with opposite signs, h/m becomes increasingly closer to -u on taking increasingly larger values of h and m. From eqn. (3), the equilibrium form becomes infinitely thin (i.e. one dimension lower than the 2D space).It also gives y’(h + ma)a dimension of energy divided by length and the other orientations (hm),with a surface of dimension zero, the dimension of energy. Note that the only ‘face’ with a dimension of energy divided by length has y‘ = 0. All other faces (hm)have an energy larger than zero. From a semantic physical point of view the application of the Wulff-Herring plot is very different from the usual application. This is caused by the dual character of the (2D and 1 D, or 3D and 2D, or 4D and 3D) crystallography employed by de Wolff, Janner and Jansen (ref. 16 of our paper). 2D space is used to provide overviews and to carry out calculations which cannot, or cannot easily, be carried out in 1Dspace. This dual character of the non- physical real 2D space and the physically real ID space is now also reflected in the new application of the Wulff-Herring plot in 2D space (and ID space).In contrast to conventional Wulff plot constructions, the construction is now also used to determine the values of y;,,, in real space. In normal 3D space the surface energy occurring on the equilibrium form is the same as the value of yhk,n. I agree with your criticism that the values of y;lm(Qhm)can be directly calculated and could be plotted in 1D space directly without using our new construction. We would have obtained the same results as now obtained from the Wulff-Herring plot presented in Fig. 2 of our paper. We are, however, convinced that a lot of mathematical and physical insight about what is going on with the morphology of modulated crystals is then lost.Moreover, in the direct procedure proposed by you, the Wulff-Herring plot procedure described above is implicitly carried out. It really makes sense to make this explicit. We note that a direct plot, as suggested by you, has already been used to study the dependence of the depths of the cusps, and the relative morphological importance, on a.A dependence on a was indeed found. 1 H. Kremers, H. Meekes, P. Bennema, K. Balzuweit and M. A. Verheijen, Phifos. Mag., in the press. Dr. K. J. Roberts (University of Strathclyde, Glasgow, UK) commented: The point is surely not whether the phase is modulated, but whether it is incommensurate or not (i.e. in the latter, the modulation vector is not an integral number of modulation lattice translations).For P-K,SO, structures three- and five-fold modulations on the a-axis are common. In the commensurate phase, the face (3kl)can be simply related to its 3-sub-cell as (lkl).Thus, for commensurate modulated crystals four Miller indices (hklm)are not needed practically. Prof. Bennema replied: In the crystallography of de Wolff, Janner and Janssen (ref. 16 and 28 of our paper) the vector q [see eqn. (4)of the manuscript] plays a key role. In the case General Discussion of one modulation corresponding to one extra internal dimension (a 4D translational invariant ‘modulated crystal’ with atoms extended to ID ‘world lines’), at least one of the factors a, p or 7is an irrational number.In the case where a, or y are all rational numbers we have a superstructure, which is translationally invariant in 3D space. From studying drawings like Fig. 1 of our paper it is found that, of a selected area in bond space [like the sub-bond space belonging to nets (Ol)], only very few discrete bonds are realised in ID space. So modulated crystals show a continuum of bonds in real space belonging to cuts (hrn).Supercrystals have only discrete bonds which are compatible with translational invariance in ID (or, more generally, in 3D) space. This is, from a morphological point of view, the essential difference between modulated and supercrystals. As an example, we present a case where Ids/= 31al (Fig. 1). Owing to the supersymmetry in the given phase of the figure three bonds out of the continuum of bonds are realised.These bonds are L, S and L’. In the case where a centre of symmetry (SC) occurs in the superspace group, which incidentally also occurs in real space, two bonds happen to be equivalent, or L = L‘. The three bonds L, S, L‘ correspond to a translation [l I] in superspace and this translation corresponds to the real translation in real space. The 2D-1D approach presented in our paper is what theoretical physicists would call a Fig. 1 Superstructure with d,I = 31al 40 General Discussion ‘toy model’. It is not possible to make a direct correlation with the morphology of real modulated crystals like crystals of the P-K2S0, structure and the mineral calaverite (AuTe, with some silver).These are the only modulated crystals on which facets (hklm)have been found to date. However, the principle of selectivity which is now found for 2D-ID space can easily be generalized to 3D-2D and 4D-3D space. In fact, we are completing a paper on this subject, where we studied a modulated Kossel-like crystal. In principle, a connection can be made with the integrated Hartman-Perdok roughening-transition theory, in which the idea of connected nets plays a key role: Assume that we have, in 4D space, a set of equidistant parallel 3D hyperplanes, cutting a real 3D crystal graph (with first-nearest-neighbour bonds for example) in slices of thickness, dhklt,,.If the cuts contain the average lowest attachment energy, it is then reasonable to assume that if the slices of the 3D crystal graph consisting of slabs with a thickness dhklrllare connected, facets (hklm)will grow as flat faces.This is, of course, provided they are growing below their roughening temperature. A study of the connectedness of slices with thickness dllklmof faces of the P-K,SO, structure and calaverite was carried out by Vogels et al. (ref. 14 of our paper). It was found that most of the observed faces of calaverite were indeed connected. This may explain the very strong tendency to form a large number of different crystal forms (hklm}.This also must be caused by the high amplitude., Concerning the relationship between the interplanar distance between hyperplanes (hklm)in 4D superspace and the corresponding dllklmin 3D space, it can be seen from Fig.2 that: The relation between d;lkim and dhkinlis given by where @ is the angle between the reciprocal vector Hhklmin superspace and the real 3D space. 1 M. Kremers, H. Meekes, M. A. Verheijen and P. Bennema, to be published. 2 K. Balzuweit, Thesis, University of Nijmegen, 1993. ’hklm Fig. 2 Hitklm corresponds to the interplanar distance. Hhkl,,,is the reciprocal vector in superspace, dilklm and d,,klmare the corresponding quantities in real space. General Discussion 41 Prof. L. Leiserowitz (Weizmann Institute of Science, Rehovot, Israel) turned to Dr. Roberts: It is possible that you could profit in your calculation of differences in attachment energy at opposite crystal faces, by making use of a more precise representation of the molecular charge distribution.For example, the representation could be extended beyond net atomic charges to atomic dipole moments (and even quadrupole moments) according to a method formulated by Hirschfeld' in his work on electron density deformation distributions. 1 F. Hirschfeld, Theor. Chim. Acta, 1977, 44,129; F. Hirschfeld, Cryst. Rev.,1991, 2, 167, and references therein. Dr. Roberts replied: I agree entirely with this point. The use of such a distributed electron density function rather than the simple point charges of the atom-atom approximation is a logical progression of the method we have used for urea. In fact, the deformation density is implicitly determined by the CRYSTAL code and so the incorporation of molecular charge distributions using this or calculations using Hirsch- feld's method into our morphological predictions should not be difficult to implement.Prof. Bennema asked Dr. Roberts: Did you check the influence of surface relaxation? Dr. Roberts replied: Surface relaxation in molecular crystals has been rather poorly studied but can be expected to be a significant effect where the intermolecular bonds contributing to the slice energy are dominated by weak van der Waals forces. In the case of urea, the dominant 3D hydrogen-bond network means that significant movement of the urea molecules is not very likely. The almost exact theoretical simulation of the observed morphology tends to support this view.Prof. Bennema added: Similar work has been done by the Ph.D. student Edo Brock, working with Prof. Feil and Dr. Briels at Twente University.' 1 E. S. Boek, W. J. Bricls, J. van der Eerden and D. Feil, J. Chem. Phys., 1992,96, 7010; E. S. Boek, Thesis, University of Twente, 1993. Dr. Roberts replied: Our work has its origins in the Ph.D. thesis' of one of us (R.D.) which was defended in 1989, thus pre-dating the work you mention by several years. 1 R. Docherty, Ph.D. Thesis, University of Strathclyde, Glasgow, 1989. Dr. M. La1 (Unilever Research, Port Sunlight Laboratory, UK) addressed Dr. Roberts: In many published studies on the molecular modelling of condensed phases, the energy calculations are based on the intramolecular charge distributions obtained for isolated molecules.In your calculations, however, you seem to have taken account of the effect of the surrounding molecules on the charge distribution using the periodic boundary approach. Would you describe the computational scheme underlying this approach and the approximations involved? Dr. Roberts responded: Basically, as described in our paper, we used the computer programs CRYSTAL and GAMESS, which are Hartree-Fock self-consistent field (SCF) quantum-chemistry codes, to calculate the charge densities over the atoms in the urea molecule for the cases, respectively, of molecules in the 3D crystal structure and the isolated molecule. These, together with studies of molecular clusters, allowed us, with confidence, to follow the changes in charge distribution that take place when the molecule transforms from its free state to being incorporated in a full solid-state structure.These results were then used to scale the experimentally derived force field of Lifson et aI. in the cases of its application to the free molecule and to those fixed on the { 11 1) and {TTT) polar surfaces. General Discussion The main approximations lie in the accuracies of the force field and of the ST03G basis set used for the SCF calculations. The excellent simulation resulting from this work tends to support the view that these approximations are not too significant for this system. Prof. van der Eerden asked: Is it true that in a crystal which contains aligned electric dipoles the total electrostatic energy of an infinite crystal should diverge, unless you have some effective screening of the Coulomb interaction? If so, a physical crystal, being finite and thus having surfaces, might use its surface degrees of freedom to compensate for this fundamental divergence.Is your adjustment of the effective charge distribution at the surface illustrating this effect, or is it independent of the electrostatic divergence problem? Dr. Roberts responded: Yes, this is the case, but urea is not a polar crystal. If we consider the 32 point groups we can subdivide these so: point groups (32) -centrosymmetric (1 1) non-centrosymmetric (2 1).. polar (1 0) non-polar (1 1) Urea crystallises with the non-centrosymmetric non-polar tetragonal space group P32,rn with two molecules in the unit cell related by the four-fold inversion axis which cancels out the individual dipole moments of molecules.Thus, there is no net polarity in the crystal, only polar surfaces which are due to different charge distributions throughout the depth of the four (1 11) growth slices. For polar crystals, such as sucrose, a-lactose hydrate, trinitrotoluene (orthorhombic phase), n-alkanes (n = odd) etc., the effects you mention do need to be taken into account when partitioning the intermolecular bonds. Prof. Leiserowitz added: The lattice energy sums in the calculations performed by Roberts and co-workers must converge. This is easily demonstrated in terms of the point- group symmetry of the crystal structure of urea, which is T2m.This point group has no net dipole moment. Prof. van der Eerden replied: To show why the electrostatic interaction energy of a polar crystal diverges consider the following: The interaction energy between two parallel dipoles is proportional to rP3, where r is the distance between the dipoles. Therefore (apart from a short distance cut-off and corrections due to discreteness of the lattice), the total electrostatic energy in an (infinite) volume V is proportional to 1 1,d3r,d3r, z “1 pd3r z V1;dr z Vln I/ V so the energy per dipole is proportional to In V, where the proportionality constant even depends on the shape of the crystal. Dr. A. L. Rohl (Royal Institution, London) said to Dr.Roberts: In your paper, you have shown that the attachment energy formalism is unable to predict polar morphologies because the attachment energy of hkl is always equal to the attachment energy of EEz If you allow your surfaces to relax, however, the surface energy of hkl is not, in general, equal to the surface energy of &Erfor polar directions, thus allowing for polar morphologies without recourse to putting charges on the atoms at the surface different to those in the bulk. Dr. Roberts responded: No, this is not the case. We have shown that an attachment energy model which takes into account the difference between the charge density at the General Discussion Fig. 3 Step/terrace growth surface model, detailing the interface kinetic mechanistic processes involved in growth below the roughening transition crystal surface and that on the free molecule replicates the polar effect rather well.In the case of urea (see above) significant surface relaxation effects due to the 3D hydrogen-bond network are not very likely. The use of surface energy for morphological modelling, whilst attractive in terms of the availablity of computer codes, is in my view lacking in a physical basis. Simply, the processes involved during crystallisation on a well defined surface have been well known, since the work of Burton et al.' (see Fig. 3), to be of an interfacial nature rather than dominated by surface effects only. The attachment energy concept is based on the formation of the stable growth slices involved in the surface terrace mediated interfacial processes.In contrast, calculation of surface energies (relaxed or otherwise) does not allow for growth-layer thicknesses, which are known to be important in our growth mechanism scheme. It is important to realise that attachment energy concepts are not restricted to molecular crystals (see, for example, ref. 2). New codes such as SURPOT3 enable attachment energy calculations for ionic systems to be carried out. Such calculations also enable the determination of the surface electrostatic potential which can, in turn, be used to predict solvent effects4 on crystal growth. 1 W. K. Burton, N. Cabrera and F. C. Frank, Philos. Trans. R. SOC. London, Ser. A, 1951,243,299. 2 C. F. Woensdregt, this Discussion. 3 C.S. Strom and P. Hartman, Acta Crystaffogv., Sect. A, 1989, 45, 371. 4 E. van den Voort, Ph.D. Thesis no. 68, University of Utrecht, The Netherlands, 1990. Dr. Rohl communicated in response: In my opinion, the case against equilibrium morphologies has yet to be proven in the literature. The case of corundum is an interesting one. Its morphology has been calculated using both relaxed surface energies' and attachment energies2 Neither model has correctly accounted for all the features in the observed morphology, suggesting either that external factors play a role or that neither model is complete. The surface energy calculation, however, does offer a possible explanation for the presence of the basal plane as the most morphologically important face, a result which is at odds with the attachment energy calculations.The predictions of the effect of solvent currently in the literature are only qualitative in nature.) The computer code MARVIN, described in ref. 4, promises a quantitative description by allowing the energies between solvent and a surface to be calculated explicitly. Furthermore, the solvent will be allowed to change conformation and orientation and the surface will relax in response. The calculation of attachment energies, allowing for surface relaxation, is very straightforward using this code, providing the opportunity for an examination of the effects of surface relaxation on the equilibrium and growth forms of organic and inorganic crystals, both in vacuum and in solvent.1 E. C. Mackrodt, R. J. Davey, S. N. Black and R. Docherty, J. Crystal Growth, 1987, 80, 441 44 General Discussion 2 P. Hartman, J. Crystal Growth, 1989, 96, 667. 3 E. van der Voort and P. Hartman, J. Crystal Growth, 1991, 112,445. 4 N. L. Allan, A. L. Rohl. D. H. Gay, C. R. A. Catlow, R. J. Davey and W. C. Mackrodt, this Discussion. Dr. Roberts communicated: It is, perhaps, useful to consider the origin of the equilibrium form which, as defined by Gibbsl and Wulff,2 reflects the morphology of a solid in equilibrium with its surrounding medium. The physical nature of this form can be derived from a polar plot of the surface Gibbs free energy as a function of the orientation of a vector centred on an origin within a 3D periodiccrystal lattice.As crystals are anisotropic, on the atomic scale, the 3D surface thus created exhibits minima. The equilibrium form is then a polyhedron constructed from surfaces defined from the normals to the radial vector where the free energy minima lie (see Fig. 4). If we neglect the entropic contribution to the free energy (and this is a big zj) we can use the surface energy (E,)to derive the equilibrium form. However, crystal growth requires chemical potential and thus cannot, by definition, be regarded as an equilibrium process. Thus the equilibrium form is not from theory a good model for the expected growth morphology of crystals. For modelling the growth process an attachment energy (EJ is usually used as it provides a recognition that the kinetics of surface step motion is the rate-limiting process involved in growth below the roughening transition.None of this theory is the subject of any serious current debate! 'Cusp" minimum in surface energy Fig. 4 Classical Wulff polar plot of surface Gibbs free energy showing well defined minima which correspond to the formation of singular habit planes Surface and attachment energies are related by the thickness of the growth step layer (dllkl)as defined by the symmetry-related Bravals-Freidel-Donnay-Harker law (see, for example, ref. 3). This inter-relationship, as expressed by Hartmaq4 is: ES = [Zdh,,/2NlrJEa,, where Zis the number of molecules per unit cell, Nis Avogadro's number and Ythe unit cell volume. As can be seen from the form of this equation the main difference is that the surface energy may unrealistically, in growth terms, predict surfaces with comparatively small layer General Discussion 45 thicknesses to be morphologically important.For example, consider a material with the rocksalt structure which has the space group Fm3m;here the effect of the face centre is to halve the importance of the crystal forms with mixed odd/even indices, i.e. the morphological importance changes from { loo}, { 1lo}, { 111>,etc. to { 1 1 1 1,{200},(220}, etc. Using surface energies to predict the growth morphology of these structures would tend to favour unrealistically, in terms of growth interface kinetics, the cubic form at the expense of the octahedron.Turning to the case of Corrundum (u-AI,~~), it maybe useful to quote the last paragraph of Hartman’s paper:4 ‘Summarising, neither the habit of natural crystals, nor that of synthetic crystals agrees with the equilibrium habit on which for corrundum the form m is so prominant. Nor do the observed habits always agree with the growth habit based on relaxed surfaces, the exception being the rhombohedral (011) habit. Therefore the discrepancy between theory and observation has to be explained by invoking external factors, of which the solvent adsorption is the most probable one as argued in ref. [8] and [21]. It is supported by the fact that the rhombohedral habit appears when the crystallising conditions are such that the effect of solvent adsorption is diminished: high temperature, low supersaturation. [8] P.Hartman, J. Crystal Growth, 1980,49, 166. [21]V. A. Kuznetsov, Sov. Phys. Crystallogr. 1968, 12, 608’. 1 J. W. Gibbs, Trans. Acad. Connecticut Acad., 1875, vol. 3; see also The Equilibrium of Heterogeneous Substances, Scientific Papers, vol. I; Collected Works, Longmans Green, New York, 1906. 2 G. Wulff, 2.Krist. Min., 1901, 34, 499. 3 F. C. Phillips, An Introduction to Crystallography, Longmans Green, London, 3rd edn., 1963, ch. 13. 4 P. Hartman, J. Crystal Growth, 1989, 96, 667. Prof. Bennema asked Dr. Roberts: Have you determined the absolute configuration so that you can actually check your predicted polarity? Do you have some idea how surface relaxation would influence the morphology? Maybe you can make a videotape with your benzophenone work.Dr. Roberts replied: Absolute configurations using diffraction methods rely on phasing using anomalous dispersion; for example, using the Bijvoet method to examine differences between the intensities of the Friedel pair I(hkZ)and Z@@. In the case of urea, examination of the crystal chemistry reveals that the surface chemistry of the { 1 1 l} and {TTi}faces differs by only a single amino hydrogen. Thus the polar effect becomes impossible to resolve by the Bijvoet method with X-rays and probably also with neutrons. Thus, for such a case, simulation offers a way to probe such slight structural effects which are impossible to examine experimentally. In terms of surface relaxation, the 3D nature of the hydrogen- bond network probably precludes any significant surface relaxation effects (see also my replies to other questions in this section).In contrast, the study you refer to on benzophenone’ shows the surface packing to involve a very small Coulombic contribution and here we know that changes in the intramolecular confirmation are very important for understanding the growth processes. The effect of intermolecular surface relaxation for this system has not, as yet, been examined, but may well also be quite significant. 1 K. J. Roberts, R. Docherty, P. Bennema and L. A. M. J. Jetten, J. Phys. D: Appl. Phys., 1993,26B, 7. Prof. Leiserowitzcommented: In order to clarify the effect of solvent on crystal growth and dissolution we assigned the absolute arrangement of the crystal structures of resorcinol,’ (R,S)-alanine,2 and the 7-form of glycine,, with respect to their (growing) crystal morphology, by employing tailor-made additives.We had also made use of crystal surface wettability measurements for the assignment of crystal polarity of alkyl gluc~namides.~ In order to assign experimentally the absolute structure of a specimen urea crystal (other than 46 General Discussion applying the method of anomalous X-ray scattering), one might consider growth or dissolution experiments in the presence of monomethylurea. H,NCONHCH,, but which must be exclusively in the cis O=C-N-CH, conformation. The additive should then be adsorbed preferentially on the (1 1 1) face, and should thus inhibit growth perpendicular to this face, and yield etch figures on this face on initial dissolution.I F. C. Wireko, L. J. W. Shimon, Z. Berkovitch-Yellin, M. Lahav and L. Leiserowitz, J. Phys. Chem., 1987, 91, 471. 2 J. L. W. Shimon, M. Lahav, and L. Leiserowitz, J. Am. Chenz. Soc., 1985, 107, 3375; L. J. W. Shimon, M. Lahav and L. Leiserowitz, Nouv.J. Chem., 1986,10,723; L. J. W. Shimon, M. Vaida, L. Addadi, M. Lahav and L. Leiserowitz, J. Am. Chem. SOC.,1990, 112, 6215. 3 J. L. Wang, M. Lahav and L. Leiserowitz, Angew. Chem., Int. Ed. Engl., 1991, 30, 696; J. L. Wang, L. Leiserowitz and M. Lahav, J. Phys. Chem., 1992, 96, 15. Dr. N. L. Allan (University of Bristol, UK)communicated: It is worth pointing out that there are many ways of dividing up a given electron density such that charges are assigned to individual atoms.In the present application it would be particularly interesting to assign point charges which reproduce the calculated electrostatic potential of an isolated urea molecule. The sensitivity of the Mulliken populations to the basis set used should also be noted. Finally, the calculated lattice energy is small. What is the likely vibrational contribution to the internal energy? Will entropic effects be important? Dr. Roberts replied: I agree that there are many ways of assigning point charges over the molecule in order to describe the molecular polarisability. In our work we have been able to use Hartree-Fock SCF calculations in using periodic boundary conditions to, for the first time, describe the growth morphology of a hydrogen-bonded solid.Our work shows much promise and opens the way for its extension to other methodologies in the manner suggested by Dr. Allan. In terms of the vibrational contribution to the intermolecular forces we add 2RT to our lattice sum (see, e.g., ref. 1) to allow for the vibrational contribution when we compare the agreement between theoretical and experimental lattice energies. Entropic effects have not been explicitly considered, but for a small, compact and rigid molecule such as urea they are likely to be small. 