The Supercooling and Nucleation of Water. A

THE SUPERCOOLING AND NUCLEATION OF WATER.

A THESIS SUBMITTED BY

INGO EDWIN KUHNS

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE

UNIVERSITY OF LONDON

September 1966. Acknowledgements.

This research was carried out at Imperial College, 1963-1966, under the supervision of Dr. B.J. Mason, F.R.S., now Director General of the

Meteorological Office. I am grateful to him, and to those other members of the Department of Cloud Physics, for those discussions which have moulded the shape of the research presented in this Thesis.

I am indebted to the Senate of the University of London for the awards without which I could not have undertaken this research.

These awards were the Granville Scholarship in Physics, 1963-1964, and two Postgraduate Studentships for the years 1964-1965 and

1965-1966. ABSTRACT.

When water freezes one of two known types of freezing is responsible for the phase change. Chemically pure water devoid of any tiny insoluble particulates will supercool to between -35°C and -41°C depending on the volume of the sample and the rate at which it is cooled. In this type of freezing, called homogeneous nucleation, the crystallisation nuclei arise spontaneously by thermal fluctuations in the bulk liquid. The second type of freezing, initiated by a crystalline substrate or by tiny insoluble particles suspended in the water, prevents such deep supercooling: the freezing temperature of the water sample is determined by the physical properties of the particle as well as those of water. This type of freezing is called heterogeneous nucleation. This thesis concentrates on a study of homogeneous nucleation.

A review of the literature relevant to this thesis is presented, with the classification of experimental research into one of these two classes. Recently, though, reports have appeared in the literature of water existing in the liquid form well below -40°C in the high atmosphere

( at the top of intense storm cells, 50,000 ft. or more in height). These observations seem incompatible with laboratory research on ' homogeneous nucleation.

A new technique has been developed to examine the freezing behaviour of airborne ultra-pure water droplets. This technique does not suffer from the disadvantage that nucleation may accidentally occur through contamination with tiny particles or the substrate itself. The measurements of the freezing temperatures of airborne droplets ( described in this thesis ) show results that are in accordance with the theory of homogeneous nucleation and other laboratory measurements. The model which has gradually been pieced together to describe the supercooling of water is shown to be essentially the same as the

' flickering cluster ' model of liquid water, which has recently had great success in explaining many of the regular and anomalous properties of liquid water. Moreover, it is possible to show that a simplified statistical- mechanical model of liquid water will predict the phenomenon of supercooling and determines the homogeneous nucleation temperature to be in the vicinity of -30 to -40c0.

After nucleation. ,the mechanism of freezing of small droplets cooled in free fall in different gases was found to be strongly dependent on the cooling gas. At atmospheric pressure these droplets shatter violently in helium and hydrogen, whereas in carbon dioxide gas many tiny ice splinters are ejected from the surface of the freezing droplet. This problem of shattering and splintering is important in the problem of ice crystal multiplication in natural clouds and the electrification of the thunderstorm.

Following this discovery, a reassessment has been made of the earlier laboratory research on this problem. INDEX CHAPTER I. Introduction to the theory of supercooling - a statement of the problem. Sec. 1.1 Homogeneous crystallisation. 1 1.2 Heterogeneous crystallisation. 11 1.3 Aims of the research in a physical context. 12 1.4 Homogeneous freezing in a meteorological context. 14 1.5 Conclusions of this chapter. 19 CHAPTER II. A survey of the literature.

Sec. 2.1 Historical introduction. 21 2.2.a Theory of the crystallisation of a pure liquid. 30 2.2.b Adaption of the nucleation equation for the homogeneous crystallisation of water. 34 2.3 Experimental observations of the supercooling of water. 46 2.4. The experimental distinction between homogeneous and heterogeneous nucleation of water. 56 2.5 Homogeneous nucleation of large volumes of water. 57 2.6 Intermediate volumes 1094- d 4 1mm. 59 2.7 Small droplet sizes < d 1004. . 61 2.8 General conclusions drawn from the literature survey. 69

CHAPTER III. The design of a new technique. Sec. 3.1 General considerations. 71 3.2 Design and construction of the freezing chamber and the optical system. 83 3.3 Particle filtration of the gas cooling medium. 86 3.4 Water purification. 88 3.5 Temperature measurement. 97 3.6 Operation of the experimental apparatus. 100

CHAPTER IV. An analysis of the experimental data. Sec. 4.1 A description of the freezing histograms. 107 4.2 An estimate of the cooling rate effect from the histograms114 4.3 An assessment of the thermal relaxation effect. 120 Sec. 4.4 Derivation of the true mean freezing temperature at a standard cooling rate. 122 4.5 Discussion of the physical model describing the supercooling of water. 130 4.6 Conclusions drawn from this section of the research. 139

CHAPTER V . The splintering and shattering of freezing droplets and their relation to the rapid glaciation of natural clouds. Sec. 5.1 Evidence for a rapid glaciation mechanism. 142 5.2 Early ideas on the glaciation mechanism. 146 5.3 Previous research on droplet splintering and shattering and some observations made during this research. 158 Summary and conclusions drawn from Ch. V . 168

APPENDICES

2.1 A derivation of Wfor a droplet in a supersat. vapour. 170 2.2 It It II be crystalline aggregate in a supercooled liquid. 171 2.3 Calculation of the radiation effect. 172 4.1 Physical constants of the gases used in this research. 174 4.2 Freezing point depression by carbon dioxide at -40°C. 175 4.3 A method for measuring the thermal relaxation time of small droplets in free fall. 176 CHAPTER I

INTRODUCTION TO THE THEORY OF SUPERCOOLING - A STATEMENT OF

THE PROBLEM.

Section 1.1. Homogeneous crystallisation

Elements and chemical compounds may exist under suitable prevailing conditions of temperature and pressure in three different physical states, viz.

(a) the solid state, generally stable at low temperatures and/or

high pressures,

(b) the liquid state, stable at intermediate temperatures and

pressures, and

low pressure.

Liquids can thus be said to constitute the 'intermediate' state of matter through which solids normally pass into vapours. These

pressure and temperature regions are most conveniently represented on the standard thermodynamic phase diagram, as in Fig.1.1.

At a given fixed pressure, for example p/ marked in Fig.1.1, the solid can be heated to a temperature Trn whereupon the solid changes to a liquid, and further, the liquid can be heated to a temperature

T b before it changes entirely to the vapour state. It appears then, 2

FIG. 1.1

PHASE DIAGRAM FOR SOLID, LIQUID AND GAS.

LIQUID

Critical Point Pc;Tc B A

Tm TEMPERATURE 3 that at a given pressure a liquid has a well defined region of temperature Tm to TB in which it is the stable state.

These 'stable' regions are characterised in the theory of thermodynamics by the thermodynamic potential q(p, T). The boundary lines dividing the states are defined by the equality

for the solid-liquid line, and cr liquid = CPsolid = cPliquid qvapour for the 1 iquid-vapour transition line. At a fixed system

m pressure such as p in Fig.1.1., up to the temperature T solid whilst at temperature greater CP liquid' at Tm )solid = 9 liquidi than Tm (5> The theory of equilibrium thermodynamics nquidzq solid' postulates that the particular configuration adopted by the system is that for which the thermodynamic potential cp is a minimum. Thus,

One would not expect to find a substance in the liquid state in that particular region of the phase diagram where CP solid liquid' With the discovery of the thermometer, Fahrenheit, 1724, was the first to observe that if a liquid (namely water) were cooled at constant pressure as from A --**B in Fig.1.1.1 it could be cooled to the point B still remaining in the liquid state! Fahrenheit had chosen for his samples of water volumes of a few cubic centimetres of boiled rainwater sealed in glass bulbs. These bulbs were exposed overnight to heavy frost; the boiled rainwater could be supercooled to -9.4°C before freezing. Since this discovery, the study of supercooling and the closely related phenomena of supersaturation of vapours and solutions has been undertaken by many various chemists.

Returning to Fig.1.1., a liquid existing in the region to the left of the line marking the fields ir which the liquid and solid 4

are stable phases is said to he 'supercooled'9 or sometimes in a

state of'suspended crystallisation The liquid in this region is

in a metastable state, that is, although a liquid can exist in the

supercooled state for very long periods of time (de Coppet, 18(2,

kept salol supercooled for six years) the laws of equilibrium

thermodynamics tell us that this is not the most stable state.

Water is not the only liquid known to be supercool. In fact,

in the light of present knowledge supercooling of the liquid state

must be regarded as a normal, general phenomenon. A short list

of liquids is given which have been reported to supercool, together

with the amount of supercooling attained where this is reported.

(i) Inorganic Liquids

Phosphorus 135 deg.0 Sulphur 160 deg.0

Sulphuric Acid 127

Water 40

Silica 513

(ii) Organic Liquids

Glycerol 120

Glucose

Ethyl alcohol

** Salol,Betol,Piperine 60 or more

iso-butyl alcohol 50

n-propyl alcohol 50 5

(iii) *** Molten Metals

Large volume Small irolume

Mercury 14 deg.0 46 deg.0 Silver 227 deg.0 Gallium 55 76 Germanium 219 Tin 31 110 Gold 221

Bismuth 30 90 Copper 236

Lead 80 Nickel 319 Platinum 37 Palladium 330

Footnote

Hildebrand et.al. Proc.Int.Conf.Crystal Growth N.Y. 1958.

** Michnevitch, Dissertation, Odessa 1941.

* * * Turnbull, D. Symposium on thermodynamics of Phys.Met.Am.Soc.

Metals. 1950, 282.

In view of the wide variety of liquids known to supercool, it would be a great step forward if a general theory of supercooling and nucleation could be developed in terms of molecular properties, so that experimental research could assist in deriving some molecular parameters. Unfortunately, such a general theory has not yet been developed, and as a 'pons exilis' we have for our use a general theory of supercooling which makes use of macroscopic parameters to describe the thermodynamic behaviour of small molecular systems, e.g. the bulk latent heat of fusion, or interfacial surface energies.

A physical model for the illustration of phase transitions is the formation of 'molecular aggregates' of the new phase arising and 6 disappearing through the influence of thermal fluctuations in the mother phase. Local energy fluctuations create suitability 'cold' regions which enables aggregates to form, and 'relaxation' occurs when the necessary energy of 'melting' becomes available, the energy exchanges occurring at the boundaries of the aggregates.

J.W. Gibbs (1876,1878) recognised that such small aggregates can exist in equilibrium with the mother phase under special conditions, though he makes no mention of the manner in which these aggregates might arise. He established from the equality of the thermodynamic potential per particle in the two phases that the work of formation of a spherical aggregate is given by

w = 167ra 3 )2 3 (p - P I where - Pi ) = 2d (1.2.) Cp where 1 is the 'radius' of the aggregate)Cris the interfacial surface tension. p is the 'pressure' of the new phasej p is the 'pressure' of the mother phase. The necessity for the two phases to exist at different pressures arises from the fact that at constant temperature the thermodynamic potentials cil° liquid (p T), C4fsolid (p, T) can only be equal if their 'pressures' are unequal. This difference of 'pressure' is provided by the capillary term ( 26). Gibbs considered the work W to be a measure r of the'stability' of the mother phase with respect to a phase transition.

Einstein 1910, and Smoluchowski 1908, were the first to clarify how these different phases may arise, though their work is restricted to the phenomenon of opalescence near the critical point. They considered that the state of a system might fluctuate about its equilibrium state, and derived very simply an expression for the

probability of a specified fluctuation in the following way:

Suppose a system is characterised by a state in which it has

W equally probable configurations. Boltzmann's relation determines o the entropy of the state to be

S = k log W + const. o o (1.3.)

If now the state of the system changes so that it can only occupy

W of these configurations, then the entropy of this new state will

S = k log W + const. (1.4.)

Evidently the entropy change will be (So - S). Now if all these

configurations are equally probable, W affords a measure of the

probability of a system being in a certain state. Then the ratio

of the probability for two different states given by (1.3.) and

(1.4.) is

P = e (1.5.) F 0

If P o o is the equilibrium state, P e.1. Now for a system whose total O total energy remains constant, at constant temperature

tesul -Follows -fro., it,. f" 14.5of-- 11-en... (S - S) = (.14, w) o reVq.-rSible 15 T chewle, re '3E4- d'r]

P = e and therefore kT (i. 6 . ) P 0

This important result shows that the probability that the system takes up a state different from the equilibrium state is proportional to the work which has to be done to disturb the system to the state of interest. If the equilibrium state is a homogeneous volume of liquid) such a fluctuation could be a state which includes one or more associated aggregates.

Volmer and Weber (1926) postulated that the growth and disappearance of these molecular aggregates are due to the local energy fluctuations predicted by Einstein and Smoluchowski. Since the Aw AcT probability of an energy fluctuation varies as e then 3 -1 the number of aggregates arising \/, cm . sec ) requiring an energy of formation 4 will be

AT K(T)e A65 (1.7.)

where K(T) may be a slowly varying function of temperature.

Equation (1.7.) is quite general and is applicable to gas-solid, gas-liquid, or liquid-solid transitions.

Fig.1.2. shows how the work of formation of formation of an FIG. 1.2

surface energy term

reduction due to cps being less than Y

n+1 vu

FIG. 1.3

L.) kT U7 kT

fluctuation freq. activation energy control. control.

