NUCLEATION OF WATER AEROSOLS BY B. J. MASON Imperial College of Science and Technology, London Received 7th July, 1960 This paper reviews recent studies of homogeneous and heterogeneous nucleation of water aerosols involving vapour-to-liquid, supercooled liquid-to-solid and vapour-to-solid transitions with particular reference to recent investigations in the author’s laboratory. THE HOMOGENEOUS CONDENSATION OF WATER DROPLETS FROM THE VAPOUR In a gas entirely free of all foreign particles and ions condensation of water vapour can occur only as the result of chance collisions of molecules. Small molecular aggregates (embryos) are continually formed and disrupted because of microscopic thermal and density fluctuations in the vapour, but only if they surpass a certain critical radius r, given by where M, OLV, PL, are respectively the molecular weight, surface tension, and density of the liquid, R the gas constant, p the pressure of the supersaturated vapour and p a the equilibrium vapour pressure at temperature T over a plane surface of the liquid, can they survive and continue to grow to become nuclei for the development of the liquid phase.The probability of formation of such a nucleus of critical size increases as the saturation ratio, S = p/pco, of the vapour increases according to the formula rc = 2M%v/P,RT In (PIP,)? (1) where 1 is the number of nuclei formed per cm3/sec, a is the accommodation coefficient of the droplet surface, N Avogadro’s number and k Boltzmann’s con- stant. This may be derived from the kinetic treatments of the condensation process given by Becker and Diiring 1 and Zeldovich.2 Fig.1 shows a plot of log I against the saturation ratio S = P/Pm for water vapour with T = 260”K, a = 1, and OLV taking the bulk value of 77.5 erg cm-2. The nucleation rate increases rapidly with increasing values of S and attains a detectable value of 1 cm-3 sec-1 only when S+5. The formulation of the theory is, however, unsatisfactory in that it treats small molecular aggregates of the condensed phase as well-defined spherical “ droplets ” having the thermodynamic and physical properties of the bulk liquid. The macro- scopic concepts of surface tension and bulk free energy become rather vague when the aggregates contain as few as 20 molecules, and are probably meaningless when applied to embryos of sub-critical size which are not in phase equilibrium with the vapour.In particular, the surface tension, to the assumed magnitude of which the calculated value of I is very sensitive, cannot be precisely defined for very small aggregates of molecules. The early stages of embryo formation need to be considered in terms of successive attachments of vapour molecules to a small existing aggregate and the probability of capture assessed in terms of intermolecular force fields, the relative orientations of molecules: etc., rather than in terms of macroscopic concepts. A quantitative theory on these lines seems remote especi- ally for a complex polar molecule like water. 20B . J . MASON 21 Despite these weaknesses, the theory, culminating in eqn. (2), has formed the basis for discussion of the condensation process and its predictions have been tested with a variety of vapours and carrier gases.With water vapour, experiments have been made by several 9 A .b ' ' I 10 ' ' 12 ' I 14 ' ' s = PlPco FIG. 1.-The rate of nucleation 1 as a function of the saturation ratio S calculated from eqn. (2). different workers in which very clean, water-saturated air has been subjected to rapid expansion in a cloud chamber and the minimum supersaturation required to produce a detectable droplet concentration determined. The results are summarized in table 1. In the literature there has been a tendency to underline an apparent good agreement between the experimental results and the predictions of eqn. (2), but these claims merit a more critical examination. TABLE 1 .-EXPERIMENTAL AND OBSERVED RATES OF HOMOGENEOUS NUCLEATION IN WATER VAPOUR author Wilson 3 Powell 4 Voliner and Hood 5 Frey 6 Sander and Damkohler 7 Barnard * 293 257 291 256.6 261 263 261 293.7 261 261 238 S Obs.droplet concn. I obs. 7-90 " cloud limit " 9 106 7-80) probably >lo3 5.03 -1 -1 02 4.36 1 102 6.60 103 106 (cm-3) 5.0 10 105 5-70 10 103 6.40 10 103 theor. > 7.0 5-42 5.90 5.42 6.40 5-68 9-50 It is rather difficult to compare the experiments of Wilson and of Powell with the theory since the rates of droplet formation corresponding to their reported appearance of a " cloud " as distinct from sparse " rain-like " condensation are uncertain. If, however, we assume that the droplet Concentration must have exceeded 103 cm-3 in order to be detected above the background condensation on ions (which were not removed) and, furthermore, that the effective sensitive time of the chamber was about 10-3 sec, we arrive at a minimum value of I = 106 cm-3 sec-1.Substitution of this value in eqn. (2) gives a lower limit for the theoretical saturation ratio of S = 7.0 to be compared with the measured value of 7.8 or 7.9. Part of this discrepancy may be accounted for by the fact that, with such high concentrations of droplets, the actual saturation ratio achieved in the cloud chamber must have been less than that calculated on the basis of a perfectly adiabatic expansion due to the abstraction of water vapour and release of latent heat by the growing droplets (see Mason 10).22 NUCLEATION OF WATER AEROSOLS Volmer and Flood, who removed foreign particles from the air by repeated expansions and ions by a strong electric field, determined a critical expansion ratio which just produced a noticeable increase in the number of droplets above the background. They estimated the minimum observable increase in droplet density to correspond to I = 1 cm-3 sec-1 for which the corresponding theoretical value of S, at a final temperature of 261"K, was 4.96 according to Volmer and Flood and 5-05 according to eqn.(2) ; these are to be compared with their measured value of 5-03. Similar good agreement was obtained for a number of other pure vapours and consequently this work became accepted as a verification of the theory and as a basis for much subsequent work in the general field of nucleation pheno- mena.However, it is very doubtful whether Volmer and Flood could have detected such a small increase in the number of droplets in a chamber whose sensitive time was only about 10-2 sec. An increase of one droplet/cm3 is just about detectable by careful visual observation ; this would correspond to I = 102 cm-3 sec-1, for which eqn. (2) gives S = 5.42. It is accordingly doubtful whether Volmer and Hood obtained such good agreement between experiment and theory as they believed. Frey 6 attempted to determine the droplet concentration more accurately by photo- graphing the transient cloud. The minimum detectable concentration was 10 droplets/cm3 and the effective sensitive time of the chamber probably not much in excess of 10-4 sec.This minimum concentration of droplets was observed at a final temperature of 261 "K and saturation ratio of 5.0; the corresponding theoretical value of S is 5.90. But Frey's method of cleaning his chamber before each experiment appears inadequate