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It is shown all along this chapter that in the case of crystal growth governed by surface kinetics, foreign adsorption on the interface may intervene to a large extent in the overall process. Moreover, the mechanism of subsequent overgrowth can be gathered from the shape of the adsorption isotherms. The growth mode in a pure system depends of the thermal roughness which can be treated as self-adsorption of molecules and molecular vacancies. Finally, foreign adsorption on a rough surface, just as strain due to structural misfit, may result in the formation of surface alloys.

The Burton, Cabrera and Frank [Phil. Trans. Roy. Soc. London A243 (1951) 299] kinetic equation governing the transport of molecules on the surface of a singular face toward growth steps is solved in a closed form in transient conditions. The fluxes of matter into or from the steps and the temporal waveform of the re-emitted flux intensity in response to long impinging pulses are calculated. In the case of a periodic modulation of the impinging flux, phase shift and amplitude of the first harmonic of the re-emitted beam waveform are explicitely given, allowing a direct comparison with lock-in measurements. The present model is compared with transient growth experiments on LiF of Dabringhaus and Meyer [J. Crystal Growth 61 (1983) 85].

The competition between advancement of monatomic steps and two-dimensional nucleation on the (111)-face of platinum during vacuum deposition in the temperature range of 130 to 400 K has been considered under the assumptions that nucleation follows a “non-classical” mechanism and layer propagation is controlled by surface diffusion. Previous experimental results [M. Alnot, J.J. Ehrhardt and J.A. Barnard, Surface Sci. 208 (1989) 285] showing that at temperatures of about 400 K the surface between steps remains atomically smooth during growth, while at the temperature of 130 K the surface becomes kinetically rough, are thus interpreted.

The purpose of the preceding two chapters was to stress the link between the “chemical” nature of the first stages of formation of embryos, and the “phase” nature of their further growth. It has been shown, through one example of homogeneous nucleation of solid or liquid clusters from the vapor in Chap. 10, that the results of both chemical and phase approaches are identical, except for the constant contribution of B = −k B T In Q rep to the classical nucleation work, ΔG cl*, due to cluster mobility in these particular cases. While similar considerations concerning the influence of entropy in other specific cases of nucleation are scarce, an agreement exists as to the calculation of ΔG cl* which may be performed currently with the methods of classical thermodynamics. This will be the subject of the next two chapters*.

Condensation of a vapor on a liquid surface is a process occurring without any difficulty even at the smallest supersaturation. If condensation takes place on a solid of the same chemical species, crystal growth may be more or less hindered depending on the type of the face, as will be discussed in the fourth part of this book.

The physically adsorbed or physisorbed layersconsidered thus far are excellent model systems for understanding the thermodynamics of phase equilibria between two- and three-dimensional phases. The relatively weak van der Waals type of bindingof the adsorbed molecules among themselves and with the substrate, as in the cases of rare gases on graphite, results in a high exchange frequencywith the vapor and assures reversibility. The substrate is flat, due to the lamellar structure of graphite, and to the strong covalent bonds within the (0001)-lattice plane, which offers one type of adsorption site.

In Sect. 2.8 we have already discussed the way of calculating the thermodynamic parameters and state functions of surface phases, which are the adsorption layers. The Gibbs definition is, however, too general to be used in many cases of practical interest. For example, it supposes that the excess of matter in the surface phase comes from both phases touching at the interface. This is not the case, e.g., of adsorption without chemical reaction of a gas on a foreign solid substrate. Furthermore, according to the Gibbs definition, the adsorption layer is not limited in thickness, although its concentration must tend to that of the bulk phase with increasing distance from the surface. In fact, as we shall see in the case of the gas-solid surface at not too high a temperature, adsorption is limited to a monomolecular layer until its completion. Subsequently this condensed layer acts as a substrate, i.e. a new monomolecular layer can be formed on top of it, and so on. It should be pointed out, however, that in other systems, e.g. aqueous solutions, where the concentration of the adsorbate in the bulk phase could be much higher than that in a gas, adsorption layers can be thicker, their diffuse parts extending far beyond the molecular dimension.