I ncreasin• su•ercoolin 10 aggregate varies with the number of molecules bound in the aggregate. (The horizontal axis could equally well represent the

'radius' of the aggregate through the relation noC r3). This work is the balance of two terms:

(a) the work needed to form a surface with associated surface energyd per unit area,

(b) a reduction or gain in work due to the aggregate becoming ever more like the thermodynamically stable phase whose potential is less than that of the surrounding phase at the appropriate conditions of temperature and pressure. 2 In short, the work needed to form a surface varies as r , and the reduction due to the thermodynamic potential difference between the phases varies as r3. Consequently, the work must have a maximum, *, for a particular radius, r*, or number of molecules n*. From Fig.1.2. it can be seen that once an aggregate exceeds a certain size r*(T), it will continue to grow through the mother phase and in this manner cause a phase change. A prerequisite, then, to a phase change is the formation of an aggregate by local energy fluctuations of sufficient magnitude such that the critical number of molecules, n*, linked together in the new phase as a result of this energy fluctuation is exceeded. This approach to the phenomenon of supercooling (or any other phase transition for that matter) will be dealt with theoretically in greater detail in

Chapter 2.

In the light of this discussion, it is apparent that the ability of a liquid to supercool can be attributed to the failure 11 of the liquid to form within itself aggregates of sufficient size

(21)i n*) capable of initiating the phase change at (or below) the temperature TM. Although it has not been stated explicitly in this section, the physical model describes the formation of aggregates in terms of the molecules of the chemically pure element or compound only. The eventual temperature at which a phase change occurs will then be a property of the element or compound only.

This type of phase transition is said to be homogeneous.

Section 1.2. Heterogeneous Crystallisation

It is clear from the equation

- aa.*/kT :77 * = K(T)e that if Acit *1 the energy barrier to the phase change is reduced, the temperature departure from the equilibrium temperature TM will be much reduced when a phase change occurs. It is well known that in the case of supercooled liquids that crystallization can be initiated at temperatures above the homogeneous nucleation temperature by the addition of so called 'nucleating agents' which 7' act by lowering the energy barrier AT*. In the case of water, condensation from the vapour is assisted by dust particles and ions. In the liquid-solid transition, the nucleating agents are solid insoluble particles whose effectiveness (the amount by which ,6(5* is lowered) will depend on the following parameters:

(i) its size, characterised by a radius 'r'.

(ii)the degree of isomorphrism, i.e. similarity of crystal symmetry 12

and lattice spacing of the insoluble particle with the solid phase,

(iii) the 'wettability' of the solid surface; by the liquid,

(iv) the existence of'sites' on the surface, such as edges, cracks, steps, which act as preferential nucleai,ion centres.

The study of heterogeneous nucleation is as much a study of

the nucleating agents as the liquids. Because heterogeneous

nucleation is not the main theme of this research a full discussion would be out of place. Instead references are given which treat

the study adequately (Chalmers 1964, Mason 1957, Fletcher 1962).

Section 1.3. The aims of this research on homogeneous

freezing of water in a physical context.

Equation (1.7.) of section 1.1. is not restricted to any liquid in particular. Referred to water, it is the equation which will be developed theoretically in Chapter II to describe the homogeneous freezing of water. Water was chosen as the liquid to be investigated for two main reasons:

(a) it can be purified to a high degree by simple physical

means,

(b) there arise some unanswered questions concerning the

freezing of supercooled water in the atmosphere.

These questions will be stated explicitly in section

1.4.

In a physical context, the aim of this research is to develop a new method by which samples of water of varying volume can be consistently frozen homogeneously. 13

Consider a volume of water v cm3 held at a supercooling

T °C (below 0°C) for a length of time t seconds. By definition s * of the quantityy (T), when

jr(T) . v. t ti 1 (1.8.)

the volume of supercooled water has a high probability of freezing.

Equation (1.8.) is only approximate; a more vigorous treatment will be given in Chapter 4.

Because of the statistical nature of the local fluctuations which produce these aggregates there will be a spread in time over which these water samples will freeze at a given fixed supercooling.

Alternatively, if many identical water samples are cooled at constant o -1 rate ( C sec ) there will be a scatter of measured freezing temperatures. It would be of theoretical as well as of practical interest to measure the extent of this scatter of freezing temperature.

Likewise, for a given duration of supercooling a larger sample of water is more likely to freeze at a diminished supercooling compared to the smaller sample, since through having a larger number of molecules it is more likely to experience a critical fluctuation.

This implies a mean volume effect for constant duration of supercooling or constant cooling rate. It should also be possible to calculate from experimental measurements the magnitude of the energy barrier * over the range of temperature amenable to analysis in this research. Having calculated 24 *, it is then possible using the theory developed in Chapter 2 to estimate the number of molecules 14

which participate in such a fluctuation. Lastly, by gathering together these pieces of information and by suitably controlling the experimental technique, an assessment will be made as to whether the water samples are freezing homogeneously or heterogeneously.

Section 1.4. Homogeneous freezing in a meteorological context.

Clouds composed of supercooled water droplets are a common occurrence in the atmosphere. The degree of supercooling is very variable: investigations by aircraft flights e.g. Peppier 1940,

Findeisen 1942, Koenig 1963, show that cloud regions at temperatures of -10°C may consist largely of ice particles, yet on occasions clouds are found to be without any detectable ice particles at temperatures as low as -32°C. Fogs of liquid water droplets at -37°C have been reported from Alaska and Verkhoyansk, Siberia.

Earlier research on the homogeneous nucleation of water suggested that pure water droplets ranging in size from 1-50 microns radius cannot freeze within the life time of a cloud at temperatures above

-35°C without being heterogeneously nucleated. These nucleating agents are tiny insoluble particles called 'ice nuclei'. Presumably, these ice nuclei enter the water droplets by first acting as condensation nuclei during the formation of the cloud, or else are picked up by collision processes actually within the cloud itself. The range of supercooling of clouds indicates the variability of the freezing properties of these nuclei; the observations of liquid water fogs at low temperatures indicate the absence of efficient ice 15

forming nuclei even close 'CO 1,h c.arth's surface.

It is well known that in middle and northern latitudes, the

freezing of supercooled water droplets often marks the onset of the

precipitation process. The mechanism, first suggested by Wegener

1911, and further developed by Bergeron 1935, Findeisen 1938 proceeds

in the following way:

In a continually rising, expanding and cooling volume of air,

a height is reached when the temperature becomes sufficiently low

that the water vapour mixed with the rising air becomes supersatur-

ated. In the presence of condensation nuclei the vapour will con-

dense out forming cloud droplets marking the level of the cloud

base. The droplets will continually grow as the air rises and cools

below 0°C. Depending on the presence and efficiency of the particles

now acting as freezing nuclei, a degree of supercooling will be

reached at which the droplets will freeze heterogeneously. Because

the equilibrium vapour pressure over ice is less than that over

supercooled water, in a maintained upward movement of air the ice

particles will grow preferentially becoming larger than the surround-

ing water droplets until they are large enough to fall against the upt coagulating with other droplets and crystals as they do so, until,

when the 0oC level is reached in fall, the frozen aggregate melts

and falls as rain.

In view of the importance of the initiation of the ice phase

in the production of rain, one of the main tasks of cloud physics

research has been to establish the reasons for the variability of 16

temperature at which ice particles are formed in natural clouds

as a consequence of the varying concentration and efficiency of

different types of ice nuclei.

Suppose now the cloud is deficient in ice nuclei. At what

temperature will the cloud droplets freeze? Quite clearly, this

will be the homogeneous freezing temperature, near -40°C, at which

supercooling a small droplet will freeze within a period of a few

seconds. If this is the homogeneous freezing temperature, then it

must be impossible to find bulk water cooled below this temperature.

Recently Simpson (1963) has reported that aircraft pilots fly-

ing on research missions have encountered riming on their windscreens

when flying into intense convective cells in which the surrounding

air temperature was below -40°C. Riming indicates the accretion

of supercooled water droplets, in these instances below -40°C.

Similar reports came from the Severe Storms Laboratory, Oklahoma,

where on a field project in 1966 I had the chance to discuss this

problem with the personnel,but unfortunately not with the pilots.

The heights attained by the aircraft are commonly around 50,000 ft.,

but the U-2, whilst being able to climb above 65,000 ft., could not

enter the storm cells because of its delicate structure. The pen-

etration aircraft are commonly the F 100 jet fighter aeroplanes, adapted for research use; they penetrate these cold cloud regions at near sonic speed. From the reports there seems no doubt that riming does occur (Simpson, private communication), but it is difficult to assess whether riming at these high speeds does imply the existence of supercooled water for the following reasons:

(a)wind shields are heated to melt the accreted ice.

(b)instrumentation and accurate temperature measurement are

still major technical problems.

(c)at these sonic speeds the kinetic energy of impact may be sufficient to melt a thin layer of the frozen particle which

quickly re-freezes to form the beginning of an accreted layer.

Thus, if all the kinetic energy of impact goes into warming the

ice particle, the elevation of temperature can be calculated since wi.V1". = M. . 6T AT I • I • --2 0 4.2 x107

If 17 f.,.=-! 700 1. ere' °II."'" 350 Ill. Stc-1 AT 15°C.

This heating will not be distributed evenly throughout the ice

particle, and this order of magnitude calculation shows that a thin

layer of ice could easily be melted to re-freeze again.

The issue at stake here is whether previous laboratory research

on 'homogeneous' nucleation of water is really homogeneous and

applicable to the atmospheric situation of airborne water droplets.

These aircraft reports are not alone in claiming the existence of

supercooled water below -40°C. Some scientists have suggested that

water can be supercooled to -70°C in_n the laboratory (Rau, 1944,

Regener, 1941), but since these reports were written it has been

found that faulty techniques were responsible or mis-interpretation

of experimental observations. 18

The irridescorce of high nirrus cllouds, or mothor-of-pearl cloud formed at heights around 27 Km. at prevailing temperatures as low as -60°C to -80°C was also once quoted as evidence that liquid water may exist et such low temperatures. The irridescenGe, the pink and green tints of the clouds, is believed to be caused by the diffraction of the sun's rays by spherical particles, indicative of the liquid phase rather than the crystalline hexagonal shaped ice particles normally found in cirrus clouds.

Mason(1957)points out that the condensate may be a vitreous ice which has not had time to grow crystalline faces at -80°C or so when the growth rate is very slow. To support this suggestion, one might mention Klinov's work, 1959, which shows photographs of small particles captured and held in the Verkhoyansk fogs at -53°C. These are spherical frozen droplets many of which show no signs of crystalline faces. Indeed, Klinov writes that when these slides are held up to the moon light, distinct diffraction rings are seen.

Whilst so very little is known about mother-of-pearl clouds, this last piece of evidence for the existence of liquid water below -40°C must be regarded as being inconclusive.

The strongest evidence so far for the existence of liquid water below -40°C is from the aircraft flights, although, as has been mentioned, the interpretation of the phenomenon of riming at sonic speeds is open to criticism. 19

Section 1.5. Conclusion of this Chapter

On both physical and meteorological grounds there is good reason for studying more closely the homogeneous freezing of water.

It would be interesting to see to what extent this freezing behaviour can be described by the theory to be outlined in the next chapter, and also to what extent current theories that have been

put forward for the molecular structure of water are compatible with the observations of these large supercoolings.

The apparatus envisaged for this research resembles more closely the conditions in the atmosphere than any other attempts so far.

Consequently, some of the questions posed by meteorological observations mentioned in section 1.4. can now be answered with more confidence than in the past. FIG. 2.1 20

H ISTORICAL 2.1 REVIEW

THEORETICAL EXPERIMENTAL 2.2 2.3 DEVELOPMENT OBSERVATIONS

DISTINCTION BETWEEN HOMOGEN- EOUS AND HETEROGENEOUS 2.4 NUCLEATION

HOMOGENEOUS NUCLEATION

LARGE INTERMEDIATE 2.5 2.6 SAMPLES SAMPLES

SMALL SAMPLES

AIRBORNE 2.7a

SUPPORTED 2.7b 2.8

CONCLUSIONS OIL SUSPENSION 2.7c CHAPTER II

A survey of the literature

As the subject of supercooling dates back nearly two hundred and fifty years there are numerous relevant reports in the literature. In order to develop a comprehensive survey of this literature,a block diagram is presented in Fig. 2.1. which shows schematically the way in which this chapter is written. Continual reference to this diagram as the relevant sections appear will make this survey more comprehensible.

Section 2.1. A historical introduction*(See footnote)

Since Fahrenheit's discovery of the supercooling of water in

1724 a great number of supporting observations have been reported.

Other than Fahrenheit many early chemists have contributed to the study, among these being the names Black (1775), Gay-Lussac (1813),

Faraday (1826), Berzelius (See bibliography), Lecocq de Boisbaudrain

(1886),de Coppet (1872) and Ostwald.

* Some of this material is drawn from "Kinetik der Phasen bildung"

M.Volmer, Dresden and Leipzig 1939, and Ostwald's"Lehrbuch der

Allgemeinen Chemie", Leipzig 1896. 22

These early chemists were often concerned with the supersaturation of salts in solution as well as with supercooling or the supersolubility of gases in liquids. However, increasing numbers of observations showed that supercooling, supersaturation and supersolubility were connected in more than one respect, e.g. the necessary absence of the solid phase before supercooling or supersaturation could be achieved, or the ability of minute particles to cause the relaxation of supersaturation or supercooling. Different observers, repeating the same measurements were unable to agree on the parameters when the solid or gas phase came out of solution, or when a supercooled liquid crystallised.

The nature of this unpredictable behaviour remained unclear.

