Heterogeneous nucleation of crystallization

4 HETEROGENEOUS NUCLEATION OF C R Y S T A L L I Z A T I O N F. L. BINSBERGEN Koninklijke/Shell-Laboratorium, Amsterdam (Shell Research B.V.) CONTENTS 1. Introduction

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2. Some Definitions

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3. Types, Causes and Inhibition of Nucleation A. B.

C.

D.

E. F.

Spontaneous nucleation Nucleation influenced by additives 1. Heterogeneous nucleation as such 2. Seeding and breeding 3. Diluents 4. Retardation of crystallization by liquid additives Nucleation determined by temperature history and rate of cooling 1. Quenching 2. Reheating from the glassy state 3. Effect of preheating conditions 4. Recrystallization after incomplete melting or dissolution of a polymer Nucleation induced by mechanical action 1. Agitation of a crystallizing solution 2. Ultrasonic or acoustic waves 3. Orientation of a polymer melt or solution Nucleation promoted by ionizing radiation Nucleation under the influence of an electric field

4. Theory of Heterogeneous Nucleation A. B.

C. D.

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General nucleation theory More detailed models 1. Nucleation on a flat substrate 2. Nucleation o n a non-flat substrate 3. Computer simulation Inteffacial free-energy parameters 1. Inteffacial free energy per unit area 2. Molecular treatment Epitaxy 1. Epitaxial nucleation on a fiat plane 2. Epitaxial nucleation at steps 189

G*

192 192 193 193 193 194 194 195 195 195 196 198 199 199 199 200 200 200

201 205 205 207 209 209 209 211 215 215 217

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A. B. C. D. E. F.

Nature of substrate materials Observedactivation free energy of nucleation Measurement of critical degree of supercooling Measurement of nucleation density Electrocrystallization Inteffacial free-energyparameters

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222 224 226 230 230 233

6. Conclusions

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References

1. I N T R O D U C T I O N

CRYSTALLIZATION iS, in most natural bulk systems, initiated by heterogeneous nucleation. Even in many laboratory experiments and in several processes this is largely the case. In practice it is impossible to purify a solution or a melt to such an extent that nucleating impurities are absent and also crystallization from the vapour occurs mostly on substrates. The nature of the nucleating surface remains obscure in most cases. Although general theories have been developed for the description of heterogeneous nucleation, many do not take specific surface morphology into account, c1) Several theories treat nucleation in continuous variables and smooth shapes. However, the relevant nuclei (or embryos, as we will define them) are so small that a treatment in molecular, i.e. discontinuous, variables is required that allows for various configurations of a rather small number of moleculesJ 2) Some advancements in this sense have recently appeared in the literature. ¢3) Many textbooks ~4-7) and review articles ts'9) treat the general nucleation theory extensively, so that we do not feel the need to rewrite all the many relevant equations. The theoretical treatment in this contribution will be devoted to some recent developments in heterogeneous nucleation, c1o) starting with an outline of only a few basic concepts. A vast complexity of crystallization phenomena is known, which occasionally leads to some confusion. It is not always clear whether, and how, heterogeneous nucleation takes place. Other modes of initiation of crystallization or of multiplication of crystals may be active. A phenomenological description of various modes of nucleation and their causes is required to make the necessary clear distinctions. Further, we will pay attention to investigations that have been directed towards unravelling the nature of the nucleating surface by studying both the kind of nucleating substances and the kinetics of nucleation. Although heterogeneous nucleation at a perfectly flat substrate is at present rejected, a full understanding of the principles underlying activity in heterogeneous nucleation has by no means been attained. We will try to give a description of several possible mechanisms remaining and show that a number of these can be ruled out on the basis of kinetic studies, leaving one or a few hypotheses for the acting mechanism. In nucleation theory, the interfacial free energy--of the interface between crystal and parent phase--is a dominating parameter. Actually, it is the main adjustable parameter of the theory. The value of the specific interfacial free energy is mostly calculated from nucleation experiments (occasionally in combination with crystal growth data) via a theoretical equation based on some nucleation model. It is very difficult to check the calculated values by direct measurement, but a few attempts have been made. A good estimate of the surface energy can often be calculated from the heat of vaporization or from interactions in ionic crystals; similar estimates for the interfacial free energy between crystal and melt based on the heat of fusion are less reliable, since in the process of melting, the heat of

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fusion is used not only in the change in molecular interaction, but also in the change in molecular mobility (translation, vibration, rotation). Moreover, the contribution of the interfacial entropy is almost unknown. A number of these problems will be touched upon in this contribution. Rather than being solved, they hamper the present theoretical discussion. This contribution is largely limited to crystallization from the one-component melt, although much of what is said is easily applied to crystallization from solution. Here, we avoid discussion of the influence of adatom population and of diffusion effects. Several experiments described are devoted to the heterogeneous nucleation in viscous polymer melts(t 1,12) where the mobility of the phase to be crystallized is very low and that of the nucleating particles is negligible.

2. SOME DEFINITIONS Crystallization comprises two processes: nucleation and crystal growth. The nucleation is the origination of a very small amount of crystalline material emerging from the parent phase by fluctuation processes in the sense that it grows to a stable crystal. It determines the crystallographic modification of the subsequently growing crystal(13) or aggregate of crystals. The number of crystallization centres determines the fineness of the crystallization pattern~f and a special arrangement of these centres which sometimes occurs influences the texture. The origination of a crystal phase from a parent phase is called primary nucleation(s) as distinct from secondary nucleation, i.e. surface nucleation on a growth plane, which is one of the known mechanisms of crystal growth. Primary crystallization causes the bulk of a system to be converted into solid or rigid material. But when a system, such as a viscous melt or a concentrated solution, has crystallized completely with respect to its volume, the crystallinity may increase further because of secondary crystallization. This may take the form of thickening of needles and platelets in crystal aggregates at the cost of the interspersed liquid, or the form of reorganization within crystallized areas to a higher degree of lattice perfection. It may result from the increase of supercooling or simply proceed isothermally with time. It is always active, e.g. in crystallizing organic polymers. Nucleation may be caused by supercooling or supersaturation only; this type differs from nucleation in the presence of additives or nucleation due to an external influence. Thus we distinguish the following main types:

Heterogeneous nucleation, i.e. nucleation due to the presence of a second phase and occurring at the interface between this second phase and the parent phase.

Homogeneous nucleation, i.e. nucleation in the absence of a second phase. If a second phase is present but does not influence the nucleation, one still speaks of homogeneous nucleation. Spontaneous nueleation, i.e. homogeneous nucleation under no other influence besides supercooling or supersaturation of the parent phase. It is sometimes used as a synonym for homogeneous nucleation, which, however, may also occur through external Except in some dendriticor eutecticcrystallizations.

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influences, such as radiation or orientation of long molecules. In the latter case we speak of: Orientation-induced nucleation, i.e. nucleation due to some degree of alignment of liquid molecules, reducing the difference between the liquid and the crystalline arrangement of the molecules. This type of nucleation is almost specific for organic polymers, where it occurs as a result of shearing the meltt 1,, 1s) or a solution.( ~4, ~7) Both nucleation and crystal growth are dependent on a number of variables and circumstances, such as time, temperature, pressure, mechanical action, pretreatments, concentration, admixtures, structural purity, and so on. In many cases the influence of crystallization conditions on the morphology is found to be the result of changes in nucleation rather than in crystal growth. Frequently a complexity of phenomena is observed. In order that clear distinctions be set, the various modes and causes of nucleation will be described before we go into the use and explanation of experimental observations. In the literature there is considerable confusion about the use of the word nucleus. We will stick to the nomenclature introduced by Volmer. (s) In a supercooled melt or supersaturated phase A, fluctuations occur which lead to the formation of embryos of phase B, aggregates of molecules having the configuration representative of phase B to be formed. As long as they are small, most embryos will decay and only a few will grow out to form a stable phase B. According to the theory there is a (supposed) maximum in free energy of formation of embryos in the course of their growth to a stable size. The embryo corresponding to that maximum is the actual nucleus or critical embryo, configurations of any other shape or size not being called nuclei. Thus, foreign particles inducing heterogeneous nucleation are termed nucleants or nucleating particles rather than nuclei. The actual crystallization nuclei are formed at the surface of such particles.

3. TYPES, CAUSES AND INHIBITION OF NUCLEATION

A. Spontaneous Nucleation Spontaneous nucleation generally requires far deeper super-cooling than crystallization due to other nucleation mechanisms, such as heterogeneous or orientation-induced nucleation. Before degrees of supercooling large enough for spontaneous nucleation are attained the material already crystallizes on heterogeneities or through external action. Consequently, spontaneous nucleation is not normally observed in the crystallization of bulk materials. In practice it is not possible to purify a substance to such an extent that heterogeneities are absent. This is illustrated by the following example: let a nucleation-promoting solid impurity, of a density of 1 g/cm 3, be present in as little as 10 -6 ppm; this amount is sufficient for 104 nucleating particles per gram of crystallizing substance, the particles being cubic in shape and having edges of 10 rim in length. In the study of spontaneous nucleation, there are some ways of avoiding heterogeneous nucleation: 1. Whereas filtration of a solution is generally insufficient, nucleating particles may be "rained out" by stepwise cooling until sufficient supersaturation is obtained for crystallization, allowing the crystallization to proceed isothermally, by which the supersaturation decreases until crystallization stops further cooling, etc.

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2. Heterogeneous nucleation may be overrun by overwhelming spontaneous nucleation as a result of extremely fast supercooling, e.g. in polymers like polychlorotrifluoroethylene(1 s) and polypivalolactone, or by very quickly attaining a high supersaturation ratio, e.g. in the fast mixing of reactants leading to the precipitation of sparingly soluble salts. (7' 19) 3. The most frequently employed method is dispersion of the liquid to be crystallized into tiny droplets, either in an inert suspension medium or as an aerosol. The droplets can often be made so small that only some of them contain foreign particles, the others being free from heterogeneities. Under such circumstances the large degree of supercooling required for spontaneous nucleation can be studied if neither the surface of the droplet nor its tiny size(2°) influence the phenomenon. This technique has been applied to metals,(21-24) alkali halides,(25'26) low-molecular-weight organic substances ( 27- 30) and synthetic polymers.( 31- 34) The experimental results have been used to calculate interfacial free energies between crystal and melt according to classical nucleation theory, assuming certain model shapes for the nuclei. Recent modifications of the theory(3'35) would lead to somewhat different values for the derived interfacial free energies.

B. Nucleation Influenced by Additives 1. Heterogeneous nucleation as such. Any available surface wettable by the phase to be crystallized facilitates nucleation in comparison with spontaneous nucleation since it reduces the interracial free energy part in the free energy of formation of embryos. A considerable reduction is only obtained when the crystalline phase adsorbs on the foreign phase in preference to the liquid phase. Such a preference may be the result of a special fit of the crystalline phase onto the substrate. The free energy of formation of embryos is then considerably reduced by the fact that molecules of the substance to be nucleated easily adsorb on the substrate in a crystalline arrangement. The corresponding adsorption energy is larger because of the shorter distance between adsorbent and adsorbate molecules than would be the case in a random configuration of the adsorbate molecules, reducing the interfacial free energy between crystal and substrate with respect to that between liquid and substrate. If there is a fit in the sense that a densely populated lattice plane of the phase to be crystallized closely fits one of the exposed crystal faces of the nucleating agent because of corresponding crystallographic parameters, one speaks of epitaxy. Other favourable types of fit have been shown to exist, e.g. a pattern of hydrogen bonding groups at one of the faces of nucleants for ice,(36) and parallel ridges in the surface of nucleating agents for linear polymers.(11) Nucleation on a flat substrate is rather improbable since steps, cracks and holes in the substrate offer a larger surface to the growing embryo. The real nature of the substrate surface, however, remains obscure in most cases, with the exception of deposition of metals from the vapour onto well-defined substrates (about which a vast amount of literature is available, see textbooks). The specific nature of nucleating substrates is the subject of further discussion below. 2. Seeding and breeding. If the particulate matter introduced into the crystallizing system are (tiny) crystals of the phase actually to be crystallized,I" one speaks of seeding. t Or isomorphoussubstances formingsolid solutionswith the substance to be crystallized.

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Seeding is not a nucleation phenomenon in a strict sense (see definitions, Section 2) but rather a matter of crystal growth on crystals produced elsewhere. It is, however, used extensively, e.g. in industrial processes to control crystal size and filterability. The number of crystals resulting may be far larger than the number of seed crystals introduced, as a result of tiny fragments breaking off from growing crystals, producing new ones. This phenomenon is termed breeding. Similarly, the number of crystals resulting from heterogeneous nucleation may be far larger than the number of particles of the nucleating agent introduced. Breeding is promoted by fragmentation of inserted or growing crystals, by segregation of polycrystalline aggregates (e.g. spherulites, dendrites) and by collisions of growing crystals with the wall of the confining vessel or with each other, and therefore by stirring, ultrasonic irradiation, etc. Occasionally, breeding is erroneously called secondary nucleation. It has, however, little to do with nucleation, but rather with further crystal growth on crystals already produced. An extensive survey of breeding is given in an excellent monograph by StricklandConstable.~37)

3. Diluents. Crystallization in organic glasses, such as glassy but crystallizable polymers, can be induced by swelling the glass in the vapour of a solvent or in the liquids (plasticizers) themselves, as has been found for cellulose triester, <3s) polyethylene terephthalate, ¢39'4°) polycarbonate t41) and isotactic polystyrene. <42) The glass transition temperature drops through swelling and the mobility of the glassy (polymer) molecules increases, which facilitates molecular reorganization. The minimum temperature required for crystallization drops by an amount specific for the liquid absorbed. ~39) The crystallization actually takes place at a very high degree of supercooling, so that the nucleation may well be of the homogeneous type. This cannot, however, be deduced from the crystallization kinetics, which are mainly determined by the diffusion of the solvent into the glass343) The crystallization centres are located at a short distance from each other and therefore a clear morphology is not found, t44) Probably the degree of supercooling attainable is even larger in a swollen polymer than in the plain polymer, because diluents often reduce the glass transition point, Tg, more than the melting point, Tm. 4. Retardation of crystallization by liquid additives. In low-viscosity systems small amounts of liquid or dissolving additives exert various influences, working by different mechanisms. Increase of the viscosity by gelatinous substances, addition of polar and hydrogen-bonding substances to melts and solutions of polar compounds, and the addition of surface-active and coprecipitating substances in general retard the crystal growth and make a higher degree of supercooling or supersaturation necessary for perceptible nucleation and may even inhibit crystallization on already present seeds. They also change the growth habit of the crystals, for instance, in simple organic compounds, t45-47) in ionic solutions,t 48- 5o) in paraffins t 51) (pour-point depressants) and in metals, t 52) It requires careful investigation to separate the effects of additives on nucleation from those on crystal growth, e.g. in multi-component systems, such as paraffin waxes. Pourpoint depressants do modify crystal growth habits; however, it is barely known whether they promote abundant nucleation (by increasing the supercooling required for an appreciable rate of crystallization) or modify the effect of microwax (i.e. high-melting point wax) seed crystals arisen at higher temperatures. Sears ¢53) indicates two classes of growth poisons for crystallization from solution. Class I poisons, among which large organic molecules are found, are taken up in the growing

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crystal and are active above a concentration of 10 -3 to 10 -4 molar. Class II poisons are specific, adsorbing materials of 10- 6 to 10- 5 molar active minimum concentration and will hardly coprecipitate. The adsorption is mainly at steps and the adsorbed impurity impedes step motion and thereby crystal growth at low supersaturation. However, surface nucleation would be promoted as a result of a reduction in step energy. Other data, however, show that very active poisons in some cases do coprecipitate and that they increase the critical supersaturation for primary nucleation35o) Very probably they inhibit the outgrowth of embryos at low supersaturation even at the surface of nucleating particles. The effects of liquid or dissolving additives as they have been mentioned above have not been observed in highly viscous systems, such as high-molecular-weight synthetic polymers. The polymer melt itself has a high viscosity which can hardly be increased by other substances, except for cross-linking agents which actually modify the polymer itself. Substances that potentially adsorb at growth faces have hardly any influence at the large degrees of supercooling required for the crystallization of polymers. Only large amounts of diluents~54- s 6) and of atactic material~4 s, 57) (i.e. non-crystallizable, but miscible polymer) reduce growth rates and modify growth habits, but still have little influence on nucleation. Addition of other polymers seldom produces any effect on crystallizationt5 s) (except sometimes heterogeneous nucleation), since as a rule two polymers, even chemically related ones, do not dissolve in one another. ~s9'6°) The only retarding effect of small amounts of additives observed in polymers is partial deactivation of nucleating agents (see also Section 5).