1 R. Docherty, G. Clydesdale, K. J. Roberts and P. Bennema, J. Phys. D:Appl. Phys., 1991, 24. 89. Dr. S. J. Maginn (ICIZENECA, Runcorn, UK)addressed Dr.Roberts: You have used different individual atomic charges in the attachment energy calculations for molecules in the surface (calculated by considering the six surrounding H-bonded molecules) and in the oncoming slice. This successfully predicted the observed polar morphology. The usual procedure when calculating such attachment energies is to use the same atomic charges for both, usually as calculated for ‘in vacuo’. In systems where strong intermolecular forces are present in the crystal, this is perhaps unrealistic. Do you think that there is a good case for routinely calculating different atomic charges for molecules in the surface and in the oncoming slice, in the code of HABIT/CERIUS? Surely this would provide a further refinement of the methodology, especially in crystals where strong intermolecular forces such as hydrogen bonds or T-T interactions exist.Dr. Roberts responded: Yes, I agree with you that the allowance for change in polarisability during crystallisation needs to be taken into account on a routine basis. The new CRYSTAL92 code? allows surface charges to be calculated routinely without the need t Details of the CRYSTAL92 code can be obtained from Dr. V. Saunders, SERC Daresbury Laboratory, Warrington, UK WA4 4AD. General Discussion 47 for the more complex approach we adopted in urea. Other codes such as PDM91,J: which empirically fit electron density to crystal structures, and other methods employing newer techniques, such as density functional methods, can also be adopted. Prof.A. A. Chernov (Russian Academy of Sciences, Moscow) turned to Dr. Frenken: Have you any data on the interactions between kinks and steps? In particular, I am interested in electrostatic, as well as elastic dipole-dipole interactions. A dipole moment of the order of 1 DI per atomic site at the step has been measured on the Ag surface from the dependence of the work function on step density. Such step dipole moments should be expected in all metals and semiconductors where there is a double electric layer associated with any crystal face, thus producing anisotropy of the work function. A step truncates the surface double layer and must therefore produce a dipole moment. Dr. J. W. M. Frenken (FOM Institute, Amsterdam) replied: The STM data that I have presented here have not been analysed to find stepstep (and kink-kink) interaction parameters.In principle, three types of interaction between steps can be imagined, namely entropic, elastic and dipole-dipole interactions. Each of these is repulsive, and the latter two should both decay similarly, namely inversely proportional to the square of the step separation. In the case of vicinal silicon surfaces, the statistical analysis of step separation distributions seems to indicate that the main interaction is not entropic and that the order of magnitude of the interaction is consistent with that estimated for elastic forces.' 1 C. Alfonso, J. M. Bermond, J. C. Heyraud and J. J. Metois, Surf: Sci., 1992, 262, 371.Dr. B. Yu. Shekunov (University of Strathclyde, Glasgow, UK) asked: Does the resolution time of the method allow the velocity of elemental steps to be measured? What kind of differences could one expect for the transition behaviour of crystallo- graphically different growth steps? Perhaps it would be important for very anisotropic vicinal hillocks. Dr. Frenken replied: In answer to your first question, the experiments described in this paper have all been performed in ultrahigh vacuum under equilibrium conditions. The roughening transition that we observe is the thermodynamic roughening rather than the growth-induced dynamic build-up of roughness. On average, our steps are static, so their velocities average out to zero.We have recently constructed a high-speed scanning-tunnelling microscope with which we follow, with atomic resolution, the equilibrium fluctuations of elementary steps and elementary kinks, with a time-resolution of a few milliseconds, at specimen temperatures between 300 and 850 K. It is possible to follow the growth in solution with the high spatial resolution of an atomic-force microscope, as is illustrated in the paper by Hillner et al. at this Discussion. Turning to your second point, the parameters that are thought to determine the transition temperature of a vicinal metal surface, are the formation energy, W,, of a kink in a step edge, and the interaction energy per atom, w,, between adjacent steps that are placed one lattice unit too close with respect to the perfect regular vicinal cut.Both energy parameters depend on the crystallographic orientation of the steps. For example, W, is much higher on a close-packed step, running, e.g., along the [lTO] direction of an f.c.c. crystal, than on a more open step, e.g. along the [loo] direction. This should, in principle, reflect itself in temperature-dependent anisotropies in growth shapes. 1 Details of PDM93 (Electric Potential Derived Mulipoles) can be obtained from Prof. D. E. Williams, Department of Chemistry, University of Louisville, Louisville KY40292, USA. 7 I D (Debye) = 3.33564 x 10-30 C m. General Discussion Prof. Chernov said: You have suggested that a strong azimuthal anisotropy of the step energy is expected including different temperatures at which the step energy vanishes.What should then be the behaviour of an isolated island formed by a loop-shaped step? Dr. Frenken answered: The case of an isolated island on a flat substrate, formed by a closed step loop, forms the 2D analogue of the equilibrium shape of a 3D crystallite bounded by a closed surface. The island equilibrium shape can be obtained via the Wulff construction, in which the familiar surface Gibbs free energies have to be replaced by the appropriate step Gibbs free energies. This procedure will, in general, lead to an anisotropic shape, characterized by straight step sections (‘facets’) and rounded regions. The straight sections correspond to cusps in the step Gibbs free energy. The rounded regions cover step orientations that are rough, i.e.for which the kink Gibbs free energy has reduced to zero. Prof. L. V. Woodcock (Universityof Bradford, UK)commented: There seems to be some confusion between the ‘surface-roughening transition’, as observed in your experiments, and the ‘surface-premelting effects’ referred to in the opening lecture by Prof. Chernov. Do you distinguish between these two phenomena and, if so, on what basis? My reason for asking this question is that 15 years ago Jeremy Broughton and myself’ found pronounced surface-premelting thermodynamic transitions on an initially smooth (111) Lennard-Jones crystal face, i.e. in a model without any ‘ledges’, ‘steps’, ‘terraces’ or ‘kinks’, etc. Would you expect to see a surface-roughening transition, similar to that in your paper, in a simple close-packed surface in addition to the thermodynamic surfzce- premelting transitions, and if so, do you think we should see it in molecular dynamics simulations of idealised low-index faces? 1 J.Broughton and L. V. Woodcock, J. Phys. C, 1978, 11, 2743. Dr. Frenken replied: There is a clear distinction between surface roughening and surface melting, which I will try to address separately (see later). The molecular dynamics simulations that you refer to in your question, indeed show surface melting on the f.c.c.( 111) face of a Lennard-Jones crystal, and not surface roughening. It is not a trivial matter to identify the surface-roughening transition in Monte Carlo or molecular dynamics simulations, without a careful investigation of the finite-size effects of the simulation unit cell.The mere occurrence of individual surface adatoms and vacancies in the outer lattice plane, might be classified as micro- or rather nano-roughness, and is not a direct indication of the step Gibbs free energy being zero. For metals, the lowest-index orientation on which surface roughening is known to occur is f.c.c.( 1 10). Equilibrium shape measurements of small crystallites seem to indicate that only the closer-packed (1 1 1) and (001) facets persist up to the melting point. Prof. Chernov commented: Phenomenologically, surface roughening and surface melting differ in their temperature behaviour (see ref. I). In the case of roughening (Fig.5) the effective width, W,of the interface of finite size, L,is finite when the surface temperature, T,approaches the melting point, T,. In the case of surface melting (Fig. 6) the thickness, H, of the discovered (molten) surface layer diverges when A T = T, -T4 0. It is known that either H z ln(A 7)or H z (A 7‘)-f. 1 A. A. Chernov, Modern Crystullogruphy III, Springer Series on Solid State Physics, Springer-Verlag, Berlin, 1984, vol. 36. Prof. Woodcock communicated: I do not think that one can so easily discriminate between these various effects on the basis of interface widths. This implies that there is a well defined division between ‘roughened layers’ and smooth layers, or, in the case of the surface pre-melting effects, between the vapour and the ‘pre-melt’, and the ‘pre-melt’ and the General Discussion I I-crystal Fig.5 Surface roughening: the effective width, W,of a rough surface of finite in-plane size, L, remains finite even at temperature, T, approaching the melting T,,, vapour 1 melt H crystal I Fig. 6 Surface melting: in contrast to Fig. 5, the disordered, molten surface layer thickness, H -+ co when T+ T,,, crystal. Even given a full knowledge of the structure and properties of an interface, it would surely be possible to define dividing surfaces only between coexisting bulk phases in equilibrium, but not in these conditions. It is difficult to see how one can use interfacial widths to distinguish these phenomena. I do accept, however, that there are distinct roughening phenomena associated with crystal surfaces that may not be explained as thermodynamic pre-melting, and the effect will therefore exhibit a different temperature- dependent behaviour.Dr. Frenken added: I agree that (complete) surface melting leads to a diverging solid vapour interface width, whereas, on a finite lateral distance scale, the solid vapour interface width remains finite in the case of surface roughening. Nevertheless, I do not regard this as the fundamental difference between the two surface-disordering phenomena. (For a more extended discussion of the differences, see later.) Dr. K. M. Robinson (Naval Research Laboratory, Maryland, USA) said: The comment that the STM results are different from X-ray diffraction results may be accounted for by the large current densities used by STM.Does the STM provide the surface with the necessary energetics to shift atomic positions? What experimental variations have been used to test the effect of the STM on the transitions? Dr. Frenken responded: I do not claim that we have obtained results that are at variance with X-ray measurements. What I would like to stress, though, is that by observing local statistics and by identifying the temperature at which the step Gibbs free energy becomes zero, we use the STM to address directly the cause of the roughening transition. With X- rays, one can measure its consequence, which is, ideally, the power-law diffraction lineshape resulting from a logarithmically diverging height-height correlation function.STM observations show that defects and impurities sometimes drastically affect the surface morphology, e.g. by pinning step-edge locations. This does not greatly influence the average local properties probed with the STM, but it probably has an important effect on the diffraction lineshapes. Your concern about the imaging currents and voltages employed in the STM, and their possible influence on the recorded surface morphologies, is certainly justified. To date, not very much is known about such effects, but field-ion microscopy studies have revealed a field effect on the activation energy barrier to surface diffusion. Our STM observations have General Discussion been obtained for a range of imaging conditions, with no obvious effect on the step morphologies. Dr.W. Dausch (BASF Aktiengesellschaft, Ludwigschafen, Germany) asked: Did you treat the silver surface at certain temperatures under the roughening temperature and then cool it by a certain amount? If so, what changes did you observe at the surface? Dr Frenken replied: Prior to each measurement, the Ag specimen was sputter-cleaned and annealed at 600 "C, i.e. well above the roughening point. After that it was cooled to the observation temperature, where it was equilibrated for several hours before scanning. Prof. van der Eerden commented: You try to use low-temperature (i.e. T<< TR)step patterns to deduce effective interaction parameters (W, and w,). If the roughening temperature predicted from these data does not coincide with the temperature actually observed, what would you expect to be the main cause (e.g. long-range interaction or temperature dependence of W, and w,)? Is your STM system capable of discriminating between some of these possibilities? Dr. Frenken responded: At this point we do not know whether or not to expect a difference between the actual roughening temperature and the value predicted from the energy parameters W, and w,, and if such a difference did occur, what the cause would be.By measuring the excursion length distributions at more than just one temperature below TR, we should directly learn how W, and w, depend on temperature. Kink-kink interactions should change the shape of the excursion length distribution.From step- separation distributions one should be able to infer the strength and distance dependence of the step interactions. Mr. L. Kuipers (AMOLF, Amsterdam) added: Recent STM results from the Lagally group (University of Wisconsin) on GaAs suggest the presence of kink-kink interactions. This would render W, dependent on the length of excursion. Prof. van der Eerden replied: Certainly, in any real system not only will the nearest neighbours interact, but also longer-range interactions will come into play. Such effects would also introduce kink-kink interactions. In this context, it may be worthwhile to note that the effect of the electrostatic interaction on surface structure is far from being understood. For example, the critical behaviour of a 2D lattice model with positive and negative unit charges and vacancies [a prototype model for the NaCl(lO0) surface] is not known.Prof. Bennema commented: It looks as if you can replace the Ag surface by a Kossel-like cell model with first-nearest neighbours. Can you calculate bond energies with pseudo- potentials from a real Ag crystal with real electrons, or will it turn out that you need more sophisticated potentials depending on distances and/or models where the interaction of steps is taken into account? Dr. Frenken said: The terrace-ledge-kink model by Villain et al., which we use to describe our Ag(115) surface, defines a vicinal type of Kossel crystal. Theoretical predictions for the kink formation energy can, in principle, be obtained from ab initio calculations or from effective-medium or embedded-atom calculations.An accurate description of step-step interactions can probably come from ah initio calculations. Prof. Chernov said: In our surface-melting experiments with biphenyl' ellipsometry has been employed. To measure the ellipticity of the reflected light, the polariser and analyser General Discussion mutual orientations should be found at the point where the outgoing beam intensity is zero. However, at ca. 0.3 or 0.45 K below the melting point (for the two different forces under consideration in our experiments) zero intensity could not be achieved. This fact suggests incoherent light scattering and, maybe a phase transition with the molten film.It might be a transition from the liquid-crystal structure of the sufficiently thick film (though still of the order of 10 A) to the conventional liquid or some other liquid crystal. 1 A. A. Chernov and V. A. Yakovlev, Langmuir, 1987,3,635. Prof. Bennema commented: You did pioneering work and this work triggered new research as, for example, the work going on at AMOLF in Amsterdam. In a joint research project very interesting surface melt phase transitions from a one-layer to a three-layer surface melt system were found. Work is now underway (at AMOLF) to study the interface crystal melt for the system caprolactum using glancing X-ray angle techniques. The Chairman invited discussion on the topics of surface roughening and surface melting, Dr.Frenken said: Historically, these two surface-disordering processes often have been confused, although they are rather different. In particular, the term ‘surface melting’ has been used many times to indicate the roughening transition.’ The microscopic cause for surface roughening2 is the vanishing, at the roughening temperature TR, of the Gibbs free energy associated with the formation of an elementary step on an otherwise flat terrace. Using renormalization group theory, one can show that this leads to an extremely weak (infinite-order) Kosterlitz-Thouless phase transition at TR. While the height-height correlation function <(h, -hJ2 >,between points with a lateral separation r, is finite at all temperatures below TR,it diverges logarithmically with r at and above TR.The macroscopic consequences of the roughening are the disappearance of a cusp in the angular dependence of the surface free energy at the terrace orientation, and the loss of the corresponding facets from the equilibrium crystal shape.It is important to realize that the disorder involved in the roughening transition is brought about by the proliferation of elementary steps, each of which leaves the original lattice structure intact. Apart from elastic lattice relaxations in the direct vicinity of a step edge, all surface atoms occupy regular lattice positions, also at and above TR. By contrast, surface meltir~g~,~ is not a true phase transition, but a continuous surface- disordering process which precedes the normal first-order melting transition at the melting point T,, and involves the almost complete loss of surface crystalline order.Surface melting is the complete triple-point wetting of the solid by its own liquid. The driving force for the phenomenon is the difference Ay = ysv-ysl -ylvbetween the specific Gibbs free energies of the solid/vapour, solid/liquid and liquid/vapour interfaces. If A y is positive, the solid surface can lower its Gibbs free energy at T, by covering itself by its own liquid. This is the case for many surfaces of metals and molecular crystals. Below T,, the liquid-like film ‘costs’ an amount of supercooling energy, which is proportional to the film thickness and to the undercooling. The undercooling energy produces an effective attraction between the solid/liquid and liquid/vapour interfaces.Since the full interfacial energy lowering, A y, would be reached only at infinite separation between the solid/liquid and liquid/vapour interfaces, these two interfaces at the same time experience an effective repulsion. The balance between these two interactions leads to a disordered film thickness which changes continuously with temperature and diverges at T,. Apart from complete self-wetting, a surface can also show incomplete melting. In that case, a disordered surface film develops which remains of finite thickness up to T,. At (and above) T,, this situation is metastable, and the melting transition is discontinuous, also at the surface. 1 W.K. Burton, N. Cabrera and F. C. Frank, Philos. Trans. R.SOC.London, Ser. A, 1951, 243, 299 52 General Discussion 2 H. van Beijeren and I. Nolden, in The Structure and Dynamics of Surfaces, ed. W. Schommers and P. von Blanckenhagen, Springer, Berlin, 1987, vol. 2, p. 259. 3 J. F. van der Veen, B. Pluis and A. W. Denier van der Gon, in Chemistry and Physics of Solid Surfaces, ed. R. Vanselow and R. F. Howe, Springer, Berlin, 1988, vol. 7, p. 455. 4 J. W. M. Frenken and H. M. van Pinxteren, in The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, Vol. VII: Phase Transitions and Adsorbate Restructuring at Metal Surfaces, ed. D. A. King and D. P. Woodruff, Elsevier, Amsterdam, in the press. Prof. D. G.Hall (Unilever Research, Port Sunlight Laboratory, UK)said: The following argument supports the view that surface melting can be expected for a solid that is wetted by its liquid at the freezing point.Consider the wetting of a solid by a different liquid. The adsorption isotherm of a vapour whose liquid wets the solid completely is shown schematically in Fig. 7(a), where r denotes the amount adsorbed, p denotes the vapour pressure and p, the saturation vapour pressure. The corresponding isotherm of a vapour whose liquid only partially wets the solid is shown schematically in Fig. 7(b). The shaded area, A, is related to the solid-liquid contact angle, 0, by A = al(l -cos O), where oIis the surface tension of the bulk liquid. I I 'PIP, Fig. 7 Absorption isotherms of a vapour on a solid (schematic) An alternative view of Fig.7(a) and (6) is to regard them as disjoining pressure (n)vs. distance (h)plots for a film of liquid separating solid and vapour, where IZ = -1 RTln(P/P,) = PLP -PI ~ ~ Vl V1 h = Tvl vI is the molar volume of the liquid and pland pp, respectively, are the chemical potentials of the liquid in the film and of the pure liquid. This alternative view has the advantage that the appropriate curves may be estimated approximately in terms of macroscopic quantities. Evidently, X q(l -cos 0) = -1 ndh hi3 Surface melting may be described by putting pI= p, in eqn. (1) and differentiating with respect to T to obtain: where sIand ss, respectively, denote the molar entropy of the liquid and solid.The variation in h with temperature is given by General Discussion 53 In the case of Fig. 7(a),which corresponds to complete wetting, eqn. (3) shows that h increases indefinitely with T, as T-+ Tf,so that a liquid film is formed. In contrast, for Fig. 7(b),which corresponds to incomplete wetting h tends to a finite value as T-+ Tf and then jumps discontinuously at Tp The relationship between disjoining pressure-distance curves and surface melting has been discussed previously by Chernov.' Clearly, the precise applicability of this kind of phenomenological reasoning to layers having a thickness of only a few molecules is debatable. Nevertheless, in the present instance, the conclusions that emerge do appear to be useful. 1 A.A. Chernov, J. Crystal Growth, 1977, 42, 55. Dr. Roberts addressed Dr. Frenken: In the overall debate on surface roughening/ melting I would like to stress the importance of being aware of molecular factors such as conformational flexibility in understanding the surface-mediated growth of molecular crystals. Recent work' on modelling the morphology of benzophenone reveals that the growth process involves a significant change in molecular conformation (Fig. 8) on growth due to crystal-field effects. The molecular packing on the prismatic (1 10) and (001) end-cap facets is such that this conformational change does not result in any substantial change (Table 1) in the surface attachment energy in contrast to the pyramidic (hkl)facets where the binding interactions during growth are much stronger.1 K. J. Roberts, R. Doherty, P. Bennema and L. A. M. J. Jetten, J. Phys. D: Appl. Phys., 1993, 26B,7. Fig. 8 Overlay of the two molecular conformations for benzophenone; A, crystal structure coordinates; B, optimised molecular structure Table 1 Normalized attachment energies compared with experimental data after scaling with respect to the { 1lo} form (hk0 calc. exp. difference (Yo) (1 10) (101) (01 1) (020) (111) (021) (002) 1.oo 1.25 1.17 1.42 1.37 1.45 1.66 1.oo 1.56 1.47 1.43 1.47 1SO 1.67 0.0 24.8 25.6 0.7 7.3 3.4 0.6 General Discussion Dr. Frenken responded: The simple complication of adding a shape anisotropy to the molecules in a van der Waals solid drastically changes the surface melting behaviour. Recent molecular dynamics simulations by Alavi and Chandavarkar' show that such anisotropy can stabilize certain surface orientations, while making other orientations melt through layering transitions. By contrast, a van der Waals solid with spherical molecules exhibits continuous surface melting on all faces. Conformational molecular flexibility adds yet another level of complexity to the problem and might well lead to additional modifications of the surface melting and roughening behaviour. 1 A. Alavi and S. Chandavarkar, preprint.