and it seems highly probable that the initial condensation which he observed occurred upon foreign nuclei or upon ions. A similar criticism may be made of Sander and Damkohler.7 In some more recent and very careful experiments by Barnard,s who was well aware of the difficulty of removing all foreign nuclei and of preventing their formation by rubber and mctal surfaces under the influence of ultra-violet light, the concentration of droplets was photographed as the expansion ratio was increased in very small steps.It was found that the increase of droplet concentration with increasing supersaturation was systematically less than that predicted by the theory. Thus if a figure for the sensitive time of the chamber (and hence for the nucleation rate I ) was chosen to make theory and observation agree for an observed droplet concentration of 1 cm-3 and S = 5-20, a droplet concentration of 103 cm-3 occurred at S = 6.60 instead of at the theoretical value of 640. However, the agreement here is quite good and probably part of the discrepancy may be attributed to the actual supersaturation being rather less than the nominal adiabatic value. The most recent work is that by Pound et aZ.9 who used purified and filtered nitrogen as a carrier gas and an electric field to remove the ions.Working with a minimum detectable droplet concentration of 10 cm-3, these authors obtained good apparent agreement between the experimental and theoretical values of the critical supersaturation when the final temperatures at the end of the expansion were only a few degrees below 0°C (see table 1). However, much larger discrepancies occurred at lower temperatures, droplets now appear- ing at much lower supersaturations than are indicated by the theory. This led Pound et al. to conclude that, unless the surface tension of water decreases rather than increases as the temperature is lowered below about -25°C (measurements show it to increase smoothly down to -23"C), these serious discrepancies at low temperatures suggest that the agreement obtained at higher temperatures may have been fortuitous ; indeed the condensation may not have been homogeneous but have occurred on nuclei produced, perhaps, by chemical reactions between the water vapour, the carrier gas and small irremoveable traces of impurities under the influence of ultra-violet light.In conclusion, it may be said that a critical analysis of the various experimental results, summarized in table 1, indicates that the Becker-Doring theory of homo- geneous condensation has not been satisfactorily verified by experiment. There are certainly unsatisfactory and unconvincing €eatures of the theory, but the dis- crepancies between the theory and the experimental results of different authors are not of a systematic nature which might be reconciled by inserting different values for such parameters as the accommodation coefficient and surface tension in eqn.(2). Much of the trouble probably lies in the experiments where there are great difficulties in excluding entirely small traces of contamination and sources of foreign nuclei, in determining accurately the droplet concentration, and the sensitive time of the chamber and hence the nucleation rate. There remainsa need for an experimentum crucis in which one can be certain of observing homo- geneous nucleation in the absence of foreign particles and in which the dropletB . J . MASON 23 concentration will be sufficiently low to ensure that the peak supersaturation can be accurately computed in terms of the expansion ratio and initial temperature for a truly adiabatic system. The experiment should probably be carried out in all-glass apparatus, with a number of highly-purified carrier gases and over a fairly wide range of temperature. CONDENSATION OF WATER VAPOUR ON FOREIGN NUCLEI (a) CONDENSATION ON IONS Homogeneous condensation of water vapour occurs only if the supersaturation reaches several 180 %; the presence of impurities in the vapour may greatly facilitate condensation.In the atmosphere there is an abundance of particles having a wide variety of size and constitution, their size and number concentration usually being such that even the most rapid cloud formation is associated with very small supersaturations, usually less than 1 %. The first experiments to demonstrate condensation of water vapour on foreign nuclei were carried out by Coulier 11 and by Aitken ;I2 they were also able to remove most of the nuclei from the air by filtration, sedimentation, and repeated cloud formation and show that the resulting clean air could sustain an appreciable supersaturation without droplets appearing in the body of the gas.Later, Wilson 13 discovered that if, after removing the nuclei in this way, the air was expanded by increasing amounts, no droplets appeared until the expansion ratjo reached a value of about 1.25, corresponding to a saturation ratio of about 4. The nuclei responsible for these droplets could not be removed by successive expansions nor by filtration through cotton wool. The fact that condensation occurred at similar supersaturations when the air was irradiated with X-rays suggested that the respon- sible nuclei were small ions.Wilson 3 therefore investigated the relative efficiencies of positive and negative ions as centres of condensation. By ionizing the air in his chamber by a very short exposure to X-rays and by application of electric fields of the appropriate polarity, Wilson was able to ensure an excess of ions of either sign. He then found that expansion from an initial temperature of 293 "I< produced condensation or some negative ions with an expansion ratio of 1.25, for which S = 4, that practically all the negative ions were involved when the ratio exceeded 1 *28 but that to effect condensation on positive ions, the expansion ratio had to exceed 1-31 ( S E ~ ) . Wilson's results have been confirmed by several Iater workers, a representative sample of results being given in table 2.TABLE 2.-sATURATION RATIOS AT WHICH CONDENSATION OCCURS ON SMALL IONS ion sign Tl('K) * TZ Vzl VI S - 293 267-8 1.252 4.2 + 293 1.31 6.0 - 293 1.236 3-7 + 293 1.31 6.0 - 267.6 1.256 4.2 - 267.8 1-253 4.1 - 291 266.5 1.245 3.98 Flood 17 - 265 1.252 4.1 Wilson 3 Przibram 14 Laby 1s Andr6n 16 Powell 4 Loeb, Kip and Einarsson 18 { i 295 1.25 1.31 295 292 1.25 414 1.28 4.87 265 3-9 Scharrer 19 {T 292 Sander and Damkohler 7 - * TI and V1 are respectively the initial temperature and volume of the air, while T,, VL refer to the values at the end of the expansion. The conditions under which condensation occurs upon ions seem to be more sharply defined and more reproducible than is the case for homogeneous nucleation ; the results obtained by different workers are in good agreement.It is interesting to compare these results with the predictions of theory.24 NUCLEATION OF WATER AEROSOLS The condition for equilibrium to exist between a supersaturated vapour and a spherical droplet of radius r, surface tension CLV, carrying a charge q in an external medium of dielectric constant E, is If daLv/dr = 0, S has a maximum value when if the embryo radius exceeds r’, it will continue to grow with a decrease of free energy