In the latter half of the last century, Redi and Spallanzani

(see bibliography) made an important step forward by noting that just as organic biological cultires are affected by minute living organisms, so might the supersaturation or supercooling be lessened by the presence of tiny insoluble particles ever present in the atmosphere. They carefully filtered and cleaned the air in contact with all solutions and found that the behaviour of supersaturated solutions changed radically. This idea was used to account for the discordance of earlier observations of supersaturation and supercooling. Although the origin, size, and constitution of these 'nuclei' were unknown, it was recognised that to be efficient nucleating agents, these nuclei ought to be isomorphous with the resulting solid phase. Ostwald, 1897, showed very convincingly how small these nuclei could be. He took a human 23 hair which, when dipped into a supersaturated solution caused no deposition, yet when gently scraped over the solid salt and then dipped into the solution produced immediate deposition, Secondly, he took drops of very dilute solution on a spatula and by evap- -10 orating the solvent produced nuclei of order 10 gm,. which he found to be effective. The freezing of supercooled water by allowing contact with atmospheric air or by shaking or scraping the container (Walker 1788, Blagden 1788, Young and Cross et.al. 1911) can naturally be attributed to infection by tiny nuclei. Early work on the dissolution of gas bubbles fell into this category.

Introducing tiny particles into supersaturated gas solutions caused myriads of tiny bubbles to be released in the wake of the falling

particle (SchOnbein 1837, Gernez 1865-1875, Schr;der 1869-1871).

There was thus another piece of evidence to connect supercooling, supersaturation, and supersolubility.

A third piece of evidence arose from the work of Lecocq de

Boisbaudran (1866) who showed that spontaneous deposition of salt from solution could take place without the introduction of artificial nuclei only if the solution was strongly supersaturated.

A similar result was obtained for the release of gas bubbles from solution. The spontaneous production of natural nuclei in strongly supercooled liquids without the deliberate introduction of artificial nuclei was not recognised until the research of Tamman (1903) and

Othmer (1915) had been reported.

De Coppet (1872) tried to introduce the idea of homogeneous 24 nucleation in quantitative form, noting that spontaneous crystallisation only occurs at strong supercooling. He came to the idea of an average lifetime for a supersaturated/supercooled liquid and concludes that on average, the stronger the supersaturation/supercooling the shorter this lifetime will be. Also, the larger the sample, the shorter will be the average lifetime. In other words, he was trying to establish the equation (1.8.). De Coppet (1907) verified these de- ductions by observing experimentally the freezing behaviour of salol.

He had, in fact, established the statistical nature of the phase change process.

Questions regarding the mechanism of this building up of super - saturation/supercooling remained unanswered until Volmer's work in 1926.

As Volmer points out, this is very surprising since the theoretical work of Gibbs (loc.cit.) had been established, yet had been overlooked or misunderstood. Because of this, much of the research following Gibbs' publication was conducted on an empirical fashion rather than on a physical approach.

Haber (1922) had the essence of a picture of a phase change, yet how his treatment applied to the phenomenon of supercooling is not at all clear. He noted that the experiments of Tamman, Othmer, and Kornfeld were of a statistical nature; this he attributed to the chance like aggregation of molecules in a disorderly group which could then transform itself into the regular crystalline phase. If the thermodynamic equilibrium temperature between the bulk solid and liquid is To, he showed that a small solid-like aggreEate of 'radius' r could exist in equilibrium at a temperature T (belowTo) defined by the equation r

2 6.51- .17:5 (2.1.)

is the molecular volume of the solid Xis the molecular where vs ) latent heat of fusion) CSIsL is the interfacial surface energy.

This equation follows from the relationships between the thermodynamic potentials of the solid (cy, ) and liquid (Ti ) phases. (ps (t)., .265e. ) 71„) = Cpx

and where the zero suffices refer to the bulk equilibrium state.

Then

= Tr) = ^r" )12 Equation 2.1. shows clearly that the smaller the 'radius' of the aggregate, the greater will be the supercooling at which the aggregate can exist in equilibrium. The failure of a liquid to crystallise is thus a consequence of the fact that aggregates of a size greater than the equilibrium radius do not form spontaneously in the liquid. By substituting the experimental value for ( To - Tr) found by Tamman, together with the other molecular quantities appearing in (2.1.), Haber showed that the critical radius for betol was of order 10 7cm, i.e. the critical aggregate consisted of a few tens of molecules.

The first fully theoretically correct interpretation of the

nature of a phase change was obtained by Volmer and Weber (1926)

for the case of condensation from vapour. They noted that the

work of formation of the critical aggregate comprised of two terms;

the work required to form a surface, and the gain in work at

constant temperature due to the thermodynamic potential differences

between the phases. This analysis (see Appendix 2.1.) yielded an

expression for the equilibrium radius of a curved liquid surface

in the presence of a supersaturated vapour

2 6 = Y M (2.2.) Aolz 13'r ID0,0 where 6:ev. is the surface energy per unit area between the liquid

-vapour phases, M is the molecular weight of the substance, /9.K.

is the liquid density; ir is the saturation ratio. This equili- 1,00 brium radius r* could in turn be substituted into the expression

A ›k L\ = 3 6,ty 11-T f >4 (2.3.)

to give the work of formation of a critical sized droplet (or

aggregate) in the known supersaturated vapour in terms of bulk known

quantities.

Volmer and Weber in the 1926 paper proceeded to develop the

theory quartitatively further by suggesting that if A * is the

work of formation of a critical sized aggregate, the number 27 becoming visible in a condensation form the vapour is proportional to the steady state concentration of critical aggregates which is /}a"* itself proportional to • cAT Thus j(T)= k(T) (2.4.)

They did not calculate K(T); rather they used the relation that 41* is a function of () to calculate the critical radius r* and hence the ratio ,6 kT

1.001 1.013 1.11 2 5 10

-4 -5 6 -7 -8 -8 r* 10 cm 10 10 1.5x10 6xi0 4.5x10 2 8.4x107 8.4x105 8.4x103 1.9x10 35 17

Volmer's idea that the number of tiny droplets appearing is proportional to the steady state concentration of critical aggregates is not strictly correct, since the conversion from critical aggregates to visible droplets is a dynamic process in which aggregates are not only converted to visible droplets but some are lost to re-evaporation also.

Although the detailed kinetics of this process are not appropriate to the liquid-solid transition, the general terms of the theory will be described here because some of the general conclusions which can be drawn are relevant to the liquid-solid transition also.

Farkas (1927) tried to evaluate K(T) by working on a suggestion by

Szilard that the number of critical aggregates becoming visible in a condensation process is the result of a dynamic process of the type,

(n) AG%) — (n41) Z (2.5.) -1 No. of aggregates passing No. re-evaporating sec . -1 critical size, sec .

-3 where Z(n) is the number of n-molecule aggregates cm .

is the number of molecules of the gas phase -1 striking unit area of curved surface sec .

A(n) is the surface area of an n-molecule aggregate.

From an elementary kinetic theory approach

?rh (2.6.) (21i 1,1 ,(T)

where m is the mass per molecule.

Farkas changed (2.5.) to differential form, and, in the final re-integration process another unknown constant appeared.

This led Farkas to the final result that — *(T) ' C .(No A(h) jole 3 VI: (2.7.) Ah 2Ti 1 2 6tv„ rw The main point here is that the number of droplets becoming visible in the condensation is indeed proportional to the number of steady state aggregates as postulated by Volmer. This hypothesis will be used again for the liquid-solid transition.

Becker and DOring (1935) show that K(T) can be evaluated com- 9

pletely without changing (2.5.) to differential form. Instead, they use a chain of algebraic relationships similar to (2.5.) for different

values of n, and, solving these equations, are led to an identical

result as obtained by Farkas except that no unknown constant appears.

Becker and Doring express their result in a modified form from Farkas

but the equivalence and the identification of the constant C can

readily be shown by manipulation of kinetic theory equations.

Kaischew and Stranski (1934) use equation (2.5.) for the problem

of evaluating K(T) not only for the formation of visible droplets from

the vapour, but for the case of vapour bubbles in a superheated liquid also. like Farkas, they changed (2.5.) into a differential form and

obtained a further undetermined constant. In a later paper (1935)

they show how K(T) can be evaluated for the growth of cubic crystals

from the vapour, but again the undetermined constant C arises as a

consequence of the method. Becker and Doring (loc.cit) evaluate

K(T) completely by their different analytical approach for this case

of growth of crystals from the vapour too.

The conclusions of this section are that the ideas of homogeneous

nucleation are quite well established, and that quantitative agreement

exists between theoretical calculations and experimental observations

for the case of condensation from a vapour, This last point is a

direct consequence of the fact that the properties of the mother phase, i.e. the vapour, are adequately known through the ideas of gas kinetic theory.

For the case of crystallisation of a supercooled liquid the molecular properties of the mother phase are not nearly so well known. For 1 example, whilst the collision frequency (cm-Zsec-1 + can readily be

30 evaluated for a gas, for a liquid this calculation is not so obvious, even though one can enumerate several of the molecular properties on which this will depend. To circumnavigate this difficulty a different type of treatment is needed for the case of crystallisation of a supercooled liquid which will be described in the next section.

Section 2.2.(a) Theoretical development of the theory of the

crystallisation of the liquid phase of a pure

substance.

The growth of crystalline aggregates in a supercooled liquid differs from that of liquid aggregates in the vapour in that the molecules in the bulk liquid must first be activated to break away from the bulk liquid before joining with the crystalline aggregate. In the vicinity of the aggregate, this activation energy must correspond to the energy of self-diffusion in the liquid. Volmer and Marder (1931) included this activation energy term in an expression for the growth velocity of crystals in a supercooled melt; the more viscous a liquid, alia aequalis, the slower would be the growth of the crystalline face.

Their expression for this growth rate is

k/kT - 6 4 *A T

1.1 (2.8.)

where U is the activation energy for self diffusion

L * is the work of formation of a critical two dimensional nucleus on a crystalline face. Becker (1938) recognised the fact that in solids and liquids diffusion controls the quantity rather than the collision frequency in the case of vapours. Borrowing 31

the result that the number of growth centres is proportional to the steady state number of critical aggregates from the previous theory, Becker wrote

- tAki (1$/k7. '°< (2.9.)

A rather naive, though useful way of understanding the control of the

diffusion term is to regard it as controlling the fraction of molecules surrounding the aggregate in a 'gas' like state. Becker did not

evaluate the proportionality constant. Frenkel (1932) arrived at the same functional form as (2.9.) for the quantity,7*, but again was unable to evaluate the proportionality constant.

Fig.1.3. shows the temperature dependence ofy* and the role

played by the activation energy. At small supercooling, when KT) U

thermal fluctuations control the rate of formation of critical aggregates. As * becomes smaller with lowering of temperature3 * increases, then, when U> KT decreases very rapidly.

Tamman et al. (1903) have experimentally established that the velocity of crystallisation does indeed show the behaviour depicted in Fig.1.3. for the liquids Benzophenon I,III, and Salipyrins

Volmer and Marder (loc.cit.) have also observed a similar behaviour with glycerine.

Besides the temperature dependence of crystal growth velocity shown in Fig.1.3., Tamman (1903) was also able to demonstrate the same temperature behaviour for the number of crystallisation centres

(Keimzahl) arising in the liquids Betol and Piperine within t seconds 32 at low temperatures. The liquids were kept at low temperature for a time t and then heated quickly to just below the melting point.

The submicroscopical nuclei which had arisen during the first period could easily be seen and counted under the microscope. Although

Tamman was unable to incorporate the liquid viscosity into any theoretical expression for the rate of formation of these nuclei, he demonstrated quite clearly that over the temperature region where the nucleation rate decreases, the viscosity of the liquids increased con- siderably with further supercooling.

This work has been repeated and confirmed by Michnevitch (1941).

A second piece of evidence for this general form of temperature dependent behaviour comes from the work of Koppen (1936) on the nucleation rate ("Keimzahl") in potassium chloride solution. He saturated

10 ml. of water with KC1 at 64°C and held this solution for 1 minute at the appropriate supercooling. He noted the number of tiny crystals appearing in this time interval and plotted this number against the supercooling. The shape of the curve was very similar to Fig.1.3., which lends to show that some form of activation energy is having an increasing effect with increasing supercooling or supersaturation.

There is sufficient experimental evidence to show that the functional form of equations (2.8.), (2.9.), with some mildly temperature dependent proportionality constant are correct. One can then write

114.T /kT * = k (7) e (2.10) 33

Turnbull and Fisher (1949) have been able to evaluate K(T) by assuming that the Eyring 'fundamental rate constant' kT is applicable to a h 'chemical' reaction of the tyre used by Farkas, viz: 9*-y h-h+1 A rather naive though useful way of understanding the control of the rate constant kT is to imagine that the liquid molecules at the h surface of the aggregate are under the influence of both the aggregate and the remainder of the bulk liquid. Such a molecule will vibrate in a potential well with a mean frequency kT at temperatures above h these for which only the first few quantum states are excited. Then, in this potential well, the probability of reaching an energy state, _ A EAT LIE above the level of some initial reference state is kT e h This treatment does not pretend to have the rigour of Eyring's statistical-mechanical treatment; the latter method makes use of the laws of statistical mechanics in the case of a reaction in which the

'energy barrier' is the activation energy of the reaction, and the reaction co-ordinate is a generalized co-ordinate in phase space. How- ever, the end product for the 'fundamental rate constant' is identical to the expression first derived, and for this reason the statistical- mechanical treatment will not be given here.

With the assistance of this fundamental rate constant one can calculate the quantity 7* by Farkas' method (see Fig.1.2.) as..7.* = _ 4fyia - 4.flAcT kT . e 1411 (41'11126). /ni (11+1)1154'4- 4)---T ki' il, 3/4 where N is the number of n-molecule aggregates cm and (a„h ) is a n factor expressing the number of molecules vibrating between the surface

of an n-molecule aggregate and the bulk liquid.

Turnbull and Fisher assumed that

24 \ (a) (an, I" an., (n+-1)2/3) for all but the smallest aggregates.

