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.