C. Nucleation Determined by Temperature History and Rate of Cooling 1. Quenching. By quenching we mean fast cooling of a melt to a large degree of supercooling. Some crystallizable materials will not crystallize at all but solidify to a glass, like glycerol, o-terphenyl, and some slowly crystallizing polymers. In others the number of crystallization centres increases rapidly with increasing degree of supercooling, and a very fine-grained crystallization morphology may result. In only two polymers has spontaneous nucleation been observed upon fast cooling, viz. in polychlorotrifluoroethylene~xs) (Tm=221°C, Tg= 52°C) and in polypivalolactone (T,, = 242°C, Tg = - 15°C). In all other eases we must assume heterogeneous nucleation. Apparently the melts contain nucleating particles of different activity, i.e. few acting at a small degree of supercooling and an increasing number at larger degrees. The reason why crystallization is much finer grained after quenching than with slow cooling is that deeper supercooling is attained before complete solidification. Little is known, however, of the nature of the differences in activity or even of the nucleating substances themselves. 2. Reheating from the glassy state. In glass-forming liquids, such as glassforming silicates, some organic hydrogen-bonding substances and organic polymers, nucleation and crystal growth do not show the same temperature dependence. Nucleation will be at its maximum rate at a lower temperature, which is probably very near to the glass transition temperature, T~, and for some substances even below it (chlorpromazinet6 ~)). Some polymers quenched below Tg without crystallization show abundant nucleation when

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heated to 10°C, or a few times that interval, above Tg. Either nucleation had already taken place during cooling, though without perceptible crystal growth (isotactic polystyrene(62)), or keeping the polymer slightly above Tg for some time was a prerequisite for abundant nucleation (poly-2,2-propylene biphenyl-4,4'carbonate(63)). In both cases further heating is then necessary to make the crystallization centres visible and to complete the crystallization within reasonable times. At large degrees of supercooling nearly all types of solid impurities become active as heterogeneous nucleants, so that the high nucleation density observed may entirely be the result of heterogeneous nucleation. For isotactic polystyrene this has been demonstrated by observing the time dependence of nucleation :(62) at isothermal nucleation in the vicinity of Tg the number of nuclei arisen soon reaches a limit, whereas at spontaneous nucleation the number of nuclei would increase continuously. Remarkably, the precipitation of impurities dissolved in a glass may occur at a far lower temperature than is required for the glass itself to crystallize. Fine-grained structures in glass ceramics have been obtained as a result of a phase separation of a nucleating agent at a temperature far below the temperature at which subsequent crystallization had to be carried out.(64)

3. Effect of preheating conditions. In several substances the crystallization has been shown to be dependent on the time and temperature of preheating above the melting point, while in others no such influence was found. For reasons of clarity we depict the usual crystallization procedure followed in most experiments (see Fig. la). The substance is heated from below the melting point, T=, to a temperature T 1 > T= for some time, and is subsequently cooled either to a temperature of isothermal crystallization T2 or at a certain rate a. Volmer (5) regarded nucleation only as a function of supercooling or supersaturation (Uberschreitung). Frenkel (6) introduced the concept of the heterophase fluctuations, according to which theory there are on one side of a phase-transition temperature (TI) fluctuations, i.e. time-dependent clusters of molecules in a configuration closely similar to that of the stable phase on the other side of the phase-transition temperature (T2). This led to the idea that fast cooling to T 2 would produce a set of embryos which represent the clusters already present at T I.(65~ The lifetime of most clusters would be long relative to • ISOTHERMAL o AMBIENT COOLING T

TIME FIG. la. Normal recrystallization procedure.

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T t = CRYSTALLIZATION TEMPERATURE T

T4 T m _~/-~ZZ,-~--.~_-~-Z2"~-'/..2~ ~__~7-._~__

/

T2

TIME

FIG. lb. Recrystallizationafter incompletemelting. the period of cooling down (if done quickly), because a series of successive fluctuations are necessary to form, or destroy, a stable nucleus. The higher 7"1, the smaller the clusters, and hence the lower T2 required for these clusters to become stable embryos. Only after prolonged keeping at T2 would enough fluctuations have taken place to form new nuclei that were not present as clusters before. Apart from this, Morgan proposed persistence of small crystalline areas above the observed melting point of a polymer, the true melting point being unknown but much higher. (66) The higher the perfection of such an area, the higher the temperature required to melt it away; in consequence, the higher the temperature of the melt and the longer the melt has been kept at that temperature, the smaller the number and size of persistent crystalline areas. Thus, a correspondence between T 1 and the number of nuclei at T2 would be explained. A general type of order in polymer melts has been postulated by Kargin(67) in relation with the clusters or swarms found in mesomorphic phases. The random coil(6°) is supposed to be an exception rather than the rule, while the polymer molecules are arranged in bundles, each keeping its neighbour over the greater part of its length. This could favour the formation of crystallization at appropriate temperatures. The existence of clusters above a phase-transition temperature has actually been demonstrated in some cases, but their relation with a phase separation is questionable. For some organic liquids the presence of clusters, which was deduced from an anomaly in the viscosity-temperature relationship, was related to the glass-forming tendency rather than to crystallization,(68'69) while in molten metals the considerable concentration present above Tin, deduced from viscosity(7 o, 71) and X-ray (72) measurements, bears little relation to the very large degrees of supercooling required for spontaneous nucleation.(2 z- 24) Only near the critical point of simple substances (N2, At) (73) and in the case of liquid-liquid separation(74,75) can the presence of fluctuations, detected by X-ray,(73) light scattering(74,7 s) and viscosity(76) measurements be associated with phase separation. Under these conditions, however, the fluctuations of measurable amplitude occur quite near the critical temperature, but not far above it. Clusters in a melt at temperatures near T m are not necessarily of crystalline order and in many instances non-crystalline clusters are in evidence, which the theory of heterophase fluctuations unfortunately does not take into account. (77) Similarly, the swarms or bundles in Kargin's theory need not be of quasi-crystalline order even if they do exist. In atactic and

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molten poly-~-olefins with long side chains, some type of order has been indicated by small angle X-ray scattering, corresponding with twice the length of the side chain, (7a) but this only means ordered side chains rather than lateral order of polymer main chains. The theories of quenched embryos (65) can hardly apply to crystallization from the melt because of the very large degree of supercooling required for spontaneous nucleation. Moreover, the time of formation and disappearance of embryos are extremely short as compared with the time necessary for quenching. (79) Persistence of small crystalline areas at temperatures far above the melting point seems improbable unless they are enclosed in holes of impuritiesJ s°) Experiments with several simple substances ta°'a~) and polymers (62's2) are in accordance with this assumption. Deactivation of nucleants by prolonged heating has been described with a rather narrow range of deactivation temperatures for each type of nucleant. (a ~) Time-dependent in lieu of instantaneous nucleation, at constant T2, need not be of the spontaneous type since heterogeneous nucleation, too, may lead to a nucleation rate dependent on T2 .(as) In other systems no effect of T 1 on nucleation density could be observed. The crystallization kinetics of polydecamethylene terephthalate, (as) polyethylene(a4) and polypropylene (s 5) were found independent of T 1 over a large range starting one or two degrees above Tm. Effects of/'1 on nucleation occasionally noticed in these and other polymers are mainly due to stresses applied in sample preparation, ( ~4) which relax more or less as a function of T1 ca 6) and it is the amount of orientation-induced nucleation that is actually influenced by the preheating temperature. Apart from the relaxation of orientation, heating at high temperatures ( > 300°C) sometimes diminished the nucleation by degradation of the solid impurities or of the polymer immediately surrounding such particles. (s 7. s s) All this leads up to the conclusion that in unstrained polymers and in low-molecularweight substances cooled down from the melt with T~ a few degrees above Tin, either heterogeneous nucleation takes place or outgrowth of crystalline material enclosed in holes of impurity particles occurs.

4. Recrystallization after incomplete melting or dissolution of a polymer. Crystallized synthetic polymers are called semi-crystalline, because they never crystallize completely. Their crystallinity is reduced as a result of polymer chains taking part in different crystallites which are generally of lamellar form--or folding back into the same lamellae.ta 9,9 o) Polymer chain parts between lamellae or in folds constitute largely the noncrystallized part of the polymer. Both lamellar thickness and lattice perfection (91) decrease at increasing supercooling at which crystallization takes place. Semi-crystalline polymers, therefore, lack a sharp melting point. Their melting range depends on crystallinity, molecular-weight distribution, lamellar thickness and the presence of inclusions, which if at all, are poorly crystallized. The same can be said throughout of the dissolution temperature. Thus we can heat the material to a temperature, T~ (see Fig. lb), at which it seems to have melted (or dissolved) completely, while it still contains many small areas of some crystalline order or at least areas which, over a small distance, have recrystallized to a state that is more strictly ordered and have a Tm higher than T 1. Further heating through one or a few °C would be sufficient to cause also these areas to melt. But if the temperature is not raised further, and if, shortly after T 1 has been reached, the specimen is cooled down again, the material starts to crystallize at the numerous small areas of crystalline order.

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This has been illustrated by Banks et a/. (92'9a~ and by the author (85~ for the recrystallization of polyolefins, but the phenomenon is observed in other polymers as well. The recrystallization pattern may be so fine that the feature of the original situation is (partly) retained or restored. This is, for instance, the case with the recrystallization after incomplete melting of ringed spherulites of some polyesters, where the recrystallization pattern is still reminiscent of the original rings. ~94~ The heat treatment mentioned above might be called seeding, as crystallization starts at the very phase which subsequently grows. The treatment is slightly different from what is called "annealing near the melting point", in which T~ is slightly below Tin, and in which the crystallinity of the polymer is increased by perfection of the crystal lattice and by lamellar thickening. It is difficult to make a clear distinction between annealing and the abundant "nucleation" by incomplete melting. Both processes comprise partial melting and recrystallization,(95-98~ but in the latter the "partial" melting is almost complete. Similar phenomena have been described for the crystallization after incomplete dissolution.t99,~ o0~ The method has been used to produce well-defined single crystals of polyethylene of uniform size.

D. Nucleation Induced by Mechanical Action 1. Agitation of a crystallizing solution. Stirring of a crystal suspension may cause fracturing of growing crystals and thereby increase the ultimate number of crystals by several orders of magnitude. As such it is not a nucleation phenomenon (see Section B.2 on "breeding").

2. Ultrasonic or acoustic w a v e s . In carefully purified solutions irradiation with ultrasonic waves has no influence on nucleation. The rate of crystallization is enhanced in the presence of nucleating impurities only.(72' 101-4~ Therefore, ultrasonic irradiation may promote heterogeneous nucleation. It is not clear whether this is an activation of nucleating impurities or just an introduction of nucleating particles, e.g. from the wall of the crystallization vessel. Long-duration irradiation of distilled water in a glass vessel produced a weak Tyndall effect in the liquid. (1°3~ The well-known induction of crystallization in a glass vessel (test tube) by scratching it on the outside has to be ascribed to the breaking loose of nucleating impurities. Once nucleation has started, the number of crystals is considerably increased by ultrasonic or acoustic irradiation as a result of fragments breaking off from growing crystals,t 104~ i.e. a case of breeding, as has been observed as an avalanche of crystals, as it were, arising from a single point. (~ 9sj Crystallization rates are enhanced and the level of attainable crystallinity is increased, e.g. in slowly crystallizing polymers crystallizing under stress (polyvinylchloride, polyethylene terephthalate) at temperatures in the vicinity of the glass transition point. ( ~0s-6~ Such effects have been ascribed to a forced mobility of polymer chain segments over short distances. Frequently, the morphology of crystallization from a melt is strongly altered by ultrasound, a much finer-grained pattern being the result. At high irradiation intensities the crystallization morphology of polymers is completely destroyed and degradation of the polymer has been observed.