(b) N and cL (A5) are smooth functions of n. = t rt. CI P,

fz. lh vz 6(1(6'1)

(c)

ki (tvt.

and were able to show that the net forward rate equation for 7could

be changed to differential form and re-integrated to give, with a valid approximation,

(2.11.)

where n in this equation is the number of molecules cm-3 in the liquid.

Section 2.2. (b) The adaption of the Turnbull-Fisher equation to the study of the homogeneous nucleation of water.

Equation (2.11.) has been developed in quite general terms and is applicable to any liquid crystallisation process. Provided U(T) and /4 (T) are known as functions of temperature, r * (T) is completely determined. By the definition of5* (T), the general condition which must be satisfied in order that a volume of liquid,11-cm3, shall crystallise within t seconds, is that

* (T). v. t 1 (2.12.)

Conversely, if one can determine the average life time for which a given volume of liquid will remain supercooled before crystallising one can calculate7* (T) and compare this with the theoretical estimate (2.11.).

U(T) is generally found from viscosity data, whilstig * (T) can be calculated (see appendix 2.2.) by standard thermodynamics in terms of molar quantities

) (2.13.) 3 I LI) di

where 675.1., is the surface energy of the liquid-solid interface

L(T) is the molar latent heat of fusion M is the molecular weight.

This derivation assumes that the shape of the crystalline aggregate is a sphere; this in general will not be the case. The modification, though, to (2.13.) is trivial - a shape factor 'L.01 is used to replace the spherical shape factor 471-. The Turnbull-Fisher equation can then be written 3f m T 651 tit/KT — 3 kTa3 A ps 14-2i) dT kT (2.14.) 36

Mason (1952) has adapted this equation to the supercooling of water by substituting all the appropriate values - except the unknown 6s1... the interfacial surface energy between supercooled water and ice. Since, of course, bulk supercooled water and ice cannot co-exist in equilibrium, this quantity cannot be measured experimentally. If 6.61...were known,the homogeneous nucleation temperature for a supercooled water droplet held for t seconds in a supercooled state could be calculated from (2.14.) and (2.12.). Since 641. appears to a third power in an exponential, small errors in 61. will radically alter the calculated? *(T). Mason has demonstrate that a 10% error in 6"sL can upset_y*(T) by a factor 6 10 , or, in a more concise way, implies that the mean freezing temperature cannot be estimated to better than 6°C. Unless an accurate value is obtained for ecsL , no worthwhile comparison can be made between experiment and theory. Before discussing the theoretical attempts to calculate 661., , a few reservations ought to be put forward about the theory put forward so far:

(i) these crystalline aggregates can be described by some characteristic 'radius' r, which, associated with an interfacial surface energy determines the pressure of the new phase to exceed that of the mother phase by 265L

(ii)the geometrical shape factor, 23, for an isometric hexagonal prism (Mason, loc.cit.) in the case of an ice like aggregate is more appropriate than that for a spherical aggregate, 471 .

(iii) the binding of molecules on the surface of an aggregate can be described by a quantity analagous to the macroscopic surface tension forces of solids and liquids. c M Donald (1953) has made a detailed survey of the temperature dependent 3 7 parameters in (2.14.). The temperature dependence of Lf(T) and U(T) are shown in Figs.2.2. and 2.3; the dashed regions of the curve are extrapolated. Owing to the temperature dependence of the latent heat of fusion, McDonald modifies the expression given by Mason _ q (,) ds 3 2 S2s L 1 -5 to the expression given by (2.13.). He also very carefully examined the theoretical attempts to calculate 65L. The earliest attempt was by Krastanow (1941), based on a casual suggestion by Volmer that

(-6 L ti ✓

L f L v (2.15.) where 6s, is the surface free energy between solid-liquid, 6" is the surface free energy between liquid-vapour. Krastanow estimated 6s, to -2 c -2 o be about 10 erg cm . M Donald modifies this to 7.7 erg cm at -40 C by allowing for the variation of latent heat of fusion with temperature.

Mason (loc.cit.) made an estimate by calculating the work needed to break a crystal of ice in the 0001 plane, breaking two hydrogen bonds per unit cell in the cleavage. Knowing the estimated strength of the hydrogen -1 bond (5,160 cal.mole ) and the surface density of broken hydrogen bonds in the 0001 plane, the surface free energy of ice with its vapour is -2 estimated to be 102 erg cm . If the surface free energy of water at

-40°C is about 80 erg cm-2, Antanov's rule Est. = 6sv 6t-v gives 6,51: 22 erg cm-2.

Clearly, these ideas will give the magnitude of 65L , but hardly ACT I VAT ION ENERGY; ERG o i / / LATENT HEAT OF FUSION ; CAL /GM / io •P. I oo 0 8 0 o I 0 •• / •• i • / • I PO 0 / -n 0 • / •• tJ

0 O

N. D O

>.-. 39 the accuracy required to compare theory with experiment. The outcome of McDonald's detailed survey shows that when all corrections have been made, the homogeneous freezing temperature of a 10/.. diameter droplet is about -26°C. It is known that these droplets can exist in liquid form at temperatures as low as -36°C ; the implication is that there is about a 15% error in the calculated values of 4LcS if -360 C is indeed the homogeneous freezing temperature. Mason (1956,1960) decided to attack the problem in another way by using Mossop's observation (1955) that small water droplets about 0.81,t diameter produced in an expansion cloud chamber froze within 0.6 sec if the temperature at the end of the expansion approached

-41.2 ± 0.4°C. At this temperature

_7* (-41.2) = . (0.4)3 x 10-12 x 0.6 3 1 12 6.5x10 critical aggregates cm-3sec-1.

This reverse attack needs some justification. Mossop reported that these small water droplets were formed by condensation on nuclei of variable composition, e.g. NaC1, MgC12, CaC12. Apparently, the freezing temperature was independent of the nature of the condensation nuclei. In view of this fact, and since the droplets were of very dilute solution, Mason regards this as indicating that the freezing of these tiny water droplets is a property of the water only, i.e. homogeneous nucleation.

Substituting this value for] *(-41.2) into equation (2.14.) gives

era = 17.2 erg cm-2 at -41.2°C. Ignoring for the time being the temperature variation of 611. , this value can be inserted into (2.14.)

40 to plot the homogeneous freezing behaviour of water.

This technique, which has since been modified only by taking into account this temperature dependence of 6 (found by e:aperimental observations of the freezing of conductivity type water), results in the homogeneous freezing line plotted in Fig.2.4. On this diagram are marked the experimental observations recorded in the literature which support this semi empirical curve; these will be described in detail in the next sections. Also in this diagram appears a line of experimental observations by Bigg (1953)7 which he proposed was representative of homogeneous nucleation of water for reasons which will be stated when his experimental work is described. To support this conclusion he developed a simple probability theory along the following lines:

Let the probability of freezing being initiated in a small volume

V. in time Sb at a temperature T °C below 0°C be ti(T ). Then the s s probability of a volume V not freezing in time t is = _ w vt/sy

(2.16.)

or log w (T ) ) e (1-- s if iffy and j t are chosen SY6b sufficiently small so that W(T5).g 1. If the sample is being cooled at a rate determined by the equation

T. = e t oC sec-1 the probability of freezing by the time a temperature Ts is reached is TS loge (1 P) = - V W(Twu ) dT (2.17.) 6) s brat FIG. 2.4. n V -1- CAME o t4OSSOP -20 LANCHAM, MASON ME LOWEST

)77131GO

.5; °

-30 F. Yj_.E 1 Z_ U FAFF _t_

L.13 PRUPPACIIM (MEAN) 0 0o0 00 0 o 0°0 0 0 0 Li t2LiEiz2- v vy ° vy 0 0 .HOFFER

-40 POUND et al.

EQUIV. DIA. Toot, FIG. 2.4. b

-25 9

-30

THEORETICAL HOMOGENEOUS NUCLEATION CURVE LOWEST -35 J7,0B1

MEANS EXPERIMENTAL POINTS (MEANS). KUHNS

EQ1.1V. DIA. 10 1000 1 crn 43

Now W(T ) can be identified with s (T), and so EV a loge (i - ID) v (2.18.) s) s 1.6

This theory is quite general for any statistical phase change process.

In the case of water, provided1*(T) is known as a function of supercooling, (2.18.) allows the probability of freezing by a temperature

T to be calculated for a cooling rate 3 o 1 s C sec -,E Note that from equation (2.18.) one can see that the larger the cooling rate, the greater will be the necessary degree of supercooling for the given probability of freezing ( ify* (Ts) is a unique function of temperature). This is called the 'cooling rate? effect, and will be in- yoked several times in this thesis .

There are three points though, which suggest his conclusion is incorrect.

(i) Meyer and Pfaff (1935) succeeded by careful particle filtration to supercool samples of water of about 1 cm3 volume to -33°C quite regularly without freezing taking place. Bigg's line determines the freezing temperature to be about -17°C. This discrepancy is beyond the statistical spread, and no cooling rate effect can account for this difference either. Mayer and Pfaff clearly state that volumes of water of order 2 cm3 can often be cooled slowly to -33°C, the water prior to freezing remaining clear and invisado, If this latter freezing is homogeneous, then Bigg must have observed heterogeneous freezing.

(ii) Taking Bigg's analysis a little further, for the probability of freezing i, the median freezing temperature,

44 js

(2.19.) log e(J.:) z = (7.0 ct is

Ts 1)=1/z. or loge (2.20.) .371-5) d Ts To

Now the earlier theoretical treatment showed that

c> 2 X --..±4,)6sITN. 3 z Tsz kT

Ts P= 1/2.. ARTS2 then (2.22.) a is to which is to be identified with Bigg's line

loge (I-) =

Equation (2.22) will only reduce to this form if 6 0<1;s making i5 P=1, r — aTs loge() *2 , SN-I 0‹, J C a Ts (2.23.)

or ctrs

However, if asi.. Qz- )s then the equation for the critical radius r* becomes independent of supercooling. This does not conform with the ideas of homogeneous nucleation put forwerd in this section. If is temperature dependent in a manner other than that which makes 45 r* constant, the plot of log (volume) against median freezing temperature cannot be a straight line.

(iii) Fig. 2.4. shows that this straight line law breaks down at small droplet sizes, yet it is these which are most likely to be homogeneously nucleated since, by virtue of being so small, the probability that they contain an active ice nucleus within, or on the surface decreases rapidly with decreasing radius.

Summary of this section

The Turnbull-Fisher equation for the homogeneous crystallisation behaviour of liquids has been adapted to the study of the homogeneous freezing of water. In the theory an unknown constant 6.61. appears which must be accurately known before the theory can predict the behaviour of pure freezing water in terms of experimentally measurable parameters. After unsuccessful attempts to calculate 06ki_ precisely, a reverse semi-empirical approach is adopted which predicts the homogeneous freezing behaviour shown in Fig. 2.4. The next sections describe experimental research on the freezing of water, and how each research fits into a scheme of either homogeneous or heterogeneous nucleation.

It will become apparent whilst reading the next sections that experimental work sadly lags theoretical progress. As a consequence, one cannot make any assessment about the validity of the theoretical ideas put forward so far; on the contrary, the semi-empirical theory put forward has to be invoked to clarify such elementary considerations as whether, in a particular research, the type of freezing observed is homogeneous or heterogeneous or even a mixture of both. 46

Section 2.3. Experimental observations of the supercooling of water.

The post-war discovery of the possibility of rain-fall control by cloud seeding stimulated more concentrated research on the supercooling of water, particularly in a meteorological context. This began with the very similar researches of Heverly (1949), Hosier (1951,

1953), Johnson (1950) and Dorsch and Hacker (1950). Each of these chose to vary the volume of water to be supercooled in order to search for any volume dependence. The rate at which the water could be supercooled was also variable (Heverly, 1-20°C min-1, 6 to 15°C min-1 Dorsch and Hacker, 0.1 to 7°C min-1 by Johnson and by an unstated amount by Hosier). Where sufficient numbers of drops had been supercooled (Dorsch and Hacker froze about 5,000 drops) the mean freezing temperature is plotted against the logarithm of drop diameter for each of these researches in Fig. 2.5.

Heverly froze his water droplets on a thermocouple tip mounted inside a cryostat. The pressure inside could be varied between 10 and 1000 mb. for the purpose of detecting any pressure dependence on the supercooling. He found that the only variable which produced a systematic effect was the volume, as distinct from the pressure and cooling rate. Surprisingly, no difference was detected in the behaviour of boiled rainwater, mountain stream water, distilled water, or a 1% brine solution.

Dorsch and Hacker chose to support their droplets on copper or platinum surfaces cooled in turn by resting them on a cold steel rod. -10

HEVERLY DORSCH, HACKER BIGG -15

-20

L -25

- 30

-35

• -40

DROP DIA. • w 10 cm. 1 1.t 10p. 100p. 1 mm. 1 cm. 48

They failed to detect any cooling rate effect, but found a spread of freezing temperature amounting to about 2°C between the 25% and 75% probability for small (i.e. 10-100/A, diameter) droplets, and 6°C for larger drops. Johnson suspended his droplets from glass fibres or paraffin coated wires inside a cryostat. His results show a scatter as much as 11°C at 500/A. diameter, but a volume effect was discernible. No systematic effect due to the varying of cooling rate was reported.

Hosler froze his water samples in glass capillaries or on platinum surfaces. For the freezing in capillaries there was no relation between the volume, surface area or length of the sample with supercooling. The only parameter which did vary systematically with supercooling was the capillary radius. This he accounted for by Weyl's theory that a double layer exists in which the molecular force field at the glass surface orients the water molecules into a position "unfavourable for a phase change". The smaller the radius, the more strongly this would affect the molecular orientation within the water true sample. (It has since been shown that this is not the tsime. freezing behaviour of water in glass capillaries). Hosler did find a volume dependence in the case of water droplets freezing on a platinum surface, and that different surfaces used to support the droplets raised or depressed the average freezing temperature with respect to platinum.