ISSN:1359-6640    

DOI:10.1039/FD9939500037

出版商:RSC    

年代:1993

数据来源: RSC

摘要:

Faraday Discuss., 1993,95, 55-64 X-Ray Surface Diffraction Studies of the Restructuring and Electrodeposition of Pb Monolayers on Au( 100) Single Crystals K. M. Robinson and W. E. O'Grady Code 6170 Chemistry Division, Naval Research Laboratory, Washington, DC 20375, USA High-intensity synchrotron X-ray radiation has increased the feasibility of studying the electrochemical solid/liquid interface in situ. Recent work on the interfacial structure of Au single-crystal electrodes has shown the need to merge appropriate electrochemical cell designs to work on standard four-cycle diffractometers. In this report, an electrochemical-based methodology for use of in situ X-ray diffraction from the electrode/electrolyte interface is applied to electrodeposition of Pb on Au(l00).The cell in this experiment, unlike traditional reflection cells, provides the necessary electrochemical control of the electrode surface, as determined by the cyclic voltammagram, in conjunction with the X-ray studies. Growth of the Pb monolayer begins with diffusion of Pb+' ions to the surface at potentials below the Au(00)-(5 x 20) P.Z.C.Electrodeposition causes a lifting of the (5 x 20) reconstruction. Pb atoms are deposited on the (1 x 1) surface in a c(2 x 2) structure. Domain sizes are small due to constriction by Au islands formed when the (5 x 20) is lifted. Irreversible surface defects are observed at -0.32 V vs. SCE. Potentials at which Pb on Pb deposition occurs result in a roughened surface and alloy formation.The interfacial structure of Au single-crystal electrodes in contact with various electrolytes is fundamentally important to the understanding of electrochemical systems. It has been shown by many different techniques that the primary faces of Au reconstruct under potential In the presence of Pb2+ ions, there is also electrodeposition at potentials anodic (more positive) than bulk Pb dep~sition.*~'~~~ The cyclic voltammagrams, CVs, of the Pb electrodeposition are very face dependent and reflect a large amount of restructuring of the interface. The Pb deposition on Au(lO0) has been studied by LEED,*,18J9STM14-16and by electrochemical methods.Io-l2J7 The results of these studies show that Pb on Au(lO0) involves a complex system of restructuring and formation of AuPb alloys.UHV studies of Pb on Au(100) show a considerable number of structures upon desorption and lateral and normal growth of alloys at submonolayer coverages. l9 The CVs of the Pb deposition on Au(lO0) contain several transitions with narrow potential widths as well as a broad transition which is assumed to be caused by Pb blocking the adsorption sites.12 Because of these transitions, any study of this system must be capable of demonstrating appropriate potential control. In situ X-ray diffraction has the capability of a surface-sensitive probe and has good potential control. Synchrotron radiation has become a powerful tool in electrochemical surface studies. Various techniques, such as surface diffraction, EXAFS, X-ray standing waves and fluorescence, have been used to characterize the electrode/electrolyte interface.20 The focus of this work has been on in situ X-ray diffraction of single-crystal electrodes.Surface diffraction has frequently been used in UHV environments.21,22 The high intensity produced by the synchrotron source has made in situ surface X-ray diffraction possible, thus allowing the potential-dependent surface structure to be probed. 55 Pb Monolayers on Au (100) Single Crystals There have been several systems studied with in situ X-ray scattering techniques which concern fundamental electrochemical systems such as Au, Pt and Cu ~xidation~~-~~ and Au( 100) surface reconstruction,' as well as electrochemical deposition on Au and Ag( 1 1 1) fa~es.'~,~~,~* all the in situWith the exception of X-ray reflectivity studies at Cu ~xidation,~~ studies used a reflectance-geometry electrochemical cell.There are two possible diffraction geometries for studying in situ electrodes which have been labelled reflectance and transmission. The X-ray scattering advantages of the transmission geometry over the reflectance geometry have been discussed previ~wsly.~~~~ In brief, the transmission geometry, in which a drop of solution is maintained on the surface by capillary action, allows for precise potential control of the sample at the expense of a slightly longer absorption length and an increased noise in the scattering, both of which can be reduced by operating at higher X-ray energy, 9.5-1 1 keV.In addition, the drop assures a constant absorption length for both large-angle and glancing-incidence angle scattering. This geometry permits all of the electrochemical experiments (CVs and a.c. impedance) to be carried out under diffraction conditions in a manner that allows comparison with what already exists in the literature and at the same time assures that a high-purity experiment has been conducted. The reflectance geometry traps a thin layer, <20 pm, of solution between the surface and a thin polypropylene or Mylar window. The assumption in this design is that the electrolyte will remain uniform in thickness, hence it will not flow when the cell is rotated on the diffractometer. The reflection geometry enhances the scattering, by allowing a larger sample and less absorption by the solution at high incidence angles.However, at angles approaching glancing incidence, this cell suffers several major disadvantages: (1) the actual glancing experiment cannot be achieved because the X-rays are reflected from the window; (2) as the glancing angle is approached, the absorption length in the solution, which is proportional to sin-' a, becomes extremely long, leading to excessive signal loss. The inability of these reflectance cells to provide well defined potential control at the surface leaves in doubt the potential at which the reported structural transitions occur. Furthermore, it is not possible to obtain a high-quality cyclic voltammagram under conditions where diffraction is being carried out and hence there are always questions about the state of the surface and the purity of the experiment.In particular, electrodeposited systems, such as Pb on Au, the quantity of material deposited cannot be precisely determined and coverages based on ex situ CVs, should not be assumed. Surface X-Ray Diffraction Surface X-ray diffraction is a well developed technique.21.22 Details concerning the reconstruction, roughness and faceting of the surface can be determined by measuring the intensity variations along the crystal truncation rods, CTRs21%22,29,30and the surface or non- integer scattering rods. The scattering vector is defined as Q =q, +q,,where q, =c*l, and q =a*h +b*k. The vectors a*, b* and c* are the reciprocal lattice vectors for the bulk crystal and h, k, I are the Miller indices.For CTRs, h and k have integer values while the vertical index, I, is not constrained to integer values due to the termination of the bulk crystal. For an abruptly terminated, or flat surface the intensity of the rods drops as 1 /Aq:, where Aq, is the distance from the bulk Bragg reflection. If the surface termination deviates from a flat surface to a rough surface the intensity of the rod drops more sharply. Changes in the surface charge or atomic density and spacing, caused by surface reconstructions, also cause changes in the intensity of the CTRS.~,~~.~~ By modelling the sum of the interferences from successive layers of the crystalline surface, a reflectivity profile based on single-layer deviations from an ideal termination can be constructed: K.M. Robinson and W. E. O'Grady The first part of eqn. (1) contains the Fresnel transmission coefficient, T(q,),for a low angle of incidence,21 atomic form factor F(q,)and a bulk Debye-Waller factor W(=+qf(u2)).The data reported within are Lorentz, area and polarization corrected. The second part of eqn. (1) contains the crystalline layer interference terms. The layer density, pm, is relative to the (loo)-( 1 x 1) layer density. The layer fluctuation parameter, o,, acts as a layer-dependent Debye-Waller factor. The layer displacement parameter, E,, allows the individual layers to deviate from the average multilayer spacing, c. A more direct approach to the reconstructed structure can be made by measuring the non-integer diffraction rods.These non-integer rods are related solely to the surface structure and not the bulk. Variations in the intensity of the non-integer rods describe how the reconstruction is arranged with respect to the bulk structure as well as determining atomic positions within the surface lattice. This latter measurement is far more difficult to perform, the intensity of the non-integer rods is small and they frequently are lost in the noise from the thermal diffuse scattering from the bulk crystal and the scattering from the solution. Transmission Electrochemical Cell A detailed description of the transmission geometry cell is discussed el~ewhere.~ Briefly, the geometry of the cell uses the traditional thin-layer cell concept, with the counter electrode directly opposite the working electrode, with the reference electrode tip situated in between.However, instead of a thin layer of solution squeezed between the electrodes, a larger drop of solution is suspended from a funnel-shaped reservoir between the two electrodes. The larger drop provides clearance for the incident and exit X-ray beams, Fig. 1. Multiple electrolyte solutions can be introduced to the crystal surface in the cell to allow tests of varying concentrations. The cylindrical shape of the cell provides 360" rotational symmetry and a 0-90" 26 range for the specular reflection measurements. The crystals used in this cell are 2.5 mm in diameter by 5 mm long and are held in a collette which seals about the base of the crystal. The collette can be mounted directly on the four-circle diffractometer.A groove in the lower half of the cell is filled with water-saturated filter paper to provide a constant 100% humidity inside the cell. This prevents the electrolyte drop from evaporating during the long exposure time needed for diffraction. In tests this mechanism kept a 2.5 mm diameter cylindrical drop stable for a period of two days even while undergoing the multiple rotations of the four-circle diffractometer. Experimental Prior studies of the electrodeposition of Pb on Au( 100) showed little more than a surface disorder in the [0, 0, 13 direction.31LEED results predict a c(2 x 2) structure for a coverage of 0.5 monolayer.8,12J8 Fig.l(b) shows the expected diffraction pattern for the c(2 x 2) pattern using the f.c.c. coordinate system and the positions at which data were collected. Because of the similarities of the atomic form factor of Au and Pb, for the c(2 x 2) structure it is expected that it is very difficult to distinguish between the surface (1 ,1) position and the (1, 1, I) CTR.32 The intensity relationship: 1 1O),, = [1 -FPb/FAu(I( 1 1 O)o,opb/I( Y~coverage)]~ developed for the c(2 x 2) at the (1, 1,0) position,32 predicts a minimal intensity variation for increasing coverage; however, comparisons of the (0,0, I), (1,1,I) and (2,0, I) CTRs and the glancing-incidence measurements of the (1, 0), (0, 1) positions of the c(2 x 2) structure allowed a determination of the Pb structure.The crystal was removed from the cell, repolished and the procedures outlined above were repeated to ensure reproducibility. The measurements were made on beamline X-23B at the NSLS at an energy of 9.5 keV, which represents a compromise between maximum beamline intensity and X-ray absorption by the electrolyte. Pb Monolayers on Au (100) Single Crystals The Au(100) crystal was cut from an Au boule, bulk mosaic <0.05" (1, 1, l), and polished down to 1 pm alumina powder. The crystals were further electropolished in a 1:1 :2 mixture of glycerol-ethanol-HCl(conc.). The crystals were then flame-annealed according to standard procedures30 and flushed with distilled water. This procedure has been shown to produce atomically smooth surfaces with a (1 x 1) surface stru~ture.~.~.~,~~ Only upon emersion at 0.4 V vs.SCE was the (5 x 20) reconstruction obser~ed.~.~~ Fresh 0.1 mol dm-3 HCIO, and 1.O mmol dm-3 Pb(CIO,), electrolyte, made from Ultrex grade HCIO,, Aldrich grade Pb(CI04),.3H,0 and 18 MSZ cm distilled water, was deoxygenated with 99.9995% pure N, and pumped into the reservoir.A small drop, 2.5 mm diameter, was brought into contact with the electrode surface with the potential held at 0.4 V vs. SCE. An Au/AuO, reference electrode was used in the measurements; however, data are reported vs. SCE. The CV from the outlined preparation technique is shown in Fig. 2. The letters indicate potential regions at which the X-ray diffraction data were measured. The result of holding the potential in a particular region will be discussed further.The CV is identical to those reported in the literaturesJ2 for Pb on Au( 100). The CV was recorded on the first cycle and is not a steady-state CV. Multiple cycles were avoided as expected alloy formation had been observed by STM.14*15 The CV clearly supports the high purity achieved in the preparation procedure and that the cell provides excellent potential control with no IR drop. In situ Diffraction Results and Discussion The integrated intensity for the (0, 0, I), (1, 1, I) and (2,0, I) CTRs for the three potential regions of Fig. 2 are shown in Fig. 3-5. There are considerable differences with the fits (a) ~ N2in,et ,,-reference electrode I-counter electrode electrolyte inlet reservoir water filter paper working electrode I X (1,2)s (220) Fig.1 (a) Transmission geometry cell for in situ X-ray diffraction. The entire cell is further encompassed in an He-filled bag. (b)The c(2 x 2) diffraction pattern. Arrows indicate positions examined by X-ray diffraction. K. M. Robinson and W. E. O’Grady 2 1 $0 -1 -2 2 1 3r=-0 -1 -2 -0.4 -0.2 0.0 0.2 0.4 EIV us. SCE Fig. 2 Au(100) cyclic voltammagram recorded in the transmission cell, (a) first CV 0.1 mol dm-3 HCI04-l mmol dm-3 Pb(CIO,), 25 mV s-I, (b)same CV except after extended period of time at potentials negative of the irreversible peak at -0.32 V vs.SCE 1 3.0 i,I3.5 2.5 -4.A I t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (090, I) Fig. 3 (0, 0, I) CTR and fits of eqn. (1) for the regions A( 0,-), B( A,-* -) and C( m, - - -) of Fig. 2 Pb Monolayers on Au (100) Single Crystals oooooo I 100 000 1000 10 000 100 10 0.0 0.2 0.4 0.6 0.8 1.0 (17 17 1) Fig. 4 (1, 1, I) CTR for the regions B (A)and C (M) of Fig. 2 10 000 1000 AU._v) gU ._ 100 inI-0.0 0.1 0.2 0.3 0.4 0.5 0.6 (2, 0, I) Fig. 5 (2, 0, l) CTR for the regions B (A)and C (U)of Fig. 2 and a calculation (-) for the ideally terminated surface in region A associated with each of the CTRs. Table 1 contains the results from a fit of eqn. (1) to the (0, 0, I) data. The (I, 1, I) and (2,0, I) CTRs are discussed qualitatively.Each potential region will be discussed separately. + 0.440 V vs. SCE The (0, 0, 1) profile was best fit with a model which depicts the (5 x 20) reconstruction and adsorption of ca. 60% (1 x 1) coverage of the Pb2+ ions. There is no significant current measured in the CV to suggest that the Pb has undergone electrodeposition. In addition, the Pb layer has a large fluctuation term which suggests that the Pb is not strongly adsorbed, which would not be possible if the Pb had been electrodeposited on the Au surface. This supports a diffusion of the Pb2+ ions to the surface as the potential is stepped below the K. M. Robinson and W.E. O'Grady Table 1 Fit parameters of eqn. (1) to the (0, 0, l) data in Fig.3 €2 (uIi2)2EIV vs. SCE 'Jl PI €1 'J? Pr + 0.40 to 0.0 0.2 0.60 -0.03 0.1 1.20 0.22 0.08 0.0 to -0.32 0.1 0.50 -0.10 0.1 1.0 0.05 0.08 -0.32 to -0.40 0.2 1.4 0.15 0.1 1.1 -0.60 0.08 P.Z.C.for Au(100)-(5 x 20).3.4 The (1, 1, I) and (2, 0, I) CTRs depict a smooth surface termination. 0.0 to -0.32 V VS. SCE. The (0, 0, I) profile is best fit with a (1 x 1) surface and 50% surface converge on Au and electrodeposited Pb layer. The (2, 0, I) profile also shows an increase in the surface roughness; however, the (1, 1,I) profile remains ideally smooth. As Pb is electro-deposited, the (5 x 20) reconstruction is lifted and forms Au-Pb surface mosaics on top of the Au (1 x 1) structure. This potential region is consistent with only 25% (1 x 1) total Pb electrodeposited.When the 20% additional Au atoms from the reconstruction are included, this accounts, within error, for the surface fit density of 50%. The (2,0, I) shows an increase in the surface disorder; however, at higher values of I, the CTR becomes nearly ideal in profile, consistent with the (1, 1, I). Rocking curves taken above and below the critical angle, Fig. 6, provide some insight into the surface disorder. Near the surface, the (200) planes are displaced f0.45" from the bulk (200) planes. There is no bulk roughening as observed by the (1 ,1,I) CTR. An exact Au-Pb exchange would require an Au: Pb ratio in a single layer of 25 : 1 to account for a 0.45" distortion, calculated for rpb/rAu= 1.2. The amount of Pb electrodeposited gives a surface ratio of Au: Pb = 4.There are two possible positions for the excess of Pb: (1) AuPb alloys on the surface or (2) a structure that slightly distorts the (1 x 1) surface. In a c(2 x 2) configuration, a Pb atom would have to be pressed only 0.02 A, based on hard-sphere calculations, into the (1 x 1) Au structure to cause the 0.45" distortion. There is no evidence, via alloy diffraction peaks, that there is any alloy formation at this potential. Secondly, there is a slight increase in the (1, 1, 0) surface diffraction, Fig. 6, consistent with the intensity predicted by eqn. (2). This suggests that there is a c(2 x 2) structure. The lack of any (1 ,0) or (0, 1) scattering, associated with the c(2 x 2), can be accounted for by the fact that there are Au islands, from the lifted (5 x 20), occupying 20% of the available surface.It has been shown that these Au atoms form small islands randomly situated on the (1 x 1) surface.33 These islands would prevent growth of large domains of c(2 x 2) and therefore weak in-plane diffraction. -0.32 to -0.40 V VS. SCE The structures reported for potentials above -0.32 V vs. SCE appear reversible in both the X-ray data and the CV. The surface distortion peaks, Fig. 6, are lost if the potential is swept positive before the CV peak at -0.32 V vs.SCE. If the potential is stepped below -0.32 V vs. SCE, the distortions in the (2, 0, I) profile are permanent, Fig. 6, and the CV is also changed, Fig. 2. The (0, 0, I) profile is best fit with a AuPb layer, possibly hexagonal, due to the high density from the fit, on top of a (1 x 1) surface, also with a slightly higher density, which could possibly be due to Au-Pb interdiffusion.A dense hexagonal overlayer has been observed in LEED for coverages >60°h.18,34There is no in-plane diffraction observed to allow a determination of the surface structure, A visual improvement to the fit would allow slight distortions to the third layer; however, with the present error associated with the intensity integration, there is no statistical improvement. Interestingly, the (2, 0, I) CTR Pb Monolayers on Au (100) Single Crystals 500 to) oo ++ 00 300 Q 100-._ 1 & v OO*v) d 00 070 00 0 0 00-00 0 OO Q? Ooe:0 Oe,0 0 oooo 0 O 0 0 t oooo + + 30 -2 -1 0 1 2 4 Fig.6 Rocking curves for the (a)(2,0,0.l), (0),(2,0,0.15),( +) positions at -0.3 V vs.SCE and (b) the (1, 1, 0.1) position at -0.3 V vs. SCE, (0)and at 0.4 V us. SCE, (+) above the critical angle and the entire (1, 1, I) CTR remain unchanged, which means the permanent distortions involve only the first two to three layers. Below the critical angle, the (2, 0, I) contains many structural features. More information is necessary to fit these complex fluctuations. -0.4 to -0.5 V VS. SCE At the potential at which Pb will deposit on Pb, the surface structure changes rapidly. Fig. 7 shows the (0, 0, I) profile in this region. The only fit to the data is a highly disordered surface.The (1, 1,l)and (2,0,1)profiles are so disordered that the rods are not distinguishable from the background. Extra peaks have appeared in the data which are powder rings of the AuPb2 alloy. An extended period, 15-30 min, at this potential increased the size of the alloy peaks, which remained after cycling the potential back to + 0.4V vs.SCE. At this point the crystal had to be removed from the cell, etched in an HNO,-HCl acid mixture, repolished with alumina powder and annealed to regain a smooth surface. Conclusions Three distinct stages of Pb electrodeposition on Au( 100) have been studied by in situ X-ray diffraction. Initially, Pb2+ ions diffuse to the surface as the potential is stepped below the K. M. Robinson and W.E. O’Grady I I I 1.0 1.5 2.0 2.5 3.0 (0,090 Fig. 7 (0, 0, I) CTR for region D, -0.4 to -0.5 V vs. SCE. The new peaks are powder rings for the AuPb, alloy. P.Z.C. for the Au( 100)-(5 x 20) surface. The reconstruction lifts as electrodeposition occurs and the Pb and Au atoms form a c(2 x 2) surface on an Au( loo)-( 1 x 1) surface. The Pb distorts the (1 x 1) surface. An irreversible deposition peak, at -0.32 V vs. SCE, causes a permanent distortion, and probable Au-Pb atom exchange, after which the AuPb forms a high-density surface, composed of two to three atomic layers, on a (1 x 1) surface. AuPb, alloy forms, probably from the Au-Pb exchange sites, at potentials favourable for Pb on Pb deposition which results in an extremely disordered crystal.These conclusions are in agreement with ex situ LEED studies on Au( 100)and STM observations of AuPb alloys on Au( 11 1). Further study on the effect of stripping the electrodeposited Pb, and of the structures observed by LEED,*-’*are underway. The authors acknowledge the Research Associate Program of the National Research Council. Additional support was provided by the Office of Naval Research. Beamtime was provided on the NRL X-23B beamline, designed and built by Dr. W. T. Elam and J. Kirkland. NSLS is supported by the Department of Energy, Division of Materials Sciences and Division of Chemical Sciences under contract no. DE-AC02-76H000 16. References 1 B. M. Ocko, J. Wang, A. Davenport and H. Issacs, Phys. Rev. Lett., 1990, 65, 1466.2 D. M. Kolb and J. Schneider, Electrochim. Acta, 1986,31, 929. 3 M. S. Zei, G. Lehmpfuhl and D. M. Kolb, Surf. Sci., 1989, 221, 23. 4 A. Hamelin, M. J. Scottomayor, F. Silva, S. Chang and M. J. Weaver, J. Electround. Chem.. 1990, 295, 291. 5 S. Strabc, R. R. Adz2 and A. Hamelin, J. Eleciroanal. Chem., 1988, 249, 291. 6 X. Gao, A. Hamelin and M. J. Weaver, Phvs. Rev. Lett., 1991, 61, 618. 7 X. Gao, A. Hamelin and M. J. Weaver, Phys. Rev. B, 1991,44, 10983. 8 P. Hagens, Ph.D. Dissertation, Case Western Reserve University, 1980. 9 K. M. Robinson and W. E. O’Grady, Rev. Sci. Instrum., in the press. 10 R. R. Adzic, E. Yeager and B. D. Cahan, J. Electrochem. SOC.,1974, 121, 474. 11 J. W. Schultze and D. Diclertmann, Electrochim. Acta, 1976, 22, 489.64 K. M. Robinson and W. E. O’Grady 12 K. Engelsmann, W. J. Lorenz and E. Schmidt, J. Electroanal. Chem.. 1980, 114, 1. 13 M. G. Samant, M. F. Toney, G. L. Borges, L. Blum and 0.R. Melroy, J. Phys. Chem., 1988,92,220. 14 M. P. Green, K. J. Hanson. R. Carr and I. Lindau, J. Electrochem. Soc., 1990, 137, 3493. 15 M. P. Green and K. J. Hanson, Surf Sci. Lett., 1991, 259, L743. 16 N. J. Tao, J. Pan, Y. Li, P. 1. Oden, J. A. Derose and S. M. Lindsay, Surf Sci. Left., 1992, 271, L338. 17 D. A. Koos and G. L. Richmond, J. Phys. Chem., 1992,96, 3770. 18 J. P. Biberian and G. E. Rhead, J.Phys. F., 1973. 3, 675. 19 E. Bauer, Appl. Surf. Sci., 1982, 11/12,479. 20 For an overview of in situ techniques see: Electrochemical Interfaces, ed.H. D. Abruna, VCH, Weinheim, 1991. 21 R. Feidenhans’l, Surf. Sci. Rep., 1989, 10, 105. 22 I. K. Robinson, in Handbook on Synchrotron Radiation, ed. G. S. Brown and D. E. Moncton, Elsevier, Amsterdam, 1991, p. 221. 23 K. M. Robinson and W. E. O’Grady, J. Am. Vac. SOC.,in the press. 24 M. Fleischman and B. W. Mao, J. Electroanal. Chem.. 1987, 229, 125. 25 C. A. Melendres, H. You, V. A, Moroni, Z. Nagy and W. Yun, J. Electroanal. Chem., 1991, 297, 549. 26 G. M. Bommarito, D. Acevedo and H. D. Abruna, J. Phvs. Chem., 1992,96, 3416. 27 M. F. Toney, J. G. Gordon, M. G. Samant, G. L. Borges, 0.R. Melroy, L. Kau and D. G. Wiesler, Phys. Rev. B, 1990,42, 5594. 28 M. F. Toney, J. G. Gordon, M. G. Samant, G. L. Borges, D. G. Wiesler, D. Yee and L. B. Sorensen, Langmuir, 199 1, 7, 796. 29 I. K. Robinson, Phys. Rev. B, 1986, 33, 3830. 30 K. M. Robinson, I. K. Robinson and W. E. O’Grady, Surb Sci., 1992,262, 387. 31 B. Ocko, J. Wang, A. Davenport and H. Isaacs, Extended Abstracts: The Electrochemical Society, Spring 1991 Symposium. 32 P. W. Stephens, P. J. Eng and T. Tse, Surfuce X-Ruy und Neutron Scattering, ed. H. Zabel and I. K. Robinson, Springer-Verlag, Berlin, 1992, pp. 79-82. 33 N. Batina, D. M. Kolb and R. J. Nichols, Langmuir, 1992, 8, 2572. 34 J. Perdereau, J. P. Biberian and G. E. Rhead, J. Phys. F, 1974.4, 798. Paper 3/00266G; Received 14th Januarj,, 1993