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3. Orientation o f a polymer melt or solution. It is long known that crosslinked, crystallizable elastomers like polyisoprene (natural rubber) and polybutadiene crystallize when strained at temperatures far above their normal melting point, oT'. A thermodynamic explanation is found in the decrease of the entropy of the crosslinked rubber liquid(6°) (decrease of conformational entropy of the polymer chains) and thereby a decrease in the entropy of melting, AS,,, while the heat of melting, AH,, is hardly changed upon straining. This causes an increase in the actual melting point, T" -- AHm/AS'. A molecular explanation is found in the increase in degree of alignment of polymer molecules upon straining, approaching the perfect alignment in the polymer crystal. Non-crosslinked polymer melts(1 o7) and even solutions(17) show a similar phenomenon, be it locally. Polymers of a molecular weight above a critical value show chain entanglements behaving as temporary crosslinks and therefore allow chain alignment upon macroscopic orientation over a certain period of time. When a polymer melt of sufficient molecular weight has been oriented by shear stresses or tensile stresses, in some cases as low as 0.01 MN/m z, nucleation is induced at a much lower degree of supercooling than is required for normal crystallization.(14) The nucleation is found in long, continuous arrays, which we call nucleation lines.(~4) These lines are parallel to the direction of the tensile strain or shear strain. The density of the nucleation lines increases with increasing strain and molecular weight. The nucleation lines are found at temperatures considerably above the normal melting point, and require high temperatures to melt away. ( ~4) Similar phenomena are observed in polymer solutions.(x 7) Orientation-induced nucleation in polymers often obscures the normal or expected crystallization kinetics and it is one of the "memory effects" besides that of a certain kind of temperature history (see Section C.4). Even repeated melting and recrystallization do not remove the nucleation lines, provided the melt temperature was not so high as to relax it. This effect explains an occasionally observed dependence of crystallization kinetics on molecular weight( ~o8-9) and indicates an advantage of following the crystallization by the microscope---where nucleation and crystal growth can be observed separately--over bulk crystallization methods. E. Nucleation Promoted by lonizing Radiation Irradiation of a substance with X-rays or y-rays causes chemical changes, thereby producing impurities and reducing the capability of crystallization. No clear evidence of promotion of nucleation from a liquid phase has been reported. Irradiation of crystallized polyethylene caused cross-linking of the polymer in such a way that many small areas were present where neighbouring polymer chains were kept in a quasi-crystalline configuration even far above the melting point. (11 o) After melting and cooling down an increased "nucleation" resulted, which phenomenon could be repeated at will in subsequent temperature cycles.

F. Nucleation under the Influence o f an Electric FieM Electric fields have a negligible influence on nucleation in general. A theoretical study showed that in case the new phase has a lower dielectric constant than the parent phase, nucleation may be promoted by very high field strengths. ( ~ 1) Electrolysis of an ionic melt or solution causes deposition of material at an electrode.

Heterogeneous nucleation of crystallization

201

Deposition of solids, e.g. metals, proceeds by crystal growth. If the electrode consists of a foreign material the crystallization starts by heterogeneous nucleation. A certain minimum overvoltage is required for nucleation, while further crystal growth requires a lower voltage. A pulse method has been developed by the Bulgarian school: ° 12-13~ a square pulse of time to and voltage amplitude E is applied which induces nucleation; afterwards there is a development voltage sufficient to promote growth of any nuclei formed but inadequate to produce fresh nuclei. This method provides a means of counting nuclei of various cathode materials and at various crystal faces. The substrate (= cathode) appears to have sites of different activity, i.e. requiring different minimum overvoltage for nucleation. In this sense the overvoltage is a variable analogous to supercooling or supersaturation in other systems. The advantages of the electrocrystallization method are found in the speed at which the crystallization conditions are attained (a matter of microseconds), in the exact establishment of short crystallization periods and in the ease with which the thermodynamic driving force, that is the overvoltage, can be varied. 4. THEORY OF HETEROGENEOUS NUCLEATION

A. General Nucleation Theory Starting from a brief introduction into general nucleation theory, we will point out some relevant differences between the various models used in the theory of heterogeneous nucleation. Most current nucleation theories are based on the general theory of Gibbs (4) and Volmer, (5) which we will call the classicaltheory. The classical concept of nucleation is based on the assumption that in a supercooled melt or a supersaturated vapour, phase A fluctuations occur which lead to the formation of embryos of phase B, aggregates of molecules having the configuration representative of phase B to be formed. Most of the embryos will decay and only a few will grow out to form a stable phase B. The probability of an embryo present is, according to Boltzmann's law, a function of the loss of entropy on its formation at constant volume and energy, or otherwise, a function of the (Gibbs) free energy of its formation at constant pressure and temperature: w ,~ exp (AS]k), w ~ exp (-AG/kT).

(1)

Since constant pressure and temperature are the general experimental condition, we will use the latter form. The rate of nucleation is then given as: I = Z exp (-AG*/kT),

(2)

AG* being the largest free energy of formation occurring during outgrowth of an embryo to a stable phase B, and Z being a proportionality constant. The free energy of formation of the embryo, AG, is considered to be the sum of a bulk free energy term proportional to its volume, and an interfacial free energy term proportional to the area of its interface with the surrounding parent phase: AG = - m(#A - PB) -~o'~'~

(3)

in which m is the mole quantity in the embryo, PA and PB are the thermodynamic potentials

202

F . L . BINSBERGEN

of the bulk phases, a is the interfacial free energy per unit area, and f~ is the surface area of the embryo. The difference in thermodynamic potential, #A--#B, is the driving force of the phase transition, while the interfaeial free energy is an impediment. Initially, as long as the embryo is small, the negative driving force term in eqn. (3), which is proportional to the volume of the embryo, is smaller in absolute value than the interfaeial free energy term, which is proportional to the surface area of the embryo. Therefore, initially the free energy of formation of the embryo is positive and later on, when the embryo grows large, the free energy will become negative and the embryo will become a stable crystal (still growing further). Therefore, the free energy of formation of the embryo has a maximum, AG*, and the embryo corresponding to that maximum is called the actual nucleus. In the classical theory, the embryo is assumed to have equilibrium shape, that is, the shape of lowest free energy at a given volume. With this assumption the maximum AG* can be calculated by putting: d(AO)/dm = 0

(4)

and inserting the result into eqn. (3), after which also the size of the nucleus is found. Two alternative descriptions have been used, which lead to similar results: 1. The size of the embryo is regarded as a continuous variable, the difference in thermodynamic potential between the two phases is given as a bulk free energy difference per unit volume: a~, = ~ ^ - ~)117,

(5)

V being the molar volume of the crystal, and an interracial free energy per unit area, a, is used. Several textbooks cs'7) and review articles c9) have treated nucleation in this way, so there is no need to rewrite the relevant derivations and we will only give the results in Table 1 for various nucleus shapes. Where the nucleation on a foreign substrate, f, is concerned, we have to consider several interfacial free energies per unit area: a~z, for the interface between crystal and parent liquid, a,z, for the interface between crystal and substrate, u~:, for the interface between liquid and substrate,

and in ease a vapour is the parent phase a transcription of subscripts is used accordingly. In addition a difference quantity is introduced: 2A~ = or,t - a~.r + cry:

(6)

accounting for the formation of a solid-liquid interface, a solid-substrate interface and the simultaneous removal of the liquid-substrate interface over the same area. In several studies the heterogeneous nucleation is treated according to the model of a spherical cap on a fiat substrate, where the contact angle of wetting of the substrate by the solid, O, is used: cos 0 = (a,.r-a,.r)/u~l.

(7)

Heterogeneous nucleation of crystallization

203

Equations (7) and (6) are interrelated by 1 - c o s 0 = 2Atr/asv

(8)

The activation free energy of nucleation (Table 1) is related to the degree of supercooling or supersaturation via Apo: (9)

AI~ ~ ASo . A T ~ AHo . AT/T=,

where ASo and AHo are the entropy and enthalpy of fusion per unit volume and AT is the degree of supercooling: (10)

AT = Tm-T ,

Tm being the melting point and T the temperature of crystallization. A#~ = ( R T / V ) In s

(11)

in case of crystallization from solution, where s is the supersaturation ratio.

TABLE 1. ACTIVATIONFREE ENERGY OF NUCLEATION AND SIZE OF NUCLEUS ACCORDING TO THE CONTINUOUS BULK TREATMENT

Model

Activation free energy G*

Size of nucleus

Spontaneous nucleation

20.

Sphere

Ap~

16n

0.3

3 (Apv) 2

Cylinder

20. . h* 40" r* ~ X-ray' = -Ap~ -

0.3 82 (A~o)~

Parallelepiped

a* ~ A"~a~ '4°*" b* = ~'~'4ab" C* ---- ~a~40.~

32 (A--~) 20.'ab0.~

217,1

Heterogeneous nucleation on a fiat plane

16n (3o,~ - 2A0.XAo) 2 3

Spherical cap

r * h~,, = Apo

Cylinder

20.,z . h* 4A0. r* = ~ , = Ap'-"~

Parallelepiped

a* ~ ~

Heterogeneous nucleation at a step

a* = 4A0. " b* = 4A0. • c* = 4a.t A'~'~ ' A'p'~ ' Ap~

Parailelepiped

4A0. b* ;

40.b . c*

= A--P~o'

(Apv) :z

0.2A0. 8n'(AP~)2 40.,

= ~

__ 0.b0.~A0.

32 (Apv)2

0..z(A0.)2 32

2. In the alternative description a molecular model has been introduced in the form of the Kossel crystal, ° 24) a cubic crystal with primitive lattice. The lattice energy was considered (x 15) to be the result mainly of nearest-neighbour interactions. This seems to be a good approximation for molecular crystals, the validity of which has been shown by Stranski and

204

F . L . BINSBERGEN

Kaischew by experiments on the development of crystal faces on a globular crystal in vapour of various degrees of supersaturation. (' ,6) In the case of crystallization from the vapour, the energy of attachment or separation of each molecule is given in terms of the nearest-neighbour interaction energy, ~b. Five types of positions of different separation energy can be distinguished (see Fig. 2), the separation energies ranging from q~ to 5~b. By averaging the separation energy for a row or a layer of molecules, Stranski and Kaischew related ~b to edge and surface free energy, respectively.

FIG. 2. Kossel crystal. The molecular treatment can be used in different ways. The numbers of atoms or molecules in the edges of an embryo can be used as if they were continuous variables. Then, the molecular theory is formally similar to the bulk theory, be it that now only crystallographic shapes rather than rounded ones are used. Lacmann ~s) wrote a compilation of the relevant equations and made the comparison between the two treatments. He extended the model to an orthorhomhic crystal with primitive lattice, with different nearest-neighbour interaction energies for the various crystallographic directions, viz. ~bo, ~b and q~. In this picture the interfacial free energies are related to the interaction energies by ~ = 2boco~a;

~b = 2aoCo~b; ~b~ = 2aobo~rc.

(12)

Here, (z~, ~b and ~, are the specific surface energies of the ~, fl and ~ faces of the embryo (see Fig. 3) and a o, bo and Co are the dimensions of a single molecule in the rectangular lattice.

FIG. 3. Three-dimensionalembryo. The free energy of formation of a rectangular embryo with edges of lengths a, b and c (Fig. 3) or of na, nb and n c molecules is then either:

or

A G = 2bc~o + 2aca~ + 2ab~c - abcApv,

(13)

AG = nbnc~a+nancdp~+nonjpc--n~nwcAIZo,

(14)

in which no = a/ao, etc. ; the bulk free energy difference per unit volume between phases

Heterogeneous nucleation of crystallization

205

A and B is A/t o and that per molecule is A/~o, while A/~o = aobocoAl~v. Now, eqns. (13) and (14) are indeed equivalent. The maximum in AG and the dimensions of the nucleus are now found by putting the partial derivatives with respect to the edge lengths zero and inserting the results into eqns. (13) or (14). The bulk parameter treatment allows all kinds of shapes, like a sphere, a spherical cap, a cylinder, etc. But these shapes are rather improbable for crystalline embryos since they correspond to a higher free energy of formation than crystal-shaped embryos do according to Wulff's rule. ° ) Moreover, since nuclei, as calculated from experimental data, often contain a rather small number of molecules, a treatment in the sense of smooth shapes is hardly justified. Therefore, below we will primarily use the molecular model.

B. More Detailed Models The molecular model can be used in a different way and it should be when at least some variables can no longer be regarded as continuous. This is the case when the dimension of the nucleus in at least one crystallographic direction is of the order of one or a few molecules. Therefore we want to compare the results of a continuous variable treatment with a treatment where the nucleus has, at least in one direction, the dimension of one or a low, discrete number of molecules.

1. Nucleation on a fiat substrate. Let us assume that the substrate is fiat and that

FIG. 4. Embryo on substrate. the bc face of the embryo will attach to it (Fig. 4). Then eqn. (14) changes into:

AG = nbncA~) + n,,nc~ob-- nanbdp~Al~o.

(15)

The parameter ~ba in eqn. (14) accounts for two missing nearest neighbours, one on each opposite side of the embryo, or in other words ~ba accounts for two contacts between a molecule of the embryo and the melt. The interfacial difference parameter, A~, accounts for one contact of a molecule in the embryo with the substrate and one with the melt. The relation with the interfacial free energy Ap is [compare with eqn. (12)]: A~b = 2bocoAa.

(16)

The dimensions of the classical nucleus and its free energy of formation are calculated using eqn. (15) and are given in Table 2, region 2.

206

F . L . BINSBERGEN TAnLE2. REGIONSOF HETEROGENEOUSNUCLEATIONON A FLATSUnSTRATE(see ref. 8) Classical nucleus Region of A4 values n~*

I I

size nb*

[ r/c*

free energy of formation AG*

A/~o

I

24c

44o4 4°

[

A/z"=-~

(Ago)2'

24c

4A4464c

J

1

> 4

(spontaneous nucleation preferred)

24. [ A/z0 I

A4< 4: 2. A4 > Ago

I

'A,o [

3. ½Ago < A4 < A/Zo 4. A~, < ½A/zo

24b

A/~o

[

A~'=~

(A~o)~'

transition region between 2 and 4 1

4b [ ~b¢ A/zo--Acp , A/zo-A¢

464c A/zo-A¢

I

4a. Secondary nucleation

A4 = o

1

4~ ~,--o

[

I I

4~ A,o

4.4° A~o

(1

FIG. Sa. Growth of a row of a new monomolecular layer on a crystal face.

Fro. 5b. Start of a new row at a monomolecular layer on a crystal face.

FIG. 5c. Growth of a row on a substrate.

A t increasing degree o f supercooling or, if A~b is already very small, the n* dimension o f the classical nucleus m a y become o f the order o f one molecule or even less. In such a case, a two-dimensional embryo has to be considered having the thickness o f one m o n o m o l e c u l a r layer: AG = nc~b +nb~c--nbnc(Al.lo--A~). (17) The dimensions o f the classical nucleus and its free energy o f formation are n o w found by

Heterogeneous nucleation of crystallization

207

applying (Table 2, region 4): aAG/anb = 0;

(18)

~AG/~n c = O.