It is hard to imagine that a crystal of macroscopic dimension, obtained in real growth conditions, could be perfect. Imperfections of different kinds, from point defects and stacking faults to edge and screw dislocations, are almost unavoidably present in a crystal lattice. They can be introduced by external factors, such as impurities or foreign particles present in the mother phase and engulfed in the crystal during growth, by thermal stress due to temperature differences or quenching, and may even be a consequence of the growth mechanism itself, as is the inclusion and condensation of lattice vacancies, the incoherent encounter of growth fronts, etc. [211, 220]. In rare cases, nearly perfect crystals are obtained by pulling material from the melt in precise crystallographic directions that favor the climb of the dislocations and their elimination at the ingot surface [221].

The condition for equilibrium in a heterogeneous system including an anisotropic solid and a vapor, may be written, by analogy with (8.1.1), in the most general form as $$ dG = {\mu ^{\left( g \right)}}dN + {\mu ^{\left( c \right)}}\frac{{dV}}{{{\upsilon _c}}} + \sum\limits_j {{\gamma _i}d{A_j}} = 0, $$ where V and v c are the volumes of the entire solid and of one molecule in the crystal lattice, respectively, while γ j and A j are the surface tension and the area of the j-th face. Capillarity approximation is implicitly admitted through the assumptions μ (c) = μ (c) ∞ and γ j = const. The expression is, therefore, valid for crystal sizes i 0 < i < ∞ (cf. Sect. 8.1).

Both Becker-Doring’s and Frenkel-Zeldovich’s treatments, considered thus far, used a theoretical artifice allowing the maintenance of a steady state, i.e. assuring a constant nucleation rate for an infinitely long period of time. In practice, when supersaturation is installed in a system, the size distribution of clusters first changes from equilibrium to that of a metastable state, which further breaks down because of the formation of overcritical nuclei. During their growth, nuclei consume matter from the mother phase, thus lowering the supersaturation. The nucleation rate decreases and finally falls to zero.

In the two preceding chapters we have shown the crucial role of surface free energy in the thermodynamic properties of large two-dimensional phases. This role is just as important, although quite different, in the case of small phases.

We shall limit our further considerations almost exclusively to the cases of growth and nucleation, from a vapor considered as an ideal gas, of a crystal considered as an Einstein solid in the high temperature range (k B T » hv) For this reason, we will focus here our attention on the equilibrium between those two large phases. Good comprehension of the phenomena in their simplest form should allow the interested reader to make the transposition to more complicated systems, for which also abundant specialized literature exists.

Although the idea of considering nucleation as a chemical reaction of polymerization followed almost immediately [12] upon the Volmer-Weber phase approach, the assimilation of polymer molecules to embryos of the new phase persists till now. The Gibbs thermodynamic analysis of the equilibria in heterogeneous systems, and the Wulff approach to the equilibrium shape of small anisotropic phases, have been too successful to be easily abandoned, even when the models, developed by Farkas [12], Kaischew and Stranski [10], and Becker and Doring [11] were similar to those for chemical chain reactions. This is perhaps the reason why the thermodynamics of these processes, developed mainly in the works of Frenkel [13] and Zeldovich [14] came later than the understanding of their kinetics. Such an anomaly had its perverse effects. As we shall see, Frenkel first found the right expression for the equilibrium size distribution of polymer molecules, and then replaced it by an erroneous one, in order to stick to the Becker-Doring kinetic result. One may regret that this error was at the origin of many lasting controversies concerning the role of entropy factors in nucleation kinetics, commonly known as Lothe-Pound replacement factors.

In Chap. 2 on thermodynamics we showed the difference between surface tension, γ, and surface free energy, f (s) [cf. (2.8.5)]. While γ represents pure mechanical work that must be done to increase the area of an interface, f (s) contains “chemical” terms, µ i Γ i , and its value at the equilibrium depends on the state of the bulk phases from both sides of the interface, via the chemical potentials. At the equilibrium between the F-face of a crystal and the vapor of only one chemical component, the chemical terms are due to the presence of admolecules and holes, as considered in Chap. 4, which build self-adsorbed layers with thickness increasing with temperature, up to a complete roughening of the surface. At temperatures much lower than that of the roughening transition, however, one can neglect in a first approximation the influence of self-adsorption on the surface free energy, and consider that the latter is identical to the surface tension. For this reason, the reader should not be confused if we use both terms to designate γ.