ISSN:1359-6640    

DOI:10.1039/FD9939500055

出版商:RSC    

年代:1993

数据来源: RSC

摘要:

Faraday Discuss., 1993,95, 65-74 Surface Roughening, Surface Melting and Crystal Quality J. P. van der Eerden Laboratory of Interfaces and Thermodynamics, Padualaan 8, 3584CH Utrecht, The Netherlands The modelling of crystal surfaces, both in equilibrium and during growth, is now at a stage where the relation between surface structure and crystal quality are accessible for future studies. The classical point of view is that the roughening transition marks the transition from step growth to rough growth with qualitative changes in the growth rate and the morphology. The surface- melting transition marks a change from solid-like to liquid-like surface layers. This enhances the kinetics and will decrease the anisotropy of equilibrium surface properties. It has been shown that surface melting may also lead to a linear growth law.Slightly speculative arguments are given to illustrate how surface roughness and softness are related to morphological instability, impurity incorporation, mother-phase inclusions, dislocation formation and stacking faults. 1. Introduction During the last international conference on crystal growth (ICCG-lo), I gave a qualitative summary of the relevance of surface roughening and surface melting for crystal growth.' This paper included both models and experimental results. In the present contribution I will use parts of that paper to summarize the essential ideas, and in addition some new results will be presented, which allow me to focus on the implications for the quality of crystals grown.I shall concentrate on model results with only a few experimental results. Surface roughening and surface melting occur in many different systems. There is no general law to determine the order in which surface melting, surface roughening and bulk melting occur. For example, for the (1 11) and (100) surfaces of lead neither surface roughening nor surface melting have been observed. For the (1 10) surface of lead and for the (100) and (1 1 1) surfaces of Lennard-Jones crystals surface roughening and surface melting occur simultaneously, below the bulk melting point. For the (0001) and (1OiO) faces of ice surface roughening is (using the equilibrium f0rm~9~) observed at a higher temperature than surface melting (viaellip~ometry~).The (001) and (010) faces of biphenyl show surface melting5 just below the melting point, but no surface roughening.The (1 10) and (201) surfaces, on the other hand, have roughening temperatures of 10-1 5 K and 20-25 K below the melting point.3 I shall start with a summary of our fundamental understanding of surface roughening and surface melting. In theoretical studies it is established that surface roughening is determined by the vanishing of the edge free energy above the roughening temperature and surface melting by the vanishing of the surface shear modulus above the surface melting temperature. Theory of Surface Roughening and Surface Melting The roughening transition for a surface with orientation no (i.e. with the unit surface normal no)is characterized by the variation, for orientations n close to no,of the specific interface 65 Surface Roughening and Surface Melting excess free energy n(n) (in J m-2).Generally we may write YyI(n)= yI(no)+ hln -nol + +Gln-no12 + @In -nol3 + (1) where terms of fourth and higher order in n -no are omitted and for simplicity cylindrical symmetry around the axis nois assumed. Here Y is the edge free energy (in J rn-l), i.e. the excess free energy associated with the addition of a (monatomic) step of height h to a smooth no surface, and for T< TR, a" and fl describe the interaction between steps. The parameters yl, y, a" and /3 depend on temperature and may behave differently above and below TR. Related to the surface tension y1 the surface stiffness a(n) is defined by: The central results of the theory of surface roughening are (i) the edge free energy is positive at low temperatures, (ii) y zz 0 above the roughening temperature TR, (iii) approaching TR from above, the surface stiffness a(no)approaches a universal value at TR and (iv) the height-height correlation length 5 is inversely proportional to Y.The analytic expressions are The dimensionless number a is of order unity, 52 is the atomic volume and 4 measures the interaction of growth units parallel to the surface.In lattice-model calculations (e.g. the Kossel model) 4 usually is taken as a constant. But when we allow it to vary with temperature, then certain important aspects of the interference with other surface transitions can be described.The equilibrium shape is directly related to the surface tension yI and can be constructed from it with the well known Gibbs-Wulff constr~ction.~ It turns out that the roughening point T,(n) is associated with the presence at T< TR of a flat face with orientation n on the equilibrium form. The size of this face is proportional to Y and the curvature of the equilibrium form in the immediate neighbourhood of the face edge is related to the step interaction parameters a" and p.The curvature of the equilibrium form at T z TR(n)gives the stiffness a(n) at TR.Thus, in principle, the equilibrium form is an excellent tool for studying surface roughening. The dynamic equivalent of surface roughening is also important.6,s On flat surfaces the positive value of the edge free energy Y implies that there is a thermodynamic barrier for growth, a barrier which can be overcome by spiral growth or by two-dimensional nucleation.Under the influence of a sufficiently large supersaturation this barrier may effectively disappear. A quantitative criterion can be formulated either as the point where the surface correlation length 5, given by eqn. (6), equals the radius of the critical nucleus,6 or as the point where the formation free energy of the nucleus equals kT.* Both methods show that surface roughening occurs when the supersaturation Ap is above a critical value Ap* which depends on the edge free energy Y: Ap2 Ap* z(Q/h)(y2/k7') (7) In the growth form, dynamic roughening is observed as the disappearance of a facet, in the growth rate it marks the transition from a non-linear to a linear dependence of the growth J.P. van der Eerden rate R on Ap. Another important point is that the surface is less stable against dendritic and cellular perturbations, owing to the decreased value of the surface stiffness. The most widely used model for surface melting is the Lennard-Jones model, a classical statistical mechanics model with the pair potential q(r)z 44(ajr)l2 -(~/r)~] where the approximate equality sign indicates that different modifications are used to obtain a smooth cut-off at a finite interaction range (usually the latter is ca. 2.5 times 0).The minimum of this potential is -E and CJ can be seen as the diameter of the atoms in this model.Extensive calculations by Broughton and Gilmer9 have shown that at the surface already well below the bulk melting point, both the particle density and the lateral diffusivity deviate from the value in the bulk. But neither the temperature dependence of these quantities, nor the temperature dependence of the surface energy, give a clear estimate of a surface melting point. Therefore we step back first to consider the phenomenon of melting itself. The fundamental difference (as is well known from real systems) between a solid and a liquid is that the shear modulus is finite in the solid phase and vanishes in the liquid phase. So far, however, this criterion has been used mainly for two-dimensional systems.’* It is unique to two-dimensional systems that the singularities at the melting point of the specific heat and of the structure factor may be very weak, and only the elastic constants vary discontinuously.With these results in mind we proposed to define surface melting as the vanishing of the local value of the lateral shear modulus at the surface. It was not immediately clear how such a quantity should be defined, but after a preliminary attempt’ we published a reliable definition.12 We found that, according to this definition, surface melting occurs, both at the (001) and at the (1 11) surfaces of an f.c.c. Lennard-Jones ~rysta1.l~ In a macroscopic picture a crystal surface above the surface melting point would have a ‘quasi’-liquid layer of thickness 6 on top of a solid crystal. The interface free energy yr = ysv is the sum of the contributions ysl of the solid/liquid interface, ylvof the liquid/vapour interface and the Gibbs free energy difference sAG(6, 7‘)between a quasi-liquid and a solid layer of thickness 6.Since the temperature is below the bulk melting point, to a first approximation AG equals AhATIT,, where Ah is the heat of melting per unit volume, and A T = T, -T is the undercooling below the bulk melting point T,. As seen in Fig. 1 the quasi-liquid layer is not isotropic, therefore correlation effects as well as the finite size effects lead to &dependent corrections to A G. The actual thickness 6 corresponds to the minimum value of ~~(6).The temperature dependence of 6 is sometimes accessible experimentally and is seen to give information on the properties of the macroscopic properties of the surface layer.Owing to surface melting the step and surface kinetic coefficients shift to larger values. Below the surfacc melting point atoms adsorbed at the terraces diffuse by ‘hopping’ to a step. The mobility of these adatoms is relatively small owing to pinning to the terrace surface, therefore this is not a very effective mechanism. Above the surface melting point adatoms are more mobile and in addition the mobility of the atoms inside the quasi-liquid on the terraces introduces a new step-growth mechanism which is similar to growth from a two-dimensional melt. As a consequence steps move faster, and also the growth rate will increase towards the value for rough surfaces.Thus the growth rate and the growth form will become less anisotropic. Interference of Surface Transitions It is important to remark that to date we do not have a single model in which surface roughening and surface melting can be accurately studied simultaneously in equilibrium. In principle Monte Carlo (MC) or molecular dynamics (MD) simulations of the Lennard- Surface Roughening and Surface Melting Fig. 1 Two equilibrium surfaces of a (1 11) surface of a Lennard-Jones crystal at T = 69 K. For this model T, = 72 K. The surface layer is disordered and mobile laterally but stratified perpendicular to the surface. Thus the quasi-liquid layer is not an ordinary three-dimensional isotropic liquid, but rather a perturbed two-dimensional liquid.Jones model could be used, but since from eqn. (3), (4) and (6) the correlation length 4 is seen to diverge at the roughening temperature, some 20 layers of 200 x 200 atoms each are necessary to reduce finite size effects to an acceptable level. In the 500-5000 atom Lennard- Jones systems which have been investigated numerically, only the qualitative onset of roughening can be investigated. For growth the situation is somewhat different. At low temperature rough surfaces can be produced by a high supersaturation, then surface melting will not be important. This allows one to study, in the Lennard-Jones model, the relation between temperature, growth rate and crystal defects.The conclusion of MC and MD calculations in the Lennard-Jones system is that surface roughening and surface melting are strongly co~pled.~?~."-'~ The physical argument starts with the observation that the bulk melting point is considerably above the melting point of a two-dimensional Lennard-Jones crystal (both at about zero pressure). The reason that nevertheless the surface layer only melts close to T,, is that the potential-energy field of the bulk layers of the crystal has a strong ordering influence. At the (001)face the surface atoms are forced in a square lattice and at the (1 1 1) face the atoms are strongly compressed. For both faces a significant lateral degree of freedom is released when a small fraction of the surface-layer atoms is promoted to an outer layer.The increased lateral mobility leads to surface melting, the incomplete occupancy of surface layers marks qualitatively the onset of surface roughening. With this picture in mind one sees that it depends on the type of interaction what precise behaviour of the surface is related to surface roughening and/or surface melting. In order to catalogue the different situations, we first adopt the point of view that at a given temperature a crystal surface can be approximately described as a Kossel crystal, but that the effective value of the lateral interaction parameter # in eqn. (3) may be temperature dependent. Structural transitions in the interfacial region may lead to rapid variations of #.J. P. van der Eerden For example, if there is a first-order transition at a temperature Tcin the surface region, then a jump in 4 = 4(T) at T = T, will result. Then it may happen that kT/+(T)is below the roughening value when T is just below T,, and above the roughening value when T is just above T,. In such a case surface roughening will appear as a first-order phase transition: y discontinuously jumps from a finite positive value below TR= T, to zero above T,. A second-order structural transition, on the other hand would lead to a power-law dependence, 4(r) -+(T,)z (T,-na.Such a dependence will keep the roughening transition infinite order, but the square-root dependence in eqn. (3) could be replaced by another power law. One possibility for such a surface-structural-phase transition is surface melting.From the theory of two-dimensional melting the thermodynamic order of the melting transition in two-dimensional systems is known10,14 to depend on details of the interaction, in particular the core energy of two-dimensional dislocations in the two-dimensional solid plays a key role.14 In the two-dimensional Lennard-Jones model melting is (weakly) first order. In the presence of a periodic substrate potential melting also depends on details of the interactions. At least lo first-order, infinite-order and two-step transitions may occur. Our MC results13 are suggestive of a second-order surface melting transition on Lennard-Jones crystals. A point which has not yet been discussed in the literature goes beyond the generalized interpretation of the Kossel model which we have used.It is easily seen, e.g.in Fig. 2 that the melting point of an incomplete layer may be considerably lower than that of a complete layer. Thus, during growth at a temperature below the surface melting point, the surface may be liquid-like in early stages of covering a flat terrace, but ‘solidify’ upon completing a layer. This should modify our description of the two-dimensional nucleation model for crystal growth considerably, since nuclei are liquid-like in their early stages of evolution. Also the dynamic roughening is influenced, since the thermodynamic barrier is poorly estimated by the edge free energy y of a step separating to solid like terraces.Growth in Pure Systems Several effects of surface roughening and surface melting on the growth of a pure crystal can be discerned. We have seen already the change over from linear to non-linear growth at the roughening point, the enhanced kinetics at the surface melting-point and the shift in the equilibrium and dynamic roughening points when the two transitions interfere. A new crystal growth mechanism can also appear. Consider the case where surface melting occurs a lower temperature than surface roughening. Then the surface structure will be characterized as terraces, separated by steps, even above the surface melting-point (but below TRof course). The atoms in the top layer of a terrace will have a chemical potential close to that of the solid part of the crystal, as solid/liquid interfaces generally are close to equilibrium.Thus, whenever growth units, arriving from the mother phase, are absorbed in the terrace layer, then they contribute to growth. The growth rate R is built up from two contributions: R = hlBstcst + Vabs(1 -acst>lo (9) where h is the step height (in m), pstthe step kinetic coefficient (m s-l), c,, the step density (m-I), Vabs the equilibrium absorption frequency of growth units (s-l), a the effective step width (usually a z h) and CT the relative supersaturation. In a very rough approximation7 Bst is given by Bst = V’adsQ (10) The absorption contribution in eqn. (9) is in principle present for solid-like terraces also but in that case the absorption frequency is negligible.When one considers the absorption Surface Roughening and Surface Melting Fig. 2 (a) Surface of a Lennard-Jones crystal, growing in a (1 11) direction from the vapour at T = 50 K at a stage where the outer layer is completed. Note that this outer layer is well ordered. (6) As (a), but at a moment where the outer layer is about half filled. Note the disorder of the outer layer in this case. J. P. van der Eerden process as an activated process in which the absorbing growth unit has to push away surrounding atoms at the surface, one gets where pLis the surface shear modulus and b is a numerical factor of order unity. The denominator in the argument of the exponent is an estimate of the elastic energy involved in the deformation of the terrace when a growth unit from the mother phase is absorbed.For solid-like surfaces this means that the growth unit has to be placed interstitially, which corresponds to a large deformation energy. For a liquid-like surface, only some liquid has to be pushed away which does not cost energy. So, for a terrace above the surface melting point, the second term in eqn. (9) is important, and may even become dominant. But this term does not depend on the step density and hence a linear dependence of the growth rate R on CJ is to be expected. So we reach the surprising conclusion that a transition from non-linear to linear growth may occur at the surface melting point, even when this is below the surface roughening point! Finally I want to mention an observation which indicates yet another possible relation between surface melting and the crystal quality.As the observation is very recent I do not yet have a full explanation. For a Lennard-Jones crystal the f.c.c. structure is energetically only slightly more favourable than the h.c.p. structure. Accordingly, in MC growth experiments for (1 1 1) surfaces from the vapour phase we often find several h.c.p. stacking faults in the f.c.c. structure. But, on one occasion, illustrated in Fig. 3, our starting configuration consisted of a thick liquid layer on top of a solid. This system was cooled at a temperature of T = 69 K. After a rapid crystallization process we found that the liquid layer was solidified in seven layers with a perfect h.c.p.structure! Whether this result is significant or accidental, it should remind us of the experimental evidence that at high supersaturations metastable crystal structures are often formed. Apparently surface melting may promote such metastable growth structures. Similarly, but less surprisingly, growth at low temperature may induce defects. Consider for example the low-temperature growth of (1 11) argon. The growth will be in layers. The energy difference between adsorption on the proper sites for an f.c.c. stacking does not differ much from the energy at the twin position which corresponds to h.c.p. stacking. Therefore coexistence of h.c.p. and f.c.c. domains in a single (1 1 1) layer occurs easily. The domain boundaries in this case are partial dislocations, surrounding stacking faults.At larger growth rates such defects proliferate and ultimately amorphous structures could occur. Here the final structure depends on the ratio q = uSta/Dbetween the step velocity uSt and the dislocation drift velocity, which is roughly estimated as D/a where D is the vacancy diffusivity in the surface layer. Surface melting increases D and hence decreases the defect density in general. Impurity Effects Below the roughening point crystals are very sensitive to traces of those impurities which are adsorbed strongly on smooth terraces between steps, but are hard to incorporate in the crystal. Such impurities lead to macrostep formation.' The point is that such impurities are quite immobile and therefore can block the motion of a step locally.It has been established that growth can be completely blocked, even when the impurity concentration in the mother phase is below the ppm level. At the base of a macrostep impurities tend to accumulate, which leads to macroscopic segregation, mother-phase inclusions etc. If surface melting occurs below the roughening point, then the step velocity increases. This reduces the blocking effect of impurities. On the other hand, when the impurities are also more mobile, then the step motion tends to collect them and macroscopic segregation of impurities takes place which is likely to be detrimental to the crystal quality. Surfaces above the roughening point, are less sensitive to such impurities, since now the Surface Roughening and Surface Melting a b C b C b C b C b a b 1 I Fig.3 Formation of h.c.p. structure during growth from the melt at T = 69 K. (a) starting from an f.c.c. crystal from which ca. seven monolayers were molten at T = 72 K x T,. The molten region solidified rapidly to seven layers in an h.c.p. stacking. Further (relatively slow) growth from the vapour led to another three layers, stacked according to the f.c.c. scheme. J.P.van der Eerden whole surface has to be covered before growth is blocked. One effect of impurities, therefore, is that when they are incorporated in the crystal, they increase the chemical potential of crystal particles and reduce the effective supersaturation.For very small impurity concentrations this only leads to an almost uniform impurity distribution. At higher impurity levels, however, morphological instability sets in, especially due to the same strongly adsorbed but poorly absorbed impurity which produce macrosteps below TR.As a result surfaces of crystals growing from impure solutions often develop surface roughness on a ‘mesoscopic’ scale, i.e.pm scale, coarsening in the form of macrosteps or a cauliflower-like morphology. Finally, as noted by Nenow and Pavlovska3J5 impurities may reduce the roughening temperature. These authors reported, for carefully purified adamantane (T, z 546 K), roughening temperatures of TR(1 1 1) z 528 K and TR(100) z449 K, whereas for less pure adamantane TR(l 11) z 488 K and TR(lOO)z 423 K.This shift of the roughening temperature can be understood as follows. Let SC#+ be the adsorption energy of impurities, S+i<O for the relevant types. Then a fraction xI,given by xiz xiLexp ( -:$) of surface adsorption sites will be covered by impurities (xILis the mole fraction in the mother phase). This will shift the broken bond energy from the value 4 for a pure mother phase to (b + Eqn. (3) then shows that the roughening point shifts accordingly. Conclusion and Outlook The roughening transition separates different growth (step growth from rough growth) and morphological (faceted from curved) regimes. Above the roughening point a crystal is sensitive to morphological instability. Below the roughening point dynamic roughening occurs above a certain value of the driving force.If the atomic (dynamic and/or equilibrium) roughness becomes too large then lattice deformations may be induced near the growing surface. Surface melting may effectively change the temperature scale, and thereby change the roughening point. The growth rate increases owing to surface melting and becomes linearly dependent on the supersaturation, the growth form is more isotropic, and the surface is less sensitive to impurities. These effects are strong when surface melting occurs at or below TR. When the molten layer becomes thick then metastable modification may grow. Finally, a better understanding of the surface transitions pays off for practical crystal growth.It is suggested that one may look for optimal conditions for the growth of high- quality single crystals along the following lines. In order to avoid dendritic instabilities, macroscopic inclusions and large dislocation bundles one wants to have a reasonably ‘stiff’ surface [in the sense of a relatively large value of the surface stiffness a, given in eqn. (2)]. This points to growth below the roughening point. On the other hand the growth should not be too slow, therefore the driving force should be chosen close to the critical value Ap* for dynamic roughening. In order to minimize uncontrollable influence of trace impurities and defect formation, one should try to be close to the surface melting point. To satisfy these three requirements is, however, not always possible.In growth from the melt at atmosphere pressure, the temperature is the only degree of freedom, and only if TR< T,,, can one meet the first two requirements. For physical vapour deposition the surfaces are usually so far below TRthat for dynamic roughening too high a vapour pressure would be required for this technique. For chemical vapour deposition the compounds can be chosen such that a dynamic roughening situation is approached, but to have surface melting as well may require conditions where e.g. temperature control is difficult to achieve. One has a large flexibility in the case of organic crystals, growing from organic solutions as chemical variation can be used to approach optimal conditions, Surface Roughening and Surface MeIting although the complexity of the system may oppose an accurate control of the purity of the system.The main point I want to stress is that at the moment we have reached the point where fundamental studies on the relation between growth systems and crystal perfection start to become feasible. To clarify this point is probably one of the main challenges for the theory of crystal growth in the next few years. I gratefully acknowledge my co-workers, T. H. M. van den Berg and J. Huinink, and the students D. Kragten and F. de Gauw for their stimulating help and interest. References 1 J. P. van der Eerden, J. Cryst. Growith, 1993. 2 D. Nenow and V. Stoyanova, J. Cryst. Growth, 1979, 46, 779. 3 D. Nenow, Prog.Cryst. Growth Charact., 1984, 9, 185. 4 Y. Furukawa, M. Yamamoto and T. Kuroda, J. Cryst. Growth, 1987,82, 665. 5 A. A. Chernov and V. A. Yakovlev, Langmuir, 1987,3, 635. 6 S. Balibar, F. Gallet and E. Rolley, J. Cryst. Growth, 1990, 99, 46. 7 J. P. van der Eerden, in Handbook of Crystal Growth, ed. D. T. H. Hurle, Elsevier, Amsterdam, 1993, ch. 6. 8 X. Elwenspoek and J. P. van der Eerden, J. Phys. A, 1987,20, 669. 9 J. Q. Broughton and G. H. Gilmer, J.Chem. Phys., 1983,79,5059,5105,5119; 1986,84,5749,5749,5759. 10 D. R. Nelson and B. I. Halperin, Phys. Rev. B, 1979, 19, 2457. 11 J. P. van der Eerden, A. Roos and J. M. van der Veer, J. Cryst. Growth, 1990, 99, 77. 12 J. P. van der Eerden, H. J. F. Knops and A. Roos, J. Chem. Phys., 1992,96, 714. 13 J. P. van der Eerden, T. H. M. van de Berg, J. Huinink and H. J. F. Knops, J. Cryst. Growth, 1993. 14 Y. Saito, Phys. Rev. Lett., 1982, 48, 11 14. 15 A. Pavlovska, J. Cryst. Growth, 1979, 46, 55 1. Paper 3/00074E; Received 5th January, 1993