If, instead of a foreign phase, the crystal itself is the substrate for the growing embryo, then A~b = 0. This is the case of the secondary nucleation (see Section 2). In Table 2 the various types of classical nuclei on a flat substrate and their free energies of formation are given for various regions of value of AS . The logarithm of the frequency o f heterogeneous nucleation will then--according to the hypothesis of nucleation on a flat substrate--be inversely proportional to the square of the degree of supercooling for low degrees, and to the degree of supercooling minus a constant for large degrees.

2. Nucleation on a non-fiat substrate (a) Nucleation at a step in the substrate. Steps, ridges, cracks, holes, etc., in the substrate offer more advantageous sites for nucleation than flat faces, because they offer a larger part of the surface of the embryo to be formed. Lacmann Ca) has treated the three-dimensional nucleation at a step in the substrate according to an equation like eqn. (15) but with Aq~ substituted also for ~bb. The result is: AG* = 4(Aq~)2~bJ(A#o)2.

(19)

When, however, Aq~ < A/~o the short dimensions of the nucleus become so small that the free energy of starting a new row becomes very important. We will show this for the case o

~b "3

4.

6

t

2

5 ,

Fro. 6. Embryo in step at substrate. q~a = ~b~ < q~c, A~b = A/~o (Fig. 6). According to eqn. (19) the free energy o f formation of the classical nucleus is 4~bc. But the free energy of formation of the embryo can become negative only after at least six rows have been formed, requiring a free energy of at least 6q~, as is shown in Table 3. TABLE 3. FREE ENERGY OF FORMATION OF EMBRYO AT STEP IN SUBSTRATE

Effect of increasing nc on free energy, if A~ = A/to

Row(s) 1 1, 2 1, 2, 3

~bc+n~(2A~b-A~o) 2~b,+nc(3A~-2A$o) 3~b,+n~(4A$-3A$o)

1-4

4 ~bc+ n,(4A ~b- 4A $o)

1-5 1--6

5~b,+n,(5A$- 5A$o) 6,b~+nc(5A~b-6A~bo)

increase from ~c up increase from 25c up increase from 35, up none none decrease from 6~bc down

F . L . BINSBERGEN

208

For higher values of A/to, i.e. for deeper supercooling, the discrepancy may become still larger, as is shown in Table 4. For the situation A~b < ½A/~o the attachment of each subsequent row is subjected to approximately the same activation free energy as the attachment of the first molecule of the first row, i.e. roughly ~bc. In this case, i.e. beyond a certain degree of supercooling, the dependence of the heterogeneous nucleation on the degree of supercooling will be negligible. It is clear that an analytical expression of AG* as a function of A/~o cannot be given for this type of nucleation at a step. A certain--rather laborious--way of plotting AG* may give an insight in the level of A~b if the experimental dependence of nucleation frequency on A/~o and the value of ~b~ are known. An envelope of the straight lines presented by the formulae in Table 3 will show a maximum. For each A/~o such an envelope can be drawn so that a plot containing a family of these envelopes will give an impression of the dependence of AG* on A/Io. This has to be done for a series of values of A~b in order to find the plot that matches the experimental dependence. TABLE 4. ACTIVATIONFREE ENERGY FOR NUCLEATION AT STEP IN SUBSTRATE

AG* according to A/to =

Lacmann

a~ 1.5A¢

44° 1.78~c

] Modelof Fig. 6 [ ]

6~c 3~c-½Aff

I

(b) Step of restricted length. A special case arises when the step has a restricted length, viz. shorter than n* = ~bc/(A#0-A~b) for low degrees of supercooling, but longer than n* for higher degrees. Let the length of the step in the e direction of the embryo be ns. Then the free energy of formation of a rectangular embryo of nb rows is A G = nbq~ c + n~2~o + nl, ns( Aq~ -- Al~o)

= nb~bc(1-nffn*)+n,Adp.

(20)

As long as n c > n~, this free energy will not become negative for any number of rows, rib, and the heterogeneous nucleation is then only possible by protrusions from the rectangular shape at the cost of a higher free energy. Hence, nucleation will be improbable. However, once n c* becomes smaller than n~ as a result of an increasing degree of supercooling (i.e. increasing A/~o) AG becomes negative at a certain value of nb. At further increasing degree of supercooling the activation free energy for nucleation will quickly approach the free energy for the deposition of the first molecule of the first row, roughly ~b~. In such a situation, nucleating particles which are inactive at some low degree of supercooling will seem suddenly to become active at increasing degree of supercooling. Restriction of step length may be the result of kinks in a step or of adsorbed "poison" molecules. A gradually increasing restriction may be the result of misfit in epitaxy (see below).

Heterogeneous nucleation of crystallization

209

(C) Nuclei of discrete number of molecules. Some small nucleus models were described t 35. ~~7) in which nuclei consisted of only a few molecules, in accordance with very large supercooling or supersaturation observed. The free energy of formation of embryos up to a certain small size was calculated. Either the largest value was taken as the activation free energy of nucleation or the nucleation frequency was calculated by considering the separate transition probabilities from one embryo to the next3 t ~7) The latter calculation was done for two-dimensional embryos and is therefore related to heterogeneous nucleation, which inevitably starts with a monomolecular layer if the nucleus is very small. The latter model gave no account of nucleation at steps which at all circumstances is more favourable than nucleation on a fiat substrate. Nevertheless, the model may indeed be valid in cases of extreme supersaturations, e.g. in vapour deposition of certain metals where experimental data point to "nuclei" of one to three molecules. ~2) In such cases the nucleation on the flat parts of the substrate may be more abundant than at steps.t

3. Computer simulation. A closer check of the validity of the various nucleation models than is possible by calculation is made by computer simulation, in which embryos grow and decay via the relevant molecular motions that we think that happen in practical systems. In such simulations there is no need to describe separate embryos nor to call a specific embryo "nucleus" (which may be a not-so-probable configuration) since the computer simulates them at random. Such simulations have been performed for condensation of a vapour on a fiat substrate, ~11 a) for liquid-liquid demixing t 1 ~9) and for nucleation and crystallization both in two- and in three-dimensional matrices, T M but not yet in the sense of heterogeneous nucleation at a step. Simulation results have shown that in current nucleation theory--where nucleation is calculated rather than performed--the simplifying assumptions required for the calculation lead to erroneous equations concerning the relation between nucleation frequency and interfacial free energy, t3) C. Interfacial Free-energy Parameters The driving force in nucleation, A/~ = /~A--#B, is a quantity that is fairly well known in that it is directly related to supercooling [eqn. (9)] or supersaturation [eqn. (11)]. The interfacial free energies, however, are hardly known. It is difficult to determine them, except for ats. Therefore, various methods have been employed to derive interfacial free energies from other physical data.

1. Interfacial free energy per unit area

(a) Solid-liquid interface. Some methods of estimation of as, are based on a determination of properties of small crystals or strongly curved crystal boundaries, such as the solubility of small crystals t 122) which is larger than large crystals according to the GibbsThomson equation. Recently, a method has been described for the estimation of the crystalliquid interfacial tension from the curvature of the interface near a grain boundary in a preestablished temperature gradient. ~12 3) t Compare with the transition in crystal growth mechanism at increasing supersaturation from spiral dislocation growth to two-dimensionalnucleation.

210

F.L. BINSBERGEN

In principle, o-,s can be derived from (droplet) experiments on spontaneous nucleation (see Section 2). The derived value will, however, depend on the nucleus model used (Table 1) and, to some extent, on the pre-exponential factor, Z, and the assumed nucleation rate, L Recently, however, considerable doubt has been cast on the nucleation models leading to TaMe 1, since in computer simulations nuclei do not appear. ° ) Different relations between AG* and A/,o have been proposed (be it for a rather narrow range of Allo[kT values). Other estimates have been based on calculations of attractive forces across interfaces. There are two major approaches: 1. The interfacial free energy between phases A and B o',b, is related to the surface free energies of the phases, o-a and o-b, by: o'ab = o'a + o ' b - - f

(21)

(o-a, o-b),

in which functionf(o.o, o.b) accounts for the attractive forces across the interface. For like substances, the attraction function would be twice the geometric mean of the surface free energies: f(o.°, o'b) = 2(oo-°o'b) :1". (22) This has proved a good approximation in case the attractive forces are London dispersion forces3124) If both dispersion forces and polar forces play a role in at least one of the phases but the attractive forces across the interface are London dispersion forces only, eqns. (21) and (22) are combined to312+) o.ab = O.a Off O,b __ 2(do.a do.b){ = Oo.a Off Oo.b Off(do.a½ --

dO'p)2

(21 a)

in which do" is the dispersion contribution and o. the contribution of other forces to the surface free energy. If more than one kind of attractive force is acting across the interface, the general form of eqn. (21) becomes: a,b = O.++ o'b--f iCto'a, lo'b)--f 2(2o-a, 2o-b)--''',

(23)

where the suffix 1,2. . . . indicates the kind of attractive force, and o.o = ~o.,+ 2o.°+ . . . . For polar and ionic forces the attraction function will depend on orientation effects and on distance of mutual approach of ions and dipoles, which means that simple functions have no general applicability. The quantity o.+is known for many substances, but o., is generally unknown. However, the dispersion part of the surface free energy do., has been determined for several solids from measurements of the critical angle of wetting by liquids. (12 #) Recently, an approach for the estimation of o.s for polar and weakly polar substances was suggested which is based on the use of the parachor, (~2s-6) P: P = Mo.~/(d,-d~), (24) where M is the molar weight and d is the density. Although the validity of the parachor for solids is questionable, the derived relation: o.t:o.~ = d~:d~

(25)

has been tried with some success, and with this knowledge o.,t could be calculated according to eqns. (21) and (22) (see Section 5).

Heterogeneous nucleation of crystallization

211

2. A second approach is based on the relation of surface energy or interfacial energy to crystal lattice energy. If the energy of evaporation, E~, is solely due to the breaking up of molecular contacts, then: Ev = (zc/2)N.e, (26) where zc is the coordination number in the crystal, N is Avogadro's number and e the molecular interaction energy. If z, is the number of molecular contacts broken per surface molecule in cutting a fiat face in the crystal, then the surface energy is given by

as = z~/2o9

(27)

where o9 is the area occupied by one molecule in the surface. Combination of eqns. (26) and (27) leads to a relation between surface energy and energy of evaporation:

z~Eo

a S = ~.~.

(28)

Only in a primitive cubic lattice is co = ~0z. A more general relation has been used in which the molecular area in the interface was put proportional to (V]N)~:

a~ = g(Ev/N~V ~) (29) g being a proportionality factor. A similar relation has been proposed (2 ~) between the crystal-melt interfacial free energy, a~, and the molar heat of fusion, AH,,: au = g(AH,./N~ V~),

(30)

although the physical basis is less clear here (see below). Equations (27)-(30) ignore entropy effects. However, Turnbull (127-s) arrived at a relation similar to eqn. (30) through considering the reduced entropy of liquid molecules at the interface. The anisotropy of the interfacial free energy has been taken into account in polymer crystallization theory. (1 s, 129) By convention, the polymer chain axis is (almost) parallel to the crystallographic c direction. Two interfaciaI free energies between crystal and melt are distinguished: the one related to chain packing, a = aa = ab and the one related to the chain-end or chain-fold faces of the polymer crystal lamellae, ac. Data on spontaneous nucleation in droplet experiments give an estimate of a 2 .a~ and data on crystal growth yield a. ac, assuming rectangular three-dimensional and two-dimensional nuclei, respectively.

(b) Interface with the substrate. In order to determine Ca, the parameter so important in heterogeneous nucleation (Table 1), a,: and as: have to be known besides a~, [eqn. (6)]. Measuring a,: is in principle possible but has seldom been tried. The substratecrystal interfacial free energy, as:, could conveniently be treated according to eqns. (21)-(23). This requires, however, knowledge of as and a:, while moreover a special fit, as in epitaxy, is not accounted for in such a treatment (see below). 2. M o l e c u l a r t r e a t m e n t . In the calculation of the free energy of formation of embryos we used--for reasons mentioned--a molecular model which required a molecular inter-

212

F.L. BINSBERGEN

facial free energy parameter. This points to a molecular treatment of inteffacial forces. We present such a treatment and put it on the basis of a lattice model similar to the Kossel crystal. Later, we will discuss the equivalence of continuum and molecular treatments.

(a) Interfacialfree energy parameter. In the case of crystallization from the vapour the surface energy of a Kossel crystal was regarded to be clue to the absence of one nearest neighbour for each molecule in the surface layer. (s's) As for the interface between a crystal and its melt, we have also to take into account the presence of liquid molecules adjacent to the surface of the crystal. The interfacial energy now is not simply related to the nearest neighbour interaction energy of molecules in the crystal. Yet, we want to keep the existing formulation of eqn. (12): ~a = 2bocoao, etc., in which tro is now an interfacial energy and ~a is a function of interaction energies between liquid and crystal molecules. In order to evaluate ~a we consider the free energy of attachment of a molecule from the melt to the crystal at different positions of Fig. 2. The change in free energy on attachment at position 3 ("Halbkristall-Lage") (5's) is - A # o : A/~o = A H o - T . A S o ,

(31)

in which AHo is the heat of fusion per molecule and ASo the entropy of fusion per molecule. The heat of fusion consists of two parts: AHe, the result of the change in nearestneighbour interactions on transition of molecules from the liquid to the crystalline state and AH~, the result of the change in intramolecular conformation and in translational, rotational and vibrational mobility. We make the simplifying assumption that the molecules of the liquid can be described to be in a similar lattice as that of the crystal, at least if they are adjacent to the crystal. We distinguish the following interaction energies, regarded as the energies needed to separate two neighbour molecules to infinite distance: between two liquid molecules

e,

between two crystal molecules along the a-direction

as=, etc.

between a liquid and a crystal molecule along the a-direction

oets,etc.

The change in interaction energy upon incorporating an adjacent liquid molecule into a row of a growing monomolecular layer (see Fig. 5a or Fig. 2, position 3) is, as stated before, -AHe. Since the added molecule originally was a liquid molecule, three liquid-liquid contacts have been replaced by crystal--crystal contacts. This together results in: - AH~

=

3e. -

~e~-

bess-- ce=.

(32)

Counting in a similar way for the start of a new row (see Fig. 5b, or Fig. 2, position 2) we find: ~b~- AH~ = 4e n - 2cel~- ~e~- b~=. (33) Then 4,c follows from eqns. (32) and (33): ~bc = 8, + ~e~,- 2eels

(34)

and ~b~ and q~b obey similar equations. The fact that we may have assumed too high a density for the liquid and too high a coordination number for the molecules in the liquid can be accounted for by an appropriate decrease of the values of e. and sty.