as long as the supersaturation is maintained and become a droplet. If the embryo carries a single electronic charge (q = 4.8 x 10-10 e.s.u.), and we assume E = 1 for air, T = 265”K, GLV to take the bulk value of 77 erg cm-2, the value of r’ is calculated to be 6 .2 ~ 10-8 cm, the corresponding value of S is 4.59, which may be compared with the experimental value of 4.1 for the critical saturation ratio required to produce visible droplets around negative ions. If one includes the doLv/dr term which, according to Tolman,20 would have a value of about 2 x 108 erg cm-2, the calculated critical value of S would be 3-90. However, this simple treatment ignores polarization of the droplet by the ion. In water, where strongly polar molecules form an oriented surface layer, the surface energy of the droplet will probably be modified to an extent depending not only on the magnitude of the charge as indicated by eqn. (3), but also on its polarity. Therein may lie the explanation of negative ions being able to promote condensation at lower supersaturations than can positive ions.Loeb, Kip and Einarsson,l8 who obtained experimental confirmation of this sign preference, argue that small ions cannot grow into water droplets by successive attachments of water vapour molecules, but rather that an ion is captured by a molecular aggregate or embryo, which is not yet large enough to possess a regular surface layer and a statistically conditioned electrical double layer which are characteristic of a true droplet. The molecules in the surface layer of the embryo are imagined to be oriented with the oxygen atoms directed outwards ; the surface force field will then tend to orient approaching vapour molecules with their protons towards the surface, so favouring H-bond linkages and propagation of the surface structure.The capture of a negative ion by the aggregate will enhance the surface force field and so favour the capture of further molecules in the correct orientation, while molecules striking a positively charged embryo would have to rearrange themselves before becoming bound. Thus, condensation may be expected to occur more readily on negative ions than on positive ions. Again, it appears that eqn. (3), which is derived in terms of macroscopic con- cepts, cannot apply to the pseudo-crystalline embryo droplets just discussed ; indeed, the existence of sign preference, which is not predicted by eqn. (3), is evidence of the breakdown of this equation in the very early stages of the con- densation process. r = r ’ = ~1~14n0,4+ ; (b) CONDENSATION ON HYGROSCOPIC, NON-HYGROSCOPIC AND MIXED NUCLEI In the presence of solid, insoluble, wettable particles, droplet formation will be much facilitated, since these particles are ready-prepared aggregates to serve as nuclei €or condensation at supersaturations which diminish with their increasing size.Eqn. (l), which is plotted in fig. 2, gives the supersaturation necessary for continued condensation to occur on pure water droplets. The degree of super- saturation required is higher the smaller the drop, some typical values being given in table 3(b). Thus a droplet of radius 10-7 cm requires a saturation ratio of 3-23 or a supersaturation of 223 % to persist, while droplets of radius greater than 10-5 cm require supersaturations of less than 1 %. For insoluble, wettable particles of the same size, the supersaturations required will be slightly less, while those required for the activation of hydrophobic particles will be higher.B .J . MASON 25 If, however, the droplet is formed on a wholly or partially soluble nucleus, the equilibrium vapour pressure at its surface is reduced by an amount depending on the nature and concentration of the solute, which means that condensation is able to set in at lower supersaturations than those required for an insoluble nucleus of the same size. This is a fact of considerable importance in cloud and fog form- ation because a high proportion of atmospheric nuclei are composed wholly or partly of soluble matter. look 95 f droplet radius (cm) FIG. 2.-The equilibrium relative humidity (or supersaturation) as a function of droplet radius for solution droplets containing the indicated masses of sodium chloride.The equilibrium vapour pressure p i over a droplet of pure aqueous solution of radius r is given by the equation (see Mason 21), - (PLIPD Pto ~ ( $ n r ~ p t imM - m> 1 ’ (4) where the primed symbols refer to the solution rather than to pure water, m is the mass and W the molecular weight of the dissolved salt, and i is van’t Hoff’s factor which depends upon the nature and concentration of the solute. Eqn. (4) can be used to calculate the relative humidity H which the air must have to remain in equilibrium with nucleus droplets of a given radius containing a specified mass of a particular salt or, conversely, to compute the radius of drops which will be in equilibrium with an atmosphere of a given humidity. We have, then, p:/pco = H/lOO, and if the humidity is nearly 100 %, eqn.(4) may be written more simply and with little loss of accuracy asTABLE 3.-cRmICAL RADJI AND SUPERSATUXATIONS FCR NUCLEI OF VARIOUS SIZES, T = 273'K (a) HYGROSCOPIC NUCLEI OF NaCl - 13 - 12 - 11 - 10 ~ ( p ) at H = 78 % 0.039 0.084 0.185 0.39 0.88 1-85 4.1 - 16 - 15 - 14 Iogm (g) yc&)* 0.20 0.62 2.0 6-2 20 62 200 r(of crystal)&) 0.022 0.048 0.103 0.22 0.48 1 4 3 2.2 Nc-lOO(= supersat. %>.F 042 0.13 4.2 X 1W2 1.3 X 10-2 4.2 x 10-3 2.3 X 10-3 4.2 x 10-4 Y at H = 100 % is approx. rc/d3. * For other nuclear substances of rnolecdar weight W, multiply by (585/ W)*. -f For other nuclear substances of molecular weight W, multiply by (W/SS.S)&.(b) INSOLUBLE WETTABLE NUCLEI log z(m) - 7 - 6 - 5 - 4 - 3 10Q)prlpco 323 112*5 101.2 100.12 100*01 - 9 8.8 620 1.3 x 10-4 4.8 2 C c, r M > C-I 0 0 - 8 18.5 '..I 2,000 4.2 x 10-5 Z 10.3 w e 4 cl M w 3 R id 0 M 0 r MB. J . MASON 27 Graphs showing how the equilibrium radii of drogdets containing specified masses of sodium chloride vary with the relative humidity are shown in fig. 2. Some growth of these hygroscopic nuclei occurs before the air becomes saturated and indeed they become droplets of saturated sodium chloride solution at 78 % relative humidity. The equilibrium radius of the droplet increases with increasing humidity until the air becomes supersaturated by a critical amount corresponding to the maximum of the relevant curve in fig.2. At this stage the solution is quite dilute, and if the droplet exceeds this critical radius given by the supersaturation required to maintain equilibrium thereafter decreases with increasing droplet size. If the supersaturation were maintained, transition from a nucleus droplet (r<rc) to a fully developed cloud droplet of r>rc would now occur very rapidly and, in theory, the droplet would grow without limit. In a cloud we are not concerned with a droplet growing in isolation under a steady supersaturation, but rather with a whole population of droplets competing for the water vapour being released in a cooling air mass. In this case, the supersaturation will not remain steady; when the vapour is being extracted at a faster rate than it is being released, the supersaturation