ISSN:1359-6640    

DOI:10.1039/FD9939500065

出版商:RSC    

年代:1993

数据来源: RSC

摘要:

Furuday Discuss., 1993,95, 75-84 Computer Modelling of Inorganic Solids and Surfaces Stephen C. Parker,* E. Toby Kelsey, Peter M. Oliver and James 0.Titiloye University of Bath, School of Chemistry, Claverton Down,Bath, UK BA2 7AY We have recently performed calculations on a number of inorganic solids and their surfaces. The emphasis of this work is to understand the role of surface defects, either intrinsic defects or additives, in modifying structure and stability at the atomic scale. The basis of the approach is to use energy minimisation to obtain the most stable configuration. In this paper we describe simulations on NiO which predict that the { 11 l} configuration is stabilised by surface oxidation at high temperatures. Further simulations of calcite model the effect of additives on morphology, and we describe the effects of lithium, magnesium and hydrogenphosphate _additives.We find that both magnesium and phosphate stabilise the { lOlO} surface while lithium stabilises the basal plane. Finally, we present preliminary work on calcite and barium sulfate which implies that these methods may provide useful insights in nucleation and crystal growth at high ionic strengths. A detailed description of surface structure and reactivity is central to understanding crystal growth at the atomic scale. This is particularly important for interpreting the role of additives which is of concern for many technological processes. However, experimental studies of the structure and stability of interfaces, particularly in polar solids, are often difficult.Atomistic simulation provides a complementary technique for probing the surface structure, stability and behaviour of defects. Atomistic simulation is now becoming well established and stems from the develop- ments of Tasker and Mackrodt, who demonstrated the potential of this technique by successfully modelling the surface properties of a range of ceramic oxides which include the cubic rock-salt oxides MgO, CaO and NiO19* and more recently materials such as L~,CUO,.~However, it is only very recently that these techniques have attempted to rationalise the effect of additives to the growth and morphology of inorganic solid^.^ The major complication with most inorganic solids is their ionic character which has a profound influence on the stability of their surfaces and hence on their morphology.However, the development of the techniques and increase in available computer power now make simulation of polar solids possible. We will attempt to highlight this by describing our current work on NiO and CaCO,. Experiments on each material show that defects and impurities significantly modify the observed morphology. Hence we aim to discover whether these simulation techniques can model this observed behaviour. In particular, we need to consider whether simulations, based on calculations of free surfaces, can be applied to the morphology of a growing crystal at either high temperatures or in the presence of a solvent containing varying amounts of impurities. However, before discussing the results we will briefly summarise the simulation approach.Theoretical Methodology The atomistic simulations are based on the Born model of ionic solids, in which the ions interact via long-range electrostatic forces and short-range forces. The latter include a representation of the repulsion between neighbouring charge clouds, and a shell-model 75 Computer Modelling of Surfaces Table 1 Interatomic potential parameters: Buckingham potential: V = A,, exp( -r,,/p,,)-CJr6 (i) calcite (shell-ion model) Ca-0 1111.0 0.30926 0.0 Ca: 2 + Mg-0 583.754 0.29654 0.0 Mg: 2 + Li-0 5 10.092 0.29654 0.0 Li: 1 +o-*.o 6959.0 0.23654 0.0 0:-0.97310c-0 527.5 0.1550 0.0 C: 0.91930 three-body potential V = 112 K(0-Oo)2 0-C-0 K= 2.6, 0, = 120.0O oxygen core-shell spring constant 74.92 (ii) barium sulfate (rigid-ion model) Ba-0 7977.5 0.2698 0.0 Ba: 2 +o.-*o 360 10.0 0.19756 0.0 0: -1.0s-0 1827.9 0.19910 0.0 s:2+0 0-S-0 K = 9.09724, 0, = 109.47O (I Ref.24. description of the ions to account for their p~larisation.~ However, for complex materials, particularly those containing polyanions, it is often necessary to include angle-dependent forces to account for the covalence, as is the case for calcium carbonate.6 All of these forces, collectively called the potential model, are specified by simple parametrised analytical functions. The reliability of the simulations are dependent on the choice of parameters. However, sometimes the reliability of the parameters for a particular compound is compromised to ensure transferability between a number of systems.In all cases considerable effort must be made to ensure that the parameters are sufficiently reliable for the particular application. The parameters for nickel oxide and calcite were obtained by fitting directly to experimental data, such as the structural, elastic and dielectric data. The parameters for the preliminary study using barium sulfate were fitted to a range of sulfates, for the latest parameters see Allan and Rohl (this Discussion). The parameters we used for NiO are those of Sangster and St~neham,~ and for calcite and barium sulfate are given in Table 1. The low energy and hence the most common surfaces of a crystal are generally those of low Miller index.These planes are the closest packed with large interplanar spacings. However, in ionic crystals other constraints also apply. If the Madelung sums are not to diverge with increasing crystal size, then the crystal must not only be electrically neutral but also have no net dipole moment perpendicular to the surface. Bertaut’ demonstrated that when there is a dipole moment perpendicular to the surface, the surface energy diverges and is infinite. Such surfaces are therefore unstable. This result provides a simple method for selecting the potentially important crystal faces for simulation. Thus the simulation strategy is first to cleave a crystal to generate a slab or block which is charge neutral and non-dipolar perpendicular to the surface. The crystal faces fall into three categories, see Fig.1. Type I surfaces comprise planes containing all component ions in their stoichiometric ratio. Common examples include the (100) and (1 lo} surfaces of rock salt which have equal numbers of cations and anions. Thus cleaving the crystal at any position will yield a non-dipolar repeat. Type I1 surfaces require more care as these must be cut at a specific plane to avoid a dipole. In the example in Fig. 1(b),the cut must produce an X-M-X repeat rather than M-X-X or X-X-M. Type 111 surfaces if cut at any plane produce a dipole moment perpendicular to the surface, and cannot occur naturally without the adsorption of foreign atoms or surface roughening.As a consequence they are usually S. C. Parker et al. .o.o.o.o] neutral non-dipolar repeat unit (b).... ~o~o~o.o 0 M2+....0.-(c) dipolar repeat unit Fig. 1 Schematic representation of (a) type I, (b) type I1 and (c) type I11 surfaces the least stable.' One apparent exception is NiO in which the morphology is dominated by the (1 11) surface when grown at elevated temperatures. The final structure (and energy) of a surface is then determined by the requirement that the system is in mechanical equilibrium. This is achieved by allowing the ions to relax to the point at which they experience zero net force. The number of relaxed surface planes is increased until the surface energy converges. Defects and impurities can be accommodated in the surface but only such that the net charge is zero.The equilibrium crystal morphology of a material can be determined by applying Wulff's Theorem,8 although it was Gibbs9 who first proposed that the equilibrium form of a crystal should possess minimal total surface energy for a given volume. Wulff proposed that a polar plot of surface energy as a function of the orientation of normal vectors would give the crystal morphology. This assumes that the crystal is small and can rearrange during growth due to the short distances which ions have to travel. Thus on calculating the surface energies we can determine an equilibrium morphology which can be compared with experiment. Results and Discussion Surface Energies and Morphology of NiO We are attempting to address the ambiguity whereby NiO expresses the notionally unstable (1 1 1) surface (type 111) in collaboration with W.C. Mackrodt'O by examining the surface energies of NiO and comparing the resulting morphology with that determined by Handwerker et aZ.l experimentally. The energies of the three lowest-index surfaces were calculated. As noted above, the {loo}and { 1 lo} surfaces can be created unambiguously by cleaving the rock-salt structured NiO as they are type I surfaces. In contrast the simple cleavage of the { 11 l} results in a dipole perpendicular to the surface, i.e. a type I11 surface. The reconstruction for stabilising the ( 1 1l}plane which maintained stoichiometry involved removing one half of the surface atoms, for example anions and adding them to the opposite cation surface. On calculating the surface energies for the { 1001, { 1101, and { 1 11) planes the results agreed with the earlier work of Tasker2 and showed that as with MgO the most stable surface by far is {loo}.Hence the resulting morphology would be predicted to be cubic. However, one major difference between NiO and MgO is that nickel oxide is known to be Computer Modelling of Surfaces non-stoichiometric. Thus we can consider a reaction in which the surface reacts with oxygen to form holes (with a charge of 1 +) and charge-compensating defects such as nickel vacancies, with an effective charge of 2 -. Thus the reaction becomes: Niki + to2-+ V& + 2h’ + NiO (1) in Kroger-Vink notation Ni& represents a Ni2+ lattice ion, V!i a doubly charged Ni vacancy and h‘ an electronic hole.There is still considerable controversy relating to the character of the hole. The most stable electronic configuration from the atomistic simulations was a hole localised on the nickel, i.e. Ni3+. The total energy of the above reaction in the bulk was calculated to be 1.6 eV, which agrees well with previous work.I2 Thus we applied the reaction to the NiO surfaces which formed neutral clusters consisting of two holes and one vacancy. The surface energies as a function of the vacancy-hole cluster are given in Fig. 2. The most obvious feature is that the variation of surface energy with defect concentration is significantly different for each surface. The { 100) plane becomes less stable, the ( 110) plane has a minimum at 50% coverage, while the { 1 1l} plane shows the most complex behaviour with a local minimum at 50% and the lowest energy at 100% coverage.However, this distinctly non-Langmuir behaviour is not particularly unusual and has been predicted for a number of systems, most notably segregation of alkaline-earth-metal cations to the surfaces of magnesia.’ The results can be explained, in part, by steric factors as (100) is the most densely packed surface and hence the addition of defects serves only to destabilise the surface, while ( 1 10) is less dense and hence can accommodate a defect concentration of 50% before steric repulsion destabilises the surface. The (1 1l} plane has the lowest density of surface ions and hence the surface energy is lowered to the greatest extent.The local minimum at 50% corresponds to the complete removal of the half-plane of surface cations which were left after the necessary reconstruction required to remove the dipole. We next calculated the morphology using the surface energies for a given surface oxidation, and compared the resulting morphologies of Handwerker et al.” We found good agreement at 75% coverage, Fig. 3. Thus we interpret the observed morphologies as being bulk NiO with a surface hole concentration corresponding to the 75% defect coverage. It is important to emphasise that a high surface-defect concentration does not imply a high bulk concentration, as even a few ppm concentration in the bulk can segregate and fully cover the surfaces.c 0.5-I I I I I I I I I0.0 0.0 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100.0 coverage Fig. 2 Surface energy vs. coverage for the (100) (O),{l10) (D) and ( 111) (A)surfaces of NiO S. C. Parker et al. Fig. 3 (a) Experimental morphology, (b) calculated morphology at 75% coverage for NiO Impurity Segregation to the Surfaces of Calcite Much of our current effort is directed towards modelling the effect of additives on surface structure and morphology. One chosen example was calcite because there is considerable experimental evidence with which to test the reliability of our approach13-15 and because there has been considerable success of Mann's group at Bath in revealing insights into the nature of calcite surfaces.I4 In this section we describe simulations modelling the influence of Mg2+, Li+ and HPOi-additives on the surface stability and morphology.The calcite structure is related to the rock-salt structure and can be envisaged as a replacement of the Na atoms by Ca and the Cl atoms by CO, groups coupled with a compression of the rhombohedron along the three-fold axis. Experimental work on calcite has shown that the low-index surfaces dominate the crystal morphology.16 A schematic representation of the low-index surfaces of calcite is shown in Fig. 4. The surface energies for the low-index faces { 1Oi4},{ lOiO}, { 1120} and (0001) were calculated using the energy- minimization method described above are given in Table 2.These results show that the { 1Oi4} surface, which has the smallest surface area, is the most stable face, while the first order prismatic faces { lOiO} and { 1 lzO} are about equal in stability. Of the low-index faces the (0001) face was found to be the least stable. This is not surprising as the basal plane is comprised of alternate layers of Ca2+ and C0:-ions in successive planes, revealing it to be a type I11 surface. The most stable reconstruction was similar to that of the NiO { 11 l} in which the surfaces were terminated with a half plane of cations Fig. 4(d). Another interesting feature of this system is that there was little surface relaxation particularly for the { 1074) surface. This indicates that the pure surfaces were simple bulk terminations.This was confirmed by the rhombohedra1 morphology of the { 10741 surface predicted from the surface energies which agrees well the experimentally observed morphology, Fig. 5(a).17We next added impurities to the surfaces, first magnesium and lithium cations and secondly, the HPOi-anion. (c) {lilO] t Fig. 4 Schematic representation of the stacking sequences for the low-index planes of calcite Computer Modelling of Surfaces Table 2 Calculated energies for selected low-index surfaces of calcite with and without defect substitution defect surface segregation energy energy /kJ mol-surface surface energy (pure) /J m-* /J m-? Mg2+ Li + Mg2+ Li + { 1oi4) { 1120) (ioio} (0001) 0.23 0.50 0.52 1.60 1.28 1.37 0.84 1.14 1.28 1.08 0.76 0.32 127.3 130.2 84.9 115.7 128.3 87.7 65.6 -208.3 The addition of Mg2+ or Li+ ions at the surfaces was carried out by substitution of the Ca2+ ions at the surface and comparing with substitution in the bulk.This is the segregation energy. The heats of segregation of Mg2+ at 100% coverage of different surfaces are shown in Table 2. A positive segregation energy indicates a preferential dissolution of magnesium into the bulk calcite crystal lattice. The results suggest that magnesium ions prefer to dissolve into the bulk from the stable faces. This result is not surprising given that magnesium and calcium carbonates can form a solid solution. However, unlike the high- temperature conditions in which NiO was grown in the previous example calcite is grown in a comparatively low-temperature solution.Thus given a high concentration of magnesium at the surface there will be little diffusion into the solid, and hence magnesium will remain at the surface. Furthermore, we have some confidence that a high concentration of magnesium ions will be present at the surface because calculations of the stability of magnesium and calcium in solution compared to the solid surface suggest that magnesium will segregate from solution to the surface, and unlike in the solid there should not be a large barrier to diffusion to the solid surface. The simulations predicted that incorporation of Fig. 5 Predicted crystal morphologies showing (a) { 10T4) rhombohedral; (b) { 1010) face stabilised with Mg2+;(c) (0001) face stabilised with Li+ and (6){ lOT0) face stabilised with HPOa- additives S.C. Parker et al. Mg2+ stabilises the first-order prismatic faces (lOTO} relative to other faces (Table 2). Thus the crystal morphology was modified to give a first-order prism capped with rhombohedra1 end faces as shown in Fig. 5(b).This morphology has similar features to that observed experimentally. 16~18 The substitution of Ca2+ ions with Li+ ions results in an effective negative charge which was compensated by the addition of a Li+ interstitial. We also considered substituting calcium with Li+ at two interstitial sites, but found in each case the lithium interstitial was destabilising. This is further exemplified by the large segregation energies from the calcite bulk to all surfaces, Table 2, by showing that replacing calcium by lithium is highly unfavourable. The calculated defect surface energies (Table 2) showed that all surfaces except the polar (0001) surface were destabilised.On cleaving the (0001) surface of pure calcite there are two coplanar Ca2+ sites where only one site is occupied. Hence by removing Ca2+, we can occupy both cation sites with Li+. This has the added effect of increasing the cation density at the surface which effectively stabilises the surface. The morphological consequence of the increased stability of the (0001) face is to predict a tabular crystal habit comprising basal (0001) and (1014) side faces, see Fig. 5(c), which agrees with recent experimental studies of calcite crystallization in the presence of Li+ ions.19 We next used the same procedure for simulating the effect of the hydrogenphosphate anion, namely substituting the surface carbonates by HPOi-, the potential model is given in Table 3.The only difference was that 100% replacement was not energetically feasible due to the size of the HPOZ- anion, but was with 50% coverage of each surface. The HPOi- anion was observed to prefer the { 10lo} surface energetically compared to ( 1 120) or { 1014). Fig. 4(4 shows the morphology predicted for the effect of HPOf- anions on calcite surfaces. The morphology is that of a first-order prism face, { lolo}, similar to that predicted for the Mg2+ additives. This leads us to question whether a monovalent anion will have the same effect as lithium and stabilise the (0001) surface.A further unexpected result on simulating the addition of hydrogenphosphate anions was the effect on the structure of thin films which, if it is a general effect, may enable simulation to be used to provide a coarse screening of potential additives used to inhibit nucleation. Applications to Nucleation Studies Simulations on thin films of ca. 25 A thick expressing the { 1070)surface (Fig. 6) show that adding hydrogenphosphate ions to the surfaces causes major disru tion to the crystal surface. However, on increasing the thickness of the film to ca. 75 K we find that after relaxation there is no disruption of the lattice. These results suggest that the hydrogenphos- phate ion inhibits CaCO, nucleation by forcing apart the molecules in the bulk and Table 3 Interatomic potential used for the hydrogenphosphate additives on calcite (i) Buckingham potential: V = A, exp( -r,/p,) -C,,/r6o.*.o 1388.773 0.362 3 1 175.0 25 P-0 9034.208 0.19264 19.8793 25 0-0, 3627.500 0.28749 3.47 this work 0-Ca 2396.800 0.29202 0.0 this work 0-H (ii) Morse potential: V &, = 5.896 eV, a = 26 Charges on P = + 3.4,O = -1.46667 and OH = -1.O.0,is the carbonate oxygen. Computer Modelling of Surfaces Fig. 6 Calcite (lOi0)thin film (25 A) with hydrogenphosphate ions (emphasised by the tetrahedra) (a) before and (b) after relaxation showing the formation of voids in the surface region preventing the crystal aggregating.This disruption of the crystal surface caused by the hydrogenphosphate additives shows the potency of the phosphate ions on the nucleation of calcite crystallites as observed experimentally. Other factors which can have a considerable impact on morphology include the degree of supersaturation, and variation in pH and ionic strength. In recent work2' Hopwood and Mann at Bath have systematically varied each of the properties in the study of the growth and morphology of barium sulfate crystals. Importance of Surface Charge to Barium Sulfate The importance of ionic strength in crystal growth is that when increased significantly it will stabilise charged surfaces. This occurs because on increasing ionic strength the charge compensatory space charge layer increases in density as it becomes thinner.This will have the effect of increasing the cohesion and hence the surface stability. There will also be a secondary effect of allowing easier access of like ions of the same charge as the surface and in the extreme case may allow crystallites to join. We have performed some preliminary calculations on BaSO,, and have attempted to correlate the simulation results with the morphological changes observed by Hopwood. We leave the detailed description of the structure and surfaces of barium sulfate to Allan and Rohl (this Discussion). We found that using the less sophisticated potential the most stable surfaces were still predicted to be the (210)and (001) surfaces; the energies are given in Table 4.However, to estimate the effect of high ionic strength on morphology we need to estimate which surfaces can support a high surface charge. Frenke122 first developed an approach for calculating surface charge from the S. C. Parker et al. differences in free energies of formation of surface anion and cation vacancies. Recently this was extended to other ceramic materials by Duffy and Ta~ker,~, who also showed that the surface charge can be calculated from the difference in vacancy interaction energies (DVIE). This is simply the difference in the segregation energy of a barium vacancy compared to the segregation energy of a sulfate vacancy. These differences, neglecting relaxation around the defects, are given in Table 4.The larger the magnitude of DVIE the larger is the surface charge; a negative value implies a negatively charged surface. Neglecting the (101) and { 11 l} surfaces the most highly charged surfaces are predicted to be (011) followed by (loo}. These two surfaces are observed to dominate the morphology at high ionic strength. The effectiveness of the space charge depends on the valence of the ions contributing to it, the charged surfaces are stabilised by adding more highly charged ions. For example, the presence of multiply charged anions should further stabilise the positively charged surfaces, namely the (01 1) and the (100) surfaces. The reason for neglecting the { lOl} and (1 1 1) is based purely on kinetic arguments.In each case there are two possible terminations with similar surface energies (denoted in Table 4 by the suffix ‘a’ and ‘b’ after the index). On the macroscopic scale we would expect these surfaces to be composed of regions of both terminations, and hence they will be atomistically ‘rough’. This roughening of the surface should lead to rapid growth since step sites are centres for growth. The correlation implies that further work including the full defect relaxation and solving the space charge problem explicitly may allow us to make quantitative predictions. Conclusion We have reported current work on the prediction of the effect of surfaces defects and additives on the morphologies of NiO and calcite using atomistic simulation.These predictions are consistent with experimental observations and demonstrate the potential of atomistic simulations in modelling the surface structure and growth of inorganic materials under a range of conditions. As with all simulations, the reliability and hence the confidence we can place in the results will continue to improve as we develop more sophisticated interatomic potentials. For example, improvement in the derivation of interatomic potentials for CaCO, will enable us to extend the range of applications. This includes determining the factors which control calcite formation over that of the other polymorphs, i.e. aragonite and vaterite. Further developments currently under investigation include the effect of temperature, particularly for ceramics such as NO, where temperatures can exceed 1500 K.In addition, studies are currently in progress to model the effect of pH, ionic strength and solvated ions on the surface stability. The preliminary results on the effect of phosphate on calcite nucleation and the correlation between surface vacancy formation energies of free surfaces and ionic strength suggest that atomistic simulation can aid in the interpretation, and perhaps the prediction, of experimental growth data. Table 4 Surface energies and differences in vacancy interaction energies (DVIE) surface surface energy/J m-’ DVIE/eV 210 0.48 1.o 001 0.61 -1.0 010 0.69 -1.2 llla 0.76 3.4 lllb 0.86 2.2 lOla 0.78 -6.1 lOlb 0.95 5.0 100 0.84 1.4 01 1 0.98 5.0 84 Computer Modelling of Surfaces References 1 P. W.Tasker, E. A. Colbourn and W. C. Mackrodt, J. Am. Ceram. Soc., 1985,68, 74. 2 P. W. Tasker and D. M. Duffy, Surf Sci., 1984, 137,91. 3 N. L. Allan, P. Kenway, W. C. Mackrodt and S. C. Parker, J. Phys. Condens. Matter, 1989, 1, SB119. 4 P. J. Lawrence and S. C. Parker, in Computer Modelling of Fluids, PolymersandSolids, ed. C. R. A. Catlow, S. C. Parker and M. P. Allen, Kluwer, Dordrecht, 1990, vol. 293, pp. 219-248. 5 M. J. L. Sangster and A. M. Stoneham, Philos. Mag., 1981,43, 597. 6 R. A. Jackson and G. D. Price, Mol. Simul., in the press. 7 F. Bertaut, Compt. Rend., 1958,246, 3447. 8 G. Wulff, Z. Kristallogr. Kristallgeom., 1901, 34, 949. 9 J. W. Gibbs, Collected Works, Longman, New York, 1928.10 P. M. Oliver, S. C. Parker and W. C. Mackrodt, Modelling Simul. Muter. Sci. Eng., submitted. 11 C. A. Handwerker, M. D. Vaudin and J. E. Blendell, J. Phys. (Paris) C.5, 1988,49, 367. 12 P. W. Tasker and D. M. Duffy, Philos. Mag. A, 1986,54, 759. 13 M. P. C. Weijnen and G. M. van Rosmalen, Industrial Crystallisation, 1984, 84, 61. 14 S. Mann, S. J. Didymus, N. P. Sanderson, N. P. Heywood and E. J. Aso-Samper, J. Chem. Soc.,Faraday Trans., 1990, 86, 1873. 15 H. Sawada and Y. Takuechi, 2. Krystallogr., 1987, 181, 179. 16 J. M. Didymus, Ph.D Thesis, University of Bath, 1992. 17 J. 0.Titiloye, S. C. Parker, D. J. Osguthorpeand S. Mann, J. Chem. Soc., Chem. Commun., 1991,20,1494. 18 H. Leitmeier, Neues. Jahrb. Miner. Abh., 1910, 1, 49. 19 S. Rajam and S. Mann, J. Chem. Soc., Chem. Commun., 1990, 1789. 20 T. Suzuki, S. Inomata and K. Sawada, J. Chem. Soc., Faraday Trans., 1986,82, 1733. 21 J. D. Hopwood, J. Crystal Growth, submitted. 22 J. Frenkel, in Kinetic Theory ofLiquids, OUP, New York, 1946, p. 37. 23 P. W. Tasker and D. M. Duffy, Philos. Mag. A, 1984, 50, 143. 24 R. A. Jackson and P. Meenan, in the press. 25 B. W. H. van Beest, G. J. Kramer and R. A. van Santen, Phys. Rev. Lett., 1990,64, 1955. 26 P. Saul, C. R. A. Catlow and J. Kendrick, Philos. Mag. B, 1985, 51, 107. Paper 3/00227F; Received 1 1 th January, 1993