The vacuum deposition of lead on a copper (110) surface in the low coverage range has been followed by monitoring in a helium scattering experiment the intensity, position and full width at half maximum of selected diffraction peaks. Up to Pb coverages θ ⋍ 0.2 the scattered intensity of the (−1,0) peak, absent on the bare Cu surface, increases as θ2, thus indicating the “decoration” of the substrate by a lattice gas of adsorbed Pb atoms. Above this coverage to the completion of the condensed layer with c(2 × 2) structure, the existence of two levels is evidenced, one being the substrate covered by lattice gas, the other that of islands of the condensed phase. The difference in height between these levels favours the assumption that the Pb atoms of the lattice gas replace the Cu atoms of the outermost layer of the substrate. The thermal behavior of the scattered intensities of the two phases gives an indirect indication in the same sense.

Cet article traite les phenomenes ayant lieu a l’interface entre un solide et son milieu, a l’equilibre et en cas de croissance du solide. Le trait commun est la rencontre, a l’interface, de deux flux de matiere de sens opposes : l’un, constitue de molecules qui s’y deposent avant d’etre incorporees dans le reseau cristallin, l’autre, de molecules qui quittent l’interface en direction du milieu. A l’equilibre, l’intensite des deux flux est egale et depend de la cinetique interfaciale, elle-meme regie par les proprietes des phases au voisinage immediat de l’interface. A la sursaturation, representee par la difference des potentiels chimiques dans les deux phases, le flux de croissance est predominant, sans qu’il existe cependant une dependance lineaire entre le flux net et la sursaturation, comme l’on aurait pu s’y attendre par analogie avec l’electrodynamique en assimilant l’interface a une resistance passive. La voie que nous nous proposons de suivre passe donc par la consideration des equilibres a l’interface entre un cristal et son milieu, qui nous revelent les mecanismes microscopiques a la saturation, avant d’aborder la cinetique interfaciale de croissance, dont la dependance avec la sursaturation peut etre assez complexe. Deux remarques decoulent de cette procedure. — L’approche utilisee est exclusivement moleculaire-statistique. Par rapport a l’approche basee sur la thermodynamique classique et sur la mecanique des milieux continus NOZIERES (P.) - * , la methode choisie a l’avantage de mieux visualiser les phenomenes physiques, au prix de quelques concessions a la rigueur mathematique. En outre, elle s’avere mieux adaptee au traitement des systemes a energie de surface elevee et fortement anisotrope, tel un cristal qui croit a partir de sa vapeur. — Il faut souligner que la cinetique globale de croissance depend autant de la cinetique interfaciale que de la cinetique du transport de matiere du milieu vers l’interface. Cependant, les mecanismes du transport dans le volume sont communs a d’autres procedes (genie chimique, combustion, etc.) et sont traites en detail par la mecanique des fluides, d’ou notre choix de ne pas depasser dans cet article le cadre des phenomenes interfaciaux.

In the first part of the paper we discuss the shape and phase shift of the reemitted beam intensity when a stepped crystal face is exposed, in vacuum, to periodic square shaped pulses of a molecular beam. Two mathematical solutions are proposed: the periodic steady state solution which allows the determination of mean stay time and scaled interstep distance on the surface from measurements of phase shift and amplitude attenuation of the reemitted beam first harmonic, and a solution valid for long incident beam pulses, according to which the same parameters can be determined from the shape of the reemitted beam intensity.

We present some results on -stability and phase transitions of binary two-dimensional phases on the surfaces of nearly perfect substrates. A simple theoretical model allows to get insight into the mutual enhancement of the condensation of the adsorption partners, resulting in the formation of two-dimensional mixture, and into the subsequent demixting of the layers, when the concentration of one of the components in the bulk phase continues to increase. Both continuous and discontinuous demixing can be expectad, and other phase transformations, like two-dimensional solidification and melting, and/or different structural transitions, are possible when the molecules of the adsorbates have different shape and the substrate is not fully isomorphic with them. Some of the peculiarities regarding the physical realizations of the proposad model are shown on the example of the coadsorption of methane and krypton on exfoliated graphite.