is forced to retreat and so growth of the droplets is restricted.Table 3 shows the critical radii and critical sapersaturations calculated from eqn. (4) for sodium chloride nuclei with masses ranging from 10-16 g to 10-8 g. We notice, for example, that nuclei of 10-15 g achieve radii of 0.62 p at a critical supersaturation of 0.13 % beyond which they may act as centres of continued condensation. The very large salt particles, of mass greater than 10-11 g, will usually not remain in the atmosphere long enough to attain their critical size. So far, we have discussed the behaviour of completely insoluble particles and of hygroscopic, soluble particles as potential condensation nuclei. But atmos- pheric aerosols are often partly soluble, partly insoluble; over the continents there is an abundance of insoluble particles coated with a thin layer of hygroscopic substance.At high humidities, these mixed nuclei react to humidity changes like wholly soluble particles of equivalent size but, at humidities below about 70 % the solution coat shrinks until the particle becomes almost completely solid. (c) THE NUCLEI OF ATMOSPHERIC CONDENSATION Atmospheric aerosols cover a wide range of particle sizes from about lQ-7 cm for the small ions consisting of a few neutral air molecules clustered around a charged molecule, to more than 10 (u (10-7 cm) for the largest salt and dust particles. Their concentrations, expressed as the number of particles per a n 3 of air, also cover an enormous range, and may exceed lO6/cm3 in the highly polluted air of an industrial city.The small, ubiquitous ions play no role in atmospheric con- densation because of the very high supersaturations (e.g. 300 %) required for their activation, while the largest particles remain airborne for only a limited time. The size distribution of aerosols measured over land is, on average, as shown in fig. 3. It is convenient to divide the particles into 3 size groups : (i) nuclei having radii between 5 x 10-7 cm and 2 x 10-5 cm (0.2~) which are called Aitken nuclei in recognition of the fact that particles of this size, but not the small ions, are detected in the Aitlcen " dust " counter ; (ii) Zarge nucZei with radii between 0 . 2 ~ and l p ; and (iii), giant nucZei with radii greater than lp.A comprehensive review of the techniques employed in determining the size, con- centration and identity of the particles and of the results obtained, is given by Mason.21 The lower limit of the size distribution is set by the fact that the smaller nuclei coagulate under the influence of Brownian motion. The upper limit is determined by the balance between the rate of production of large particles at the earth's surface, their upward transport and the rate at which they are removed by sedimentation, by precipitation, etc. The number concentration of Aitken nuclei may vary from only a few per cm3 over the oceans and in the upper air to, perhaps, a million or more per cm3 in industrial cities.28 NUCLEATION OF WATER AEROSOLS Over the continents, the size distribution of large and giant nuclei obeys a law of the form IZ = A/m = B/r3, where n js the concentration of particles of mass greater than m or radius greater than r.Typically, the concentrations of particles o€ radius greater than O.lp, l p and l o p are respectively 103/cm3, 1 /cm3 and l/litre, compared with averagc Aitken-nucleus counts of about 40,00O/cm3. Over the oceans, the abundance of large nuclei, which originate in the foam of breaking waves, increases with increasing wind speed and roughness of the sea surface. In winds just strong enough to produce " white horses 'I, sea-salt particles of m> lO-I4g (about 0 . 2 ~ radius at 78 % humidity)occurinconcentrations of about 10/cm3, the numbers falling as the particle size increases, to reach about l/litre for nuclei larger than 10-9 g.In winds of hurricane force, the numbers of giant nuclei may increase IOO-fold, while over a calm sea they may be only one hundredth of these quoted figures. FIG. 3.-A Aitken nuclei - 2 10' I 0 -5 10 -6 10 + iclei zr itme . radius (cm) generalized representation of the size distribution of natural aerosols polluted air over land. in heavily In the absence of rain, which washes out the larger particles, appreciable concentrations of sea-salt nuclei may be carried some hundreds of miles inland and, conversely, nuclei of land origin are carried out to sea. The fact that the Aitken-nucleus counts over the oceans do not change appreciably with windspeed and the state of the sea surface indicates that they are not mainly of maritime origin but that they have been transported from land sources.Their mean life in the atmosphere is several days. Atmospheric aerosol originates in three ways : (i) by condensation and sublimation of vapours during the formation of smokes and in reactions between trace gases through the action of heat, radiation, or humidity. (ii) the mechanical disruption and dispersal of matter at the earth's surface, either as sea-spray over the oceans, or as salt and mineral dusts over the continents. (iii) by coagulation of nuclei which tends to produce larger particles of mixed constitution. Typical substances formed in large quantities by condensation following combustion include ashes, soot, tar products, oils, as well as sulphuric acid and sulphates in cases where the fuel contains sulphur.A great variety of particles is formed in this way by industrial operations and by domestic fires. The sizes of these particles cover a wide range, but are primarily within that of the Aitken nuclei.B . J . MASON 29 Chemical reactions between the nitrogen, oxygen and water vapour of the air and various trace gases, for example, sulphur dioxide, chlorine, ammonia, ozone and oxides of nitrogen, are also important sources of nuclei ; here solid particles may play an important role by adsorbing the gases and water vapour and thereby concentrating the substance-perhaps in solution. Examples are : the formation of NH4Cl in the presence of HN3 and HCl vapours ; the oxidation of SO2 to SO3 and the conversion of the latter to H2S04 in the presence of water vapour and, more important, the oxidation of SO2 by sunlight in the liquid water of cloud and fog droplets to form HzSO4 ; the reaction of sulphuric acid and ammonia to produce ammonium sulphate ; and the production of the higher oxides of nitrogen by the action of heat, ozone, or ultra-violet radiation.The mechanical disintegration, by wind and water, of rocks and soil produces particles mostly of radius greater than 0.1~. Rather little is known about the role which soil and mineral-dust particles play in atmosphere condensation but Twomey 22 suggests that the rupture of crusts of salt adhering to the surfaces of soil particles is an important continental source of cloud nuclei. Until recently it was generally assumed that wind-borne sea-spray was the main source of sea-salt nuclei.But Woodcock et al.,23 using high-speed photography, have shown that the numerous small air bubbles bursting in the foam of waves each produce a small water jet which shoots into the air and breaks up in 5 or so droplets of diameter about one-tenth that of the air bubble