On the Nucleation of Crystals

N. Fuchs, Moscow

At the present time it may be regarded as firmly established that all physical and chemical processes leading to the formation of a new phase in an initially homogeneous medium—such as the melting and solidification of crystalline substances, boiling and condensation, crystallization from solutions, allotropic transformations, dissociation of solid compounds, etc.—always begin at individual points of the system, the so-called “nuclei,” and from there spread further, proceeding exclusively at the interface between the two phases.

Owing to this circumstance, the kinetics of all the processes listed above, though so different in essence, has a very similar character. In all cases the emergence of a new phase is the consequence of two successive processes—the formation of nuclei and their growth. There is a profound difference between these processes. The growth of nuclei, if only it is thermodynamically possible, always does in fact take place; moreover, in the case of physical transformations it almost always occurs at a finite rate, which, to a first approximation, is proportional to some quantity (“supersaturation,” “supercooling,” etc.) indicating how far the system is from the equilibrium state. Finally, the rate of growth of nuclei is usually a definite quantity, constant under given conditions and readily determinable from experiment.

The situation is quite different with the rate of formation of nuclei. At small supersaturation (supercooling, etc.) it is practically always equal to zero, but, beginning with a certain value of the supersaturation, it increases sharply. The more rapid increase in the rate of formation of nuclei with increasing supersaturation, as compared with the rate of their growth, is responsible, among other things, for the increase in the dispersity of the system being formed as supersaturation increases—a very general and important phenomenon, though it is often masked by secondary processes: coagulation or coalescence of droplets or crystallites, recrystallization, etc. Furthermore, the rate of formation of nuclei in all cases proves to be a quantity that is highly variable, poorly reproducible, and dependent on a whole series of circumstances that often escape the experimenter’s attention.

It follows from what has been said that, in studying the kinetics of phase transformations, it is necessary to investigate separately each of the two above-mentioned stages of the process. In the present article we shall dwell on one of them—on the formation of nuclei, and we shall restrict ourselves to one of the most interesting and practically most important, and therefore most thoroughly studied, cases: the formation of crystalline nuclei in liquids. The literature on this question is extremely extensive; however, on reading it one cannot avoid the thought that in no other field of physical chemistry has so much labor perhaps been expended with such modest results. This is explained by two causes: first, the majority of investigators who worked in this field dealt with it “incidentally,” without considering its study their principal task, and therefore, as a rule, were poorly acquainted with previous theoretical and experimental work on this question, so that between individual groups of works there is almost no continuity or connection. Secondly, in almost all these works far too little attention was paid to secondary circumstances (of which we shall speak below), which in fact play a decisive role here.

The extreme dispersion and mutual isolation of the numerous works on the crystallization of liquids, the abundance of experimental data, often contradictory and unreliable, and the impossibility of using them to test one theory or another because of the presence of various “side” circumstances—all this makes a systematic exposition of the question of the nucleation of crystals a very difficult and thankless task. Perhaps for this reason there is not a single more or less complete review on this question, unless one counts the review compiled more than 30 years ago by W. Ostwald in his Lehrbuch der allgemeinen Chemie,* to which we also refer readers wishing to become more closely acquainted with the older literature on the crystallization of liquids. The reviews found in the well-known books of Freundlich, Illinicke, Tammann, and others are so incomplete, especially in their theoretical part, that they do not even give a correct idea of the present state of the question.

In view of the extensiveness of the literature, only the most important experimental results and theoretical views could be presented in the present review; however, even this selected material is still so large that, in order to avoid making the review excessively bulky, we had to give it a very condensed, and in places even synoptic, character. We have chosen the following order of presentation: in the first part of the review are given (with the most necessary com-

* Until recently it was assumed that at least the melting of crystals can occur simultaneously throughout their entire mass; however, in the unpublished work of Follmer it has been possible to show that the melting of crystals always begins from corners and edges and then proceeds into the interior of the crystal, so that here too we encounter no exception to the general rule.

(with commentaries) the most important experimental data are given; in the second, the theory of the question is set forth.

I. Experimental Data

As has already been indicated, at a small supersaturation of a solution (or supercooling of a liquid) the rate of formation of crystalline nuclei is practically equal to zero—spontaneous crystallization is impossible. The fact of the existence of supersaturated solutions and supercooled liquids is of exceptional importance for clarifying the mechanism of crystal nucleation. Therefore the discovery of the phenomenon of supercooling (of water) by Fahrenheit^2 in 1724 may be regarded as the beginning of the scientific investigation of the question discussed here. It is interesting that already in this first work on the supercooling of liquids an important observation was made, the significance of which, however, was understood only 150 years later—namely, that in sealed glass bulbs, especially if the air had been pumped out of them, it was much easier to supercool water than in an open vessel. If, however, air was then admitted into the bulbs, the water contained in them crystallized immediately.

The existence of supersaturated solutions was discovered by Lowitz^3 (1785), who at the same time found the following important facts: 1) a supersaturated solution crystallizes immediately if a crystal of the substance contained in the solution is thrown into it, whereas crystals of other substances have no effect whatever. Lowitz established an analogous phenomenon for supercooled liquids as well. 2) Crystallization of solutions always begins at separate points in the liquid, from which it then spreads in all directions.

According to the experiments of Gay-Lussac^4 (1813), access of air to supersaturated solutions causes their crystallization, as in the case of supercooled water. Shaking the solutions leads to crystallization only in open vessels, so that here too one may suppose the action of air. Protecting solutions from contact with air by means of a layer of oil, etc., produces the same effect as sealing the vessel. Gay-Lussac also notes the very poor reproducibility of experiments with supersaturated solutions and the dependence of the results on a whole series of circumstances. Ziz^5 (1815) observed that in the presence of air crystallization always began at the surface of the liquid, which likewise pointed to the crystallizing action of air.

The observations of Löwel^6 (1855), who showed that upon filtration or prolonged standing of air it loses its crystallization activity, seemed to indicate clearly that the whole matter lay not in the air itself but in particles suspended in it; however, this conclusion was drawn only 10 years later by Violette^7 and Gernez^8. The impetus for the work of these authors, who arrived simultaneously and independently of one another at almost identical conclusions (1865), was the bacteriological technique of sterile cultures developed shortly before by Pasteur. The analogy

The analogy between the infection of a nutrient substrate by bacteria suspended in the air and the crystallization of a supersaturated solution under the action of dust particles is obvious, and the methods of studying these two phenomena must have much in common. In a series of ingenious experiments, Violette and Gernez proved that the crystallization of supersaturated solutions of sodium sulfate is caused exclusively by the entry into them of the tiniest crystals of the decahydrate of this salt suspended in the air; specks of dust of any other kind have no effect.

Under conditions excluding the entry of decahydrate crystals—for example, in sealed vessels—solutions (with slight supersaturation) can remain for years without crystallizing.