Quasi-periodic closed growth lamellae are observed on the solid-liquid interface of pulled NaCl crystals doped with lithium. A phenomenologic explanation is given which assumes a long range interaction between growth steps.

The microscopic approach to the elementary processes taking place at the interface between a crystal and its mother phase (vapor, solution, melt), due to Kossel and Stranski, has always been a powerful tool in resolving problems of equilibrium shape, growth and dissolution morphology and overall interface kinetics. However, it is only during the recent decade that experimental techniques have become available which allow one to check and enrich many of the Kossel-Stranski ideas through direct observations on the atomic scale of both surface morphologies and surface processes. The present work, dedicated to the memory of I.N. Stranski, reviews some of these new results.

The specific free energy γSL of the crystal-melt interface for pure KCl, NaCl and mixed KCl-NaCl, is determined from contact angle measurements. The comparison with values estimated by the current hypothesis as well as with those issued from droplet freezing experiments show that these both methods give dispersed but rather lower values for γSL.

The increase of the threshold for crystallization in gels, compared to that in pure solutions, is interpreted as resulting from the nucleation in small isolated volumes. The depletion of the solution inside those volumes can result in an absolute ‘nucleation limit”, the experimental determination of which allows to calculate activation energies for nucleation (and interfacial tensions crystal-solution) independetly of the value of the pre-exponential factor entering the nucleation rate equation. The feasibility and the advantages of droplet experiments (crystallization in dispersed solutions) are discussed.

The specific heat of immersion of calcite in water depends on the powders used. For ground crystals, the heat of immersion is very high because of the breaking of surface bonds during grinding, and, also, because of a partial surface decomposition of about 30 % during thermal treatment. Calcite crystals, limited only by singular faces, have a low heat of immersion in water because their surface is of high crystallinity and does not dissociate (below the temperature of bulk decomposition).

The pH value of saturated calcite solutions obtained from ground powders indicates that their surface becomes dissociated to some 30 % when the samples are outgassed in vacuum. Treating them in a CO2 atmosphere regenerates the carbonate on the surface. On the contrary, the preceeding treatment does not have any effect on the pH value of saturated calcite solutions obtained from crystalline powders (rhombohedral crystals).

Following Mayer's and Frenkel's statistical treatment of condensation, two difference equations are proposed, that could be considered as most general forms of the Thomson‐Gibbs formula relating the equilibrium vapor pressure of small phases with their dimension. The first equation (18) assumes the knowledge of the complete ``cluster integral'' for particles of each size. The second one (19) takes into account the internal partition function of the most stable configuration only. Both equations, being discontinuous, give supersaturation ranges in which particles could be considered as critical nuclei in the condensation process. The computer calculation of the cluster integrals for (2‐ to 14‐atomic) particles of a monatomic substance with central forces, using the Lennard‐Jones 6–12 potential, shows a negligible difference between the numerical values of the supersaturations obtained by Eqs. (18) and (19). On the other hand, from the comparison of these data with the Thomson‐Gibbs formula for the same model, it follows that the latter is a fairly good approximation even for particles of only a few atoms. A discussion of the most stable structure of small particles and of the smoothing effect of their entropy terms on the vapor pressure curve, enables us to affirm that such particles have a pronounced liquid character even at low temperatures.