and that this is a much more effective source of nuclei. Furthermore, Mason 24 discovered that, besides the large salt nuclei produced by the breaking jet, about oiie hundred much smaller particles, as small as 10-15 g, are formed by the rupture of the bubble cap and believes that this is the main source of the more numerous, smaller salt nuclei. The most important feature of the coagulation process is probably the capture of small hygroscopic nuclei by larger insoluble particles to form mixed nuclei.It seems likely that, during cloud formation in the atmosphere, all the available large and giant nuclei become centres of condensation together with a proportion of the Aitken nuclei which will vary with the particular circumstances. The more efficient, hygroscopic nuclei may originate from the sea, or result from combustion, or from chemical reactions in the atmosphere. Clouds formed over the middle of large oceans usually have rather low concentrations of droplets whose formation may be accounted for by condensation of the available sea-salt nuclei. Over the continents, the droplet concentrations are generally 10 times larger so the majority of the nuclei must be supplied either by natural and man-made combustion or by particles originating from the land surface.( d ) DROPLET GROWTH BY CONDENSATION We shall discuss the growth of a small isolated single droplet about a hygro- scopic nucleus in a n infinite atmosphere maintained at constant temperature and supersaturation. The rate of mass increase of a droplet of radius r is given by dmldt = 4nrD(p -pr>, (6) where D is the diffusion coefficient of water vapour in a i q p is the vapour density at distances remote from the droplet andp, the corresponding value at the surface of the droplet. Since dm/dt = 4nr2p~dr/dt and p = pM/RT, where p~ is the density of the droplet, p the vapour pressure, M the molecular weight of water, R the gas constant and T the temperature of the air, eqn. (6) may be rewritten as dr DM DM 1 r- = - ( p -pi) = - (Sp - p:>, dt pLRT PLRT (7) where p i is the vapour pressure at the droplet surface, p a the saturation vapour pressure at air temperature T, and S = p/pa the saturation ratio of the environ- ment.The growth rate of the droplet is controlled not only by the rate at which water vapour can diffuse to its surface, but by the rate of condensation? which is limited by the rate at which the liberated heat of condensation can be dissipated. Nearly all this heat is lost from the drbplet surface by conduction through the air according to the equation30 NUCLEATION OF WATER AEROSOLS K being the thermal conductivity of the air, L the latent heat of condensation and T, the surface temperature of the droplet which is higher than that of its surround- ings.We also have eqn. (5) for the equilibrium vapour pressure p: over the surface of a dilute solution droplet and the Clausius-Clapeyron equation for the variation of the saturation vapour pressure with temperature l d p LM pdT ==z* By making only slight approximations, the last four equations may be combined to give dr dt r- = [(S-l)----] 2oLVM 8*6m p,RTr Wr3 which shows how the growth rate of the droplet is determined by the size and nature of the nucleus ; the supersaturation of the air, the rate of diffusion of water vapour to, and the conduction of heat from, the droplet. The times taken from droplets arising on sodium chloride nuclei of various masses to grow to specified radii may be calculated from eqn. (10) ; some specimen results are given in table 4.TABLE 4.-RATE OF GROWTH OF DROPLETS BY CONDENSATION ON SALT NUCLEI temp. T = 273'K pressure = 900 mbar nucleus mass 10-14 g 10-13 g 10-12 g supersaturation = lOO(S- 1) % 005 ' 0.05 0.05 radius (p) 1 2 5 10 20 30 40 time (sec) to grow from initial radius of 0 . 7 5 ~ 2.4 0.15 0.01 3 130 7.0 0.61 1,000 320 62 2,700 1,800 870 17,500 16,800 1 4,500 44,500 43,500 41,500 8,500 7,400 5,900 The formation of a cloud involves the growth, in a cooling air mass, of a population of droplets growing on condensation nuclei covering a wide range of sizes. In this case the supersaturation varies with time in a manner determined by the rate at which water vapour is released for condensation by the cooling of the air minus the rate at which it is condensed on to the droplets.The problem is now specified by dBerentia1 equations expressing the rate of cooling of the air, the time variation of the supersaturation and the equations of droplet growth for nuclei of differing sizes. A discussion of this complex problem is given by Mason.21 THE NUCLEATION AND FREEZING OF SUPERCOOLED WATER DROPS Interest in the supercooling and freezing of water, matters which have been extensively studied since the early work of Fahrenheit iu 1724, has greatly increased during the last fifteen years, nlainly because of their essential,importance in the physics of clouds and precipitation. Although large quantities of water such as lakes and ponds do not usually supercool by more than a few hundredths of a degree, the tiny droplets composing atrnos- pheric clouds may exist in the supercooled states down to temperatures as low as -40°C.In cloud physics one is concerned with the temperatures at which airborne drops, varying in diameter from a few microns to about 5 mm for the largest raindrops, will freezeB. J. MASON 31 and how the attainable degree of supercooling may depend upon the drop size, the rate of cooling and the purity of the water, Although there was some indication in the extensive writings of earlier scientists that the attainable degree of supercooling tends to increase when the volume of the water sample is reduced, there was so much variability in the results, with serious discrepancies between those of different workers, that no clear-cut relationships could be deduced. It appears that the earlier work may have failed to provide the required information for three main reasons.First, the water samples used by the different investigators varied greatly in their origin and purity; secondly, they were usually contained in glass tubes or supported as drops on variously-treated metal surfaces so that freezing may often have been initiated by the solid boundaries ; thirdly, in any one investigation, the volume of the sample was not usually varied sufficiently to establish clearly how this might be related to the degree of supercooling, particularly as there was usually a considerable spread in the freezing temperatures recorded for specimens of the same volume. For these reasons, the whole subject has recently been examined afresh. (a) THE F'REEZMG OF WATER CONTAINING FOREIGN NUCLEI-HETEROGENEOUS NUCLEATION The great majority of experiments have been concerned with the study of heterogeneous nucleation in that the water used almost certainly contained foreign particles which initiated crystallization.A considerable improvement in the technique of investigating the supercooling of such water was made in the author's laboratory by Bigg,25 who eliminated the influence of solid supporting surfaces by suspending the water drops at the boundary between two immiscible liquids having different densities, where they were also protected from infection by airborne particles. He also investigated a wide range of drop sizes-varying in diameter from about 20 p to 2 an-and thus volumes which differed by a factor of 109. The use of five pairs of supporting liquids, the members of a pair being practically immiscible with water and with each other, established that the observed freezing temperatures of the drops were a property of the water and not of the surrounding media.Bigg determined the freezing temperatures of large numbers of drops of various sizes, cooled at a constant rate. Fig. 4 shows the distribution of freezing temper- atures of more than 1000 drops each of 1 mm diam. The drops were made from distilled water from which gross impurities had been removed but which still FIG. 4.