The crystallization of solutions upon contact with various bodies turned out, in reality, to be due to the same decahydrate crystals that had settled on the latter from the air. After washing, recrystallization, and so on, these bodies lost their activity. Similar observations were also made on solutions of certain other salts.

According to Gernez’s experiments, some salts also crystallize when their solutions come into contact with crystals of other substances; however, it is possible that in the latter the corresponding salt was present as a negligible impurity. An unquestionable nucleating action, as Lecoq de Boisbaudran[^9] (1866) showed, is possessed by crystals of substances isomorphous with the dissolved substance, though only at not very small supersaturation.

A very important observation of Gernez is the following: salts, even those readily soluble, sometimes adhere so firmly to various solid bodies (for example, the walls of a vessel) that they can be washed off only with great difficulty. Because of this, it is easy to fall into error and attribute the crystallizing action to the solid bodies themselves, as indeed happened with some authors who tried to refute the conclusions of Violette and Gernez. The circumstance indicated above also explains the long-known phenomenon that certain precipitation reactions, used in analytical chemistry for the detection of elements, succeed well only in the vessels in which they have been carried out before.[^10]

Wilhelm Ostwald[^11] set himself the task of clarifying a very interesting question—the minimum size of crystals still capable of causing the crystallization of supersaturated solutions (1897). Unfortunately, the method employed by Ostwald made it possible to determine only the minimum weight of a crystalline seed; and since the number of individual crystals in the seed was not measured, these experiments have only qualitative significance—they show that very small crystals indeed do not cause crystallization of weakly supersaturated solutions.

Already in the works of Violette and Gernez the fact had been noted that, at sufficiently high supersaturation, crystallization of solutions occurs even under conditions excluding the possibility of nuclei entering from outside. The author who investigated this question more thoroughly de—

De Coppet[^12] (1872) came to the conclusion that, for solutions of every substance, there exists a certain critical supersaturation above which they crystallize spontaneously, i.e., without any external influence. An exact determination of this supersaturation proved, however, to be impossible, since the results of individual experiments differed very greatly from one another. Of enormous importance in these experiments, as de Coppet emphasizes, is the time factor; namely, with the exception of two limiting cases—very large and very small supersaturations—one cannot simply say whether a given solution crystallizes or not, but must deal with the mean lifetime of this solution, i.e., with the time after which the first crystal is formed. Since this time, as subsequent investigations showed, increases continuously, though sharply, as the supersaturation decreases, it is of course impossible here to speak of any critical supersaturation, and Ostwald’s theory of “metastable” and “labile” states separated by a sharp boundary proved inapplicable to the phenomena under consideration. Nevertheless, owing to the specific form of the dependence of the mean lifetime or the probability of crystallization of solutions on supersaturation—namely, the fact that in a certain “critical” concentration region the probability of crystallization of a solution increases extremely rapidly with supersaturation—it seems expedient to retain the notion of a “metastability limit” or “critical supersaturation” situated within the above-mentioned region.

However, even such an interpretation of the concept of the metastability limit requires one further, very substantial reservation. As Jaffe[^13] showed (1903), repeated filtration of solutions through a layer of cotton wool considerably increases their stability, i.e., increases the critical supersaturation. According to the experiments of Füchtbauer[^14] (1904), filtration of supercooled liquids produces an analogous effect. Thus there is no doubt that, in the experiments of de Coppet and other investigators, “spontaneous” crystallization in fact occurred under the action of some solid particles suspended in the liquid, which served as nuclei. This is also indicated by the extremely poor reproducibility of the experiments. Since it also occurred in filtered liquids, one can hardly doubt that in these latter as well crystallization took place on particles that had passed through the filter.

We see that, after the work of Jaffe and Füchtbauer, the situation became to a certain extent analogous to what it had been after Löwel’s discovery of the crystallizing action of atmospheric dust. The further course of the investigation should naturally have been directed toward the careful removal of particles suspended in the liquid and the clarification of their nature. Unfortunately, the investigation took a different path—the path of completely ignoring the role of suspended particles. The cause of this was a scientist whose services in the field of the study of phase transformations are, generally speaking, enormous—we mean Tammann.

Tamman and his co-workers studied chiefly the crystallization of certain organic liquids: salol, benzophenone, piperine, etc., which possess a strongly pronounced capacity for supercooling; they used the following method, developed by Tamman[^15] (1898). The supercooled liquid is kept for a definite interval of time at a temperature considerably below the melting temperature, after which the submicroscopic crystalline nuclei that have formed are “developed,” i.e. grown at a temperature so close to the melting temperature that new nuclei can no longer form at it. The dependence is determined between the number of nuclei and the magnitude and duration of the supercooling (the “exposure”) of the liquid. It turned out that, for all the substances investigated, the rate of formation of nuclei at first increases with supercooling, reaches a maximum value, and then gradually falls to zero. The addition of various substances, soluble and insoluble in the liquid, can very strongly change the rate of formation of nuclei both in the direction of its increase and in the direction of its decrease.

The number of nuclei increases with the time of “exposure,” and the rate of their formation either remains constant or increases with time.

In addition to the circumstance mentioned above—the complete neglect of the role of solid particles suspended in the liquid—Tamman’s method gives rise to another serious objection: nuclei of sufficiently small size, during “development,” should not grow but should melt, since the melting temperature of crystals depends on their size. Since at considerable supercoolings the rate of growth of crystals is enormously slowed, it is quite possible that the majority of nuclei formed at such supercoolings undergo this fate; Tamman’s observations are thereby explained.

Be that as it may, thanks to Tamman’s enormous authority, the method he applied and the results achieved with its aid received universal recognition, entered all textbooks, and undoubtedly greatly impeded the investigation of the question of the mechanism of the nucleation of crystals.

The following observation made by Tamman and his co-workers[^16] is extremely important: the capacity of liquids for supercooling depends to a high degree on their previous history, namely—preliminary heating of the liquid increases this capacity, and the more strongly, the higher the temperature and the longer the duration of heating. The same effect was found by Tamman[^17] and Webster[^18] in the case of molten metals (a circumstance apparently of great importance in the heat treatment of metals) and by Richards[^19]—in supersaturated solutions. This phenomenon apparently has a very general character, covering all cases of formation of a new phase.

A significant step forward was the work of Hinshelwood and Hartley[^20] (1922), who investigated the crystallization of p-toluidine by the follow-

...following method: a large number of sealed ampoules with the molten substance were immersed in a thermostat, and the number of samples crystallized over a definite interval of time was recorded. The results of this work are shown graphically in Fig. 1, where the abscissa gives time in minutes, and the ordinate the percentage of samples crystallized at various temperatures. The melting point of p-toluidine is 43.3°C.