Heterogeneous nucleation of crystallization

213

Here we see that ~b is not directly related to the heat of fusion. Whereas in case of crystallization from the vapour AHvap,o = 3ess = 3~b when the internal contribution AHwp: is zero, a comparison of eqns. (32) and (34) shows that such a relation does not exist for crystallization from a liquid.

(b) Value of the "interfacial difference parameter, a4,". The parameter A~ accounts for the difference between the energies of deposition of a molecule on the substrate and a molecule deposited on the crystal itself (compare Figs. 5a and 5c). In order to evaluate A~b we express it first in nearest neighbour interaction energies according to the scheme of the previous paragraph. To this end we introduce, in addition to the interaction energies mentioned, the interaction energy: between a liquid molecule and the substrate

4e~:,

between a crystal molecule and the substrate along the a direction of the crystal

oss:.

Then, the growth of a row on the substrate by one molecule (Fig. 5c) results in a change in interaction energy of A ~ p - A H e ~- 3ett-l- e,tf--aSls--a~,s$-- be,ss--cSss.

(35)

Combination with eqn. (32) yields: At~ = e t S - oct,- oe,S+ ~e,~.

(36)

(c) London dispersion interactions only. In the case of crystallizing hydrocarbons we may assume that most interaction energies are due to London dispersion forces. Indicating the dispersion part of the interaction energy between like molecules as a quantity with a single suffix, we state for the liquid e u - e~.

(37a)

The favourable packing of crystal molecules and the fact that they are confined to the force field of their neighbours make the interaction energy between molecules larger than just a London interaction energy determined by a given distance e~ = e~+oe s,

(37b)

in which es is the ordinary London interaction energy and oe~ accounts for the special packing of a molecule between its neighbours in a crystal lattice. A similar reasoning applies to the molecules of the substrate where, moreover, polar interactions may play a role: 8:: = e / + o~:, (37c) e / b e i n g the London interaction energy between substrate molecules. We now apply the well-known approximation for the London interaction energy between unlike molecules according to Hildebrand and Scatchard: ~12 = (/~1~2) ~"

(38)

In the systems under investigation the interaction energies between unlike molecules are of the London dispersion type only, so that for these interactions no other terms arise than stated in eqn. (38). H

214

F . L . BlrqSBB~GEN

Combination of eqns. (36)-(38) inclusive yields: ½ ½ = (=g, -- 5½ l)(=e~--e/)+0s,.

(39)

At first sight, the only quantity on the right-hand side of eqn. (39) directly related to the substrate is 8:, and since for an efficient nucleating agent a low value for Aq~ is required, we expect this only to be the case if e / i s substantially larger than =~=.Another cause for a low A~b value is found, however, when the substrate is capable of accommodating either the lattice of the crystal (epitaxy) or the long-chain molecules of a polymeric substance to be crystallized. When we used es: = (~,g$)~ (40a) in the derivation of eqn. (39), we now replace this by

8=s = (8:s)~ + ~. : , ,

(40b)

being an accommodation coefficient of value 0 < ~ < 1. Now, eqn. (39) turns into A~ = ( : ~ -- ~ ) ( : ~ -- 8)) + (1 -- ;)oSs.

(41)

F r o m this equation we see that Aq~ can have a l o w value as a result o f a large accommodation

coefficient even if ~: is not much larger than ,~,.

(d) Equivalence of molecular and continuum treatments. The interfacial difference parameter A~b and the interaction energies can be related to physical quantities of greater accessibility, viz. the surface energies of the various phases. This is done by multiplying the latter by the molecular cross-sectional area of the crystal molecules along the bc plane of the crystal: Aq~ = 2bocoAa, (16) =e, = 2boc0°a,, etc.,

(42)

so that eqn. (41) is converted into Aa = (,a~ - a~)(=aJ- a}) + (I - ~)oa,.

(43)

In this equation, each a is the London dispersion contribution to the surface energy of the phase in question, except 0a,, which has a similar meaning as oe,. Although eqn. (42) is a crude approximation for the relation between nearest-neighbour interaction and surface energy, a fairly accurate calculation of this relation by de Boer and Hamaker (~ 3o) led to a similar result. Equation (43) follows also from eqn. (6) with the use of eqns. (21) and (22) when the accommodation coefficient, ~, is taken into account.

(e) Other types o f interaction. When strongly polar interactions play a role (ionic, dipolar, hydrogen bonds) it is not so easy to derive a relation for the interfacial difference parameter. The interactions depend on the distance of mutual approach, on asymmetry of arrays of ions or dipoles on either side of the interface, on degree of orientation of dipoles and on arrangement in required patterns. Moreover, other than nearestneighbour interactions must be taken into account.

Heterogeneous nucleation of crystallization

215

In the case when the substrate is polar but the crystallizing substance is not, eqn. (41) still applies. We see that in such a case mainly the degree of accommodation of the crystal lattice onto the exposed face of the substrate plays a role.

(f) Orientational effects. Specific orientation of the crystal lattice of the growing embryo with respect to the substrate is a major factor in reducing A~b,i.e. in increasing the accommodation coefficient. There are various causes for such orientation: 1. Epitaxy, i.e. the lattice of the embryo almost matches to that of the substrate in two dimensions. The accommodation coefficient is mainly determined by the degree of mismatch or disregistry and varies from one molecule to the next. 2. Adsorption of elongated molecules in ridges of the substrate surface. 3. Polar interactions between embryo molecules and the substrate which are maximized in specific orientations of the embryo molecules. Examples are matching patterns of hydrogen bonding in ice and the nucleating substrate(36) and minimum potential configurations of dihalobenzenes on alkali halides. (1 a t)

D. Epitaxy In the description of epitaxial nucleation, spatial requirements have to be taken into account in addition to the energetics of the local special fit, since the lattice mismatch determines over how large an area the fit will still be reasonable. Let the substrate lattice parameter be yao, then the disregistry is defined as t5 = 1 la°-j'a°l

(44)

fao This is illustrated in Fig. 7. If molecule number n fits exactly onto the substrate, then molecule number n + 2/6 is just at the worst adsorbing position.

FIG. 7. Epitaxial deposit of crystalline monolayer on substrate with slight misfit.

1. Epitaxial nucleation on a fiat plane. For epitaxial nucleation on a flat substrate at low degrees of supercooling a low disregistry, c5, is required. The substrate will become active from a certain critical degree of supercooling, ATe, onwards. Theories of oriented overgrowtht 132-3) have considered the interfacial energy related to the interface between substrate and crystal over semi-infinite areas. Local regions of relatively good fit are separated by a grid of dislocations. The strain of either phase or of both is taken into account and the energy per dislocation is calculated. A spherical-cap nucleus model then leads for low values of the disregistry to a relation for the critical degree of supercooling according to Turnbull and Vonnegut :(1 a 2) ATc = (Tm/AHo)c62,

(45)

where c is an elastic constant to be determined. However, in view of the low degrees of supercooling generally observed in epitaxial nucleation, the dimensions of the nuclei in directions parallel to the substrate are relatively

216

F.L. BINSBERGEN

large. Now, in order that the free energy of formation remains within acceptable values (see Tables 1 and 2) the thickness of the nuclei must be very small, of the order of one molecule (see calculation below). As a result, we can regard epitaxial nucleation in general as a kind of two-dimensional nucleation. This concept infers that epitaxial nuclei are not larger than ao/6 in linear dimension and probably quite a bit smaller. Layers larger in size than ao/6 are cut in separate parts as a result of dislocations developed and cannot be considered to be coherent embryos. Probably the crystal growth proceeds initially by deposition of subsequent monomolecular layers on top of the first rather than by immediate expansion of the first layer beyond the size ao/6. However, the deposition of the first layer is the actual nucleation phenomenon, since nucleation of subsequent layers requires a lower activation free energy. We consider here the epitaxial nucleation of a Kossel type monomolecular layer on a similar substrate. The free energy of formation of a square embryo, of an edge n molecules long, is [see eqn. (17)]: AG = 2n~b-n2(A/to-Aq~). (46) As we did in eqn. (39), we split the interfacial difference parameter, AS, into two parts: A~b -- A~b'-A~b s

(47)

of which A~s is the part dependent on the fit of crystallized molecules onto the substrate. According to eqn. (41): AqUae= (1 -- ~)o8~ (48) in which [ is the accommodation coefficient, which varies, on an average, between 0 and 1. Locally it may be lower than zero. A single molecule deposited on the substrate is assumed to have the best possible fit: A~bs- = 0, ~ = 1. The next adjacent molecule will not fit exactly but will be slightly outside the position of lowest energy on the substrate. Let ~ be a sinusoidal function of position on the substrate--as has been suggested before by others (134) such that ~ = 0: --- ½ cos 2ny/.rao + ½ cos 2~zZ/.rao

(49)

where y and z are the crystallographic directions in the plane of the substrate. The misfit may partly be compensated by the development of a strain in the layer embryo. The strain is then the result of a derivation from equilibrium distance of adjacent molecules as determined by, for example, a Lennard-Jones potential well. Since such a potential leads to forces per unit strain about two orders of magnitude larger~f than a potential as given by eqns. (48) and (49) does, the strain energy will be small as compared with the misfit energy as long as n < ½3. (Of two springs in series the weaker one takes up the larger part of the deformation energy !) The free energy of formation of an embryo is then: AG = 2n~b- nZ(A/z0 -- A~b')+ o8,.f (n,6),

(50)

in which (n-- 1)/2

f(n, 3) -- 2n ~ (1--cos 2zrrnr) for n = odd,

(51a)

m=l

hi2

f(n, 3) = 2n ~ {1 - c o s 2rc(rn-½)6} for n = even.

(51b)

m=l

t For crystallization from melt or solution; in crystallizationfrom the vapour the differenceis about one order of magnitude.

Heterogeneous nucleation of crystallization

217

A good approximation o f f ( n , 6) in the range 2 < n < 2/36 is:

f(n, 6) = 1.662n 4, so that eqn. (50) now reads AG = 2n~b-n2(A#o-A~b')+ 1.6oSs62n4.

(52) (53)

For low values of A/~o, i.e. for low degrees of supercooling, this is a monotonically rising function of n. Epitaxial nucleation requires a maximum in AG, i.e. as a first condition: aAG/On <=O. At the moment when this condition is reached the function shows, at the position where this condition applies, an inflexion point, which gives a second condition: aZAG/On 2 = 0. Substitution of eqn. (53) in both conditions gives equations which when solved for n lead to: n* = 0.5386~/o8,÷6~, (54) AUo = 2.785~bto~6~+A~b '.

(55)

This is a necessary but not a sufficient condition for epitaxial nucleation, since further lateral growth of the embryo would lead to an increase in AG, which means that the embryo is unstable at all sizes and is therefore likely to disappear again. A stable embryo requires AG =< 0, which condition is reached for: Ago > 3.52oe~6~b~ + A~b'.

(56)

However, before a stable embryo has developed, a second monomolecular layer has been initiated and therefore the critical value of A#o is between those of eqns. (55) and (56). With eqn. (9), the value of the critical degree of supercooling is found between 2.785q+Aq~' < (ATc/Tm)AHo < 3.52qq-A~b',

(57)

in which q = oe~6~ck~. Experimentally observed values of ATc < 20°C for organic substances and ice nucleated on various substrates would indicate that the disregistry is only one or a few percent, as is illustrated by the following example. With the assumptions AHo/T,, = 0.01qb--which applies within a factor of 3 for many organic substances and ice--A~' = 0 and oe~ = 0.2~b we find 12° < AT < 15°C for a disregistry 6 = 0.02.

2. Epitaxial nucleation at steps. Experimental values of ATe are frequently much lower than calculated ones. Moreover, observations indicate specific, active sites on nucleating surfaces (see Section 5) which are supposedly steps, cracks, etc., offering a larger surface to growing embryo than a fiat plane does. Epitaxy may still be important, but embryos now tend to be elongated along the ridge in which they form (see Fig. 6) and probably the disregistry along this direction only plays a role. The extension of the embryo in the perpendicular direction is small and the disregistry in that direction has therefore little energetic consequence. In fact, this is a case of nucleation at a step of restricted length [see eqn. (20)], the restriction itself depending on supercooling. For the build-up of embryos along a step, the free energy of formation can be calculated per row according to a procedure similar to that used in the previous pages, be it that disregistry along one direction only is taken into account. However, the game of counting how many rows are involved in the formation of a critical embryty--as done before, which led to Tables 3 and 4mbecomes very casual because this depends on at least four variables,

218

F.L. BINSBERGEN

oe,, t~, ~, and Apo. An analytical relation between nucleation probability and degree of supercooling cannot be given. Moreover, other conditions play an important role as well, such as the perfection of the step itself, epitaxial fit either at both intersecting planes forming the step or at only one. Hollow ranges, cracks of any shape, holes may be even more important. Little is known in detail about all this. Certainly, the low abundance of active sites observed and their repeatability in successive heating-cooling cycles require other explanations than the plain availability of steps (see below).