ISSN:1359-6640    

DOI:10.1039/FD9939500075

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年代:1993

数据来源: RSC

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Faraday Discuss., 1993,95, 85-95 Monte Carlo Studies of the Interdependence of Crystal Growth Morphology, Surface Kinetics and Bulk Transport Rong-Fu Xiao, J. Iwan D. Alexander* and Franz Rosenberger Center for Microgravity and Materials Research, University of Alabama in Huntsville, Huntsville, AL 35899, USA The evolution of crystal growth morphologies was explored using a Monte Carlo model. The model combines diffusive transport in the nutrient with thermally activated and local configuration-dependent steps for attachment, surface diffusion and detachment. The solid pair interaction energy, $J, temperature, T, and chemical potential difference between nutrient and surface, Ap, were used as input parameters. For a given set of simulation parameters, we found that, due to the competing effects of bulk diffusion and interface kinetics, there is a critical size beyond which a crystal cannot retain its macroscopically faceted shape.This critical size scales linearly with the mean free path in the nutrient, a. The stabilization of the growth shape by anisotropies in the kinetics is reduced through thermal and kinetic roughening. Thus, further increases either in Tor Ap,and/or decreases in 4, cause successive transitions from faceted to compact dendritic and side-branched dendritic morphologies. On interfaces with emerging screw dislocations the spiral growth mechanism dominates at low Ap. When a is comparable to the lattice constant, the combination of bulk and surface diffusion reduces the terrace width near the centre of a growth spiral.At elevated Tand Ap, 2D nucleation-controlled growth can dominate in the more readily supplied corner and edge regions of a facet. while spiral growth prevails in its centre. The Monte Carlo method has been used to model equilibrium and growth morphologies of surface-roughening transition^^$^ and growth rate dependence on temperature and ~upersaturation.~.~ These studies primarily focused on interface kinetics and in some cases included surface diffusion. Bulk transport of growth units to the interface has only recently been included in Monte Carlo model^.^ l2 These models are based on a modification of the Witten-Sander diffusion-limited aggregation m0de1.I~ Diffusive transport of the growth species to the crystal surface is modelled using a random walk process and interfacial attachment, diffusion and detachment kinetics on a lattice-solid that grows from a prescribed seed are considered.Transition probabilities for surface processes are calculated from neighbour interaction energies, growth temperature and the chemical potential difference between the nutrient and crystal. The morphology of the growth surface will be determined by the interaction between the surface and bulk (diffusion) processes, but will also be affected by macroscopic effects such as the large-scale geometry of both the growth surface and the source of the growth species. Adsorption and the kinetics of attachment and evaporation occur on molecular and lattice parameter length scales.On the other hand, the transport kinetics, for instance in ;ivapour, are associated with the length scale of the mean free path, which is at least three orders of magnitude larger than the molecular dimensions. Furthermore, surface diffusion can occur at intermediate length scales. A continuum approach cannot capture the microscopic details of these processes. However, as we will demonstrate in the following, a 85 Interdependence of CrystaL Growth Morpho Logy Monte Carlo approach allows for the incorporation of such disparate length scales into a single model. Model and Simulation Procedure The vapour is assumed to be isothermal and composed of an inert gas B and a highly dilute growth species A. Convection, latent heat of crystallization and A-A interactions in the vapour are ignored.Diffusion is described using a random walk of growth units that are released from a boundary source region. The source is either spherical or planar for 3D growth, and circular for 2D growth. After moving from the source region, particles continue to walk until they leave the system or reach the crystal surface. Upon arrival at the crystal surface, thermally activated configuration-dependent surface processes are deter- mined by impingement, surface diffusion and evaporation rate equati~ns.~-llThe impingement rate is assumed to depend on exp (AplkT),where Tis the growth temperature, and the chemical potential difference Ap represents the supersaturation of the nearby nutrient.It must be emphasized that this chemical potential difference is solely that which drives attachment of particles once they have been transported to the interfacial region, and does not represent the bulk nutrient supersaturation typically measured by the experimen- talist. In general, Ap will depend on the local growth morphology. In this model, as a first approximation, we have assumed that we can take Ap to be fixed. The evaporation rate is assumed to depend on the local surface configuration of the site from which a unit is to be dislodged. It is proportional to exp( -E,/kT)where the configurational dependence is expressed in terms of the site energy, El.This is calculated from the sum of the products of pair interaction (bond) energies, 4,with nearest and next-nearest neighbours.Similarly, the surface diffusion rate is assumed to depend on the local configuration of the site occupied by the particle and on the configuration of the potential jump site. The probabilities for attachment, evaporation and surface diffusion follow from the rate equations. The details of the model and simulation procedures have been reported el~ewhere.~ Results and Discussion Growth Morphology and Nutrient Concentration Distribution at Mean Free Path of One Lattice Constant First we illustrate the dependence of the growth patterns and associated nutrient concentration distributions on bond strength (or temperature) at a fixed supersaturation Ap/kT = 0.69. The 2D simulations on a square lattice were terminated when the growth pattern had gained 3 x lo3 particles.Fig. 1 shows the results for various qh/kTvalues. The contours represent three subsequent groups of 1000 particles each. At the lowest temperature, the growing ‘crystal’ retains a compact, faceted habit up to the terminal size [Fig. l(a)].The growth form is surface kinetics dominated, despite the finite supersatu- ration. With increasing T, even the smaller crystals develop depressions of a few lattice constants in the middle of the facets [Fig. l(6) and (c)], where growth eventually stops. This causes ‘macroscopic’ depressions. On further increase of the temperature [Fig. l(d)], dendritic growth sets in and subsequently side-branching occurs. This morphological evolution can be interpreted in terms of the concentration distribution that develops about the crystals.Fig. 2(6)-(d) exhibit the concentration profiles along the radius vector through a corner and centre of a facet, respectively, for the terminal shapes of Fig. l(a),(c) and (4.As a general feature, the magnitude and gradient of the concentration in the corner region are higher than in the centre of the facet. This reflects better nutrient supply by bulk diffusion to protrusions, and agrees with numerous experimental findings.I4-l6 The non-uniformity in interfacial concentration increases as more open morphologies evolve. Note that in Fig. 2(c)and (d)the concentration gradient in R-F. Xiao,J. I. D. Alexander and F. Rosenberger (c) Fig. 1 Effect of bond strength (or temperature) on growth patterns of 2D crystals on square lattice at fixed supersaturation (Ap/kT= 0.69): (a) +/kT= 4.6; (b) +/kT= 3.9; (c) +/kT= 2.30; (4 +/ kT = 0.69.From ref. 10. 71 .g- 0.4 E c 0.2 n 0 40 80 0 40 80 normalized radius, r/ b Fig. 2 Nutrient concentration profiles at growing crystals. Concentration in the nutrient along radius vectors through corner (0)and centre (0)of facet: (a)for a circular crystal, run as test case for the algorithm; (b)-(6)for Fig. l(a),(c) and (4,respectively. From ref. 10. Inte rdependen ce of Crysta1 Growt h Morphology the ‘facet’ centre has vanished due to screening by the rapidly growing branches. Hence, no further growth occurs in this region. Note that even for the macroscopically stable case of Fig.l(a), the concentration gradient, and thus the nutrient flux at the facet centre, is significantly less than that at the corner. This contrasts with the widely used stability requirement for a planar interface that the nutrient flux be uniform.I7 I9 Our assumption that la1 = b results in the redistribution of growth units by surface diffusion over distances comparable to the volume diffusion length. This compensates for the non-uniform flux to the surface. However, when a is increased (see next section), the effect of surface diffusion for face stability is reduced, and the condition that the flux be uniform to preserve a growth shape is approached. Fig. 1 and 2 provide insight into the stabilizing effect of anisotropic surface kinetics.The formation of depressions is associated with the exposure of higher step densities and, consequently, higher sticking probabilities which, in turn, leads to better utilization of the leaner growth unit supply and stronger surface diffusion at the face centre regions. The ability of interface kinetics to stabilize the face reaches its limit as, with increasing depression, the step density becomes sufficient to accommodate the attachment of most growth units that arrive at the face centres. This interplay between destabilizing bulk diffusion and stabilizing interface kinetics was first proposed by Chernov in his theory for shape-preserving growth of faceted crystals.20,21 The above simulations also reveal the destabilizing effect of an increase in growth temperature, which induces a transition from faceted to non-faceted (thermally roughened) surface^.^^,^^ This is illustrated by Fig.3(a) for Ap/kT= 0.69, and is associated with a 1.o 0.8 4-9 -5 0.6 ._3 sri on 3 0.4 ._c1 0.2 0.0 012345 12345 bond strength, 4/kT supersaturation, Ap/kT Fig. 3 Dependence of sticking probability on (a) normalized bond strength (or temperature) and number of occupied neighbour sites at fixed Ap/kT= 0.69; (b) normaiized supersaturation (temperature) and number of occupied neighbour sites at fixed +/kT = 3.9. From ref. 10. R-F. Xiao, J. I. D. Alexander and F. Rosenberger reduction in anisotropy of the attachment kinetics.The sticking probabilities for the three possible surface site configurations of the square lattice (one, two and three nearest neighbours, respectively) converge to a common value with increasing temperature. Furthermore, it is evident that an increase in supersaturation is also morphologically destabilizing. Formally, this results from the dependence of the local sticking probability on the impingement rate. Fig. 3(b) shows for the relatively high +/kT = 3.9, where thermal roughening is negligible, that the anisotropy of the sticking probability is reduced with increasing supersaturation. This is analogous to kinetic r~ughening.~~-~~ The dependences of crystal morphology on normalized supersaturation and on bond strength/temperature are, of course, tightly coupled.Fig. 4 summarizes our results for growth patterns of up to 3 x lo3 particles on a triangular latti~e.~ In general, as indicated by the right shaded curve, the supersaturation above which the compact faceted form is no longer stable decreases with decreasing bond strength and increasing temperature. The same holds for the transition from compact branched to open side-branched morphologies (left shaded curve). Note that the specific location of the boundaries between the different morphological regions in Fig. 4 are only representative for growth patterns containing 3 x lo3 particles. For larger sizes, loss of morphological stability would set in at combinations of lower supersaturations and higher bond strength than in Fig.4. Even for the most favourable growth conditions, the system will eventually reach a size where destabilizing bulk diffusion will cause the loss of morphological stability, i.e. the anisotropy in interfacial kinetics can no longer compensate for the increasing non-uniform nutrient supply. The close coupling between surface roughness and anisotropy in growth morphology is further illustrated by Fig. 5, which was obtained for three-dimensional growth on a seed of 60 x 60 lattice constants, with periodic lateral boundary conditions.ll To reduce the simulation times, both bulk and surface diffusion were neglected for this case. Particles ‘ballistically’ arriving at random locations of the crystal surface either stick or are 0, I I I -0 1 2 3 4 normalized bond strength, +,/ kT Fig.4 Crystal growth morphology (3 x lo3 attached particles) as a collective effect of normalized supersaturation and bond strength (or temperature) for a triangular lattice and mean free path of one lattice unit. Three regimes: compact faceted, compact branched and dendritic multiple side-branched. Regime boundaries (shaded curves) from Monte Carlo results judged as ‘boundary cases’. From ref. 10. Interdependence of Crystal Growth Morphology ........ ........... . ..' =. 1 2 3 4 number of attached particles (full layer) . Fig. 5 Normalized surface area vs. number of attached particles (in units of a full layer, i.e. 3600 particles) for Ap/kT = 0.35 and various $/kT values (marked on the figure).The normalization is based on the area of a perfectly smooth surface. Insets represent growth morphologies at $/kT = 0.35 (normal growth) and $/kT= 3.9 (layer spreading following 2D nucleation). After ref. 11. discarded. Fig. 5 shows plots of the surface area, normalized with the area of a perfectly smooth surface, as a function of number of attached particles (in units of a full layer, i.e. 3600 particles) for Ap/kT= 0.35 and various +/kT values. At the lowest temperature, growth proceeds anisotropically, layer by layer (lower inset). Thus, the surface area oscillates periodically, with minima on layer completion and maxima at about half-filled layers. At higher T and, thus, a rougher surface, the distinction between half-filled and completed layers becomes blurred.Then growth on the delocalized surface proceeds without a nucleation barrier in an isotropic, continuous fashion. Effect of Mean Free Path on Growth Pattern The above results correctly describe the growth conditions that lead to the loss of shape stability. However, the crystal sizes at which morphological breakdown occurs (i.e. the simulated critical sizes) are orders of magnitude smaller than those observed, for instance, in condensation from vap~urs.~~-~~ Yet the above results are based on la1 = b, while the a prevailing in typical experiments is 103-105b. Hence, we have investigated the effect of a on the critical size. The crystal shape insets in Fig. 6, with contours for each consecutively attached group of 2500 particles, were obtained for Ap/kT= +/kT= 0.69 and various mean free path lengths.Most noteworthy, an increase of a from 5b to 50b causes a transition from side- branched dendritic to compact-faceted shapes. This stabilizing effect is to be expected. With increasing a, a given interface site can directly draw growth units from a larger sector of the nutrient, and the destabilizing screening by protrusions is less pronounced. For quantitative evaluation, we define a critical crystal size based on the ratio of the distance of a corner and face centre, respectively from the crystal's centre. For a perfectly faceted square crystal, this ratio is J2. We consider, somewhat arbitrarily, a crystal as marginally stable when this ratio is 1.6. Fig. 6, which summarizes our results, shows that the critical crystal size scales linearly with a.In addition, for a given supersaturation, the critical 91R-F. Xiao, J. I. D. Alexander and F. Rosenberger 1 mean free path (b) Fig. 6 Dependence of critical size on mean free path at ApikT = 0.69 and various $/kTvalues for both 2D and 3D systems. Insets illustrate the morphologically stabilizing effect of increasing mean free path, with the four contours in the patterns corresponding to additions of 2500 particles each. After ref. 11. ~~ ~ 0.69 + 2.3 A n 3.9 0 0 4.6 size increases with decreasing growth temperature. This again reflects the stabilizing effect of anisotropy in kinetics, which increases as the temperature is lowered due to a reduction in thermal roughness.Furthermore, Fig. 6 shows that 3D cases are more stable than their 2D counterparts. This results from the fact that, on average, there are more solid neighbours associated with surface particles in three dimensions, which reduces the surface roughness further. Computational time limitations prohibit the direct simulation of the critical size for the a values used in morphological stability experiment^.*^-31 However, Fig. 6 encouraged us to extrapolate linearly to these conditions. This leads to an order-of-magnitude agreement between experimental and modelling results for the critical size. The 3D results approach the experimental findings closer than the 2D results.1° Surface Diffusion and Facet Stability In general, surface diffusion tends to reduce surface roughness and, thus, increases the anisotropy in attachment kinetics.This has been studied both theoreti~ally~,~>~~>~~ and e~perirnentally.~~The effects of surface diffusion and its interaction with bulk diffusion are implicitly contained in the results of Fig. 1 and 2, and the insets in Fig. 4 and 6. For a more Interdependence of Crystal Growth Morphology explicit demonstration of these effects, we have performed 3D simulations for 4lkT = 1.6 and Ap/kT = 0.69, without consideration of bulk diffusion. Inset (b)in Fig. 7 shows that with surface diffusion the surface is much smoother than without surface diffusion [inset (a)].As we have shown previously,ll this stabilizing contribution is effective only at the length scale comparable to the surface diffusion length, which, itself, is strongly influenced by the surface roughness.Hence, as Fig. 1 reveals, the terrace widths decrease in response to an increase in temperature. Fig. 7 shows explicitly that the average surface diffusion length decreases with increases in both T and Ap. This was not accounted for in earlier simulation^,^ in which the surface diffusion length was considered as an externally adjustable parameter. In contrast, the surface diffusion length in our model is an outcome of the simulation. Surfaces with Dislocations The effects of growth temperature, supersaturation, and surface and bulk diffusion on the growth morphology of surfaces with emerging single and paired screw dislocations have been investigated in detail." The most significant results of this work are described below.Growth morphologies of a dislocated surface with contributions of both bulk and surface diffusion are depicted in Fig. 8. At fixed $/kT= 5.3, the supersaturation was increased from Ap/kT = 0.69 in (a),to 2.0 in (b)and 3.0 in (c).It is evident, in particular from Fig. 8(c) which shows lateral depressions in the steps, that bulk diffusion also destabilizes steps. Probably the most interesting new feature revealed by this simulation is the variation of the terrace width. Narrower terraces are found near the centre of the spiral, particularly at higher supersaturation. This is at variance with classical theorie~~~.~~ 37 and the results of earlier Monte Carlo simulations that did not account for bulk diff~sion,~~~~,~~ but was anticipated by Chern~v,~~ who analysed the effect of anisotropic surface kinetics on the stability of shallow vicinal hillocks using perturbations of the form exp( -p/h),where p is the radial distance from the hillock centre.! 0.1 0.2 0.3 0.4 0.5 kTl4 Fig. 7 Dependence of interfacial morphology on surface diffusion. Average surface diffusion length vs. (temperature)/(bond strength) for Ap/kT= 0.69 (0)and 3.0 (A), la1 = b. Inset: morphologies at Ap/kT = 0.69 and +/kT= 1.6 (a) without surface diffusion, and (b) with surface diffusion. After ref. 11. R-F. Xiao, J. I. D. Alexander and F. Rosenberger Fig. 8 Cooperative effect of surface and bulk diffusion on interface morphology with a single screw dislocation for +/kT = 5.3, la1 = b.Planar source parallel to interface. (a) Ap/kT = 0.69, (b)Ap/kT = 2.0, (c)Ap/kT = 3.0. From ref. 11. This decrease in terrace width toward the spiral centre can be understood as a cooperative effect of bulk and surface diffusion. Since the above simulation sequence was based on a = b, terraces and steps near the tip of the growth hillock that protrudes into the nutrient are better supplied with growth units by bulk diffusion. This leads to tighter winding of the spiral in this region.22 On the other hand, the surface diffusion fields of terraces at the periphery of the spiral overlap less than those near the centre. Overall, this leads to higher spreading velocities of the outer turns of the spiral and, thus, to an increase in terrace width with distance from the spiral’s centre.Based on the above model assumptions, we expect that this behaviour is more likely to occur in growth from condensed phases rather than from vapours where a >> b. The above results for dislocation-assisted growth were obtained with a planar source parallel to the interface. To emphasize the importance of 3D transport for the growth morphology, we have also carried out simulations in a spherical source; for details see ref. 1 1. Fig. 9 presents the results obtained for a face with a pair of dislocations emerging at its centre, for otherwise the same conditions as for Fig. 8. In all three cases shown, the morphologies are the result of the addition of 20000 particles.Note that, in spite of the better supply at edges and corners, growth at low supersaturation [Fig. 9(a)] occurs only through attachment onto steps that originate at the central dislocation pair. However, the higher supersaturation at the corner destabilizes the growth step loops, leading to lateral protrusions towards the corners, which is similar to the growth patterns obtained in the 2D simulations of Fig. 1 (c).Such star-shaped step growth has been observed experimentally; see Fig. 60 in ref. 40. With increasing supersaturation, the controlling effect of the Interdependence of Crystal Growth Morphology Fig. 9 Growth morphologies of face with pair of dislocations of opposite sign, spherical source, +/kT= 5.3, bulk and surface diffusion, jaJ= b.(a) Ap/kT = 0.69, (b)Ap/kT = 2.0, (c) Ap/kT = 3.0. From ref. 1I. dislocations decreases and 2D nucleation becomes significant at the corners [Fig. 9(b)].At an even higher supersaturation [Fig. 9(c)], growth is essentially dominated by 2D nucleation at the corners, in spite of the dislocations at the face centre. Of course, at higher temperatures i.e. at higher surface roughness, this cross-over and loss of shape stability occurs at lower supersaturation. The authors are grateful for support by the Microgravity Science and Applications Division of NASA under Grant NAG8-790, and by the State of Alabama through the Center of Microgravity and Materials Research at the University of Alabama in Huntsville. References 1 A.A. Chernov, in Crystal Growth, ed. H. Peiser, Pergamon, Oxford, 1967, p. 25. 2 A. A. Chernov and J. Lewis, J. Phys. Chem. Solids, 1967, 28, 2185. 3 V. 0.Esin, L. P. Tarabaev, V. N. Porozkov and T. A. Vdovina, J. Crystal Growth, 1984,66, 459. 4 G. H. Gilmer and P. Bennema, J. Crystal Growth, 1972, 13/14, 148. 5 H. J. Leamy, G. H. Gilmer and K. A. Jackson, in Surface Physics of Materials I, ed. J. B. Blakely, Academic Press, New York, 1975, p. 121, and references therein. 6 R. H. Swendsen, Phys. Rev. B, 1977, 15, 5421. 7 G. H. Gilmer and P. Bennema, J. Appl. Phys., 1972,43, 1347. 8 V. 0.Esin and L. P. Tarabaev, Phys. Stat. Solidi A, 1985, 90, 425. 9 R.-F. Xiao, J. I. D. Alexander and F.Rosenberger, Phys. Rev. A, 1988,38, 2447. 10 R.-F. Xiao, J. 1. D. Alexander and F. Rosenberger, J. Crystal Growth, 1990, 100, 313. 11 R.-F. Xiao, J. I. D. Alexander and F. Rosenberger, Phys. Rev. A, 1991,43, 2977. 12 Y. Saito and T. Ueta, Phys. Rev. A, 1989, 40, 3408. R-F. Xiao, J. I. D. Alexander and F. Rosenberger 13 T. A. Witten and L. M. Sander, Phys. Rev. B, 1983,27, 5686. 14 W. F. Berg, Proc. R. Soc. London, Ser. A, 1938, 164, 79. 15 C. W. Bunn, Discuss. Faraday Soc., 1949, 5, 132. 16 J. C. van Dam and F. H. Mischgofsky, J. Crystal Growth, 187, 84, 539. 17 A. A. Chernov, Modern Crystallography IZI,Crystal Growth, Springer Series in Solid State Sciences, ed. M. Cardona, P. Fulde and H.-J. Queisser, Springer-Verlag, Berlin, 1984, vol.36, and references therein. 18 A. Seeger, Philos. Mag., 1953,44, 1. 19 W. R. Wilcox, J. Crystal Growth, 1977, 37, 229. 20 A. A. Chernov, J. Crystal Growth, 1974, 24/25, 11. 21 A. A. Chernov and T. Nishinaga, in Morphology of Crystals, ed. I. Sunagawa, Terra, Tokyo, 1987, ch. 3, pp. 207-267. 22 W. K. Burton, N. Cabrera and F. C. Frank, Philos. Trans. R. Soc. London A, 1951,243, 299. 23 J. D. Weeks, in Ordering in Strongly Fluctuating Condensed Matter Systems, ed. T. Riste, Plenum, New York, 1980, p. 293, and references therein. 24 A. A. Chernov, Annu. Rev. Mater. Sci. 1973, 3, 397. 25 C. E. Miller, J. CrystalGrowth, 1977, 42, 357. 26 G. H. Gilmer and K. A. Jackson, in Current Topics in Materials Science, ed. E. Kaldis and H.Scheel, North-Holland, Amsterdam, 1977, vol. 2, p. 79. 27 D. Nenow and V. Stoyanova, J. Crystal Growth, 1977,41, 73. 28 D. Nenow, V. Stoyanova and N. Genadiev, J. Crystal Growth, 1984, 66, 489. 29 C. Nanev and D. Iwanov, J. Crystal Growth, 1968, 314, 530. 30 C. Nanev and D. Iwanov, Crystal Res. Technol. 1982, 17, 575. 31 M. Staynova and C. Nanev, Crystal Res. Technol., 1988, 23, 1061. 32 R. L. Parker, in Solid State Physics, ed. H. Ehrenreich, F. Seitz and D. Turnbull, Academic Press, New York, 1970, p. 151, and references therein. 33 P. von Blanckenhagen, in Structure and Dynamics of Surfaces II, Phenomena, Models and Methods, ed. W. Schommers and P. von Blanckenhagen, Springer-Verlag, Berlin, 1987, p. 73, and references therein. 34 N. Cabera and M. M. Levine, Philos. Mag., 1956, 1, 450. 35 R. Kaishev, Crystal Growth, 1962, 3, 29. 36 T. Surek, J. P. Hirth and G. M. Pound, J. Crystal Growth, 1973, 18, 20. 37 H. Muller-Krumbhaar, T. W. Burkhardt and D. M. Kroll, J. Crystal Growth, 1977,38, 13. 38 R. H. Swendsen, P. J. Kortman, D. P. Landau and H. Miiller-Krumbhaar, J. Crystal Growth, 1976,35,73. 39 G. H. Gilmer, J. Crystal Growth, 1976, 35, 15. 40 I. Sunagawa and P. Bennema, in Preparation and Properties of Solid State Materials, ed. W. R. Wilcox, Marcel Dekker, New York, 1982, vol. 2, p. 1, and references therein. Paper 2106353K; Received 27th November, 1992