These results, above all, introduce a substantial correction into Tammann’s assertion concerning the continuous increase in the number of nuclei with time, which proves to be valid only for comparatively brief observation. In reality, the total number of nuclei contained in, or forming in, a given volume of liquid at a definite supercooling is limited. This fact

Fig. 1.

by itself compels one to suppose that the crystallization of p-toluidine is caused by some particles suspended in the liquid. This is also indicated by another observation: by prolonged heating in a sealed tube one can deprive p-toluidine of the ability to crystallize spontaneously; however, when the sample is left in the air this ability is rapidly restored (dust). As we shall see below, Hinshelwood and Hartley were able, on the basis of the curves they obtained, to carry out even an interesting analysis of the dispersion of the dust contained in a sample of p-toluidine. The action of preliminary heating is explained by these authors as the gradual dissolution of the dust.

Bielmann and Kitt21 (1933) found that the destruction of the ability to crystallize spontaneously is also possible by centrifuging the liquid (piperonal and allocinnamic acid), which clearly indicates the decisive role of suspended particles in this phenomenon. Still more convincing are the experiments of Meier and Pfaff22 (1934), carried out with a series of organic substances (salol, benzophenone, acetophenone, etc.). It turned out that the same action as centrifuging is produced by filtering liquids through a fine-pored Schott glass filter with an average pore size of 1.5 μ. In those cases where such filtration proved insufficiently effective and the liquid, under sufficient...

with sufficiently prolonged and strong cooling, nevertheless crystallized, success was invariably achieved by supercooling the liquid for a short time before filtering, i.e. allowing crystals to grow on the dust particles, crystals already incapable of passing through the filter.

Liquids treated in this way and placed in sealed tubes could not be made to crystallize by any means. Neither shaking nor prolonged cooling down to the temperature of liquid air led to the desired result. When the ampoule was opened, however, the liquid crystallized rapidly.

The question whether the impossibility of spontaneous crystallization in the absence of suspended particles is a general property of all supercooled liquids remains, for the time being, open.

The following observation by Meier and Pfaff is extremely instructive. They “inoculated” a filtered liquid with crystals of the same substance, known not to contain suspended particles. After melting the crystallized mass, there was again obtained a liquid incapable of spontaneous crystallization. Hence there follows the important conclusion that the suspended particles that promote crystallization cannot be freely floating microscopic crystals of the same substance in the liquid, as Bloch, Briggs, and Kuhn suppose²³.

Fig. 2.

From what has been said above it clearly follows that Tammann and his school, strictly speaking, investigated not the spontaneous crystallization of supercooled liquids, but the crystallizing action of certain particles of unknown origin. This circumstance deprives Tammann’s work on the nucleation of crystals of a significant part of its scientific value.

A special study of the crystallizing action of solid particles is the subject of the work of Richards¹⁹ (1932), who used the following procedure: he ground crystals of salol (or benzophenone) with a small amount of one or another substance, placed the molten mixture in sealed tubes, heated them for 2 hours to a definite temperature, cooled them to 25°C, and determined the percentage of samples that crystallized within 2 hours. The results of one series of Richards’s experiments are shown in Fig. 2, where along the abscissa is plotted the amount of superheating of the benzophenone samples (the difference between the heating temperature and the melting temperature of the substances), and along the ordinate—the percentage of crystallized samples. Curve A refers to commercial benzophenone without any additives, B to benzophenone with an admixture of charcoal, C to an admixture of crushed quartz. Crushed glass, mica, etc., also exert a strong effect.

Thus, the presence of certain solid substances considerably increases the duration of heating necessary for the “deactivation” of specimens, i.e., in order to render them incapable of spontaneous crystallization. It is very significant that, when these substances are mixed into the molten liquid, they produce no effect. If, however, the mixture is then made to crystallize and is melted again, the effect appears, although it is somewhat weaker than when the solid mixture is ground. It is curious that vigorous shaking leads to the immediate crystallization of specimens of the mixture which have not been heated very long and have not yet lost the capacity for spontaneous crystallization, and has absolutely no effect on specimens that have lost this capacity. The action of shaking is apparently based on the fact that it leads to collisions and friction between solid particles.

Evidently, any solid body immersed in a supercooled liquid or a supersaturated solution may play exactly the same role as suspended particles, including the walls of the vessel in which it is contained. The correctness of this assumption is confirmed by the above-mentioned observations on the influence of the vessel on the success of certain analytical reactions and, especially, by the work of Roginsky, Sena, and Zeldovich^24 (1932) on the crystallization of nitroglycerin. In general, it is rather difficult to make nitroglycerin crystallize. It has, however, long been observed that, if this has somehow been accomplished, then after the crystals are melted, recrystallization is achieved very easily; moreover, the same crystalline form is always formed from which the melt was obtained (nitroglycerin crystallizes in two forms). At the same time, the properties of the melts obtained from the two forms proved to be completely identical. Therefore the above-named authors quite rightly attributed this curious phenomenon to the influence of the walls of the vessel and proved their supposition by pouring molten nitroglycerin into another vessel—in the majority of cases the transferred nitroglycerin did not crystallize. Those cases in which crystallization nevertheless occurred can quite well be explained by the action of suspended particles (not taken into account by the authors of that work).

Upon more prolonged heating of nitroglycerin, it ceases to crystallize even in the same vessel in which it had been melted. In this case crystallization is achieved only by rubbing the walls with a glass rod. The same operation, however, does not produce the desired result in a beaker that has not been in contact with crystalline nitroglycerin. The analogy between these observations and Richards’s experiments is, as we see, almost complete.

Let us also mention the experiments of Krotov,^25 who found that a polished surface of metal or glass does not act on supercooled salol or benzophenol, whereas one treated with emery paper causes crystallization. Apparently, with such treatment, difficult-to-remove particles of some organic substance, serving as nuclei, become lodged in the scratches.

The contact action of various minerals on supersaturated solutions of certain salts was first investigated quantitatively by Vollmer and Weber^26 (1926), who used very pure solutions (filtered through an ultrafilter). It turned out that, on the surface of all the minerals investigated, crystallization begins at a considerably lower supersaturation than within the solution; moreover, the action of different minerals, and even of different faces of one and the same mineral, is highly specific. The angles and edges of crystals are especially active. Stranski and Kuleliev^27 (1929) showed that the crystallizing action of minerals is the stronger (i.e., crystallization begins at the lower supersaturation), the greater the similarity in crystal structure between the mineral and the dissolved salt. Closely connected with the crystallizing action of the surface of solids is, of course, the phenomenon—described many times in the literature—of a definite orientation, relative to the latter, of crystallites formed on the surface.