-The distribution of freezing temperatures of 1127 water drops of 1 mm diam. (after Bigg). contained very small foreign particles. The most frequent freezing temperature was -24°C with half of the drops freezing below this temperature.Thus, if a large volume of water is sub-divided into many smaller samples of equal volume, the freezing temperatures of the latter show a simple probability distribution as illustrated in fig. 4. Because of this statistical character of the nucleation events, it is necessary to determine the freezing points of large numbers of samples in order to obtain characteristic and significant relationships. Such an important relationship is revealed when one plots the median freezing temperature of a large32 NUCLEATION OF WATER AEROSOLS group of drops, i.e. the temperature below which half of the drops freeze, against the logarithm of the drop diameter (or volume). This produces the straight-line relationship shown in fig. 5 which may be represented by the equation log V = A-B(273 - T ) = A - BT,, (11) where V is the drop volume, T the freezing temperature in OK, and A and B are constants for the particular sample of water under test.Bigg's work has recently been checked and extended by Langham and Mason.26 The results obtained with water varying in purity from that of rain water to that produced by multiple distillation showed the same general trends; plots of the median freezing temperatures of groups of drops against the logarithm of their diameters produced straight lines parallel to those of Bigg, but displaced towards lower or higher temperatures depending upon whether the water used was more, or less pure than that investigated by Bigg. Although in these experiments the grosser particles were removed from the water, it still contained large numbers of small particles which can be removed only by taking extreme measures.Eqn. (1 1) therefore represents nucleation by foreign particles-heterogeneous nucleation. This relationship between droplet volume and degree of supercooling may be explained on the assumption that the water was contaminated by particles which were, at one time, airborne, and whose efficiency as freezing nuclei increased exponentially with decreasing temperature in the manner observed for atmospheric aerosols.21 ' The activity of the contained freezing nuclei may thus be represented by n = no exp (aT,), where n is the concentration of nuclei which becomes effective at temperatures be- tween 0°C and - T'"C, and no and a are constants.For drops of volume V being cooled to temperature - TsoC and containing a randomly-distributed population of nuclei, the probability P of a drop containing at least one effective nucleus on reaching the latter temperature is or Now Bigg's empirical relationship shows the value of Ts for which drops of different volumes have a 50 % probability of freezing, i.e. P = 0.5, and for these, eqn. (14) becomes P = l - exp(-Vn), (13) (14) In (1-P) = - Vn = - Vno exp (aT,). Vno exp ( a q ) = const., or In V = In C - a x , which is Bigg's relationship. The value of the constant fixes the shape of the freezing-nucleus distribution, while the value of C depends upon the total con- centration of nuclei, i.e. upon the purity of the water. (b) THE PROPERTIES OF ICE-FORMING NUCLEI In the absence of foreign surfaces, nucleation of the ice phase may occur only by the chance orientation of localized groups of water molecules into an ice-like configuration.A suitable solid particle, however, may cause water molecules to become " locked " into the ice lattice under the influence of its surface force field. Such a molecular aggregate will not only be bound to the surface of the particle, but will have only one exposed surface; on both counts it will be less vulnerable to thermal bombardment than will a spontaneously formed aggregate, and will therefore have a higher probability of attaining the critical size at which it may nucleate the ice phase. The formation of a stable ice nucleus must be largelyB . J. MASON 33 determined by the temperature of the supercooled liquid: and the configuration of the surface force field of the substrate which, in general, will not be uniform, but contain some specially favoured sites for nucleation.The ice nucleating ability of a wide variety of both natural and artificial sub- strates has been tested, usually by dispersing them as fine dusts, smokes or sprays into a cloud of supercooled water droplets, and determining the highest tem- perature at which about 1 in 1.04 of the particles produced an ice crystal. In an attempt to discover the nature and origin of the atmospheric ice nuclei responsible for initiating natural rainfall, Mason and Maybank,27 and Mason 28 tested the nucleating properties of 35 different types of soil and mineral-dust particles.Twenty-one of these, mainly silicate minerals of the clay and mica groups, were found to produce ice crystals in supercooled clouds at temperatures of - 15”C, or above, and of tb.ese, 10 were active above - 10°C (see table 5). The most abundant of these is kaolinite with a threshold temperature of - 9°C ; the kaolin minerals together with the illites and halloysite are considered to be the most important natural sources of efficient ice nuclei. The possibility of inducing rain by the introduction into clouds of artificial nuclei has stimulated many investigations of the ice-nucleating ability of a wide variety of chemical compounds, but there has been little agreement in the results published by different workers. Careful tests in the author’s laboratory indicate that many of the published results are spurious because of the presence, in the air or the chemicals, of small traces of silver or free iodine, leading to the formation of silver iodide which is the most effective of all substances which have been studied so far.If all such trace impurities are removed, many of the substances that have been suggested are found to be quite ineffective. There remain those TABLE 5.-sUBSTANCES ACTIVE AS ICE NUCLEI subs tame coveilite vaterite P-tridymite magnetite ltaolinite anauxite illite metabentonite glacial debris hematite brucite gibbsite dickite halloysite volcanic ash dolomite biotite attapulgite vermiculite phlogophite nontronite natural nuclei crystal symmetry hexagonal hexagonal hexagonal triclinic monoclinic hexagonal hexagonal monoclinic monoclinic hexagonal monoclinic monoclinic monoclinic artificial nuclei crystal threshol d temp.