ISSN:1359-6640    

DOI:10.1039/FD9939500085

出版商:RSC    

年代:1993

数据来源: RSC

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Faraday Discuss., 1993,95,97-107 Hartman-Perdok Theory: Influence of Crystal Structure and Crystalline Interface on Crystal Growth Cornelis F. Woensdregt Institute of Earth Sciences, Utrecht University, P.O.Box 80.021, 3508 TA Utrecht, The Nether lands Hartman-Perdok theory enables classification of crystal faces as F, S or K faces. Only F faces should be present on growth forms as they grow slowly according to a layer mechanism. Such a classification can be quantified for ionic crystals by the calculation of attachment energies in electrostatic point- charge models. The attachment energy of a crystal face (hkl)el,is the energy released per mole, when a new elementary growth layer, called a slice, with a thickness of dhklcrystallizes on an existing crystal face.Theoretical growth forms can be constructed by making use of the supposition that ek'is directly proportional to the growth rate. Often these growth forms are very similar to the morphologies of crystals grown in nature or in laboratory experiments (zircon (ZrSiO,), alkali feldspars ((K,Na)AlSi,O,), natural silicate garnets [A:' B;+(Si0,)3]}. The structure of the crystalline interface is very important for the crystal growth processes. The derivation of F faces provides the atomic topology of the crystalline interface as well. Sometimes there is more than one possible surface of a slice with a thickness of dhk/, e.g.zircon (01 I), garnets (1 10) and (1 12), which may lead to growth via slices of thickness 3dhk/,causing a significant increase of the growth rate.Ordering of the ions which are situated on the slice boundaries may reduce the growth rate (alkali feldspars and YBa,Cu,O, ,). Experimental crystal growth of PbClz shows that adsorption of ~ OH- and H,O+ ions on special sites of the crystalline interfaces may cause habit modifications. This effect can be explained for PbClz in terms of adsorption of Pb(0H)Cl on {21 l}. The influence of the solvent on the growth can also be explained in terms of impurity adsorption on the crystalline interface. The growth of flattened PbCl, crystals from HC1 due to the reduction of the growth rates of (010) and { 121) is such an example. Hartman-Perdok Theory Crystalline materials are often bounded by planar surfaces called crystal faces. In the early days of crystallography these crystal faces were already connected with the internal crystal structure.In 1784Hauyl concluded from the cleavage planes of calcite that the crystals are composed of molecules intigrantes after what Steno2 in 1669 had already formulated as the law on the constancy of interfacial angles. Crystal faces are crystalline interfaces during the process of crystal growth. Sometimes habit changes occur due to alterations of the external growth conditions such as supersaturation and/or impurity concentrations. The surface topology of the crystalline interfaces during the growth determines the surface diffusion, incorporation of the growth units into kinks or steps, adsorption etc. Hartrnan and Perd0k~3~ described the relation between the internal crystal structure and the external crystal morphology.They classified crystal faces either as F faces, which contain at least two periodic bonds (PBCs) in a slice of thickness dhkl,as S faces parallel to only one PBC, or as K faces not parallel to any PBC at all. The F faces are the slowest to grow according to a layer mechanism and are therefore the only important ones for the crystal morphology. 97 Hurtmun-Perdok Theory The PBC is an uninterrupted chain of strong bonds between the crystallizing units, such as ions, molecules atoms or clusters thereof, formed during crystallization. Assuming that strong bonds are confined to the first coordination sphere, a limited number of PBCs can be distinguished, so the identification of PBCs and subsequent classification of crystal faces as F, S and K faces can be of great value in understanding the crystal morphology (see for a review hart mar^^.^).The slice with an F character defines, however, not only the elementary growth layer in case of layer growth, but also the topology of the crystalline interface during the crystal-growth processes on the scale of atoms, ions or molecules. Hence the influence of the crystal structure on the surface configuration of the crystalline interface can be derived with atomic precision by the Hartman-Perdok method. Electrostatic Point-charge Calculations Crystals grown in nature and in the laboratory are mostly not the equilibrium forms, which satisfy the thermodynamic conditions of the minimum specific surface free energy formulated by the Gibbs-Curie law.They are growth forms of which the habit is controlled by kinetic processes and prevailing growth conditions (impurity concentrations, degree of supersaturation etc.). Only under special conditions, such as a closed system in which Ostwald ripening can take place, will equilibrium crystals be formed. The attachment energy, which is assumed to be directly proportional, at least for the slowly growing F faces, to their growth rates' can be computed in an electrostatic point- charge model. Hence calculations of attachment energies can provide data about the growth rates of individual crystal forms {hkl).The attachment energy (E.,)is defined as the energy per mole released, when a new slice of thickness dhk[crystallizes on an already existing crystal face (hkl).dhklis equal to the period with which the same surface configuration is repeated, so it is subjected to the same extinction conditions as X-ray reflections. In order to calculate the electrostatic attachment energies of cubic sphalerite, Hartman* introduced the formulae given by Madelung9 which describe the potential induced in a point P by an infinite row of equally spaced point charges. Further details about the computation of the attachment energies in an electrostatic point-charge model have been published elsewhere, lo Theoretical Growth Forms In the following section examples of theoretical growth forms will be given.They have all been obtained by a three-dimensional plot of the attachment energies according to the principles of the Wulff p1ot.l' In this plot the distance from the central point to {hkois taken to be directly proportional to the absolute value of the attachment energy ,$'I. The innermost closed surface is the theoretical growth form. Zircon (ZrSiO,) In the tetragonal zircon (ZrSiO,) crystal structure the silicon-oxygen tetrahedra are bonded to six different zirconium ions. On the other hand each zirconium ion is surrounded by six silicate tetrahedra. When only Zr and silicate ions are taken into account as crystallizing units, four different PBCs can be described,I2 viz. [OOl], (loo), (H)and (+,+,1 +).F faces are (100)and {01l}.Calculation of attachment energies has been performed in electrostatic point-charge models: I, Zr4+Si4+Oi-; model 11, Zr4+Si2+0i.5-;and model 111, Zr4+SioO:-. The theoretical growth form is short prismatic following { 100) and is terminated by the dipyramid (0111. The lower the oxygen charge, the more elongated is the crystal parallel to the c axis (Fig. 1). Detailed study of crystalline interfaces reveals that sometimes two different surfaces can be distinguished, which differ from each other in height by one submultiple of the slice C. F. Woensdregt Model I Model II Model IllOOl0010 1oi Fig. 1 Theoretical growth forms of zircon (ZrSiO,). Growth via slices d,,a: (a)-(c);growth via slices 420: (4-0. thickness.l3.l4 The difference between these slices is caused by their distinct surface configurations.Examples are the slices dollof barite15 and dollof ADP.16 If more than one slice configuration can be traced, the growth may also take place by slices with a thickness of a submultiple of dhkrprovided that they have also an F character. These slices have a higher negative attachment energy, which results in much higher growth rates than would be expected if the growth proceeded by means of slices with normal thickness dhkl. The slice do,1, which can be defined either as slice dtl bounded by Zr ions or as slice d:l bounded by oxygen ions of the silicate tetrahedra (Fig. 2), is such an example. The slice d:, is not only undulated in the plane of projection, but also in the third dimension.This has been indicated by the shaded areas of Fig. 2, which contain two boundaries. One with silicon ion Si(l), linked to oxygen ions O(1) and O(2,OiO) that are all situated at height 0, and another with Si(4) bonded to O(13) and 0 14,OiO) all at height 0.5. The slice configuration with the lowest attachment energy is 6oil, e.g. model I: -1479 kJ mol-l. The attachment energy of d:, is much higher ( -2 165 kJ mol- l). As these slice boundaries differ in height a half slice with thickness d022,and these half slices dt12and are F faces, the growth of (01 1) may also take place by slices of thickness In that case the growth rates of (01 1) increase and the growth models are even more prismatic. Alkali Feldspars [(K,Na)AISi,08] The reduction of effective charges changes the absolute values of the attachment energies, and sometimes also the relative order of magnitude, as has been shown for zircon.In the same manner ordering of A1 during the growth could have an effect on the theoretical growth form of potassium feldspar (KAlSi,O,). The alkali feldspar [(K,Na)AlSi,O,] crystal structure consists of a corner-sharing three- dimensional network of SiO, and AlO, tetrahedra. The electrostatic imbalance resulting from the replacement of Si4+ by A13+ will be restored by the incorporation of monovalent cations, such as K+and Na+, which occupy the large cavities in the structure. In the triclinic 100 Hartman-Perdok Theory Zr= 0 . SI = 0 0.1 nrn Fig. 2 Projection of the zircon crystal structure along the a axis ordered structure there are four non-equivalent tetrahedral sites, i.e.T1(0), Tl(m), T2(0) and T2(m). In this structure all the A1 is located on T1(0), while Tl(m), T2(0) and T2(m) are occupied by Si. In the monoclinic high sanidine A1 is randomly distributed over all T1 and T2 sites. In the monoclinic ordered orthoclase A1 is divided in equal amounts over the Tl(0) and Tl(m) sites. The F, faces of alkali feldspar,17.18 which are at least parallel to two PBCs completely built-up from (Si,Al)-0 strong bonds, are { 1lo}, {OOl}, {OlO), (301) and {ill). The F, faces which are parallel to only one of these PBCs and to another having both (Si,Al)-0 and additional weaker (K,Na)-0 bonds, are {130}, {021}, {221), (7121, (100) and {TOl).The growth forms of the models with the formal point charges are strongly dependent on the degree of ordering of silicon and aluminium. When the Si,Al is completely disordered the growth form is prismatic following { 1 lo), with additional (001) and {OlO), while (021) is almost invisible [Fig. 3(a)].The partial ordering of Si,A1 yields the presence of ('ZOl) and { 1If)as additional almost invisible crystal forms on the growth form of ordered orthoclase [Fig. 3(b)].The fairly triclinic growth form of low microcline [Fig. 3(c)] is thick prismatic following { 110) and shows the influence of the completely ordered Si,Al on the crystal morphology. Garnets In the cubic silicate garnet structure of A:+B:'(SiO,), the tetrahedrally coordinated Si ion shares each of its oxygens with the adjacent B06 octahedra. Each B06 octahedron shares six of its twelve edges with adjacent AO, dodecahedra. Natural garnets can be classified into two distinct groups according to their chemical composition.The first group has the chemical composition Ca,B,(SiO,),, where B stands for trivalent cations, such as A13+, Fe3+ and Cr3+. The chemical formula of the second group is A,Al,(SiO,),, where A stands for divalent cations, such as Fe2+, Mn2+and Mg2+. Synthetic garnets have the general chemical formula A:+B:+(B3+O:-)3. As trivalent A ions, Y,Gd and other rare-earth-metal elements, and as trivalent B ions, Fe, Gd, Al and Ga may C. F. Woensdregt Fig. 3 Theoretical growth forms of potassium feldspar polymorphs.Disordered surface configu- ration: (a) high sanidine, (b)orthoclase (c) low microcline. Ordered surface configuration: (d) low microcline. occur. Yttrium ion garnet (YIG), gadolinium gallium garnet (GGG) and yttrium aluminium garnet (YAG) are examples of synthetic garnets. The F forms of all garnets are { 1121, { 1lo}, { 123}, {OOl), { 120) and {332}.19 Boutz and WoensdregtZ0 analysed the surface structure of the slices of Mg,Al,(SiO,), (pyrope). For the crystal form { 112) there are also two different slice configurations as is shown in Fig. 4. The slice d, contains only complete silicate tetrahedra and its boundaries are occupied by alternating ions of A1 and Mg. It differs from the slice configuration of dlI2*in height by half the slice thickness of dIl2.The slice d1122cuts through silicate tetrahedra and its boundaries are occupied with alternating Si and A1 boundary cations. The slice dzz0can also be traced in two different ways (see also Fig. 4).The slice d220,is bounded by Si and Mg ions, while that of d2202is bounded by A1 ions only. The difference in height between the former and the latter is exactly half the thickness of slice d,,,. '/ Fig. 4 [1TO)projection of pyrope I02 Hartman-Perdok Theory The attachment energies of the F faces have been calculated in different electrostatic point-charge models in which the point charges of the ions forming Si-0 tetrahedral bonds vary from the completely ionic model with formal charges, [Si4+Oi-], to the covalent model with reduced charges, [SiOO;].In all models the formal point charges of A1 and Mg have been used. The theoretical growth (see also Fig. 5) form based on the formal charges shows only { 112). However, on the more covalent-like models, with lower effective charges of oxygen, qo,(loo}and { 1lo} appear, while at the same time, (21 1) disappears slowly. If the effective charge of qo = -lei, the growth form is merely bounded by (I lo}. Crystalline Interfaces Influence of Two Different Coexisting Surface Structures The possibility that two different surface topologies could sometimes be present on one and the same crystalline interface must not be