The phenomenon of crystallization of supersaturated solutions when the walls of the vessel are rubbed with a glass rod is also well known; in the chemical literature hundreds of cases are described in which this operation caused crystallization, and hundreds in which it produced no effect whatsoever. Undoubtedly, the effectiveness of rubbing must depend not only on the nature of the dissolved substance and of the solvent, but also on the degree of supersaturation, the preliminary treatment of the walls of the vessel, etc. Unfortunately, this phenomenon has never yet been investigated quantitatively (i.e., with account taken of the degree of supersaturation).

Fricke^28 (1932) established that, instead of a glass rod, the wall of a glass vessel can with equal success be rubbed with any solid body, and that the action of rubbing consists in exposing a fresh surface of the glass that has not been in contact with air, water, etc. Therefore the same effect is produced by fracturing the glass under the solution. On contact with air or on washing, the newly formed surface of the glass becomes inactive.

Apparently based on the same principle is the crystallization of a solution sometimes observed upon vigorous shaking; in this process, pieces of glass are probably torn from the walls.

Finally, in recent times a considerable accelerating action on the crystallization of supercooled liquids has been discovered for strong electric fields,^29 rapidly alternating currents, X-rays,^30 etc. Unfortunately, in all these works unfiltered liquids were used, so that it is unknown whether new nuclei were formed or only the growth of existing ones was accelerated.

In conclusion, let us mention an interesting observation by V. Fischer^31 (1922): according to the rate of crystal growth and of nucleus formation, all salts may be divided into two groups. To the first, rapidly crystallizing group belong salts formed by monovalent—

ions and crystallizing without hydrate water, i.e., in other words, salts formed by weakly hydrated ions; to the second—the slowly crystallizing salts—the remaining salts. Unfortunately, the method used by Fischer made it possible to obtain only qualitative results.

Theory

Before proceeding to set forth the modern theory of the nucleation of crystals, it is necessary to dwell briefly on the views expressed on this question by de Coppet and Tammann, since these views have become widely disseminated and are presented in some modern manuals of physical and colloid chemistry without any indication that other theories exist.

According to de Coppet32 (1872), only those molecules of the liquid whose kinetic energy lies within certain, comparatively narrow limits can take part in the formation of crystals. Taking into account the Maxwellian distribution of velocities, from this follows the dependence found by Tammann between the magnitude of supercooling and the rate of formation of nuclei, characterized by the presence of a maximum of this rate at a certain supercooling. Apart from the absolute groundlessness of de Coppet’s hypothesis, we now know that Tammann’s curves have nothing in common with spontaneous crystallization.

Tammann’s own theory (1902), which he continued to defend until very recently,33 is based on the assumption that in every liquid there exist two kinds of molecules—“isotropic” and “anisotropic”—and that crystals can form only from the latter. The rate of transformation of one kind of molecules into the other increases with temperature, but, generally speaking, is small. An increase in temperature shifts the equilibrium toward the formation of isotropic molecules, and this, according to Tammann, explains the deterioration of the crystallizing ability of liquids upon prolonged heating.

If Tammann’s theory were correct, then, depending on the percentage content of molecules of one and the other kind in the liquid, i.e., on its preceding history, not only the ability of the liquid to crystallize would have to change to some extent, but also its other properties, which in reality is not observed. Therefore we consider discussion of Tammann’s theory superfluous.

The modern theory of the process of formation of a new phase begins with Gibbs34 (1875–1878), whose views on this question reduce to the following.

An isolated system is, as is known, stable if, for any possible infinitely small change of its state in which its energy remains constant, the increment of entropy is \(< 0\). If this condition is also satisfied for any finite change of the state of the system, then the stability may be called absolute. If, for some finite changes, the increment of entropy is greater than zero, we may call the system relatively

stable.* Such, for example, are supersaturated solutions, whose entropy increases by a finite amount upon crystallization.

From the point of view of classical thermodynamics, a system can be brought out of such a state only by means of an external action, i.e. with the expenditure of a certain work \(A\), which, in Gibbs’s opinion, may be taken as a measure of the stability of the metastable system.

To clarify the physical meaning of the quantity \(A\), let us consider the simplest special case of the formation of a new phase—the condensation of supersaturated vapor. For each degree of supersaturation there exists a definite size of liquid droplets at which they are in equilibrium with the vapor; moreover, for not very large supersaturations, i.e. for droplets whose radius is large in comparison with the radius of action of molecular forces, this size is given by the well-known Kelvin equation:

\[ RT \ln \frac{p}{p_0} = \frac{2\sigma M}{r\gamma}, \tag{1} \]

where \(p\) is the pressure of the supersaturated vapor, \(p_0\) that of the saturated vapor, \(\sigma\) the surface tension, \(M\) the molecular weight, \(\gamma\) the density of the liquid, and \(r\) the radius of the droplet.

In order to condense the vapor, it is sufficient to form within it a liquid droplet with a radius exceeding \(r\) by even an infinitesimal amount, since such a droplet will thereafter grow spontaneously. Therefore \(A\) is equal to the work of isothermal reversible formation, from the supersaturated vapor, of an “equilibrium” droplet of radius \(r\). In the case where the mass of vapor is large in comparison with the mass of the droplet, this work, as Gibbs showed, is equal to one third of the free surface energy of the droplet:

\[ A = {}^{1}/_{3}\, S\sigma = \frac{4}{3}\pi r^2\sigma \tag{2} \]

(where \(S\) is the surface of the droplet).

At infinitesimally small supersaturation \(r\), and consequently \(A\), are infinitely large—the system is absolutely stable. As the supersaturation increases, \(r\) and \(A\) decrease extremely rapidly; the relative stability of the system falls.

The situation is entirely analogous in the formation of a new liquid or gaseous phase inside a liquid, with the difference that, instead of the Kelvin equation, the radius of the nucleus is determined by other equivalent equations. In the case of formation of a crystalline phase, the crystallite in equilibrium with the liquid must, as Gibbs showed,^35 satisfy the condition that its free surface energy have the minimum value—

* Since Gibbs does not give such systems any special name, we shall henceforth call them, following Ostwald, metastable.

for a given volume). This condition, obviously, determines the crystalline form of the nucleus. The work of its formation is expressed by a formula analogous to formula (2):

\[ A = {}^{1}/_{3}\sum S_i \sigma_i, \tag{2′} \]

where the summation extends over all faces of the nucleus.

Gibbs also considers the process of formation of a new phase on the interface between two phases.^36 Here the nucleus has the form of a lentil (Fig. 3). In this case too the work of formation—

Fig. 3.

of an equilibrium nucleus is equal to \(^{1}/_{3}\) of the change in the free surface energy of the system, i.e.

\[ A = {}^{1}/_{3}\,[\sigma_1 S_1 + \sigma_2 S_2 - \sigma_3 S_3]. \tag{2″} \]

The meaning of the letters \(S_i\) and \(\sigma_i\) is clear from the drawing.