Collective Summary

(i) When the supercooling is plotted against log (V), as in

Fig. 2.5., the typical slope of the line may be 7-10°C for a factor 49

10 in volume.

(ii)the distribution of freezing temperature for a given droplet size is typically about 2-5°C between 25% and 75% probability of freezing.

(iii)the absence of any cooling rate effect, even though this was varied by a factor 70 in Johnson's research. The discordance of the results well outside the limits of experimental error,with Hosler's observations of the raising or lowering of freezing temperature with respect to the platinum surface strongly suggest heterogeneous nucleation by the substrates. It is strange that no cooling rate effect was reported. This was perhaps first shown in the data of Tamman and gichner (1935), which showed a significant trend to lower freezing temperature for an increased cooling rate. This effect was so small compared with gross differences by differing water treatment that they did not recognise this as being of fundamental importantg.,particularly as the theory had not been developed at that time. Careful inspection of their data suggests a change of 2.2°C for a 10 - fold change in cooling rate. Bigg (1953), Gokhale (1965), Vali and Stansbury (1965) have found a rate effect which is given as 2.8°C, 2.5°C, 0.65°C per factor 10 in cooling rate. Bigg 1953 proposed to work on the same lines as the above mentioned authors by choosing the same variable parameters, but selected a method first used by de la BZche (1822) to avoid the nucleating influence of the substrate. This method invokes the short range structure of liquids, in that the weakly ordered arrangement of molecules in the supporting liquid is unable to influence the corfiguration of the water 50 molecules. Bigg chose to suspend the water drops at the interface between two liquids which were practically immiscible with each other and with water. A further advantageous feature of this technique was that the water samples could not pick up any airborne contaminants through being completely surrounded by liquid. As Bigg mentioned in his Thesis, the chief argument against this technique was that a slight solubility could alter the freezing behaviour. He showed,though, that within the limits of experimental error, no systematic difference was obtained with the following pairs of liquids:

(i)Carbon tetrachloride - paraffin oil/toluene

(ii)Ethylene dichloride - paraffin oil/toluene

(iii)Carbon tetrachloride - amyl acetate (iv)Drifilm surface - toluene/amyl acetate.

The two liquids were contained in a shallow dish, the drops being supported at the interface where a thermocouple was situated. The droplet size could be changed from 10/A, diameter to 1 cm. diameter, a factor 109 in volume. Bigg established the following facts:

(a)freezing is a statistical phenomenon. There is a spread of

4°0 between the 25% and 75% probability of freezing.

(b)Plotting logs(i-) against Ts for P4, a good straight line is obtained over a factor 107 in volume. Its equation is

loge (v,I ) a Ts + b (2.24.)

This equation breaks down below 30 diameter, as can be seen in 51

Fig.2.5.

(c) The rate of cooling did affect the mean freezing temperature, amounting to ^12°C per factor 10 in cooling rate. This Bigg regarded as real since there were enough samples for a statistical study and the effect was larger than experimental measurement errors.

(d) water droplets of given size froze at exactly the same median freezing temperature regardless of the source of water, be it freshly distilled or specially prepared conductivity water from the

National Physical Laboratory. This fact, together with the self con- sistency of his measurements led him to conclude that the freezing was homogeneous.

That this conclusion is erroneous was shown in Section 2.2. What agency, then, caused his water samples to be nucleated at a temperature other than their natural freezing temperature? One is led to look more closely at particulate contamination by the supporting liquids, a view supported by Bigg too (private communication).

Langham and Mason (1958) developed an arguement that can be used as a tentative explanation. Suppose the supporting liquids are air contaminated and communicate this contamination to the droplets as a result of which the droplets are infected with an ice nucleus population per unit volume active between supercooling TS and (Ts +4 T.)

T + AT) =be aTs (2.25.) m (TsI s S

Then the number of particles active at supercooling less than Ts in a volume')." cm3 is

52 Ts b e a T = it s as (2.26.) mT s

If mT, 4 Poisson statistics show that the probability of not containing an ice nucleus is

P e. (IlTs` J e (2.27.) 01.

Then the probability of a droplet freezing by a temperature Ts below

0 oC is

0.1s or, (2.28.) "CX? a -Vb PLs =

for the median freezing temperature, PT . simple algebra shows

log = a T + b . e ( s

The assumption that the supporting liquids contain an active ice

nucleus population described by (2.25.) is to some extent drawn from the cold expansion chamber measurements of airborne ice nuclei popul-

ations which do typically show this behaviour (Findeisen and Schulz

(1944), Aufm Kampe and Weickmann (1951), Workman and Reynolds (1950).

The constant 'a' characterises the nucleating ability of the ice nuclei

and 'b' their concentration. Provided the 'shape' of the ice nucleus

distribution remains the same, the varying concentration of motes moves

the line 53

log 1.1) =aT b parallel to itself. This is e 1 8 to some extent reflected in Fig.2.5., where one can generally say that the displacements of the lines of the various authors are due to a varying concentration of ice nuclei in their water samples. Thus,

Bigg water droplets had a smaller concentration of tiny insoluble impurities than had Dorsch and Hacker's.

This detailed discussion of Bigg's work is not meant to belittle his contribution to the study of homogeneous nucleation. For its time it was a valid and careful piece of research which pointed out a method of general approach to this phenomenon. His results were within them- selves consistent with a simple probability theory which suggested homogeneous nucleation, though the main reason for his premature conclusion was the lack of any good supporting theory for direct comparison. Secondly, this discussion will clarify the reasons for such careful water handling techniques undertaken in later research work,

particularly the avoidance of contamination by room air.

Mossop (1955) realising that in the literature there were records of much lower supercooling than attained by Bigg, decided to repeat the freezing observations of large water samples taking great care to see that airborne contaminants were excluded from the samples. This was ensured by boiling re-distilled water in a sealed flask with one outlet through a pyrex capillary tube. These pyrex tubes were internally coated with water repellant surfaces (dimethylchlorosilane and'Drisil 29') to prevent sites on the internal glass surface from nucleating the water.

After about 15 minutes steam flushing, the capillaries were sealed by a hand torch at two points to trap a little condensed water. The trapped 54 water was used as a sample to be frozen by immersion in chilled acetone. Temperature measurements were stated to be ± 0.2°C. By plotting the number of samples freezing within a given temperature interval Mossop was able to distinguish between three types of behaviour:

(a) those samples which still contained nuclei and froze at smaller supercoolings isolated from their neighbours (b) on this plot,

(b) Specimens which froze at temperatures intermediate between

(a) and (c),

(c) Deep supercooling independent of the nature of the inner surface of the capillary, be it glass, fused quartz or hydrophobic coated glass. This type he suggested froze homogeneously.

From his survey of the literature and from his own results,

Mossop drew the line shown in Fig. 2.6. The difference of slope between this line and those of Fig.2.5. is immediately apparent. In fact it runs very close to the semi-empirical homogeneous nucleation curve of Fig. 2.4. There are some reservations to be put on this line though:

(i) it was shown in the discussion of Bigg's work that a straight line over an extended volume range is inconsistent with the premises of homogeneous nucleation theory.

(ii) the cooling rates were not always known. It may be that this effect is small, in which case the results are more or less comparable.

(iii) there is always the possibility of deeper supercooling with another substrate. 55

FIG. 2.6

Meyer, Pfaff A -34

-35 • • Mossop -36 E ID -37 •• 0) •• -38 Bigg (mean values) ID

-39

Lafargue

Mo s s

1? 100".1 1mm 10? droplet diameter

FIG. 2.7

distribution predicted by Bigg's line 0)1 . N a) 1.0 by Mossop's line 0.8 O 0.6 2 a., 0.4

0.2

. . It 21 0 2 ji 6 C temperature; median centered on zero

cooling rate-06 °C min"' 56

(iv) homogeneous nucleation is a statistical phenomenon and it is not strictly correct to take only the lowest freezing temperatures.

Mossop also demonstrated that if the line of Fig.2.6. is representative of homogeneous nucleation a simple probability consideration of the type used by Bigg predicts that the statistical distribution of freezing temperature is much less than observed previously; this is shown diagramatically in Fig. 2.7.

Section 2.4. Conclusions of this discussion and the distinction between heterogeneous and homogeneous nucleation.

Several points were raised about the validity of Bigg's work.

Theory predicts a new type of behaviour which is supported by observations to be discussed in the next sections. The evidence points to a

distinction in freezing behaviour which is summarised here.

Heterogeneous freezing Homogeneous freezing

(i)Scatter of freezing temp., 4°C. 1 C. (ii)Slope of log(1-)a:T , 7°C per 3°C per factor 10 V B factor 10. around 50pn diameter. (iii)Cooling rate effect: several 0.5°C per factor 10. degrees per factor 10 in cool-

ing rate.

(iv) Deep supercooling attained

when strict cleanliness pre-

cautions are taken. 57

Section 2.5. Homogeneous nucleation of large samples of water.

Meyer and Pfaff (1935) followed a suggestion by Tamman and

BUchner (1935) that water can be supercooled below -20°C by carefully removing particulate matter. They came to this conclusion by comparing the behaviour of water with other liquids, both organic and inorganic, such as methyl alcohol, salol, sulphuric acid etc. which became viscous before crystals of the solid phase appeared in the process of cooling. Water had not been supercooled to this extent.

Secondly, boiling filtered water to expel the dissolved gas caused a o reduction of 2 C of freezing temperature by the destruction of some active ice nuclei. A third point was that Lilienthal (1934) had observed liquid water at -21.5°C, and Wegener had seen small liquid water droplets at -30°C in fogs in Greenland.

Meyer and Pfaff followed up this suggestion in the expectation that if the nuclei are removed, water, like other liquids, ought to supercool to a viscous or glaesy state. Soda and borosilicate glass were found to nucleate the ice phase more readily than quartz glass.

These vessels were carefully cleaned with alcohol and ether before running in distilled or collodion filtered water. Sealing in the sample was effected by closing the tubes with a hand torch, or covering the water with cleaned paraffin wax. They found that occasionally a sample remained inviscid and unfrozen for a long time at -32 to -33°C.

Because the experiments with other liquids showed that 'nests' of crystals formed just after nucleation and that the solid end product remained opaque, the totally different behaviour of freezing supercooled water suggested that nucleation was not typical of pure liquids. 58

Further experiments were designed for conditions of ultra-purity.

Water was distilled quiescently up to five successive times in vacuo in small inter-linked glass tubes which had been cleaned by prolonged steam flushing. The final volumes of water of order 2 cm3 could be slowly cooled to temperatures often as low as -32°C to -33°C, the water prior to freezing remaining inviscid and clear. Meyer-and

Pfaff concluded that the water was probably nucleated from the glass, and that it may be possible to supercool water below -33°C.

Mademoiselle Rnyardelle 1954 placed doubly distilled drops of water

• at the interface of vacuum distilled mercury and scintered glass filtered silicone oil. Ten successive freezings of the same drop of volume 0.05 cm3 on five successive days showed a gradual lowering of lowest freezing temperature, the lowest being -30.6°C, the highest o -18.5 C. After several months the same drops froze at -33.5 ± 0.6°C when cooled at 0.05°C min 1. From these measurements she calculated from equation (2.14.), 18.8 erg cm-2 Z 651. 4 19.4 erg cm-2.

Wylie 1953 was aware, as were Meyer and Pfaff, of the effect of atmospheric dust on the extent of attainable supercooling. His method was essentially similar to that of Mossop, and again he found three modes of behaviour:

(a) those samples at which ice dendrites greWfrom the same

point on the glass wall.

(b) intermediate supercooling (15-20°C) with a scatter of 7°C

on repeated freezing,

nucleated by the glass.

Class (c), though rare, he thought to be homogeneous nucleation. 59

A collective discussion of the research described in section 2.5.

There is not enough data given to make a complete statistical survey of these results. This, of course, stems from the great difficulty of removing tiny insoluble nuclei from such large quant— ities of water, or cooling the samples without nucleation taking

place at the surface.

Mlle. Bayardelle reported a scatter of about 0.6°C for her water drops, but not much confidence can be attached to this because it is not known how many observations were made to obtain this result, or the experimental temperature measuring error. The gradual lowering

of mean freezing temperature she observed is probably accounted for

by the larger impurities settling out under gravity or by the ice interface pushing out the impurities into the silicone oil. It is

possible that filtering the silicone oil removed the particulates

which contaminated Bigg's similarly treated water samples.

In all these cases it is difficult to assess the type of freezing

observed, but in view of the strict cleanliness precautions taken and

the closeness of the lowest freezing temperatures to the homogeneous

nucleation curve of Fig. 2.4. it appears that nucleation could have

been homogeneous in those rare instances of deep supercooling.

Section 2.6. Intermediate size samples 100,. d G 1mm.

Representative of this group are those observations of Mossop already described.

The establishment of the semi-empirical curve of Fig.2.4., 60 encouraged Langham and Mason (1958) to try to produce pure water in bulk so that homogeneous nucleation could be systematically studied for as wide a range of sample size as possible. The technique used was identical to that of Bigg (loc.cit.) except that the suspending liquids were cleaned by re-distillation. Of the several methods of water purification they tried, the best results were obtained by first ion-exchanging the water and then re-distilling it quiescently into a cleaned glass storage flask. After continued operation of the still for several days, the water purity was of conductivity 6 standard ( 10 mhos.cm.). The results of the supercooling of this water are shown in Fig. 2.4. Drops of diameter less than 250/, could be consistently supercooled to regions close to the homogeneous nucleation curve with a small spread of freezing temperature measured to be 0.8 °C between 25% and 75% probability. The larger drops of. size

250/, to 1mm. diameter showed a median freezing temperature much higher than one would expect from the theory. Obviously, the water had not been kept sufficiently free from foreign impurities to enable con- sistentlow freezing temperatures to be achieved. However, the lowest freezing temperatures within each size group lay quite close to the curve indicating the probable absence of foreign nuclei.