5. EXPERIMENTAL OBSERVATIONS AND MEASUREMENTS

A. Nature of Substrate Materials Substrate materials or nucleating agents are selected according to the nucleation density they produce either (a) on a large, almost flat substrate face or (b) by a fine dispersion of the nucleator in a melt or solution of the substance to be crystallized. Method (a) allows a good comparison either between substrates or between various substances to be crystallized on a single substrate. However, a blank experiment, i.e. crystallization without a substrate, is generally absent. In method (b) the comparison with the blank is readily made. But, what is the blank? Frequently, the effect of a purposely introduced nucleating agent in a melt (or solution) is compared with the crystallization of that melt as such, i.e. on adventitious impurities of unknown source. Little is known about the nature of naturally present impurities on which the crystallization in bulk systems starts. Frequently, the number of active particles increases with increasing degree of supercooling, which suggests that materials of different critical degree of supercooling, ATc, become successively active. (s a) Therefore, the cooling rate is important in the comparison between a nucleated melt and the blank, the clearest differences being observed in isothermal crystallizations. A clear distinction has to be made between solid materials that promote persistence of crystalline material above the normal melting point and others whose nucleating activity is not related to the temperature history above Tm prior to crystallization. Materials that promote persistence of the crystalline phase above its melting point in cracks or inclusions are frequently themselves poor nucleating agents as has been shown for powdered quartz and activated black in phenylsalicylate(s 1) and for impurities of unknown type in antimony,(s o) polyoxyethylene(s2) and isotactic polystyrene.(62) Therefore, we will distinguish as real nucleating agents those substances that promote nucleation independently of the previous thermal history of the melt in which they are dispersed. Many examples of oriented overgrowth have been reported: metals on metal substrates, on oxides and on inorganic salts, (1 a 5) paraffins and polyethylene on NaCl, (1 a6-7) organic substances on oriented polymer films,(1 a 7) organic substances on alkali halides, (1 a 1) alkali and ammonium halides on various types of mica, (la a-o) polyacrylonitrile on quartz ( ~4o) and various polymers on alkali halides. (141) Epitaxy requires a two-dimensional fit of the lattice of the crystallizing substance onto that of the substrate. Frequently, a fit in only one dimension appears already to be sufficient for oriented overgrowth, such as polyethylene on NaCl. (la6-7) In other cases, oriented overgrowths are found without a clear indication of any fit. Apparently the surface of the

Heterogeneous nucleation of crystallization

219

substrate exerts some directional effect on the adsorbed molecules for reasons of packing (polyolefins, see below), of hydrogen bonding patterns (ice, see below) or of minimizing potential energy of polar interactions. (1 a 1) It is not surprising, therefore, that frequently nucleating activity--expressed as ATe--is unrelated to lattice misfit. (142-a) And quite often a nucleating effect has been ascribed to epitaxy on grounds of similarity of lattice parameters but without demonstration of the occurrence of oriented overgrowth. (142,144-5) On the other hand, when the crystallization is forced to occur on the substrate, e.g. in the case of deposition from the vapour or from dilute solution, minor energetic effects will promote unidirectional over randomly oriented crystallization. On some substrates the metastable crystal modification crystallizes where otherwise the stable modification grows, e.g. the fl-modification of polypropylene on y-quinacridone (146) and on the aluminium salt of 6-quinazarin sulphonic acid (as) and the fl-modification of polypivalolactone on various minerals having a common surface geometry, such as muscovite, diaspore, microasbestos, and clay minerals of the attapulgite and kaolin types. (94) Many nucleating agents for the crystallization of ice have been discovered, mostly in the widespread search for substances that promote precipitation from rain clouds. A long-known epitaxial nucleating agent is AgI. Several organic substances capable of hydrogen bonding are good nucleating agents in evident relation to the pattern of hydrogen bonding groups in exposed crystal planes. (a6'147) Probably some hydrated silicas are active according to a similar mechanism. Some inorganic salts composed of polarizable ions are said to derive their ice-nucleating ability from a dipole orienting action of surface ions aiding in ordering the water molecules. ( ~4 a) Since the crystallization of silicate glasses is generally slow, the production of glass ceramics requires nucleating agents. Colloidal dispersion of heavy metals, such as silver and gold, is achieved by the (photo) chemical decomposition of metal salts dissolved in the glass. Other nucleating agents are titanates, of which a fine dispersion is obtained upon precipitation at temperatures far below the temperature at which the glass is crystallized. (t 49) Several searches for effective nucleating agents have been carried out in the field of crystallizing polymers. The size of the spherulites--into which polymers usually crystallize-is reduced by such agents resulting in better mechanical and optical properties of moulded objects; moulding times are reduced as a result of faster solidification; and in some cases a substantially higher level of crystallinity is obtained than without nucleating agents (e.g. in polyethylene terephthalate mouldings). In polymers, catalyst remnants have been said to be the typical, naturally occurring, nucleating impurity. However, metal salts, oxides and hydroxides as remain after workingup of the polymer in the plant hardly show any nucleating effect when purposely added. Especially the non-polar polyolefins and isotactic polystyrene are insensitive to inorganic substances (contrary to what has been claimed in many patents (t s o-1)) except to very-highenergy surfaces, e.g. of heavy metals, such as silver and gold (x 1) (in very fine dispersion). Several alkali halogenides have been reported to act as epitaxial nucleating agents for polyethylene crystallizing from dilute solutions. ( ~a 6-7,14 z) In bulk polyethylene, however, they have only a weak effect, the degree of supercooling at which they become active ~z s 2) being larger than that for the impurities naturally present in polyethylene, or at least at the same level. For polypropylene, two inorganic nucleating agents have been found active in promoting crystallization from the melt, i.e. ZnCI2 and As203(a°); they are probably epitaxial ones because of close similarity of crystallographic parameters.

220

F . L . BINSBERGEN

Various inorganic materials have been found active nucleators for polar polymers, e.g. quartz, aluminium hydroxide, graphite, carbon black titanium dioxide, alkali halides and Pb3 (PO4)2 for nylons, c153-6) divalent metal sulphates and carbon black for polyethylene terephthalate t157) and alkalihalogenides for polypivalolactone, whereas certain mineral muscovites promote the //-modification of the latter polymer, probably in an epitaxial way.(94) Higher-melting semi-crystalline polymers can act as nucleating agents, e.g. nylon-66 and polyethylene terephthalate for nylon-6 and polypropylene for polyethylene as long as the dispersed nucleating polymer is crystalline. When the higher-melting polymer remains molten in the temperature region of crystallization of the lower-melting polymer, a nucleating effect is absent (e.g. polypropylene, Tm ~ 170°C, dispersed in high-density polyethylene, T, ~-. 132°C). After both components have crystallized and subsequently have been heated to a temperature between the melting points of the two polymers and cooled again, the nucleating effect is observed. ~158) Efficient nucleators for polyolefins and isotactic polystyrene have been found in many classes of metal salts of organic carboxylic, sulphonic and phosphonic acids c1a, 159) and of organic pigments, ca1' 146,160-2) A fair number of nucleating agents for polyolefins were found active in polypivalolactone, ca63) too, while salts of aromatic phosphinie acids were claimed as nucleating agents for nylon-6 c164) and some epoxidized carboxylates for polyethylene terephthalate.C a 65) However, for a good effect, the nucleating agents must be well dispersed in the polymers, which is often a major problem. Particle sizes far below 1 gm are required in most polymers in order to raise the nucleation density sufficiently over that of the base polymers. Frequently, several members of a class of otherwise good nucleators failed to show a good nucleating effect, probably for one of the following reasons. Since the incorporation into the polymer comprises a melt processing step of temperatures generally above 200°C, some substances may suffer a polymorphic transition, loss of water of crystallization, (superficial) decomposition, reagglomeration, or adsorption of additives (stabilizers) dissolved in the polymer or may chemically attack the polymer and become covered with degraded polymer. Especially stearates, used as neutralizers for the acidic catalyst remnants in polyolefins (and the resulting stearic acid), have a detrimental effect on the nucleating activity of most organic metal salts. Stearates and metal salts of other acids carrying a long aliphatic chain dissolve in the molten polyolefins and are not nucleating agents. They solubilize other nucleating salts by the formation of mixed salts and cause disappearance of the nucleating effect.~11) A few organic nucleating agents were found to dissolve in the molten polyolefin but to reprecipitate upon cooling before the polymer recrystallized. Among these were the sodium salts of cyclopentane carboxylic acid and of cyclopentyl acetic acid and a few non-salt type substances: the salicylhydrazones of benzaldehyde and salicyl aldehyde, cl 1) In so far as their physical state is concerned, it is almost certain that all nucleating agents act as such in their crystalline state. As regards the nucleating effect claimed for liquid droplets in polymer melts, ca66) we have to make a clear distinction between the overwhelming nucleation caused by a finely dispersed material and relatively large droplets acting as crystallization centres without bringing about fine-grained crystallization. The droplets might easily contain enough solid impurities for the low level ca 66) of nucleation observed. Various additives that dissolve in the melt or solution and remain dissolved may promote nucleation by adsorption at or reaction with particulate material. Carboxylic acids, such as

Heterogeneous nucleation of crystallization

221

p-tert, butyl benzoic acid, react with catalyst remnants (probably only superficially) in polyolefins to yield nucleating agents. (11) The nucleating activity of copper dust for the precipitation of barium sulfate is strongly enhanced by addition of isopropylalcohol to one of the reagents. (1 o 3) And the nucleating ability of alkali halogenides in nylon-6 is enhanced by fatty amines or calcium and zinc soaps. (t55) Nucleating effects of impure substances have therefore to be regarded with some suspicion. The actual effect may be due to the impurity in the additive, e.g. when large amounts of cheap filler material are applied in crystallizing polymers. Otherwise, adsorbing additives sometimes spoil the nucleating effect by altering the surface characteristics of the nucleating agent. The nucleating effect of quartz powder in nylon is reduced by rendering the powder hydrophobic by cationic detergents. (167) The nucleating agents for polyolefins (and for other polymers) probably have a common surface characteristic related to their nucleating activity. Epitaxy as a mechanism for the nucleating effect can be ruled out, since the nucleating agents are active in all polyolefins tested and occur in homologous series of organic acids suggesting gradually increasing lattice dimensions. A number of crystal structure determinations have been reported for metal salts of organic acids of chemical structures resembling those of nucleating agents. The crystals are built up of polar (p) and non-polar (n) layers. The polar layers contain the carboxylate, sulphonate, phosphonate, etc., groups and the non-polar ones the hydrocarbon groups of the substance. Sandwich structures (n-p-n-n-p--n) are found when the cross-section of the hydrocarbon group is equal to, or larger than, that of the polar group, e.g. in potassium hydrogen phenylacetate (~6a) [(C6HsCH2COO)2KH], in ammonium hydrogen phthalate, (169) in potassium hydrogen benzoate, (17°) in potassium hydrogen p-chlorobenzoate (17~) [(C1C6H4COO)2KH], in calcium ~-naphthyl phosphate trihydrate (172) [Ca(C1 oHTHP04)2 • 3H20], and in dipotassium phenylphosphate sesquihydrate (17 a) [K203POC6Hs. 1½H20]" Alternating layer structures (n-p-n-p-n-p) are found in strongly hydrated substances or hydrated substances carrying other polar groups besides the acid group, such as hydrated potassium hydrogen salicylate (174) [(HOC6H4COO)2--KH-H20], hydrated magnesium benzenesulphonate (175) [(C6HsSOa)2Mg" 6H20], and potassium p-aminobenzoate trihydrate (176) [NH2C6HsCOOK" 3H20]. Here, the expansion of the polar layer by the water molecules is compensated for by the hydrocarbon groups of adjacent layers meshing together like the teeth of gears. Probably, however, little water of hydration will be left after a salt has been heated in the polymer melt, so that alternating layer structures may be converted into sandwich structures by the heat inevitably produced on the incorporation of the salt into the polymer. Condensed aromatic compounds, such as flavanthrone, ~177) violanthrone, (17s) and copper phthalocyanine~179)--all good nucleating agents--show a regular packing of parallel stacks of the flat ring systems. The evidence presented on nucleating agents for polyolefins suggests that they have the following features in common: 1. The crystals of these nucleating agents expose faces of hydrocarbon groups. The hydrocarbon groups, often bulky, are arranged in rows, leaving shallow ditches in the exposed faces. 2. Although the hydrocarbon groups resemble moderate to good solvents for polyH*

222

F . L . BINSBERGEN olefins--i.e, aromatic, chlorinated aromatic, cycloaliphatic and branched-chail aliphatic hydrocarbons--the nucleating agents are, nevertheless, insoluble in th~ polymer melt owing to a polar group attached to the hydrocarbon group or to .' condensed aromatic structure.

The nucleating effect is probably linked up with the occurrence of parallel ditches in thq substrate surface, forcing adsorbed polymer molecules to assume a stretched conformatiol over some distance, which makes crystallization much easier. This picture is in agreemen with the observed orientation in the crystalline texture of a polymer crystallized on orientet crystal needles of a nucleating agent. (1 ~, 16 ~) In this way the accommodation coefficient, may approach the value of one [see eqn. (34)], while the factor (a,s*- es ~) in eqn. (34) is ver'. small as a result of the interaction between like materials. Since steps offer a larger surface to the growing embryo than a flat crystal face, even if i contains shallow ditches, we must assume that steps and other distortions of the faces of th, substrate are the actual nucleating sites. This has frequently been demonstrated in the so called decoration techniques, whereby metal is deposited from the vapour onto a substrat~ to an incomplete (i.e. low degree of) coverage. The deliberate introduction of cracks ant lattice defects in silver iodide by photochemical decomposition considerably enhanced it nucleating ability for the crystallization of ice. (x s o) The critical degree of supercooling wa decreased from 4°C to 0.4°C.

B. Observed Activation Free Energy of Nucleation In the theoretical section we have considered the activation free energy for nucleation The ratio of the pre-exponential factor, Z in eqn. (2), to the observed nucleati01 frequency, I, will tell us what level of nucleation free energy barrier is encountered il practice, i.e. AG*/kT = In Z/L In most experiments on heterogeneous nucleation, the nucleation frequency, I, is no determined as such, but a supercooling is determined at which I becomes appreciable, i.¢ nucleation is observed. That supercooling is then called the critical supercooling fo nucleation, ATc. Depending on the cooling rate, I will be of the order of 0.1-1s- 1 in mos experiments, the range of values which relates to the small part of the crystallizing systen that is usually observed. The pre-exponential factor Z is composed of a factor related to the molecular mobilit) expressed as a jump frequency, j, and a factor indicating the number of molecules availabl for nucleation in the system considered Arm. Turnbull and Fisher tl 81) suggested the use of the jump frequency description accordin to the theory of absolute reaction rates, ° a2) AG*.

i - - - -kTF** T - 7 exp

(~k-T) -,

(58

where h is Planck's constant, F is the partition function of the liquid molecule and F** th same for the molecule in its activated jump state, and e~ is the activation energy for viscou flow or rotational diffusion. Estimates of F**/Ffor low-molecular-weight liquids Oa 2) are in the region of 0.05. Th value will be a little lower for polymers. The activation free energy for viscous flow range

Heterogeneous nucleation of crystallization

223

from 8 kJ/mol for some low-molecular-weight organic liquidst to 50 kJ/mol for some crystallizing polymers. We will use here about average values of 11.5 kJ/mol for lowmolecular-weight substances and 30 k J/tool for polymers. Deviations from the exact values o f / , F**/Fand 8~ have a minor effect on the value of AG*[kT, except in the case of polymers. The jump frequency has then, at temperatures of the order of 350°K, a value of j = 109.5 s - 1 for low viscosity systems and o f j = 106s - 1 for polymers. The number of molecules available for nucleation, Nm, is determined by the experimental system and by the mode of nucleation. For spontaneous nucleation N , = oN/V, where v is the volume considered for a nucleation event, N is Avogadro's number and V is the molar volume, which we take to be, on an average, 50 cm 3. For heterogeneous nucleation on a large, flat substrate Nm ~ ( N / V ) ~ per unit area, which is approximately 5 x 1014 cm -e. Frequently, experiments are performed in a hot-stage microscope where the area of view is about 10-4 cm z, resulting in Nm = 5 x 101 o. In case of a finely dispersed nucleating agent, however, the area per particle may be only 1/am z, reducing Nm to 5 x 106. In case of nucleation at steps one has to assume a step density or interstep distance. This may vary widely among substrates. If we assume an interstep distance of 20 nm, the average total step length per cm z is 5 x 105 cm. With a molecular diameter of 0.5 nm the relevant number of molecules in contact with steps is N m = 1012 cm-2 In this calculation the number of active sites is 10 ~z per cm 2 or 104 per small nucleating particle. Experimental observations, however, indicate a much lower number of sites of probably high activity. Presumably a nucleating particle may have just one of high activity, the level of activity being different among particles. This means that Arm = 1 per particle. The values of A G * / k T observed in various modes of nucleation and in various systems are listed in Table 5. In experimental circumstances, other than assumed above, the corresponding values of Nm and e~ can generally be estimated easily, after which the value of A G * / k T according to Table 5 is adapted accordingly.