Gibbs’s ideas concerning the process of formation of a new phase shared the fate of his other creations—they remained incomprehensible and unnoticed by his contemporaries* and only 50 years later were discovered in Gibbs’s works by Volmer.

A major step forward in understanding the mechanism of crystal nucleation was the work of Kuster^37 (1903), who attempted in the following way to provide a theoretical basis for the existence of a hypothetical “limit of metastability.” The higher the supersaturation of a solution (or the supercooling of a liquid), the smaller the size of crystals in equilibrium with the liquid. However, this decrease in crystal size is possible only down to a certain limit, which Kuster calls the “primitive crystal,” i.e., one that contains the minimal number of atoms necessary for the construction of a single crystal cell. Obviously, the nucleation of any crystal must begin with the formation of such a “primitive crystal,” which, according to Kuster, occurs spontaneously. The further fate of the “primitive crystal” depends on the concentration of the solution. If it is greater than the solubility of the “primitive crystal,” it will grow further; if it is less, it will dissolve. Thus the limit of metastability is determined simply by the solubility (or the melting temperature) of the “primitive crystals.” However, in Kuster’s opinion, crystal nucleation is also possible in metastable solutions, since, owing to fluctuations, the concentration at individual points may

* Among them also W. Ostwald, who translated Gibbs’s works into German, despite the fact that Ostwald himself, as we have already seen, worked extensively on the study of metastable systems.

exceed the limiting value, but the probability of such an event, and consequently also of the appearance of a crystal, must rapidly fall with decreasing concentration. An analogous effect in supercooled liquids is produced by temperature fluctuations.

The weak point of Küster’s theory is that it entirely bypasses the question of how the “primitive crystallites” themselves are formed. Meanwhile, the formation of these latter is undoubtedly also a process possessing a certain probability, increasing with supersaturation, and from this in turn follows the absence of any boundary of metastability whatever. Nevertheless, it must be acknowledged that Küster’s views already constitute a certain approximation to the modern views on the question considered here.

An exact repetition of Küster’s views is found in Haber’s well-known work[^38] on the conditions for the formation of amorphous and crystalline precipitates (1922). Since the rate of formation of primitive crystallites in the labile region must increase with temperature, i.e., with the mobility of the molecules of the liquid, it follows obviously from the above that at the boundary of metastability the rate of formation of new crystals is maximal. Thus Haber arrived at an explanation of the results obtained by Tammann (see above). This part of Haber’s work now has only historical significance. Of much greater interest are Haber’s views on the process of formation of the solid phase in highly supersaturated solutions—namely, that the character of the precipitate (colloidal or crystalline) depends on the relation between the rate of accumulation (Häufungsgeschwindigkeit) of molecules and the rate of their “ordering” in the aggregates that have formed (Ordnungsgeschwindigkeit). Unfortunately, Haber did not return to this topic and did not give his views a quantitative formulation.

As has already been indicated, Volmer was the first to discover in Gibbs’s works the theory, expounded above, of the stability of metastable systems. At the same time, Volmer gave this theory a new and very fruitful direction.[^39] From Gibbs’s classical-thermodynamic point of view, metastable systems were indeed stable, i.e., a finite external action was necessary in order to remove them from this state.

The kinetic theory of matter, namely the theory of fluctuations developed by Einstein and Smoluchowski,[^40] leads to an entirely different result. According to this latter theory, the entropy of an isolated system is not a strictly constant quantity corresponding to its maximum possible value (at the given energy of the system), but undergoes a continuous series of very small random changes, remaining at all times below this maximum value, though very close to it. Correspondingly, the parameters determining the macroscopic state of the system—for example, concentration, temperature, pressure, etc.—in separate parts of the system do not remain constant, but fluctuate about certain mean values corresponding to the maximum of the system’s entropy.

The probability of such a state of the system, in which the deviations of these parameters from their mean values lie within the limits \(h_1—h_1+dh_1;\ h_2—h_2+dh_2 \ldots\), is equal to:

\[ W = Ce^{-\frac{S_0-S}{k}}\,dh_1\cdot dh_2\ldots, \tag{3} \]

where \(C\) is a constant, \(S_0\) is the maximum value of the entropy (for \(h_1=h_2=\ldots=0\)), and \(S\) is the entropy corresponding to the deviations \(h_1,h_2,\ldots\). Since the energy of the system is constant, in the case of “isothermal fluctuations” we may put:

\[ S_0-S=\frac{A}{T}, \]

where \(A\) is the work expended in transferring the system from the state with maximum entropy to the state under consideration. Consequently,

\[ W = Ce^{-\frac{A}{kT}}\,dh_1\cdot dh_2\ldots \tag{4} \]

Thus, as in all problems of statistical mechanics, the probability of a given fluctuation is determined by: 1) the energy factor \(e^{-\frac{A}{kT}}\), and 2) the statistical weight of the fluctuation, i.e. the magnitude of the elementary phase volume \(dh_1\cdot dh_2\ldots\), or by the sum of such volumes extended over all values of the parameters corresponding to the state (fluctuation) of the system under consideration. Determining this statistical weight is a difficult and, often—as in the case that interests us, a fluctuation leading to the formation of nuclei of a new phase—insoluble problem. However, according to Einstein’s important observation, changes in the magnitude of the phase volume are reflected in the value of \(W\) much less than changes in the magnitude of the work \(A\). Therefore, if it is necessary to know not the exact value of the probability \(W\), but only its order of magnitude, then one may altogether neglect the statistical weight of the state and regard the probability of the latter as simply proportional to \(e^{-\frac{A}{kT}}\), leaving open the question of the magnitude of the constant \(C\):

\[ W = Ce^{-\frac{A}{kT}}. \tag{5} \]

However, when applying the theory of fluctuations to the process of formation of a new phase, still another difficulty arises. Equation (5) gives an expression for the probability of finding the fluctuating system in the given state, i.e. the probability that at an arbitrarily chosen moment the system will be in this state. But we are interested in the probability of formation of a nucleus during a definite interval of time, i.e. the probability that during this interval of time the system will come into the given state. For nuclei with a size smaller than the critical (equilibrium) size, and therefore again disintegrating, these two probabilities differ from one another

...simply a factor expressing the lifetime of the nucleus. In the case that interests us, however, of stable nuclei that continue to grow further, such proportionality, though probable, is not obvious. As we shall see below, this proportionality has been proved with the aid of molecular-kinetic reasoning.

The most important conclusion to which Volmer’s theory led was the clarification of the exact physical meaning of the concept of the metastability limit. From equations (1) and (2) we find the work of formation of a stable nucleus from a supersaturated vapor:

\[ A=\frac{16\pi\sigma^3 M^2}{3R^2T^2\gamma^2\left[\ln\frac{p}{p_0}\right]^2}. \tag{6} \]

Hence, for the magnitude of the exponential term \(e^{-\frac{A}{kT}}\) in equation (5), we may compile the following table (water vapor at room temperature):

As is known, water vapor at room temperature, in the absence of dust particles and ions, condenses at approximately an eightfold supersaturation. For this range of supersaturations, the curve of the dependence of the probability of formation of nuclei on the magnitude of the supersaturation has the form shown in Fig. 4.