(“C) symmetry temp. (“C) threshold substance - 5 -7 -7 -8 -9 -9 - 10 - 10 -11 -11 - 12 - 12 -13 - 14 - 14 - 14 - 15 -15 -15 silver iodide lead iodide cupric sulphide mercuric iodide silver sulphide silver oxide ammonium fluoride cadmium iodide vanadium pentoxide iodine hexagonal hexagonal hexagonal tetragonal monoclinic cubic hexagonal hexagonal or thorhomb i c orthorhombic -4 -6 -6 -8 -8 -11 -9 - 12 - 14 - 14 listed in table 5 ; the first six, which are only slightly soluble, are active to the extent of about one particle in lo4 producing an ice crystal and the indicated threshold temperature when introduced into a supercooled cloud formed in a cloud chamber.They also cause highly purified bulk water to freeze at these same temperatures. NH4F, CdI2 and 12, being soluble in water, are inactive in a water-saturated atmosphere but produce ice crystals in an environment supersaturated relative to -34 NUCLEATION OF WATER AEROSOLS ice but sub-saturated relative to water, at the temperatures indicated. More de- tailed accounts of these experiments are given by Mason and Hallett,29* 30 Mason and van den Heuvel.31 Although there is a tendency for the more effective nucleators to be hexagondly- symmetrical crystals in which the atomic arrangement is reasonably similar to that of ice, table 5 shows that there are a number of exceptions ; but all substances which are active above - 15°C possess a low-index crystal face in which the degree of misfit between the ice and substrate lattices is less than 15 % (see Bryant, Mason and Hallett 32).However, there is not, in general, a high correlation between the threshold nucleation temperature and the degree of misfit, indicating that nucleating ability is only partly determined by geometrical factors. This is also strongly indicated by observations that epitaxial deposits of ice crystals on large single- crystalline substrates grow preferentially on the growth and cleavage steps, on etch pits, and other imperfections on the surface 32 (see fig. 6(a)). (C) ? h E i HOMOGENEOUS NUCLEATION OF SUPERCOOLED WATER A number of experiments (see Mason21) have confirmed that micron-size water droplets formed by condensation in very clean, particle-free air may be super- cooled to about -4Q"C before freezing occurs. For example, Mossop 33 reported that, in a cloud of droplets of diameter about 1 p produced in an expansion chamber, the number of ice crystals increased rapidly as the temperature fell below -4O"C, and all the droplets were estimated to have frozen after being maintained at - 41.2 f0.4"C for 0.6 sec.As solid particles are excluded from these tiny droplets formed in clean air, nucleation can occur only by small groups of water molecules becoming oriented by chance into an ice-like configuration. Such molecular aggregates will con- tinually arise and disappear as a result o€ thermal agitation but the lower the temperature, the greater will be their size and frequency of formation until, eventu- ally, they attain it critical size above which they survive and continue to grow, forming nuclei for the ice phase.An expression for the rate of formation, fcm-3 sec-1, of such nuclei has been derived by Turnbull and Fisher 34 viz. : nkT I f i ~ exp - [( U -t- Wc)/kT] ,, where n is the number of molecules per cm3 of the liquid, k is Boltzmann's constant, la Planck's constant, U the activation energy for self-diffusion of a molecule in the liquid, and Wc the work of nucleus formation. Now where OSL is the specific surface free energy of the crystalJliquid interface and A is the total surface area of the nucleus. The free energy of formation of an ice embryo containing g molecules is where ps, , u ~ are the Gibbs (or chemical) potentials per molecule in the solid and liquid phases.For (metastable) equilibrium between the embryo and supercooled liquid AG must be a maximum and d AG/dg = 0, when dA dAdV $9 SLdVdg' pL--ptJ = as,-- = B -- where Y is the volume of the nucleus. But d L -(pL-h) = - ( s L - s s ) = - dT T'B . J. MASON 35 where SLY S, are the entropies per molecule and L is the latent heat of fusion. and A4 dA TLdT osL-- - Np,dV where TO is the thermodynamic freezing point (= 273"K), My ps the molecular weight and density of the solid phase and N is Avogadro's number. If the critical nucleus is assumed to be a hexagonal prism with height = short diameter = 2rc, and where L' is now the latent heat per g. Therefore and The probability P that a droplet of volume Y will freeze within a time t is given by or Idt, and dPldt = VI, (26) 1 dPdT I =--- VdT dt' Eqn.(27) may be used to determine 1 at a given temperature from experimental data on the frequency with which a population of droplets each of volume V freezes when cooled at a constant rate. But because the parameter CTSL cannot be calculated with cogfidence, and because a 10 % error would lead to an error of 106 in the calculated value of I, it does not appear profitable to make a close comparison between the nucleation rate calculated from eqn. (25) and that which may be deduced from experimental observations of freezing drops. However, one may use the experimental data of MOSSOP,~~ Carte,35 and of Langham and Mason,26 who studied the freezing af pure water droplets in the diameter range 1-5Qpy to deduce the values of I gt different temperatures in the range - 41°C to - 35"C, substitute these in eqn.(23, and calculate the correspond- ing values of (TSL. Computed in this manner, QSL = 19.0 erg cm-2 at -41"C, 20.0 erg cq-2 at - 38"C, and 20.3 erg cm-2 at - 35°C. Using these and extra- polated values of: OSL, eqn. (25) can be used to predict the temperatures at which water drops will crystallize spontaneously in terms of their volume and the time. The lower curve of fig. 5 indicates the temperatures at wJajch drops of various diameter should freeze when held at these temperatures for a period of 1 sec. This curve,, representing hamogcneous nucleation, differs markedly from that of Bigg's line which represents heterogeneous nucleation.36 NUCLEATION OF WATER AEROSOLS Also in fig.5 are plotted the observations of a number of workers who have been able to supercool small droplets (10-3Op diam.) down to temperatures ap- proaching -4O"C, and also a few exceptional cases, recorded in the literature, of much larger volumes being supercooled to unusually low temperatures. The close grouping of these observations about the theoretical curve strongly suggests that, in most cases, the nucleation of the water samples was homogeneous and that foreign nuclei were not involved. It is not possible to make a more exact com- parison between the experimental data and the theory because the cooling rates employed are not always stated in the original papers, but they are unlikely to have differed by more than tenfold from those used by Bigg, Carte and MOSSOP, in which case, the temperature corrections would always be less than one degree.equivalent drop diameter FIG. S(a).-Bigg's relationship between the median freezing temperatures and the diameters of water drops containing foreign nuclei (heterogeneous nucleation). @).