TABLE 5. OBSERVED AC'I'IVATIONF a r 2 ENERGY OF NUCLEATION, AG*, IN VARIOUS TYPESOF EXPERIMENTS

AG*/k T observed

Mode of nucleation

Size of system for which I = 0.Is-1

Spontaneous nucleation Melt, in droplets Heterogeneous nucleation on a flat plane Large substrate, on microscope hot stage Nucleating agent Nucleation at steps Large substrate, on microscope hot stage Nucleating agent Single site of high activity Nucleating agent

t For liquid metals even much lower.

I

Low viscosity systems I Viscous polymers

I0/tm 3

50

42

10- 4 cm2 1/tin z

49 40

41 31.5

10- 4 cmz 1 gm 2

43 33.5

34.5 25

1 particle

24

16

224

F . L . BINSBERGEN

C. Measurement o f Critical Degree o f Supercooling At very large degrees of supercooling or supersaturation nuclei are expected to be very small (see Section 4), i.e. of the order of a few molecules. It is, therefore, not surprising that in deposition of metals from the vapour at relatively low temperatures, nuclei were coneluded to be smaller than 1 nm (*s3) and in some cases to consist of 1 to 3 atoms. (2) On the other hand, multilayers of adatoms on field emission microscope tips appeared to be required for the nucleation of some vapour-deposited metals at low temperatures. Surprisingly the critical adatom population for nucleation appeared to be independent of tip temperature. (1 s4) Here, some problems remain to be solved. A critical degree of supercooling or supersaturation has been determined in various systems where epitaxial nucleation is presupposed. Alkali halides have been crystallized from aqueous solution on freshly cleaved mica (139) and ammonium iodide on various types of mica showing slightly different lattice parameters. (13s) The data did not fit the dislocation theory of epitaxial nucleation, (~ 3s) while a test of the monolayer theory (Section 4) is almost impossible since the values of as, or ~bfor the relevant crystal faces in solution are lacking. Some disagreement on the critical supersaturation has been ascribed to the experimental problem of exactly assessing the temperature of the substrate, a poor heat conductor. (x3s) The critical degree of supersaturation depends strongly on saturation temperature, even for eases of large disregistry, and would surprisingly disappear completely upon extrapolation to temperatures of 50°C and higher. A study has been made of the oriented crystallization ofp-dichloro, p-chlorobromo and p-dibromo benzene on various alkali halides. (~ a i,~ s s) Whereas little relation was found between lattice disregistry and critical degree of supercooling, ATe, a fair correlation was found between observed ATe and values predicted by means of the cylindrical heterogeneous nucleus model (Table 1). The Aa value was related to the calculated energy of oriented adsorption of the crystalline phase onto the substrate, Usf: U,,~ = 2(a,,-Aa),

(59)

while the a,~ values were determined from spontaneous nucleation in droplet experiments. Unfortunately, U~s was taken to be solely the summation of the energies of interaction across the crystal-substrate interface, in the sense of (in our nomenclature): U~f = e,.r/2boco where it should be"

2bocoU,.r = dp-Aq~ = eu-Stf-e~l+~s

(60)

according to combination of eqns. (34), (36) and (59). As pointed out in Section 4, the calculation of interfacial parameters should concern both formed and broken molecular contacts. Especially, the liquid-substrate interaction energy, g~j, in eqn. (60), may have an important influence on the calculated differences in ATe, where the spread in calculated values was less than in observed ones for the various substrates. (1 s s) The critical degree of supercooling of tin has been determined for nucleation on various substrates.C, 44) The data are listed in Table 6. We want to illustrate the limitations of an evaluation of such data according to a nucleation theory, although various other sets of data could have been used just as well. In the reference, Aa values were calculated according to the spherical cap nucleus model using a,~ = 58.3 erg/cm ~, which was taken from a study on spontaneous nucleation in droplet experiments (~ s6) in which a spherical nucleus was assumed. A rather different set of h a values is obtained when the parallelepiped nucleus model

Heterogeneous nucleation of crystallization

225

(Table 1) is used and consequently also a cube nucleus model for the evaluation of data on spontaneous nucleation, leading to a,~ = 47.1 erg/cm 2. Both sets of Aa values have been listed in Table 6. Here, we see that the interfacial free energy parameters are actually the adjustable parameters of the theory and that, without further evidence, a single set of ATe determinations does not give conclusive evidence in p r o o f of any theory. Moreover, both sets of Aa values in Table 6 are based on a model of nucleation on a flat plane, whereas nucleation at steps or grain boundaries in the substrate or at specific active sites is much more probable, certainly in the experiments mentioned, in which large substrate crystals were used. TABLE6. NUCLEATIONOF TIN ON VARIOUSSUBSTRATES(144) Aa, erg/cm 2 Substrate

Observed To, °C

Pt Y (prism face) Ag Y (basal face) TiC MoS AI

4 6 7 8 15 17 18

According to I spherical cap modeF144) I

1.4 2.05 2.4 2.8 5.15 5.85 6.2

According to parallelepiped model

I

0.75 1.7 2.3 3.0 10.6 13.6 15.3

The sensitivity of the derived interfacial free energy parameters to the value of various other parameters such as the pre-exponential factor, Z, is illustrated on the basis of determined critical degrees of supercooling of polyethylene droplets on freshly cleaved alkali halide crystals. (1 s 2) In the reference, a parallelepiped nucleus model (Table 1) has been used for the calculation of Air while a correction was applied accounting for the temperature dependence of the heat of fusion in eqn. (9), the correction of which has a minor influence on A~. The nucleation frequency was assumed to be I = 107s - 1 c m - 2 , the area of contact between a droplet and the substrate being roughly 10-7 cm z. The pre-exponential factor was taken to be Z = l0 al s - 1 c m - 2 , ~b = 10 erg/cm 2 and a c = 68 erg/cm 2. The resulting values for Aa are given in Table 7. As pointed out in Subsection B, a more sensible value o f Z [ I for heterogeneous nucleation of a viscous polymer on a flat substrate in an area of 10- 7 cm 2 would be 5 x l0 t4 leading to A G * / k T = 34. Use of this value reduces the Aa values for a parallelepiped nucleus in Table 7 by a factor of 1.63, as a result of which the thickness of the classical nucleus would be of the order of one molecular diameter. This illustrates the inadequacy of the three-dimensional nucleus model in this case. Therefore, we now use the data mentioned in an equation for a monolayer nucleus. The free energy of formation of such a nucleus is according to Table 2, using eqns. (12) and (16): AG* = 4ao b{TslctTsl A#v - 2Aa/ao

(61)

leading to Aa = 0.5ao{(AHvATflTm)-(4ao bast co-~,/AG*)}.

(62)

226

F . L . BINSBERGEN

The evaluation according to this equation depends strongly on the value of the product basz casl used. A most probable value, derived from a correct evaluation of crystal growth data,(J 20) is 1500 erg 2 c m - 4 (see also Subsection F, p. 230). Using this value and AG*/kT = 34 we obtain: Aa = 0.14(ATe- 19.5). (63) This means that ATe could not be less than 19.5°C, which is at variance with the experiments. We see that neither the model of a parallelepiped-shaped nucleus nor that of a rectangular monomolecular layer nucleus on a fiat plane can explain the observed nucleation phenomena. Upon testing the formula for nucleation at a step according to a parallelepiped nucleus model one finds a nucleus of less than four chains; this makes the formula very inaccurate, as pointed out in Section 4. The molecular model, however, allows the calculation of the minimum number of chains in the nucleus especially in this case, where very probably A~b ,~ ~c. For nucleation at steps the observed value of AG*/kT is approximately 27.5 (see Table 5) for an area of 10 -7 cm 2. The value of ~bc is 10 k T i f we take ~s~ = 150 erg/cm 2. Then the number of rows in the nucleus is between 2 and 3, which requires: A~b ~ 0.7A/~o.

(64)

The corresponding values of Aa are presented in Table 7. We see that in this case the difference in result between the continuous variables treatment and the molecular treatment is small, although there is a world o f difference in method of evaluation. However, the molecular treatment indicates that the nucleus is approximately o f the same size on all four substrates. TABLE 7. NUCLEATION OF POLYETHYLENE ON

ALKALIHALIDES(t 52) Aer,erg/cm3

Substrate LiF NaCI KC1 KI

sao

10- s cm

ATe, °C

(a)

4.01 5.63 6.28 7.05

15.2 16.0 18.1 19.2

1.45 1.6 2.05 2.3

I I I I I

(b) 1.7 1.8 2.0 2.1

(a) Evaluation according to ref. 152 with Z/I = 10"4. (b) Evaluation according to eqn. (64).

D. Measurement o f Nucleation Density The nucleation of ice on monodisperse AgI particles dispersed in water droplets has been studied.(1 s 7) The degree of supercooling at which 50 Yoof the water droplets of about 50/~m solidified was determined as a function of number of AgI particles per droplet and of particle size. A cooling rate of 2°C/min was employed. Fairly smooth plots of ATe versus number of particles per droplet were obtained for each particle size, which plots could be reasonably combined to a single, linear, plot of ATe versus the logarithm of the total surface area of AgI available per droplet. A nucleation theory was not tested in the reference. The presentation of the data allows

Heterogeneous nucleation of crystallization

227

us to do it. For instance, the curve presented for 0.75 #m particles (ref. 187, Fig. 3) can reasonably well be approximated by I = 2.1 x 107 exp-[790/(AT)2]s - 1 cm-2,

(65)

which equation also accounts for the observation that all droplets were frozen within a temperature range of 4°C. The form of eqn. (65) is similar to the equations for heterogeneous nucleation on a flat plane or at a step via a parallelepiped nucleus. However, in both cases the pre-exponential factor is many orders of magnitude too low (see Table 5). The test for monolayer nucleation on a flat plane (see Table 2, region 4) would require the data of ref. 187 to fit an equation of the form AG* p k-'T-= AT--6.1Aa' (66) in which p is a constant and A~ is in erg/cm 2. In order that Aa be positive, a value of AG*/kT lower than 18 would be required (at a supercooling AT = 13°C for the 0.75/am particles) which again is improbably low in comparison with the values mentioned in Table 5. A test of the cumulative row model for nucleation (see Section 4, and Fig. 6) is difficult. The number of rows for which AG is at its maximum is about 21 for AG*/kT = 33.5 (see Table 5) and asl = 31 erg/cm 2, giving ~b/kT ,~ 1.6. For such a number of rows, however, the model does not deviate much from the parallelepiped model, which above was shown to be unfit. Moreover, the required value of A~ would be about 5 erg/cm 2, which seems rather high for a good nucleating agent. The remaining model is that of a restriction of embryo length along the step because of misfit in epitaxy. However, little quantitative interpretation is possible in this case as has been pointed out in Section 4. The authors of ref. 187 used the term "density of nucleating sites". Perhaps, nucleating activity is mainly due to sites of specific activity. The strong increase of nucleating activity of Agl after damaging by u.v. radiation indeed points in this direction. Moreover, similar indications have been found in other studies. In the study of crystallization of p-dihalo benzenes on alkali halides, t 1a 1) low-angle grain boundaries appeared to be the preferential nucleation sites. The "epitaxial" crystallization of alkali halides on mica was in most cases marked by the appearance of only one crystal at a low degree of supercooling; the second and subsequent crystals appeared at larger degrees. (1 a 9) The isothermal crystallization kinetics are been determined for polypropylene containing various amounts of either the sodium or the aluminium salt ofp-tert, butyl benzoic acid. (12) The salts were so finely dispersed into the polymer that the size of the salt particles was below the limit of resolution of the optical microscope, i.e. <0.2 pm. For this reason, the number and size of the particles were unknown. After the specimen had been fast cooled from the melt to the crystallization temperature, an almost instantaneous nucleation density was observed. During the relatively slow outgrowth of the nuclei to spherulites little further increase in nucleation density was observed. The nucleation density increased by a factor of 10 over an increase in supercooling by 4°C over the temperature range of 152-137°C (see Fig. 8); at the latter temperature the crystallization was too fast to be wholly isothermal. The nucleation density was dependent on the content of nucleating agent but not proportional to it, possibly as a result of a difference in effectiveness of the dispersing technique at different ratio of polymer to nucleant. Quenching

228

F . L . BINSBERGEN • 0.03% o O.t% N, ©rn-I

A 0.3%

I0 It

ii 4

~. /

/

•

/ /

1011

A

•

/

,

/ /,

•

/

, ,OlO

/

/

/

o/

• •

/

A /~

•

~/

•

/

/

^/

•

/

/

/*

/

/

A ,o'

/

/

,/

~

/

•

i

I

/ /

/

/

o

.

/ A

I0 a 155

150

t45

140

T,°C

FIG. 8. Number of crystallization centres versus temperature of crystallization for polypropylen¢ containing various amounts of sodium p-tert, butylbenzoate.

of a microscope specimen from the melt in cold water resulted in a nucleation density of 2x 1013 cm -3, indicating this to be a lower limit of the number of nucleating particles present. The nucleation density was independent of polymer molecular weight over a fair range of molecular weights used (see also ref. 188). Various explanations of the observed behaviour of the nucleating agents in polypropylene

Heterogeneous nucleation of crystallization

229

have been tried. The increase in number of active particles at successively deeper supercooling might be related to a particle size distribution, the more active particles having the larger surface area. Let/(AT) be the nucleation frequency per unit area of substrate and N x the distribution of particles with respect to surface area x per particle. The fraction of particles with area x that has started the crystallization at time t is 1 -

exp ( - xIt),

(67)

so that the total number of crystallization centres at time t is oo

N(t) = S Nx{1 - e x p ( - x l t ) } d x .