As can be seen from the table and from the drawing, the probability of formation of nuclei increases enormously with supersaturation. Therefore condensation can in practice occur only beginning at a certain supersaturation, or more precisely within a certain narrow range of supersaturation, which is the practical limit of metastability.

All that has been said above remains valid also for any other process of formation of a new phase; however, let us note that the true position of the metastable limit has so far been reliably determined only for the case of bulk condensation of water vapor (under adiabatic expansion) and certain other volatile liquids, and perhaps for the case of condensation of metallic vapors on cooled surfaces. In the latter case it was also found that the position of the metastable limit (as was to be expected from the above) depends on the duration of observation; specifically, as the latter increases, the magnitude of the supersaturation necessary for condensation decreases.\(^{41}\) Finally, recently Volmer and Flood\(^{42}\), by varying the initial temperature of the condensing water vapor, succeeded in showing that equation (6) also gives good quantitative agreement with experiment.

Thus, although at large supersaturations the Kelvin equation...

... and, consequently, equation (6) becomes inapplicable; however, at present there is hardly any reason to doubt the fundamental correctness of the Gibbs–Volmer theory.

Volmer^43 especially emphasized the important role played, in all processes of formation of a new phase, by solid walls or other interfaces in the initial system. It is not difficult, using the example of vapor condensation, to convince oneself that such surfaces must greatly facilitate the formation of stable nuclei.

Fig. 4.

If by \(\Theta\) we denote the contact angle formed by the liquid on the surface of a solid wall (Fig. 5), and by \(r\) the radius of curvature of a drop in equilibrium with the vapor, then from equation \((2'')\) it is easy to obtain:

\[ A=\frac{1}{3}\left[2\pi r^2(1-\cos\Theta)\sigma_1 -r^2\sin^2\Theta(\sigma_2-\sigma_3)\right]. \]

But according to a well-known theorem of the theory of capillarity,

\[ \sigma_2-\sigma_3=\sigma_1\cos\Theta; \]

therefore

\[ A=\frac{1}{3}\pi r^2(2-2\cos\Theta-\sin^2\Theta\cos\Theta). \tag{7} \]

Thus the work of formation of a stable nucleus on the surface of a wall decreases together with the magnitude of the contact angle \(\Theta\), but for any \(\Theta\) it nevertheless remains less than the work of formation of a nucleus in free space. At \(\Theta=0\) the work \(A\) is equal to zero—on a wettable surface condensation begins already at an exceedingly small supersaturation.

In addition to the magnitude of the contact angle, determined by the physicochemical nature of the liquid and the wall, another factor of great importance here is the geometrical factor not taken into account by Volmer, namely the form of the surface. It is not difficult to see that, in condensation in pits, depressions, etc., present in the wall, the work \(A\) has a smaller value than on a flat surface, and in sufficiently narrow crevices it may even become negative, i.e., condensation will take place already in unsaturated vapor (capillary condensation).

The action of walls must, of course, also manifest itself in all other cases of formation of a new phase. It is easy to realize that at small supersaturations nuclei may in general form

only on walls, dust particles, etc. Experiment fully confirms, as we saw above, this conclusion.

Furthermore, since the surfaces of solid bodies are never homogeneous, their individual regions obviously possess different activity with respect to the formation of a new phase—an activity determined both by the physico-chemical nature of the region and by its geometrical form and, finally, by the size of the region.

At small supersaturations the formation of nuclei will again occur only on separate, most active regions. This conclusion is also confirmed by experiment.^44

The application of the Gibbs–Volmer method, which is very convenient in the case of formation of a liquid or gaseous phase, becomes difficult in the case of crystalline nuclei. Here it is more convenient to use the molecular-kinetic method first proposed by Kossel^45 and Stranski^46 in their theory of crystal growth and later also applied by the latter to the case of the formation of crystalline nuclei.^47 In presenting this method, however, we shall prefer the derivation of the equation for the rate of formation of droplets in a supersaturated vapor, which was mentioned on p. 511. We shall give here only the idea of this derivation, proposed by Farkas^48 and corrected by Kaischew and Stranski,^47 and the final result.

Suppose, for simplicity, that the condensation process is stationary, i.e. a constant pressure of supersaturated vapor \(p\) is artificially maintained in the system, and the nuclei are removed as they form, so that the state of the system does not change with time. Let \(Z_n\) denote the number, contained in \(1\ \mathrm{cm}^3\), of molecular aggregates (droplets) consisting of \(n\) molecules, and let \(F_n\) denote the surface area of such an aggregate. During time \(dt\), on \(1\ \mathrm{cm}^2\) of surface there fall

\[ \frac{pNdt}{\sqrt{2\pi MRT}} \]

vapor molecules; consequently, on the surface of one aggregate,

\[ \frac{pNF_n dt}{\sqrt{2\pi MRT}} \]

molecules, and on average

\[ \frac{pNF_n Z_n dt}{\sqrt{2\pi MRT}} \]

aggregates with \(n\) molecules attach one more vapor molecule and are transformed into aggregates with \(n+1\) molecules. The quantity

\[ {}_n V_{n+1}=\frac{pNF_n Z_n}{\sqrt{2\pi MRT}} \]

therefore represents the rate of transformation of aggregates \(n\) into aggregates \(n+1\).

The rate of the reverse transition (from \(n+1\) to \(n\)) is expressed, obviously, by the analogous formula

\[ \frac{NF_{n+1}p_{n+1}Z_{n+1}}{\sqrt{2\pi MRT}}, \]

where \(p_{n+1}\) is the vapor pressure of aggregates \(n+1\).

Fig. 5.

The number of aggregates \(n\) disappearing per unit time owing to evaporation and condensation is equal to \({}_{n}V_{n-1}+{}_{n}V_{n+1}\); the number newly formed is equal to \({}_{n-1}V_n+{}_{n+1}V_n\).

Since the process is stationary, i.e. \(Z_n\) is constant, we have

\[ {}_{n}V_{n-1}+{}_{n}V_{n+1}={}_{n-1}V_n+{}_{n+1}V_n \]

or, rearranging terms:

\[ {}_{n-1}V_n-{}_{n}V_{n-1}={}_{n}V_{n+1}-{}_{n+1}V_n=\ldots=J. \tag{8} \]

It is easy to see that \(J\) represents the number of nuclei formed per unit time.