Comment Later experiments by Pruppacher (1963) showed that redistillation re-introduced more insoluble impurities through the action of the water leaching out particles from the glass surface. Pruppacher, using the same technique but with pure ion-exchanged water, managed to supercool approximately 75% of a number of 2 mm. diameter water drop- o lets to within - 3 C of the curve of Fig. 2.4. 61

The useful size range covered by Langham's research over which

homogeneous nucleation was consistently active was disappointingly small.

Hoffer (1961) again used an oil immersion technique to study the

freezing of droplets 50 to 200/A, diameter. From the freezing histo-

grams those few droplets freezing heterogeneously are easily picked out. The mean freezing temperature he measured as -36°C j-°C, with a spread of 1.5°C for those droplets freezing 'homogeneously'. From

the data, he calculated 651. to lie between 18.6 and 18.9 erg cm-2 at -36°C.

Section 2.7. Small droplet sizes 1,A.. 4 d L 100 An. / /

Observations in this region of droplet size freezing at low temperatures to -38°C are numerous; the reason is that the more often an initial sample is divided the smaller will be the number of impurities it will contain. For example, soppose 1 cm3 of water containing 103 insoluble motes is subdivided into droplets of 101A diameter. Then only 6 1 in 10 of there tiny droplets will contain an insoluble particle - an insignificant proportion that cannot affect the statistical treatment of a few hundred droplets. Thus smaller water samples are much

purer from insoluble content. Also, because of the minute surface area of such a small droplet, the chance of being nucleated by substrates or substrate imperfections becomes smaller too. It would be impossible to give a detailed discussion of all the reports relevant to this section here; where it is possible several reports will be grouped together according to the nature of the experiments. 62

Section 2.7.(a) The freezing of small airborne droplets.

One of the earliest observations of the deep supercooling of small droplets was that of Rumpf and Geil (198), who tried to measure the equilibrium radius of small droplets of salt solution of known concentration in a low temperature air stream of 100% humidity. As a side line of this research they noted too that small o droplets of radius 2-3/A. could easily be supercooled to -35 C,which was the lowest temperature attainable with their apparatus. The onset of rare freezing by natural nuclei active at below -100C was easily seen by the twinkling of the particles in the field of view.

There are more observations of the deep supercooling of small droplets in air which stem from low temperature expansion chamber measurements. The earliest and best of these was by Sander and

Damkohler (1943), whose work was once interpreted as evidence that water could exist in liquid from below -40°C. For this reason, and to offer evidence that this is not the case, their research will be discussed more fully than other references to similar work.

They had as a cold pressure chamber a glass bulb of volume

10 cm3 fully immersed in cold methanol. This bulb could be highly evacuated and refilled with carefully filtered and cleaned air at the required overpressure for the expansion. In this manner it was possible to calculate the critical supersaturation necessary to produce a visible condensate in the temperature range from +35°C to -75°C by noting the expansion ratio when a few particles per cm3 became visible. Viewing was through a double walled Dewar vessel (which housed the cold liquid surrounding the glass pressure bulb) by a

63 collimated beam from a mercury arc lamp.

When the air temperature at the end of the expansion fell below

-62°C the condensate particles loam distinctly seen to glitter - this has been interpreted as either the onset of the ice phase by freezing or by the spontaneous condensation into the crystalline phase. There are two pieces of evidence which suggest the latter is correct:

(a) that in the plot of 1"/ against temperature for the onset of condensation a distinct inflexion is seen at -62°C in an other- wise straight line towards higher vapour pressure. Now equation (2.13) shows that , and it may be that at these low temperatures L.

kT kTT -2 At temperature not far below 0°C, doisv "." 102 erg cm (Mason, loc.cit) -2 and d ^, 80 erg cm ; one would therefore expect condensation to ttt the liquid form first until the temperature is reached which determines the relationship

3 3 Cobs

Sub (2.29.)

This means that the liquid phase would indeed precipitate first below

-40°C but would instantaneously freeze. A certain amount of support for this comes from the continuity of the line ( 1/)17, )04. T through -40°C to eventually innex at -62°C. Sander and Damkohler estimate the sensitive time of their chamber to be about 0.3 seconds, in which time the frozen droplets may not have had time to grow crystalline faces and 64 show specular reflection.

(b) contrary to the case when liquid is the first condensate, the presence of ions had no effect on the saturation ratio when glittering first appeared. This suggests another deposition process is occurring.

Krastanow (1940) completed the calculation outlined on the assumption that the ice crystals are cubic shaped and that 65y . He found (it appears fortuitously) that sublimation is the preferential form of deposition below -65°C. If the crystals are assumed to be hexagonal shaped and o'sv 1.1 6.0r , this sublimation temperature is below -100°C, but little confidence can be attached to these estimates in view of the uncertaintities involved. It may be that the discontin- uity found is the sublimation temperature.

A similar research by Madonna, Pound et al. (1961) has also served to locate this transition temperature at -65°C, though there is slight disagreement on the absolute supersaturation necessary to form a visible condensate at this temperature.

Cwilong (1947),Fournier d'Albe (1949), and Mossop (1955) have reported the onset of the ice phase in low temperature expansion chambers utilising primary condensation on a nucleus. Their apparatus differed from that of Sander and Damkohler, and Pound et al., in that nuclei were necessary at low temperatures because the apparatus could not pro- vide a large enough expansion for homogeneous condensation. Their water droplets were therefore larger than those of the other researches, and the onset of the ice phase could be seen by glittering in the condensation fog near -40°C, below which there was a sudden increase in the number of ice crystals appearing. It is not very clear why Sander and Damkohler did o not see this glittering below -40 C. It may be that either the particles 65 were too small to show this glittering, or else it could be the great difficulty in deciding whether a tiny isolated particle glitters without a comparison with neighbouring particles. One might also note that frozen droplets do not 'glitter' unless they have had time to form crystalline faces. This will probably not occurr within 0.3 sec. below

-40°C.

The conclusions of this discussion of the expansion chamber observations are as follows:

(i) the glittering observed by Sander and Damkohler, Pound et al. at -63°C does not necessarily imply the existence of liquid water in a state of metastable equilibrium down to these low temperatures. It is more likely that the glittering is caused by crystals growing directly from the vapour.

(ii) The experiments of Fournier d'Albe and Mossop may have shown true homogeneous freezing since the nature of the condensation nuclei does not affect the ultimate freezing temperature and the droplets were of very dilute solution.

Schaefer (1946) produced a supercooled fog at -20°C by breathing into an ordinary commercial refrigerator unit. He discovered that a tiny piece of solid carbon dioxide, when allowed to fall through the chamber, left myriads of tiny glittering ice particles in its wake. The dry ice chilled the immediate surrounding air to -78°C or so, causing the fog droplets to freeze. In order to establish this freezing temperature more precisely Schaefer allowed a small pellet of frozen mercury to be passed through the cloud. Whilst the pellet was frozen crystals appeared; immediately melting occurred this property was lost. This fixes the freezing temperature near -38.9°C. 66

Later, in 1952, in the Langsdorf type diffusion chamber he

measured the freezing temperature to be -38.5°C. The base of this

chamber was strongly cooled by solid carbon dioxide to about -75°C,

whilst the top, in which there was a warm water vapour source, was main-

tained at room temperature. The water vapour, diffusing downwards,

cooled under the influence of the strong temperature gradient until it

became saturated. Initial condensation occurred on dust particles, but,

when all of these had been 'rained' out the supersaturation ratio rose

to such a value that cosmic ray tracks could be seen. Droplets which

grew spontaneously from the vapour under these conditions fell at a rate -1 of 1-2 cm. sec through the freezing level. They could be photographed

freezing by the appearance of a 'twinkle' under the correct illumination

conditions.

These diffusion chamber measurements are extremely important in our

understanding of homogeneous nucleation since they pertain to conditions

of ultra-cleanliness.

Schaefer has also reported some field observations of the behaviour

of condensed water in air colder than -40°C in Yellowstone National Park,

Wyoming (1962). It happens that in the Rocky mountains in mid-winter

the air temperature can fall below -40°C: the air is also noted to be

very clean owing to the absence of industrial pollution centres, housing

etc. His reports describe the eruption of geysers into this very cold

air, the spontaneous condensation forming myriads of tiny droplets which

were found to freeze just before the ambient temperature reached -40°C.

There was no evidence of a phase transition taking place below this

temperature. Similar reports have come from Verkhoyansk, Siberia(Klinov, loc.cit.) and from Fairbanks, Alaska (Kumai, loc.cit.) There is no 67 field evidence for the existence of liquid water below -40°C.

Section 2.7. (b) Small droplets freezing on a substrate.

Bazikailo (1941) was perhaps the first to report that the deposit obtained by condensing water vapour onto a cold glass surface remained liquid until -36°C temperature was reached. Weickmann (1947) obtained a similar result by condensing water vapour onto a metal plate. The freezing temperature of the condensate was -40°C.

The most valuable research undertaken by systematically examining the effect of physical parametis in this size category comes fromthe work of Jacobi (1955) and Carte(1956). Their techniques were very similar.

The water droplets were formed by condensing water vapour from moist air onto various types of surfaces, e.g. highly polished nickel, gold,

silver, and silicone, collodion, pyroxylin varnish covered surfaces.

These surfaces were in good thermal contact with a cold metal block cooled by immersion in liquid air or oxygen, the rate of cooling being controlled by an electric heating coil. They both found that metallic surfaces elevated the mean freezing temperature above the level determined by non-metallic surfaces. However, as Carte points out, the lowest freezing temperatures are independent of the surface used. Again, this is another observation which shows that metallic surfaces act as nucleating agents.

Carte observed three types of behaviour whilst repeatedly freezing the same droplets: 68

(a)a smaller supercooling to -33°C with n scatter of 5°C or so. (b)medium freezing temperature, -35°C, which becnme less with successive freezing.

(c)low freezing temperature, -37°C, with a 1°C scatter independent of the surface used. He concluded that type (c) froze homogeneously by the following argument:

Equation (2.18.) shows that for a fixed probability, all sized droplets freeze at the same temperature provided the ratio V remains P constant.

For example, from Fig.20. the homogeneous freezing temperature for -1 a 10/4^- diameter droplet cooled at 1°C sec is -38.0°C. If now the cooling rate is increased by a factor 10, for constancy of R the droplet size will now be 29 diameter and will freeze, for the same probability, at -38.0°C. One can now read from Fig. 2.4. that the freezing temperature of 20p-droplets cooled at 1°C sec-1 is -37.5°C. The 'cooling rate effect' therefore is about 0.5°C per factor 10 in cooling rate which agrees with Carte's measurements. His results were consistent with a simple probability argument and the semi-empirical homogeneous nucleation curve.

Jacobi 1955 Carte 1956

Size range: 1-300/A, diameter 9-35/^ diameter

Scatter: 0.2°C below 50 v. dia. (a)Deep supercooling 1.0°C

1.0°C above dia. (b)Elevated supercooling 5.0° 50f- Cooling rate: 2-20oC min-1

Cooling rate effect: i°C per factor 10. 69

Lafargue 1950 used a method developed by Dessens to capture tiny water droplets on spiders' threads at the exit of a cold air flow.

Small droplets, 1-10//v-diameterl of pure water, NaC1, KT, MgC12, 7112 solutions froze at -40.5 t 1.5°C in cleaned, filtered air.

Section 2.7. (c) The freezing of small droplets in oils or other

insoluble liquids.

Lafargue (loc.cit.) cooled an emulsion of water droplets in silicone oil and noted the temperature at which these froze. He found that freezing took place in less than 1/10th sec, resulting in a sometimes perfectly transparent (though more often opaque) spherical droplet.

Pound et al. (1951) report identical observations. Lafargue quotes a freezing temperature -40.5 t 1.5°C, whilst Pound reports -40 t 2°C with a 70% probability of the droplets freezing between -36 and -42°C.

Bigg's research (1953) also comes within this category.

Section 2.8. General conclusions drawn from a search of the literature.

There is a vast body of experimental research on the supercooling of water presented in the literature. Any further research requires considerable justification. Other than the aircraft observations mentioned in Ch.1., no recent evidence has come forwardbfor the existence of a phase change below the temperature region in the vicinity of -40°C, which is generally accepted to be the region of homogeneous nucleation.

Earlier claims to the existence of liquid water to temperatures as low o as -80 Cin laboratory experiments have been discussed by various authors, 7 0 e.g. Mason "The Physics of clouds" O.U.P. p.137-142, and it appears that in some instances the technique was at fault (Rau) or certain assump- tions made to calculate end temperatures after an adiabatic expansion were incorrect (Regener), or in other instances mis-interpretation of experimental observations (Sander and DamkOhler). Whilst one cannot so far rule out the possibility of a phase change below -40°C, the evidence for this is very weak.

Fig.2.4. shows that the limit of supercooling of water samples in the laboratory comes close to the semi-empirical homogeneous nucleation curve; it has therefore a certain amount of support, though how much of the 'volume effect' is due to leterogenous nucleation, and how much due to a cooling rate effect is in many instances difficult to evaluate, especially with larger sample sizes. Other than the work of Jacobi

(and again the possibility of surface nucleation arises here) no single technique has yielded a conclusive 'volume effect'. The inability of the experimental physicist to make a more detailed test of the predictions of homogeneous nucleation theory stems from the fact that avoidance of nucleation by a substrate or a surrounding liquid and precision temperature measurement are incompatible. A new research technique must include the following features:

(a) the use of a single technique over as wide a sample size range as possible.