(68)

x

The fact that about 70 ~o of the particles active at a temperature Tis not active at T + 2°C and is not seen to become active during crystallization could be reconciled with eqn. (68) by an extremely wide particle size distribution only, at least much wider than would be possible for particles of sizes below 0.2 pm. Residual crystallinity above the melting point of the polymer in cracks and holes in the nucleating particles can be ruled out, since the temperature of melting, T 1 > Tm (see Fig. la), does not influence the recrystallization kinetics. The activity of the particles might be limited by the requirement that active particles should at least have a size large enough to initiate a heterogeneous nucleus that would form a stable lamella, i.e. a lamella with a thickness, L, larger than L* = Con* (see Table l, region 4a):

,

L > L * = con~ -

Co~c

Co~c

Apo

AS°A T .

(69)

Let us denote the number of active particles by N`,. According to Fig. 8 this number can be represented by N`, = No exp (0.58AT). (70) Na can also be presented as a function of the frequency distribution of particle sizes with respect to the L dimension: oo

(71)

N`, = ~ NLdL. L*

Then dN`, dL*

aN,, d(AT) d(AT) clL*

and with eqns. (69) and (70):

eo~c

N L = 0.58No ( L ~ e

xp ~[0"8 Co~'x .3 L--g~ASo) .

(72)

The value of the exponent: 0.58 c°dpc = 0.58AT L*ASo

(73)

is, for instance, approximately 10 for T2 = 153°C (with AT = T m - T 2 and T m = 170°C) and even higher at lower temperatures of crystallization. This means that in the region of L* values corresponding to a crystallization temperature, T2, around 153°C a decrease in L*

230

F . L . BINSBERGEN

by 10~o would cause NL to increase by a factor of more than 3. For a further decrease in L* the factor by which NL increases becomes even progressively larger. This shows that eqn. (72) represents a highly improbable particle-size distribution. Moreover, in specimens containing poorly dispersed aluminium salts of, for example, succinic or adipic acid, all well-visible particles were found to be active at temperatures below 135°C, but for the larger part inactive in crystallization experiments above 148°C. These facts make a nucleating activity controlled by a minimum particle size improbable. The evaluation leading to eqn. (72) applies equally well to the--similarly to be discarded-idea that particles should have at least the size of the heterogeneous nucleus according to Table 2, regions 2 or 4. Models predicting a nucleation frequency proportional to available area of substrate do not explain the observed almost instantaneous nucleation of a small fraction of the particles present. A distribution of steps of limited length might explain both the dependence of nucleation density on degree of supercooling and the instantaneous character of the nucleation. Experimental evidence of such a distribution is lacking, however. A distribution of steps of limited length is in fact equivalent to a distribution of active sites of varying activity, i.e. critical degree of supercooling.

E. Electroerystallization There is a vast literature on the electrodeposition of metals onto conducting substrates. Few studies, however, were concerned with the testing of nucleation theory. The deposition of cadmium~1s 9) and of liquid mercurycl 9o) on platinum cathodes was studied according to the pulse method. ~t 13) Whereas the nucleation kinetics formally followed the classical theory, the application of the spherical cap-shaped nucleus model led to nuclei of a few atoms in contradiction to the predictions of the theory. Probably the activation free energy for nucleation is somewhat less than predicted, which again suggested the presence of certain a c t i v e sites. (I 12)

F. Interfacial Free-energy Parameters A better view on the validity and usefulness of nucleation theories would be possible if independently determined values of the interracial free energies were available. Most of the trot values reported have been derived from droplet experiments on spontaneous nucleation (see Table 8). For various classes of substances the relation with the heat of fusion, eqn. (30), has been examined. The approximate value of g appeared to be 0.45 for metals, except aluminium,~23) 0.32 for semi-metals, ~23) 0.33 for simple organic substances and water t2 a) and 0.09 for paraffins.~29,191) However, the factor g, although characteristic of a class of substances, is dependent on the nucleus model used, the trot values being a factor 1.24 larger in the evaluation according to a spherical nucleus than according to a cube-shaped nucleus. The crystal face to which the derived osz value belongs has not been specified. Some trs~values have been derived from experiments on fast precipitation from sparingly soluble substances from aqueous media where spontaneous nucleation must be assumed.~7't9) The derived values depend again on the evaluation model used and a most

Heterogeneous nucleation of crystallization

231

TABLE 8. CRYSTAL-MELT INTERFACIAL FREE ENERGY AND ITS RELATION WITH HEAT OF FUSION(23)

Substance

a, erg/cm 2 from spontaneous nucleation, spherical nucleus assumed

g from eqn. (30)

Hg Mn Fe Co Ni Pd Pt Cu Ag Au Ga Sn Pb Bi Sb AI H20 Benzoic acid Methyl chloride NH3 1,2-Dibromo ethane Benzene Naphthalene Diphenyl White phosphorus

24.4 206 204 234 255 209 240 177 126 132 56 54.5 33.3 54.5 101 93 32 35 21.4 29.1 21 20.4 30.1 24.0 12.6

0.53 0.48 0.445 0.49 0.445 0.45 0.455 0.44 0.46 0.435 0.435 0.42 0.385 0.33 0.30 0.365 0.32 0.36 0.39 0.33 0.33 0.35 0.31 0.31 0.68

probable value t192)is quoted in Table 9. The value found for SrSO4 was in fair agreement with the value derived from the increased solubility of small crystals. °22) In polymer crystallization where the difference between tT(= oas~ = ~rst) and ac(= cast) has been regarded, the derived values resulted from a combination of data on crystal growth TABLE 9. CRYSTAL-SOLUTION INTERFACIAL PREE ENERGIES OF SPARINGLY SOLUBLE SUBSTANCES( 192)

Substance

a~,, erg/cm 2 most probable value

BaSO4 BaCrO4 BaMoO4 BaWO4 BaCO s PbSO4 PbCrO4 PbCO3 SrSO4 CaWO4

135 120 103 94 115 100 170 125 85 151

232

F . L . BINSBERGEN

rates and spontaneous nucleation in droplet experiments. The results depend strongly on the method of evaluation of the experimental data. Data on the temperature dependence of the rate of crystal growth o f polyethylene led to a a c = 6 x 10 2 erg 2 cm - 4 on the assumption that the crystal growth is governed by two-dimensional nucleation of successive monomolecular layers. (~s) When, in addition, the lateral growth of initiated monomolecular layers is taken into account, offering an increasing area for the nucleation of the next, the derived value is approximately (~2°) aa~ = 15 x 10 2 erg 2 cm -4. Droplet experiments on spontaneous nucleation of polyethylene in which Tc was determined as a function of the rate of cooling led to (a2) aZac = 1.55 x 104 erg 3 cm -6 on the assumption of a cylindrical nucleus. Evaluation of the same data on the assumption of a parallelepiped shaped nucleus while the temperature dependence o f the heat of fusion was taken into account led to t ~s) a2ac = 7.32 x 103 erg s cm-6. Determination o f solidification half-times in isothermal experiment led to a2a¢ = 1.5 x 104 using a parallelepiped nucleus model. (s4) The latter experiments required for a good fit a pre-exponential factor many orders higher than is reasonable for a viscous polymer (see Table 5), the estimate being r o u g h l y Z = 104~ cm - a s -a Combination of the values ofaa~ and a2a~ mentioned leads to a range of 4.9-25.8 erg/cm 2 for a and o f 23-305 erg/cm 2 for a C. Selections of a value of a out of the range indicated have been based on a comparison with the value found for spontaneous nucleation of paraffins. However, in the calculation of the latter the anisotropy of the nucleus has not been taken into account, which would be certainly necessary since the calculated dimensions of a paraffin nucleus would otherwise be less than the length of a paraffin chain. Moreover, the polyethylene nucleus as calculated has a thickness of only two chains which makes a treatment of nucleation data according to a continuous variable model rather uncertain. A discrete number of chains has been assumed in the nucleus in an evaluation of spontaneous nucleation data of polypropylene, taa) The free energy of formation of an embryo has been calculated for embryos of up to five chains. Unfortunately, the free energy of deposition of the first segment of a new chain onto the cluster has not been taken into account as an important part of the activation free energy of nucleation (as indicated by Becker and Dtiring). The present uncertainty in a and a~ values might be resolved by a computer simulation taking full account o f all possible configurations and additions of single segments. ° 2o7 As long as trst values are inaccurate, the estimation of Aa values remains uncertain. Nevertheless, we give as an example the calculation of Aa values using assumed trst data for a few substances where mainly London dispersion forces play a role across the interfaces, i.e. for polyethylene and polypropylene nucleating on various substrates, in order to illustrate the importance of (partial) accommodation of the crystallizing molecules onto the substrate. Schonhorn determined the surface free energies of amorphous and about 100 ~o crystalline surfaces of various polymers, c19 s) Since the method o f determination was the measurement of the critical surface tension of wetting he determined for the hydrocarbon polymers mentioned the dispersion contribution to the surface energies, at and aas. Applying the Fowkes relation, eqn. (21a), to the interfacial free energy between crystal and liquid we can determine oa~, supposing oat to vanish: acr~t = 0%+ (~as~ - a~) 2"

(74)

For the moment we assume ~a,t = 12 erg/cm 2 as a mean value of the range indicated

Heterogeneous nucleation of crystallization

233

above for polyethylene. Assuming the ratio a~JAH,,, to be the same for polyethylene and polypropylene, aa~l = 9 erg/cm 2 for polypropylene. We can now calculate oO~ and subsequently the value of Aa (by means of eqn. (43)) for various values of ~:, i.e. for various substrates. The results are presented in Table 10. TABLE 10. INTERFACIAL FREE ENERGY COMPONENTS FOR POLYETHYLENE (PE) AND POLYPROPYLENE (PP) IN e rg/ c m 2

al .tr. dr, ~ - tr~~ otr~ otrs

PE

PP

36 70 2.35 12 6.5

28 39.5 1.0 9 8

=

A~, for

{

~: a~ = ~00 tr¢ at 121

7.4 5.0 2.7 0.3

•

=1

¢

v-

0.37 < 0 < 0 < 0

56..4.3 3.3

¢

< < < <

1

00 0 0

According to the theoretical evaluation of the interfacial difference parameter in Section 4, we can expect heterogeneous nucleation at low degrees of supercooling only if Ark ~ ~b or Aa ~ a~s~. Apparently this criterion is, according to Table 10, not satisfied without a certain degree of accommodation of the chain molecules on most substrates; indeed, substances like graphite (atr: ,.~ 120 erg/cm2), BaSO 4 (atrf = 76 erg/cm 2) and Fe20 3 (atrf = 107 erg/cm2) t 124) do not act as nucleating agents for polypropylene. It is not at once understood from Table 10 why K I (da: = 67 erg/cm2), KCI (aa: = 77 erg/ cm 2) and NaCI (dtr: = 88 erg/cm2) O 94) are nucleators for polyethylene, be it modest ones, but not nucleators for polypropylene. This may be due to the bulkiness of the polypropylene chain, allowing only p o o r accommodation, or to the uncertainty in the oa, t values, a slight shift in the latter easily providing an explanation.

6. C O N C L U S I O N S In the practical study of crystallization, it is not always certain if, and how, heterogeneous nucleation takes place. Various modes of nucleation must be distinguished which seemingly lead to similar results but on careful examination prove to be quite different phenomena. A good set of definitions is, therefore, required to translate experimental results into terms of crystallization theory. Nucleation theories treating the dimensions of embryos as continuous variables are inadequate for the description of heterogeneous nucleation since at least one of the dimensions of the nucleus is of the order of one molecule. Therefore a molecular theory is required which treats the addition of molecules or at least of rows to a monomolecular layer. Preferably, any nucleus model should be avoided and the nucleation be treated along a random addition of

234

F . L . BINSBERGEN

molecules to the growing embryo; this requires a computer simulation, however, which as yet has not been applied to heterogeneous nucleation. Heterogeneous nucleation on a flat plane must be regarded as an improbable phenomenon since nucleation at a step requires a lower activation free energy and will have happened already at a lower degree of supercooling or supersaturation. Even a description according to nucleation at a step is not able to explain experimental data on heterogeneous nucleation in several cases and nucleation sites of specific activity but of unknown nature must be assumed. Therefore, the relevant details of the surface structure of the substrate remain vague. Highenergy surfaces are not necessarily well-nucleating substrates. There are many indications that the substrate causes a specific orientation or arrangement of the adsorbed molecules promoting the crystal lattice of the latter, be it with a limited degree of lattice accommodation. In this respect true epitaxy is an exception rather than the rule, where epitaxy requires a lattice fit in at least two dimensions. Measurement of the critical degree of supercooling, ATe, or of supersaturation at a single cooling rate does not supply sufficient data for the testing of theories of heterogeneous nucleation. An isothermal, kinetic measurement of the nucleation frequency is required, or at least a measurement of ATe as a function of the cooling rate or of the particle size of the dispersed substrate. Only kinetic measurements can provide separate values of the pre-exponential factor, Z, and the activation free energy of nucleation, AG*. Evaluation of kinetic droplet experiments on spontaneous nucleation led to values of Z m a n y orders higher than expected from theory. On the other hand, some experiments on heterogeneous nucleation lead to low values of Z, indicating sparsely present nucleation sites (of specific activity). Some studies have shown that the formation of nuclei stops after a certain time, i.e. no real nucleation rate emerges and there is a nucleation density rather than a rate. This again points to sites of specific activity. The interfacial free energy parameters (ast and An, and ~b and A~b) are actually the adjustable parameters of any nucleation theory. Therefore, trst values as obtained from droplet experiments on spontaneous nucleation have to be regarded with suspicion, the values being dependent on the nucleation model used. Where only London dispersion forces act across the solid-melt interface tr,z may be calculated from data on the surface free energy of the solid, tr,, and the liquid, o'~.If the data on as have been obtained from measurements of the critical surface tension of wetting of the solid then the tr,t value calculated does not contain the part 0a,, the latter being the result of the special packing of molecules in a crystal lattice. Calculations of An must remain inaccurate where accurate data of trot are lacking and the degree of accommodation of the lattice of the adsorbed molecules onto the substrate is unknown. It is still hardly possible to make a comparison between Atr values from kinetic nucleation data and independently calculated Aa values (as a check of the theory), because of the abovementioned uncertainty about both the right nucleation model and the separately calculated Atr values. As shown above, the total picture of heterogeneous nucleation is rather less luminescent than some authors want us to believe. Much carefully directed experimental work has still to be done and theoretical calculations should be checked by computer simulation, the latter being fairly easy where fast computers with a large memory are available.

Heterogeneous nucleation of crystallization

235

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