Substituting into equation (8) the expressions written above for \({}_{n-1}V_n\), \({}_{n}V_{n-1}\), etc., and expressing \(F_n\) and \(p_n\) (the latter by Kelvin’s equation) through the radius of a droplet consisting of \(n\) molecules, one can set up a differential equation; solving it gives for \(J\) the expression (\(C'\) is the constant of integration, \(N\) is Avogadro’s number):

\[ J=C'\frac{pN}{\sqrt{2\pi MRT}}\,e^{-\frac{S_3}{3kT}} \tag{9} \]

in agreement with the Gibbs–Volmer theory [cf. equations (2) and (5)]. Instead of the unknown constant \(C\) of that theory we now obtain the expression \(C'\dfrac{pN}{\sqrt{2\pi MRT}}\); the constant \(C'\) again remains indeterminate.

The reason for this lies in the impossibility of extending the reasoning just considered down to aggregates consisting of 2, 3, etc., molecules, since the elasticity of their vapor is unknown. Therefore equation (9) gives only the relative rate of formation of nuclei from the vapor of a given substance at different supersaturations.

In Volmer’s opinion, partly confirmed by experiments on the condensation of various vapors,\(^{42}\) the order of magnitude of \(C'\) is approximately the same for all substances. If this is indeed true in the case of vapor condensation, it is only because the elementary process of condensation of a molecule that has struck the surface of a nucleus encounters no difficulties, i.e. requires no special activation energy. Therefore, if in any two substances the size and mass of the molecules, as well as the forces acting between them, are the same, then the rate of formation of nuclei at equal supersaturations of the vapor will also be the same.

The situation is entirely different in the crystallization of solutions, where each dissolved ion or molecule, before entering the crystal lattice, must free itself from its solvation shell, which requires a certain expenditure of energy. Obviously, the greater this energy, the more slowly the process of crystal growth will proceed, and likewise the formation of crystalline nuclei, as we saw in Fischer’s experiments (see p. 507). In this case the magnitude of the constant in equation (9) will evidently vary strongly from one substance to another.

Turning to the molecular-kinetic theory of the formation of crystalline nuclei, let us note that here too the guiding idea is the one first expressed by Gibbs: that the growth of crystals proceeds by jumps, in contrast to the growth of isotropic drops.^49 According to Gibbs, the formation of a new atomic plane in a growing crystal resembles, to a certain extent, the process of formation of a new phase—for this a certain finite supersaturation is required. Just as in the formation of a new phase, there must first of all be formed a two-dimensional nucleus stable at the given supersaturation; after this, its further growth, ending with the formation of a new atomic plane, proceeds without any difficulty. If the degree of supersaturation is small, then the rate of formation of two-dimensional nuclei is so small in comparison with the rate of their growth that the formation of a complete atomic plane will have time to be completed before the appearance of a new nucleus, as a result of which the growth of the crystal takes on the character of a regular layering of one atomic plane upon another. This also explains the fact that crystals are bounded by very perfect planes. Let us note that this theory has recently received direct experimental confirmation: it turned out that at very small supersaturations the rate of growth of NaCl crystals perpendicular to the faces of the cube is zero.^50 It is necessary, however, to bear in mind that the supersaturations in question here are negligibly small in comparison with those required for the nucleation of a new crystal.

The further development of Gibbs’s views was carried out in the works of Kossel,^45 Volmer,^51 Brandes,^52 and especially Stranski.^53

Let us first consider Stranski’s interpretation of the dependence of the vapor pressure of crystals on their size.

If we “disassemble” the crystalline atomic lattice in the proper order, i.e. layer by layer and row by row, then, if the atoms lying at the edges of the crystal are not counted, all the remaining atoms will be removed from position 1 in Fig. 6.

Obviously, the work of detaching an atom from this position is equal to the lattice energy $\varepsilon_0$ per 1 atom. The vapor pressure of the crystal (if one neglects the terms depending on the heat capacities) may be set equal to $Ce^{-\frac{\varepsilon_0}{kT}}$, where $C$ is a certain constant. In doing so, in the case of large crystals we neglect the circumstance that the detachment of atoms lying at the edges of the crystal (from position 2) will require a somewhat smaller work $\varepsilon_1$. For very small crystals, however, this can no longer be neglected. Here the vapor pressure will be determined by the mean work of detachment $\varepsilon$, falling to the share of one atom in the atomic layer; moreover, it is obvious that the smaller the crystal, the greater the percentage of atoms of the layer will be removed from position 2, and consequently the smaller will be the mean value of the work of detachment, i.e. the greater the vapor pressure.

Stranski showed that, starting from any model of a crystal, one may, using these considerations, obtain a molecular-kinetic derivation of the Kelvin equation, as well as of Gibbs’s theorem on the form of very small crystals (see p. 506).

In exactly the same way one can find the elasticity of a pair of a two-dimensional nucleus; namely, it is determined by the mean work of detaching an atom in the atomic row constituting one side of the nucleus.

Hence Stranski and Kaischew \(^{47}\) derive an expression for the rate of formation of crystalline nuclei by the method already considered above (p. 514). The formula obtained by them differs from formula (9) in that, in addition to the exponential term corresponding to the work of formation of a three-dimensional nucleus,

\[ C e^{-\frac{S_3}{3kT}}, \]

the formula contains a term corresponding to the work of formation of two-dimensional nuclei on the faces of the three-dimensional one.

Fig. 6.

However, since this latter, as we have already indicated, is apparently in all cases very small in comparison with the former, this additional factor may in all probability be neglected. Unfortunately, the equation for the rate of formation of crystalline nuclei can at present be used to an even lesser extent than equation (9) for the formation of droplets, since, in addition to the uncertainty of the basic constant \(C'\), the magnitude of the free surface energy of crystals is also unknown. In other words, applications of the modern theory of crystal nucleation can be only qualitative in character, although the theory itself is quantitative.

As regards the fundamental possibility of determining the constant \(C'\), it is clear that this problem is closely connected with the question of the content, in metastable systems, of unstable amorphous or crystalline aggregates. As soon as a theoretical or experimental method is found for determining the number and size distribution of these aggregates (whose existence cannot be doubted) as a function of supersaturation, it will be possible to determine also the magnitude of the constant of interest to us.

Let us now consider, from the molecular-kinetic point of view, the formation of crystalline nuclei on a solid wall. Sa-

On the Nucleation of Crystals

The simplest case is the crystallization of an electrolyte on a wall having an ionic structure. Here the ions contained in the solution are more or less firmly adsorbed on the oppositely charged ions of the surface layer of the wall.

If the arrangement of these latter ions and the distance between them are the same as on one of the faces of the salt present in the solution, then the formation of a crystal on the wall will in principle not differ from the growth of an already existing crystal.