-Median freezing temperatures for groups of droplets of very pure water having diameters <500p and the lowest freezing temperatures recorded for drops of d>500p. These experimental data lie close to the curve calculated from the theory of homogeneous nudeation (eqn. (25)). An earlier analysis of this kind encouraged Langham and Mason 26 to try and produce, in appreciable quantity, water entirely free from foreign particles and to study systematically the homogeneous nucleation of large numbers of drops varying in diameter from about 10 p to 1 mm.Purification of water to this degree proved difficult but, using a multiple distillation technique in which extreme pre- cautions were taken to exclude room air, to remove particles from the inner walls of the Pyrex still, and to prevent ebullition, they produced drops of up to 2 lzlpn diam. which could be regularly supercooled to temperatures very close to limits indicated by the theory of homogeneous nucleation (see fig. 5). Altogether the agreement between the theoretical curve and the experimental data of the several different workers, who together covered a volume range of more than 1012, is rather impressive. There is, therefore, a substantial body of experi- mental evidence to suggest that Bigg's result, expressed by eqn.(11) and the line in fig. 5, represents the relationship between the volume of a water drop and the depth of supercooling for heterogeneous nucleation, and that the curve of fig. 5, which is based on eqn. (25), represents the corresponding relationship for homo- geneous nucleation.FIG. 6(a).-ice crystals growing epitaxially at growth steps on a basal surface of cadmium iodide. FIG. 6(h).-Oriented deposit of ice crystals growing on a surface of freshly-cleaved mica. [To face page 36. B . J . MASON 37 FORMATION OF ICE CRYSTALS DIRECTLY FROM THE VAPOUR There has been considerable controversy as to whether ice crystals may form by direct deposition from the vapour without the intervention of the liquid phase.In heterogeneous nucleation, Mason et aZ.31~32 have shown that, in order to form ice crystals on various crystalline substrates at temperatures only a few degrees below WC, it is necessary to cool the air to the dew point but, at lower temperatures, ice crystals appear when the air is sub-saturated relative to water but supersaturated relative to ice to a degree which is independent of the tem- perature. In this latter regime, no visible deposit of water droplets appeared and since the criterion for ice-crystal formation was the degree of supersaturation rela- tive to ice and not the relative humidity (which sometimes exceeded 95 % without ice crystal formation), it may well be that crystals were formed by a direct vapour- to-solid transition. In precipitating water vapour by rapid expansions in cloud chambers, Sander and Damkohler,7 and also Pound et aZ.9 reported that, until terminal temperatures of - 62°C to - 65°C were reached, the condensation appeared as spherical particles but, at lower temperatures, as a cloud of angular, glittering ice crystals.Sander and Damkohler suggested that either the crystals were formed by direct homo- geneous sublimation of ice from the vapour, or that - 62°C represented the crystal- lization temperature of what, at higher temperatures, had been liquid droplets. Pound et a/. suggested that because homogeneous nucleation of liquid water droplets at -40°C had been well established, the particles appearing between this temperature and - 65°C were frozen droplets which had not yet developed crystal- line faces large enough to cause specular reflection.A similar suggestion was made earlier by Mason 41 to account for the appearance of iridescent mother-of- pearl clouds at temperatures of about -80°C in the stratosphere. Certainly, substitution in eqn. (2) of the parameters which are relevant to a vapour-solid transition suggests that, during the cooling of water vapour, the first condensation product would be liquid droplets rather than ice crystals although, at temperatures below -4O"C, the droplets would freeze within a fraction of a second. At very low temperatures, around - 140"C, the condensate might be in the form of amor- phous, non-crystalline ice as observed by Blackman and Lisgarten.42 Experimentally, Maybank and Mason43 have shown that during the rapid expansion of clean, moist air, ice crystals appeared only when the saturation ratio exceeded 4 and the terminal temperature fell below - 4.O0C, suggesting a two-stage process.At lower temperatures and correspondingly higher supersaturations the numbers of ice crystals increased smoothly and no evidence was obtained for a sudden increase or change of mode of formation at any temperature down to about -100°C. Accordingly, there appears to be no convincing evidence that water molecules condense homogeneously into the ice lattice without first going through the less-ordered liquid structure. 1 Becker and Doring, Ann. Physik, 1935, 24, 719. 2 Zeldovich, J. Expt. Physics (Russ.), 1942, 12, 525. 3 Wilson, Phil. Trans. A , 1899, 193, 265. 4 Powell, Proc. Ray. SOC. A , 1928, 119, 553. 5 Volmer and Flood, 2. physik. Chem. A , 1934,170, 273. 6 Frey, 2. physik. Chem. B, 1941, 49, 83. 7 Sander and Damkohler, Naturwiss., 1943, 31, 460. * Barnard, Ph.D. Thesis (Glasgow Univ., 1954). 9 Pound, Madonna and Sciulli, Airforce Cambridge Research Centre, Geophys. Res. Pap., no. 37, 1955. 10 Mason, Proc. Physic. Sac. B, 1951,64,773. 11 Coulier, J. Pharrn. Chim. Paris, 1875,22, 165. 12 Aitken, COIL Scientijic Papers (ed. Knott) (Camb. Univ. Press, 1923). 13 Wilson, Phil. Trans. A, 1897, 189, 265.35 NUCLEATION OF WATER AEROSOLS 14 Frzibram, Sitz. Akad. Wiss. Wien, 1906, 115, 81. 15 Laby, Phil. Trans. A, 1908, 208, 445. 16 Andrkn, Ann. Physik. (Lpz.), 1917, §2, 1. 17 Flood, see Volmer, Kinetik der Phasenbildung (Steinkopff, Dresden), 1939, p. 132. 18 Eoeb, Kip and Einarsson, J. Chem. Physics, 1938, 6, 264. 19 Scharrer, Ann. Physik, 1939, 35, 619. 20 Tolman, J. Chem. Physics, 1949, 17, 333. 21 Mason, The Phy,sics of Clouds (Clarendon Press, Oxford), 1957. 22 Twomey, Bull. de I’Obs. de Puy de D h e , 1960, p. 1. 23 Woodcock, Kientzler, Arons and Blanchard, Nature, 1953, 172, 1144. 24 Mason, Nature, 1954, 174, 470. 25 Bigg, Proc. Physic. SOC. B, 1953, 66, 688. 26 Langham and Mason, Proc. Roy. SOC. A, 1958,247,493. 27 Mason and Maybank, Quart. J. Roy. Met. Soc., 1958, 84, 235. 28 Mason, Quart. J. Roy. Met. SOC., 1960, in press. 29 Mason and Hallett, Nature, 1956, 177, 681. 30 Mason and Hallett, Nature, 1957, 179, 357. 31 Mason and van den Heuvel, Proc. Physic Soc., 1959,74, 744. 32 Bryant, Hallett and Mason, J. Physic. Chem. Solidr, 1959, 12, 189. 33 MOSSOP, Proc. Physic. SOC. B, 1955, 68, 193. 34 Turnbull and Fisher, J. Chem. Physics, 1949, 17, 71. 35 Carte, Proc. Physic. SOC. B, 1956, 69, 1028. 36 Jacobi, J. Met., 1955, 12, 408. 37 Pound, Madonna and Feake, J. Colloid. Sci., 1953, 8, 187. 38 Meyer and Pfaff, 2. anorg. Chem., 1935, 224, 305. 39 Wylie, Proc. Physic. SOC. B, 1953, 66, 241. 40 Bayardelle, Compt. rend., 1954, 239, 988. 41 Mason, Quart. J. Roy. Met. Soc., 1952, 78, 22. 42 Blackman and Lisgarten, Proc. Roy. SOC. A, 1957, 239, 93. 4 3 h M ~ ~ n and Maybank, Proc. Physic. SOC., 1959,74, 11.