discarded. Especially, when the specific surface -2.0 -1.75 -1.5 -1 .o Fig.5 Theoretical growth forms of pyrope C. F. Woensdregt energies of the two surfaces are very similar, they are both present separated by steps which have a height equal to a submultiple of the slice thickness dhkl.If these slices with reduced thickness also have an F character, the layer growth proceeds at higher growth rates than under normal circumstances. Examples of these crystalline interfaces are (01 1) of ADP,I6 zircon,12 barite,15 aragonite2' and (112) and (1 10) of garnet.20 Ordering Boundary Ions Sometimes ions are located exactly on the boundary of two adjacent F slices. Such an F face can be defined as one with statistically disordered boundary ions. In this situation the n boundary ions are randomly distributed over 2n sites, which means that the occupancy of all the sites is 0.5.Another possibility is that only half of the boundary sites are occupied by ions. In the latter situation an ordered configuration of ions and vacancies is energetically the most stable situation. Such an ordered surface can be present only if the growth is proceeding slowly. Alkali Feldspars The oxygens of the alkali feldspars on the boundaries of the slices dm,, d2,,,,and d20jcould be ordered. The same ordering can also take place for the potassium ions on the surface of d020.Particularly when Si,Al is fully ordered, the effect is very strong for (001)and (010) [Fig. 3(d)]. They are morphologically much more important than on the correspond- ing model with disordered surface configuration [Fig.3(c)]. YBa2Cu,0, -(YBCO) YBa2Cu30,-(YBCO) has for x = 1, the following F forms:22 (OOl), (01 l},(0131,{ 112) and ( 1 14). The theoretical growth form of this tetragonal phase is tabular following {OOl), with (01 1) as lateral forms. There are two different slices for (OlO), i.e. dolo,and dolob[Fig. 6(a)]. Their surface energies are exactly identical owing to the presence of a symmetry centre at the origin. The slice with thickness of d020is, however, an S face as the [loo] PBCs are not connected parallel to [UOW] within this slice. When x = 0, YBCO becomes orthorhombic. The slice d020[Fig. 6(a)],in which the [loo] PBCs are connected parallel to [301] and [30l]in the case of Yf3a2Cu,0,, determines the Fcharacter of (010).This slice has in its centre Y ions and half of the copper oxide layer (Cu3+Cu:'Oi-) on either side.The copper and oxygen ions in the outermost layer show presumably an ordered arrangement of which the (1 x 2) structure is shown in Fig. 6(a).Calculations22 of the attachment energy indicate that (010) with an ordered boundary structure could be present on the theoretical growth form. Exactly on the slice boundaries of (001) are situated Cu+ ions (x = 1) or Cu3+ and 02-ions (x = 0). The most stable configuration is that of a quadratic lattice with sides 2a, of which the Cuf ions occupy the nodes and centre of the lattice [see Fig. 6(b)].When ordering of the Cu+ ions can take place, the growth rate of (001) is reduced by a factor of 0.74 in the case of YBa,Cu,O,.For YBa2Cu,0, the reduction is even higher (0.53). Adsorption Zircon Caruba et al.23proved experimentally that in the crystal structure of zircon oxygens are partly replaced by OH due to the adsorption of protons on the crystal surfaces when they grow from hydrothermal solutions. There are four equivalent 0-0 distances of 0.2840 nm in the zircon crystal structure, which are short enough to comply with the conditions for a hydrogen bond O-H-*-O.These relatively short bonds between oxygens are drawn in Fig. Hartman-Perdok Theory I I 09 0 0,,@0 occupancy : 112 0 occupancy : 1 w '(4 I 00 I 0[OOlT + 0 0 0 + 0 + cu *Y 0 Ba 00 additional oxygen Fig. 6 (a) [loo] projection of YBa2Cu,0,-,, (6) ordering of (001).Left: Disordered surface; right: ordered surface. 2 for the oxygen O(3).They could provide the additional strong bonds parallel to (110) in order to establish the F character of { 110)and to reduce its attachment energy. In addition they could change the character of (001).Hydrogen bonds are, however, much weaker than ionic bonds. So the F character of { 1 10)and of (001) will not be so pronounced as in the case of the genuine F forms (100) and (011). According to solubility studies of quartz in H,O-CO, and H,O-Ar systems24there are indications that Si(OH),.2H20 is the dominant aqueous silica species in the supercritical regions of these systems. In this complex four hydroxy groups are tetrahedrally coordinated to the silicon atom with two water molecules attached by hydrogen bonding.Their internal Si-OH and O-H***O bonds coincide with the bonding scheme of zircon. Once incorporated in a kink site, these uncharged solvated silica species could act as impurities and reduce the growth rate of the slices dl and dm1.They are neutrally charged and poison the special kink positions on (001) and { 1 lo), which have already been described as possible hydroxy-group adsorption sites. C.F. Woensdregt I05 Instead of silica complexes water molecules could be adsorbed on the surface. In that case the hydrogen bonds of the water molecule react with the crystalline interface in a similar manner to those of the silica complex. Cotunnite (PbC1J The F forms of cotunnite (PbClJ are25 (110}, (0201, (Oll), (120}, (2001, (1 1l}, (121}, (201) and (21 1).This analysis is also valid for Pb(OH)Cl, Pb(0H)Br and Pb(0H)I which have similar crystal structures. For the closely related SbSI structure the same F faces as for cotunnite are present with exception of (1201, which is an S face. The theoretical growth form of PbC1, is short prismatic following { 1 lo} and has (01 1) as terminal form. The effect of the relative supersaturation, 0,in combination with the presence of impurities has been investigated26 for PbCl,, which crystallizes from an aqueous solution to which an impurity in the form of KC1, NH,C1 or CdC1, has been added. The experimental growth habit is isometric with dominant (21 1) as terminal form for low 0and low levels of impurity concentrations.Higher (T and higher impurity concentrations lead to less OH- ions and more C1- ions. The effect is a more acicular habit with dominant (01 1) terminal forms instead of (21 1). The presence of (21 1) can be explained by the adsorption of OH- ions in the form of a two-dimensional adsorption layer of Pb(0H)Cl. The slice dj, is characterized by sheets of Pb and C1 atoms in its centre and two clearly protruding C1 ions. Other slices do not show such a structure. The supposition is that these C1 ions situated at the slice surface have been replaced by OH-ions. The face (2 1 1) belongs to the zone [1 1 I], ofwhich the period is 1.268 1 nm. The same zone has in case of Pb(0H)Cl an identity period of 1.2665 nm, which is the smallest misfit (0.13%) of all possible lattice rows.In fact, deposits of Pb(0H)Cl have been found besides the crystals of PbC12. Hence the habit with (21 1) as terminal form can be explained by the adsorption of one of the solvent components (OH-ions) or lead hydroxyl complexes such as Pb(OH)+ or Pb2(0H)3+, which also may be present in these solutions. Crystallization from HCI-containing solutions shows26 that the habit depends both on (T and the concentration of HCl. The effect of the presence of HCI is twofold, both the growth rate of { 121) and (010) decrease. When the supersaturation increases, the habit becomes elongated with dominating (010) and (1001. The very pronounced decrease of the { 121) and {OlO) growth rate could be explained by the adsorption of H30+instead of Pb2+ ions, which have similar ionic radii.The Pb ion marked by an arrow in Fig. 7 is situated both in slice d,z, and As its adsorption energy is the highest of all possible adsorption sites, adsorption of H,O+ is very likely to occur and would cause the observed habit modification. Fig. 7 [lOf] projection of PbCl, showing the most suitable site for H,O+ adsorption 0,Pb; 0,C1 Hartman-Perdok Theory Discussion and Conclusions Hartman-Perdok theory provides theoretical growth models which can be compared with crystal morphologies observed in nature or obtained during crystal-growth experiments in laboratory. In the present paper the theoretical growth models derived for zircon, alkali feldspars and garnets illustrate that the relative morphological importance of the crystal faces can be derived not only qualitatively but also quantitatively.In the quantitative electrostatic point-charge model computations the surface of a slice is assumed to be ideally crystalline. It represents the flat part (terrace) of an elementary growth layer. In fact such surfaces are obtained by splitting the ideal bulk crystalline structure. The influence of surface reconstruction or relaxation is not considered. This process of crystal growth cannot be understood properly without a thorough knowledge of the crystalline structure of the interface. Hartman-Perdok theory describes the surface configuration of the crystalline interface with atomic precision.Ordering of boundary ions will always reduce the attachment energy and will sometimes change the relative morphological importance as well. Examples of ordering have been given for the ordered polymorph of potassium feldspar and YBa2Cu307. In the latter case ordering of the boundary ions is absolutely necessary, if not (010) would not be an F face. The probability of the coexistence of two different surface topologies on the same face (hkl) has been describedt3,14 as a Boltzmann distribution of the specific surface energies. Even, if this relation is not completely true, the following observations are still valid. If the difference between the two specific surface energies is small, each of the two surfaces will occupy about one half of the total crystalline surface.If the difference increases, the probability decreases, but will also depend on the crystallization rate. Only when the crystallization proceeds slowly will the crystal tend to have energetically the most favourable crystalline surfaces. If such a submultiple of the slice dhklhas F character the growth rate will increase substantially. In the case of zircon, which always grows in a very impure environment, the negative surface occupied by silicate ions will directly be covered by impurity cations. Therefore the difference between the two specific surface energies will become smaller and thus the probability that two surfaces exist together will increase. During crystal growth the crystalline interface is in direct contact with the solution or melt from which the crystals grow.Factors such as impurities and supersaturation must be the cause of the diversity of habits and crystal forms observed in nature and laboratory experiments. Adsorption of impurities including the solvent can produce additional strong bonds and change the crystalline interface. Although the classic Hartman-Perdok theory takes into account only the strong bonds and the corresponding PBCs derived from the bulk crystalline structure, the effects of adsorption of impurities and solvents can be explained in many cases (e.g. zircon and PbCI,) by taking into consideration their interactions with the slice boundaries that are known with atomic precision. The introduction of scanning tunnelling microscopy (STM) and atomic force microscopy (AFM) enables imaging of the real structure of the crystalline interface to be achieved.On the (001) surface of silicon, which is following the principles of the Hartman- Perdok theory, a K face, surface reconstruction has been observed.27 This results in F character of the crystalline interface. In future, observations made by these and other methods must be taken into account in order to adjust and refine the Hartman-Perdok theory and the computation of attachment energies. The author is very much indebted to Professor P. Hartman for his stimulating interest for many years and his very helpful comments on the manuscript. C. F. Woensdregt 107 References 1 R. J. Haiiy, Essai d’une thtorie sur la structure des crystaux appliqute a plusieurs genres de substances crystallistes, Paris, 1784;read in B.G. Escher, Algemene Mineralogie en Kristallografie, J. Noorduyn & Zn, Gorinchem, 1954. 2 N. Steno, De solido intra solidum naturaliter contento dissertationis prodromus, Florentiae, MDCLXIX; read in B. G. Escher, Algemene Mineralogie en Kristallografie, J. Noorduyn & Zn, Gorinchem, 1954. 3 P. Hartman and W. G. Perdok, Acta Crystallogr., 1955, 8, 49. 4 P. Hartman and W. G. Perdok, Acta Crystallogr., 1955, 8, 525. 5 P. Hartman, in Crystal Growth: An Introduction, ed. P. Hartman, North-Holland, Amsterdam, 1973, ch. 14, pp. 367-402. 6 P. Hartman, in Morphology of Crystals, ed. I. Sunagawa, Terra Scientific Publications, Reidel, 1988, part A, ch.4, pp. 269-3 19. 7 P. Hartman and P. Benema, J. Crystal Growth, 1980, 49, 145. 8 P. Hartman, Acta Crystallogr., 1956, 9, 569. 9 E. Madelung, Phys. Z, 1918, 19, 524. 10 C. F. Woensdregt, Phys. Chem. Mineral., 1992, 19, 52. 11 G. Wulff, Z. Krist. Mineral., 1901, 34, 499. 12 C. F. Woensdregt, Phys. Chem. Mineral., 1992, 19, 59. 13 P. Hartman and W. M. M. Heijnen, J. Crystal Growth, 1983, 63, 261. 14 W. M. M. Heijnen, Geol. Ultraiect., 1986, 42, 11 (Ph.D. Thesis, ch. 2). I5 P. Hartman and C. S. Strom, J. Crystal Growth, 1989, 97, 502. 16 M. Aguil6 and C. F. Woensdregt, J. Crystal Growth, 1987, 83, 549. 17 C. F. Woensdregt, Z. Kristallogr., 1982, 161, 15. 18 C. F. Woensdregt, Z. Kristallogr., 1992, 201, 1. 19 P. Bennema, E. A. Giess and J. E. Weidenborner, J. Crystal Growth, 1983, 62, 41. 20 M. Boutz and C. F. Woensdregt, J. Crystal Growth, submitted. 21 W. M. M. Heijnen, N. Jahrb. Mineral. Abh., 1986, 154, 223. 22 B. N. Sun, P. Hartman, C. F. Woensdregt and H. Schmid, J. Crystal Growth, 1990, 100, 605. 23 R. Caruba, A. Baumer, M. Ganteaume and P. Iaconni, Am. Mineral., 1985,70, 1224. 24 J. V. Walther and Ph. M. Orville, Am. Mineral., 1983, 68, 731. 25 C. F. Woensdregt and P. Hartman, J. Crystal Growth, 1988, 87, 56 1. 26 M. van Panhuys-Sigler, P. Hartman and C. F. Woensdregt, J. Crystal Growth, 1988,87, 554. 27 P. E. Wierenga, J. A. Kubby and J. E. Griffith, Phys. Rev. Lett., 1987, 59, 2169. Paper 2/06649A; Received 14th December, 1992

ISSN:1359-6640    

DOI:10.1039/FD9939500097

出版商:RSC    

年代:1993

数据来源: RSC