(b) Precision temperature measurement to t 0.2°C or better.

(d) ability to study the smallest sample sizes possible where the 4 volume effect is most pronounced (See Fig.2.,..)

The experimental research described in the following chapters was designed on these considerations. 71

CHAPTER III

The design of a new technique

This chapter is divided into six sections. The first section describes the principle on which a new experimental technique is based; in the main these stem from the considerations of desirable features summarized in Section 2.8. It contains also those limitations to be expected with the use of this technique, together with mention of the care necessary for the success of the measurements.

The remaining five sections describe in detail the design of those main parts of the apparatus, in so far as they can be considered separately, with a final report on the method of operation of the entire system.

Section 3.1.

The roots of this new method stem from the diffusion chamber principle which Schaefer used to measure the 'spontaneous' freezing temperature of water. It had several attractive features, viz: firstly, by allowing the water droplets to fall through a temperature gradient in very clean air the problem of accidental nucleation of the droplets by a supporting substrate or impurities in a surrounding liquid is avoided.

Secondly, nucleation of the water droplets can readily be seen by eye or photographed without having to disturb or touch the droplet in its free 72 fall. Thirdly, by growing the water droplets by condensation from the vapour in an absolutely particle free environment the droplets cannot contain any insoluble impurity. They must, therefore, freeze homogeneously.

However, this system had its disadvantages which led Schaefer to believe that freezing was 'spontaneous'. The very strong temperature gradient about the freezing level with the associated sharp level of freezing, together with a very limited droplet size range combined to produce what appeared to be a freezing level at which all droplets very suddenly froze. The theory of droplet growth by condensation from the vapour shows that

s af (3.1.) at-

where r is the instantaneous radius,

is the environment saturation ratio.

For a given saturation ratio,

Gtr (3.2.)

In practice, under the conditions prevailing in the diffusion chamber, a droplet can grow within a few seconds to a radius of up to

8 microns; thereafter growth is much slower, with a limit in the maximum droplet radius, of about 10 microns radius in the time commonly taken to fall to the freezing level. The droplets originate in those 7 3 levels of the diffusion chamber where the supersaturation is strongest.

This imposes a limit on the smallest size droplet which can fall through the freezing level because the fast initial growth ensures a diameter of at least 10 microns before the falling droplet falls through the freezing level. This size range is not sufficient to detect any variation of mean freezing temperature with volume as can readily be seen from Fig. 2.4. (a) or (b).

This type of chamber can be more suitably used for the study of homogeneous nucleation by injecting water droplets of a much greater range of size without having the chamber supersaturated. In this research this has been achieved for droplet diameters ranging from about

5-130 microns. Reference to Fig.2.4. (a) shows that over this size range the variation of mean freezing temperature with droplet diameter is most pronounced, and is thus the ideal region for detailed study.

This achieves another desirable feature mentioned in Section 2.8.

Also, through elimination of the necessity of maintaining a supersaturation, the chamber top need not be kept warm; the temperature gradient at the -40°C level can be tstretched'to a gradient of about o -1 -1 1 C cm from 5-10C cm which charaterises a diffusion chamber of the type used by Schaefer. If there is a statistical range of freezing temperature this will be shown more readily by the corresponding height range of freezing in the chamber with the smaller temperature gradient.

For example, if the range of freezing temperature for a given droplet sine group is about 1°C, then the corresponding height range in a o -1 o -1 gradient of 1-0 C cm will be about 1 cm. In a gradient of 10 C cm the height range will be only 1 mm., which is with difficulty detected by photography let alcre by eye. With a well calibrated photographic system 7 4

outside a chamber of this type it is possible to detect variations in

freezing levels separated by 0.5 mm. These two points, viz: the small

droplet size range and the strong temperature gradient probably account

for Schaefer's conclusion of 'spontaneous' nucleation.

Stretching the temperature gradient too far will cause the air to

lose its static stability, and thus the gradient becomes less well-

defined. In the course of construction of a chamber suitable for this

present research several prototypes had to be discarded for this reason.

Small heat leaks through viewing and lighting windows (with the inner -1 temperature gradient less than 1°C cm ) caused the air inside the

chamber to convect slowly.

Fig.3.1. illustrates how the final technique has been arranged. A

more detailed diagram and photograph are presented in Figs. 3.14. and

3.16.

The base of the chamber is strongly cooled to about -50°C by the

evaporation of dry ice placed on ledges surrounding the chamber. Lining

the walls inside the chamber is a thickness of copper plate chosen so -1 that the base cooling develops a temperature gradient of order 1°C cm .

in the viewing region, at which level there are also thermocouple probes

A,B,C,D for temperature measurement. This region is strongly illuminated

by a collimated and carefully heat filtered light beam from a 100 p.p.s. mercury discharge lamp. A and B are two thermocouple junctions to measure the ambient air temperature at a set height. C and D are the two junctions of a single thermocouple used to measure the temperature gradient at the level of AB, set exactly 1 cm. either side of the line

Water droplets are injected into the chamber through a small aperture FIG. 3,1 7 5

DROPLET INJECTION POLYSTYRENE HEAT INSULATION

/// //

TOP — 25°C / Z

/ /

/ 0 f reezin _Lty_el _ BEAM p.p.s . 100

/CD_ICE { / Z / V / Z COLD BASE —50°C / 7

//////z/zzzzzzzz4 7 6 in the top and are allowed to fall through the light beam just in front of the thermocouple points. A camera is set to photograph the region

ABCD to record the exact level at which the 'brightening' on freezing occurs. A typical eventl indicated on the photograph, is shown in Fig.

3.2. The fall trace on the photographic negative appears as a dashed line as a consequence of the intermittent light pulses.

Let the temperature at the level of A,B, be -T0°C and suppose a droplet is observed to freeze at a distance z cms. above AB. Then the freezing temperature is given accurately by the equation

T z(dT;) (3.3.) ° \ot provided the temperature varies linearly with height between the thermocouple points C and D. This assumption was found to be correct by re- placing C and D with 'absolute' junctions and comparing the temperature obtained by interpolation at AB with the experimentally measured value.

The purpose of the pulsed light source is to enable a measurement of the fall velocity to be obtained by counting the number of pulses visible per cm. on the calibrated photographic trace. The terminal velocity is known empirically as a function of droplet radius in air at 0°C; it t has the general form 11-1:ht for small droplet sizes up to 31C)AAr where k, is a constant and V is the terminal velocity. Clearly the terminal velocity is a good quantity from which to extract the droplet radius by virtue of its square law dependence. The fact that terminal velocities are measured at -40°C poses no difficulty since a correction is easily made as follows: FIG. 3.2.

THE BRIGHTENING SEEN 1HEN WATER DROPLETS FREEZE

NEAR - 37 c

7 8

At terminal velocity the weight of the droplet is balanced by the

drag force of viscosity. Using Stokes' equation

opo.(ni-i'vro (3.k.)

CD -40 (-40 -40

where C .D is the 'drag coefficient' introduced to correct for deviations from Stokes' law at Reynolds number greater than unity,17 and '7 o -go are the viscosities of the gas at 0°C and -40°C, Iro and Ar_40 are the

terminal velocities at 0°C and -40°C. Now C1, itself is of order unity

for the droplet sizes used in this research, and it is not expected to

vary strongly from this value as the temperature is altered. One can

justifiably write, then,

flov-0 = -140 -sto

Or,

11-0 Yo (211.). '6. (3.5.) 233 7-40 Fig.3.3. shows how the terminal velocity of a droplet in air

varies with radius at 0°C and -40°C at 760 mm Hg. pressure.

In the manner outlined the freezing temperature, droplet size and

cooling rate can be measured for each droplet. Repeating such a set of

measurements for a large number of droplets enables the statistical

temperature range as well as the variation of mean freezing temperature

with droplet size to be measured. This is the data needed to verify the FIGURE 3.3. 7 9

TERMINAL VELOCITY OF WATER DROPLETS IN AIR AT

-40°C AND 0 °C

32 -40°

24 (/) 22 U 20

2 MICRONS RADIUS 0 10 20 30 40 50 60 70

200 THERMAL RELAXATION TIMES OF WATER DROPLETS IN AIR AND HELIUM AT -40°C, ASSUMING NO EVAPORATION.

100

AIR

-50

HELIUM

MICRONS RADIUS

10 20 30 40 50 60 70 predictions of homogeneous nucleation theory.

This technique is limited in its working range of droplet diameter not by growth considerations as in the diffusion chamber, but by the thermal inertia of the freely falling droplets. Measuring the ambient air temperature will give the droplet temperature precisely only if the thermal inertia of the droplet is small enough for it to be considered in thermal equilibrium with it environment at all stages in its cooling.

What is this limiting droplet size?

Fig.3.4. shows the thermal relaxation time as a functioncf droplet radius for freely falling, non-evaporating droplets in air and helium at -40°C. The thermal relaxation time, -1 , is defined as the time taken for a droplet to come to (1 - k) of the temperature difference to which it is suddenly subjected. A criterion to estimate this limiting size is to calculate that size droplet whose thermal lag in temperature behind the environment temperature is of the order of imprecision of the temperature measurement. This thermal lag is found by solving the differential equations

T I

cri d-z. , at- dz. at- dZ (3.6.)

where T o is the environment temperature, v is the terminal velocity. The steady state lag of temperature is

die = 01 °C dz. (3.7.) 82 adopting the criterion mentioned for a temperature measuring accuracy o o -1 of - 0.1 C. Now for a temperature gradient (41° ) of order 1 C cm ,

the product IV is of order 0.1 cm. The droplet diameter in air which

satisfies this condition is between 40 and 50 microns diameter, whereas

in helium this is satisfied by a droplet of 60 to 70 microns diameter.

This is a consequence of the fact that "to(-1- 1 where K-is the thermal k. conductivity of the surrounding gas. For the purpose of this study

helium is the most suitable gas in view of its high thermal conductivity

and relatively high viscosity compared with hydrogen. This still rather

small range of from a few microns to 60 microns diameter was increased

to 120 microns by making a calculated correction for this thermal lag.

Another way to overcome the relaxation problem is to make the

temperature gradientl dI / very small. The limitation, though, is the

loss of static stability of the enclosed gas as mentioned earlier.

Extreme care was necessary in designing the lighting system to prevent

any heat input from the powerful lamps causing the gas to convect under

this weak stability.

Freezing begins when the droplet has reached a nucleation temp-

erature of about -37°C, then, very suddenly, its temperature rises to

0°C as the latent heat of freezing is partly absorbed by the droplet

itself and partly communicated to the surrounding air. Continued rel-

easing of such droplets into the cold stable environment will eventually

cause convection of the air at the freezing level due to absorption of

the latent heat of freezing. Another requirement, then, is a droplet

producer which produces the smallest number of droplets possible of controllable size. 83 No device has been constructed so far which will produce a single droplet at a time with a wide working size range. Of the droplet

producing devices constructed, the most versatile and reliable was the ordinary atomiser operated intermittently to give a few droplets at a time. In this way, the air or gas inside the chamber could be left to settle down (i.e. dissipate the latent heat absorbed from the droplets) before the next few were photographed.

Summary of this section.

In this section the principles of the technique developed have been put forward, and some limitations considered. The partial success of earlier research on the supercooling of water has been used as a guide in determining the requirements of any new research. The next sections describe in greater detail the individual components of the apparatus asfar as they can be considered separately.

Section 3.2. The design and construction of the freezing chamber

and the optical system.

Foremost in the design of the cooling chamber were the optical requirements.

When a supercooled droplet freezes the dissolved gas, being insoluble in ice, comes out of solution in the form of tiny bubbles which are sufficiently small to act as light diffracting centres. These centres cause more light to be scattered or diffracted towards the eye of an observer than the water droplets alone. Now a small water droplet scatters least light in a direction at right angles to the line of ill- 84 umination; when a droplet freezes the sudden brightening is therefore most pronounced in the direction at right angles to 4he line of illumination. The overall light scattering is so weak, that, when photographed against an absolutely black velvet background, the phenomenon can just be recorded using a very fast lens and high speed film. The chamber was therefore made quite large and lined with black velvet.

Fig.3.5. is a photograph of the chamber.

An intense parallel light beam from a mercury discharge lamp was shone through the chamber along the solid perspex light guides seen in the photograph. The advantage of using solid viewing and lighting windows is that the problem of misting and heat conduction becomes less severe since perspex is a poor heat conductor. It can readily be machined and polished to give an excellent overall viewing transmission with very little distortion. In the photograph the viewing windows are seen to be in a five-fold multi-layer glass arrangement: this was later replaced by a solid perspex block.

With such intense illumination and very black background, the tiny droplets brightened most noticeably on freezing; it appearias if a tiny light had been switched on inside them. A photographic record of this phenomenon is shown in Fig.3.2. In the background grazing the edge of the light beam and slightly out of focus are the four thermocouple points A,B,C, and D supported on fine nylon fishing wire. The photograph was taken on Agfa Record hypersensitive film (1,200 A.S.A.) at an aperture of f.2.8, exposure sec.

Several prototype chambers made of brass were found to be totally unsuitable for this technique, because, although the air inside was im- parted with an extremely small temperature gradient, the slightest dist- FIG. 3. 5.

THE COLD CHAYBaz 86 urbance caused the system to convect. Secondly, sealing leak-tight windows onto metal proved extremely difficult as they had to with stand temperature variations of up to 70°C. Hardwood made impervious to gas by several coats of lead paint was eventually found to be ideal.