The main purpose ol this paper is to underline the fundamental structural as well as interfacial energy differences existing between the crystal-melt interface and other interfaces such as crystal-vapour, liquid-vapour and crystal I-crystal II. After a short review of the existing theories of liquids and melting process, an approach towards understanding the crystal-melt interfacial structure based on the Bernal random-liquid model is suggested. Finally, some preliminary experimental results on the measurement of contact angle and adhesion energy between the densest crystal plane and the melt for close packed metals are presented., Le but principal de cet exposé est de souligner la différence fondamentale qui existe entre l'interface cristal-bain fondu et les interfaces cristal-vapeur, liquide vapeur et cristal I-cristal II, tant du point de vue de la structure que de l'énergie libre interfaciale. Après une brève revue des théories des liquides et de la fusion, nous proposons une voie d'approche de l'étude de la structure de l'interface cristal-bain fondu d'un corps monoatomique fondée sur le modèle des liquides de Bernal. Les premiers résultats d'une mesure de l'angle de mouillage et de l'énergie d'adhésion entre les faces denses des métaux à structure compacte et leur bain fondu sont également évoqués., Mutaftschiev Boyan. L'interface cristal˗bain fondu. In: Bulletin de la Société française de Minéralogie et de Cristallographie, volume 92, 6, 1969. Réunion annuelle de l'Association Française de Cristallographie, Marseille, 21-23 avril 1969.

Not available, La caractérisation thermodynamique de notre système krypton-méthane sur graphite exfolie a été effectuée par volumétrie d'adsorption entre 72 et 86 k. Cette étude nous a permis de mettre en évidence les phénomènes d'entrainement et de déplacement qui s'effectuent selon une transformation du second ordre. D'autre part, une comparaison de nos résultats expérimentaux avec les prévisions théoriques établies dans le cadre du modèle champ-moyen a montré un bon accord entre les deux. De plus, elle nous a permis de confirmer le caractère quasi-idéal de la solution bidimensionnelle observé expérimentalement. L'étude structurale du même système a été effectuée par diffraction de neutrons. Elle nous a permis de mettre en évidence l'existence d'une limite de solubilité du krypton dans la couche mixte et qui varie en fonction de la température. Enfin, l'étude de la fusion de deux compositions du film mixte nous a permis de confirmer l'existence de la solution bidimensionnelle, ce qui rejoint les conclusions des études théoriques et thermodynamiques

Not available, La caractérisation thermodynamique de notre système krypton-méthane sur graphite exfolie a été effectuée par volumétrie d'adsorption entre 72 et 86 k. Cette étude nous a permis de mettre en évidence les phénomènes d'entrainement et de déplacement qui s'effectuent selon une transformation du second ordre. D'autre part, une comparaison de nos résultats expérimentaux avec les prévisions théoriques établies dans le cadre du modèle champ-moyen a montré un bon accord entre les deux. De plus, elle nous a permis de confirmer le caractère quasi-idéal de la solution bidimensionnelle observé expérimentalement. L'étude structurale du même système a été effectuée par diffraction de neutrons. Elle nous a permis de mettre en évidence l'existence d'une limite de solubilité du krypton dans la couche mixte et qui varie en fonction de la température. Enfin, l'étude de la fusion de deux compositions du film mixte nous a permis de confirmer l'existence de la solution bidimensionnelle, ce qui rejoint les conclusions des études théoriques et thermodynamiques

Not available; Les données thermodynamiques du mouillage tridimensionnel, appliquées à l'adsorption, considéré comme un phénomène de mouillage bidimensionnel, permettent de schématiser simplement le comportement d'une couche mixte adsorbée. Il apparait que la condensation et le déplacement d'un gaz dans un alliage binaire adsorbé dépendent des potentiels chimiques des phases gazeuses à l'équilibre avec les constituants du mélange et des conditions de condensation des mêmes corps mais en phase pure. La résolution d'un modèle thermodynamique statistique s'appuyant sur l'approximation de Bregg-Williams et la simulation numérique de la coadsorption par la méthode de Monte Carlo confirment et affinent ce phénomène. On montre notamment que plus l'affinité électrostatique entre les gaz est forte, plus la lacune de miscibilité est étendue. L’étude expérimentale, menée sur les systèmes Krypton-Méthane adsorbés sur graphite exfolié et MgO, a permis, en confirmant les différents résultats des modèles, de mettre en évidence quelques points remarquables. Ainsi, sur le graphite, l'augmentation de la pression partielle en méthane, induit-elle la démission d'une phase liquide dans un alliage solide constitué du mélange Krypton-Méthane. C’est la retro-fusion bidimensionnelle. Sur le MgO, la nature des phases reste mal définie, mais il a été montr2 que la solution bidimensionnelle formée s'écarte notablement de l'idéalité