The magnitude of the necessary supersaturation will, as before, be determined by the work of formation of a two-dimensional nucleus on the surface of the wall—in other words, by the work of detaching an ion from the surface, i.e. by the magnitude of the adsorption forces. In the case where the wall is a single crystal, there may be continuous accretion of the crystalline substance onto the wall, as is usually observed in crystallization on isomorphous crystals. In the case of a polycrystalline wall, crystallites will evidently form only in those regions of the surface where the ions are arranged in the most favorable manner. The definite orientation of crystallites with respect to the wall, often observed during crystallization on walls, also becomes understandable.

The case considered above, in which the structure of the flat ionic lattice on the crystal face coincides with that on the surface of the wall, is, of course, very rare. However, a considerably lesser degree of geometrical correspondence between these structures also has a favorable effect on crystallization. For example, according to Royer,^54 two flat cells combined in the plane \((1,0,0)\) of the salt \(\mathrm{MgSO_4\cdot 7H_2O}\) coincide in form and size with 7 adjacent cells of the cleavage plane of mica, and indeed magnesium sulfate crystallizing on mica is always oriented with respect to it in the position \((1,0,0)\). It is very significant that the work of formation of a two-dimensional ionic nucleus in the middle of the face of an ionic crystal, as Brandes^52 showed, is considerably greater than at corners or edges. Here we find an explanation for the above-mentioned observations of Vollmer and Weber on the crystallization of salts on the surface of minerals, and also for the well-known fact that crystals form especially readily along cracks, scratches, etc.

We encounter a very important case of the crystallizing action of a wall in the crystallization of organic substances whose molecules contain polar groups and, upon adsorption, orient themselves in a definite way with respect to the surface of the adsorbent. The orientation of the adsorbed molecules may be so perfect that the adsorption layer is a two-dimensional crystal, the further growth (thickening) of which will proceed with the same ease as the growth of a three-dimensional crystal. In this case a particle of the adsorbent will exert on the liquid the same action as an equal-sized ready crystallite of the substance. This is precisely how Hinshelwood and Hartley (p. 502) conceived the action of suspended dust on a supercooled liquid, and determined the size

*

of the largest dust particle contained in the ampoule with the liquid, by supercooling at which the liquid crystallized.

The fact that preliminary heating of the liquid deprives it of the ability to crystallize is explained by Ginshelwood and Hartley by the fact that the dust is an organic substance and, on prolonged heating, dissolves in the liquid. In individual cases this interpretation may be correct; however, it cannot explain the action of the vessel walls, the addition of inorganic powders, and a number of other observations. Bilmann and Kitt[^20] attribute the influence of preliminary heating of the liquid to the disorienting action of high temperature on the adsorption layer. This explanation, according to which one would have to admit that the orientation or disorientation of adsorbed molecules are processes lasting tens of minutes, is, of course, unacceptable.

Richards (see p. 503) gives an explanation built on views analogous to Polanyi’s well-known theory of adsorption: the role of the compressed layer is played, in Richards’ case, by an adsorbed crystalline layer which, owing to adsorption forces, has a melting point higher than normal; the role of the vapor is played by the liquid in equilibrium with the crystalline layer. The thickness of the latter is determined by the magnitude of the superheating, decreasing as the temperature rises. Finally, at a certain temperature, depending on the nature of the adsorbent, the crystalline layer disappears completely—the dust particle is “deactivated.” The gradual deactivation of the liquid upon heating is explained by the difference in the nature of the dust particles contained in the liquid.

In order to explain, according to his theory, why crystallization on dust particles that have not been fully deactivated sometimes occurs after many hours and even days, and sometimes almost immediately, Richards has to assume that the rate of growth of the crystalline layer in the adsorption space can have the most varied magnitude—from the normal rate of growth in the free volume down to a rate tens of thousands of times smaller. This assumption is a weak point of the theory.

Of interest is the explanation given by Richards of the so-called Ostwald step rule, according to which, from metastable systems, the less stable modification is first separated out, which then passes into the more stable one. In Richards’ opinion the whole matter here is that, in preparing a supersaturated solution or a superheated liquid, they are usually heated above the transition point of the modification stable at low temperature into the unstable one, so that the crystalline layer in the adsorption space represents the unstable modification, and it is this, consequently, that crystallizes out upon cooling. Let us note, however, that the step rule can be derived, as Stranskiĭ showed,[^55] from the theory of spontaneous crystallization.

It seems to us that many phenomena observed in crystallization on the surface of solids can be explained mo—

more satisfactorily than had been done up to now, if one takes into account the presence in all solid bodies of cracks and channels of almost molecular size, the existence of which may now be regarded as firmly established. It is easy to see that a crystallite sitting in such a crack, whether formed there or introduced there from the surface by mechanical means, can be removed from it only with very great difficulty. Washing out the crystallite, even when its solubility is good, requires a very long time because of the extreme slowness of diffusion in very narrow channels. For the same reason, crystallization of a supersaturated solution under the action of such a crystallite begins only after a long interval of time, during which the crystallite “grows through” the channel to the free surface. Rubbing the surface with some solid body, i.e. removing the surface layer, may expose the crystallite and cause immediate crystallization. Thus we see that many phenomena associated with the crystallization of supersaturated solutions, and, above all, the enormous role played here by the time factor, can be satisfactorily explained if one accepts that the principal carriers of nuclei are cracks. It is somewhat more difficult to do this for the crystallization of supercooled liquids. For this case it is necessary to make several assumptions that do not contradict modern molecular-theoretical conceptions, but nevertheless require experimental proof, namely: 1) the linear rate of growth and melting of crystals (at least organic ones) in capillaries whose diameter exceeds the diameter of the molecules of the substance by only a few times is considerably less than the normal rate; 2) crystallites located in such capillaries in a semi-adsorbed state can be superheated considerably above their melting temperature, since their edges and corners, from which according to Volmer melting begins, are protected by adsorption forces. Therefore the melting of these crystals occurs only at the interface with the liquid phase. Let us note that these assumptions are not the only ones by which the observed facts can be explained on the basis of the “crevice hypothesis.” On the other hand, the possibility of a satisfactory interpretation of the observations of Roginsky and Richards set forth above without the aid of this hypothesis seems to us doubtful.

Let us summarize all that has been said above. If we exclude the cases of crystal formation at enormous supersaturations—for example, in the preparation of colloidal solutions, etc.; in other words, if we restrict ourselves to that region of supersaturation (of solutions) in which crystals of at least microscopic size are obtained, then it may be said with a large measure of certainty that in this region spontaneous bulk crystallization of solutions does not exist—crystals originate only on the surface of solid bodies immersed in the liquid: the walls of the vessel, dust particles, etc. To an even greater degree this apparently applies to the major-

of supercooled liquids. It is precisely this circumstance that explains the overwhelming majority of the complex and confused phenomena observed during crystallization. The mechanism of the very process of crystal nucleation on the surface of solid bodies must remain very little investigated and obscure, and the clarification of this question constitutes one of the most urgent problems of modern physical chemistry.

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