In order to check that no carbon dioxide gas had diffused into the chamber a small quantity of internal gas was examined by the chemical lime water test, and every few weeks the chamber was subjected to a slight excess pressure from inside and connected to a manometer to test for leaks. These checks are very necessary, because it was later discovered that the behaviour of freezing water drops in carbon dioxide is quite different from that in air. This will be discussed in greater detail in Chapter V. Fig.3.5. also shows the prespex droplet entry tube and the gas inlet and outlet taps.

Section 3.3. The particle filtration of the gas cooling medium.

Through operating the chamber in an unsaturated condition one loses the convenience of being able to 'rain out' all the minute dust particles suspended in the gas. To avoid the embarrassment of the water droplets collecting dust particles as they fall, these particles were removed by a very simple and efficient technique.

The air emerging from a compressed air cylinder is relatively clean compared with room air. Passing this air through 10 ins. of hard packed glass wool leaves the air so clean that a Pollak counter is unable to detect any remaining particles. A measure of this cleanliness is afforded by the ratio , where

I is the photoelectric current fafter he expansion. _Lo {_before FIG. 3.6 g7

DETAILS OF THE AIR CLEANING PROCESS,

INITIAL FINAL % TEMP. OVERPRESSURE / l o DEFLECTION, I, DEFLECTION, I 143) Room air

17°C 99±1 mm. t 6.8 1:.05 cm. 2.4 c„,. 0.35

(b) Compressed air

17 148 6.8 5.85 0.86

17.5 102 6.5 6.5 1.0 140 6.75 6.75 1.0 165 6.3 6.3 1.0 202 6.95 6.95 1.0 270 7-2 7.1 0.99 278 7-1 6.9 0.97 280 7•1 6.55 0.92

At- 0,n- ok•fr e.55 urn- of 270 w, ft. 4j Ke txrcu,sion rAkio u 1.361 corrf spott 4[61 iv a- skiAyA,-, ok, rc,1-149 of 7-I. ' 11 /4.4:4 5 0,. h.crA.,h...ok yod-i4 how,b3tKeba,,5 co Kaerts cti4,0K frowk, ike ii4:;41Ar oscAA.r s i a_ 0,Gcount for Itn.e cLilAt,i1..es it al. .5 a. LVa.vuokle4 kr readicA.. (e) Compressed, filtered air after flushing chamber

0 Au, 117 3'0 0.9 0. 30 15 122 3.0 1.35 0'45 30 124 3.0 1.45 0.48 2 ivi.;41,. 118 3.0 1. 95 0'65 4 118 3.0 2.2 0.74 9 11 8 3.0 . 2-6 0.87 11 120 2. 8 2.5 0.90 13 120 2.8 2.6 0'94 88 Details of the 'cleanliness' of various air samples together with the

cleanliness of the air emerging from the chamber after a certain duration

of flushing are given in Table 3.6. The last tabulation shows how the

air emerging from the cold chamber gradually becomes cleaner with in-

creased flushing time. After about 20 mins. of slow flushing, the

change in galvanometer current from the photoelectric cell of the Pollak

counter was hardly detectable. A compressed air line from this same

cleaned gas was tapped for the operation of the atomisers too.

Section 3.4. Details of the water purification process.

Pruppacher (loc.cit.), 1963, has carried out an exhaustive set of

tests on the cleanliness of water purified in different ways, viz:

distillation, ion-exchanging, ion-exchanging and redistillation, with and without gas removing cartridges etc. A good measure of the particulate cleanliness is the extent and regularity with which a deep supercooling can be attained. He found that straight ion-exchange filtration gave by far the best particle-free water. Subsequent distillation, then, appeared to be responsible for the re-introduction of particulates.

Using the results of Pruppacher's research, a simple yet efficient particle filtration resin was used in a manner depicted in Fig.3.7.

Experimentally it has been confirmed that there is an optimum flow rate for optimum particle filtration. For a cross-section of C104 Elgastat end-polishing resin, this flow rate is about 17 litres/hour.

From the end polishing resin the water is conducted in the absence of any room air through the glob bottle of the ultra microscope section and then to the twin atomisers shown in Fig. 3.16.

to dust trap 9 FIG. 3.7

SCHEMATIC DIAGRAM OF THE WATER . PURIFICATION PLANT. distilled water storage bottle

to atomisers

end polishing resin

flow line

glob bottle of scintered glass support the ultramicroscope section 90

TIT WATER PURIFICATION ASSEMBLY. BEHIND TH3 POLLAK COUNTER

ARE THE TWIN ION EXCHANGE COLUWNS. 91

FIG. 3.8.

PARTICULATES RENDERED VISIBLE IN TAP WATER. 92

FIG. 3.9.

PARTICULATES RENDERED VISIBLE IN DISTILLED WATER.

•

• • 0. • 10 • I . 0 •

• 93 RIG. 3.10.

PARTICULATES RENDERED VISIBLE IN ION-EXCHANGED dAnll 94

A GREATLY DILUTED SUSPENSION OF POLYSTYRENE LATEX COLLOID EX-

AYINED BY THE SAYE TECHNIQUE. THE PARTICLE DIAYETER IS 0.557 ± O. Olog.“ 95 THE ULTRA- MICROSCOPE ASSEMBLY. FIG. 3.11

water inlet

water outlet

large aperture condensing doublet telescope adjustable mercury arc sighting lamp 1500 cc. aperture glob bottle

THE THERMOCOUPLE ARRANGEMENT. FIG. 3.12

differential thermocouple

I 1.0cm nylon fishing thin perspex support line support frame

potentiometer silicone oil

thermos flask with crushed ice 96 Before using the water plant all the glassware (including glass stoppers) was rinsed with hot chromic acid, washed in distilled water and flushed with several tens of Winchesters of ion-exchanged water to remove the dust and loose particulates caught on the glass surface.

The inter-connecting tubes were of P.V.C. plastic.

Figs. 3.8., 3.9., and 3.10., show the particulates in tap water, distilled water, and ion-exchanged water respectively. The scale of the photographs corresponds to an area of dimensions 3jcm. x 2icm.

These water samples were photographed by passing an intense parallel beam of light from a mercury arc lamp through the bulk liquid contained in a 1500 cc. glob bottle. This bottle was situated inside a black velvet-lined, light-tight box depicted in Fig.3.11. After the eye had become accustomed to the darkness, these tiny particles could be seen like yellow stars twinkling in a sky blue background. In Fig.3.10. only, the sky blue beam background appears as a faint grey scattering because of the longer exposure given to the film. These photographs were taken on 1200 A.S.A. film at an aperture of f.16, the duration of the exposure being 15 seconds for the ion-exchanged water, and 1 sec. for the distilled and tap water.

Presumably the 'twinkling' is caused by Brownian fluctuation of the orientation of the particles within the light beam. An estimate for the size of these particles can be obtained from the Einstein equation for the root-mean-square angular rotation of particles much larger than the molecules of the supporting medium:

(3.8.) 97 where V is the rotation in time t, 47 is the liquid viscosity, 'r' is the 'radius' of the particle. The frequency of the 'twinkling' is -1 of order 3-5 sec , caused by rotation presenting a different area of scattering cross section to the light beam. Suppose the rotation necessary to present a different cross section is of order V2: radians. Then - _16 2. If 0,3 . . 3 x 4 . to-?- . r which shows that r 0.37,

To check this calculation, polystyrene latex spheres of diameter

0.;^ , 0.5 N , and 0.9 ,with very small dispersion (for example, electron diffraction photographs show the dispersion of the 0.5 de‘ particles to be about - 0.01A^ ) were examined by the same technique. The 0.5/ diameter spheres could be picked out quite easily, but the 0.1/0-spheres could only be detected in large quantities by the effect they have in changing the colour of the scattered beam from sky blue to deep purple.

One can conclude this section by stating that the insoluble impurities, of size greater than about 0.5/Adiameter, amount to no more than a few per cubic centimeter in the ion-exchanged water. Compared to the particulate content of tap water, these photographs show that ion-exchanging is also an efficient particle removing process.

Section 3.5. Temperature measurement

Fig.2.4. (a) or (b), shows the expected 'range' of mean homogeneous freezing temperature as a function of droplet size. The statistical range for a fixed droplet size is likely to be about 1°C. To be 98 able to resolve features of this freezing range requires a temperature + resolution of at least - 0.2°C or better still 0.1°C. This is not easy in air at -40°C; the system finally chosen requires careful calibration.

Very small bead thermistors looked promising, having an additional feature that no ice bath is required. They have a resistance variation of about 2202°C-1 at -35°C; this is very easily measured in a resistance bridge. Unfortunately, they do not hold their calibration very well as was evident when re-checking the calibration at the mercury point, -38.9°C, after a period of one month. Thermistors were abandoned in favour of the more stable, though less sensitive, thermocouple.

Figs. 3.12. and 3.13., shows the method of mounting the thermocouple junctions. The thin black perspex frame had nylon fishing line stretched across at 1 cm. intervals to which the thermocouple junctions were attached. This method minimises the effect of any thermocouple support on the local temperature gradient.

Two copper-constantan thermocouples in series develop an E.M.F. of 727V per °C with cold junctions in an ice bath at 0°C. The precision to which the thermocouple potentiometer could be read was - V; this gives a theoretical resolution of at least t 0.1°C. Calibration to this precision demanded great care. Mercury had an excellent calibration point at -38.9°C (its melting temperature) because of its good thermal conductivity and large latent heat per unit volume.

In the vicinity of -30°C to -45°C no other liquids have this useful property; N.P.L. calibrated thermometers which register to t 0.1°C are only calibrated to -30°C. 9 9

FIG. 3.13.

THE TiMITCOUPLE ARRANG.,IENT. 100

In this case the most reliable way to calibrate the thermocouples is to locate the -38.9°C experimentally and use the tabulated temperature - E.M.F. slope for copper-constantan thermocouples for variations up to 5°C on either side of this point. For the experimental determination of the mercury point, the 40 s.w.g. welded cold junctions were carefully wrapped in thin walled polythene tubing and submerged in a melting vessel containing about 1 kg. of distilled mercury. The warm junctions were similarly immersed in thin walled glass tubing containing silicone oil; this glass tube was itself deeply immersed in a Dewar of distilled water and crushed ice stirred at regular intervals.

Knowing this calibration point any neighbouring temperature can be estimated accurately enough by the following relationship provided the temperature departure from the calibration point is not large:

38.9±41 -381 dT

Similarly, knowing the resistance of the galvanometer and the gradient-measuring thermocouple, the calibration at -40°C can be more readily calculated than measured empirically.

Section 3.6. The operation of the entire assembly

Fig.3.14. depicts schematically in block sections the interconn- ections of the complete apparatus. The photograph of Fig.3.15. shows FIG. 3.14 101

BLOCK DIAGRAM OF THE COMPLETE ASSEMBLY. distilled water,

atomisers

ion exch.; particle removing. !selector

ultra— microscope freezing chamber

filtering, Pollak monitor. thermocouple ice potentiometer. bath.

compressed gas. 102

FIG. 3.15.

THE COMPLETE APPARATUS. THE FREEZING APPARATUS FIG. 3.16 103

comFressedd gas line

water line

to pressure release, water overflow and dust trap.

leak test manometer

open/ close valve

gas taps It gas tight freezing chamber heat filters

solid perspex light guides

multiple glass windows mercury arc lamp

cold metal base n.b. the heat insulation chamber, inside which the freezing chamber was situated, is not drawn here. 104

A CLOSER LOOK AT THE CENTRE SECTION OF THE APPARATUS. 105

how it actually looked in practice. To the left top, the large glass

distilled water storage bottles and the twin ion-exchange columns can

be seen; in front of these is the Pollak counter and the photoelectric

cell galvanometer. In the centre there is the outer heat insulation

chamber with the atomisation chamber and droplet settling column on

top. Supported in a laboratory clamp to the left of the outer chamber is the leak test manometer. (Fig.3.16. shows this section in diagrammatic

form). To the right side of the outer chamber the thermocouple potentio-

meter and recording galvanometers can be seen; the mercury discharge

lamp and associated heat filters are located underneath the galvanometer

stand, but partially obscured by it in this photograph.

Immediately after loading with dry ice the chamber was flushed with

filtered gas for 15 minutes and the water system set running to load the

water line with fresh ion-exchanged water. Approximately two hours later after a slow cooling a test run was carried out to find where the mean freezing temperature is located for a given droplet size. This level was then adjusted to co-incide with the level AB so that the whole statistical range is lovered between the differential junctions C and D.

After the temperature levels in the chamber had settled down a few drops were injected by operating the atomisation trigger and leaving the shutter open for the appropriate time. Temperature measurements were continuously recorded in progression with the photographic film. It would have been impossible to print the thousands of 35 mm. frames taken in this way; instead a microfilm reader was used to take these results straight from the negatives.

Before concluding this chapter it might be appropriate to dispel a criticism that has often arisen in the discussion of the research,

106

namely the excess temperature of a droplet over its surroundings due to

a radiation effect. A calculation on the following lines shows the effect to be negligible.

(a)Division of the chamber into sectionsr, each with a characteristic black-body radiation temperature.

(b) Net budget of radiant energy received by a droplet of radius r. (c) An 'equilibrium' is set up in which the droplet, by virtue of an excess temperature over the surroundings, conducts this excess radiant

energy to the surrounding gas. This equilibrium equation will give a measure of the excess temperature. The detailed calculation is given in Appendix 3 - may it suffice here to present the results of this calculation.

Droplet radius (/(.): 10 20 30 40

o C excess temperature: 1.4x103 2.8x10 3 5.2x10-3 5.6x10-3 7.0x10 3

It can be seen from this tabulation that the effect of radiation is unimportant, and does not affect the temperature measurements. 107

CHAPTER IV

An analysis and discussion of the experimental data.

Very necessary in the analyses of this chapter are the physical

properties of the six gases used in the course of this research, namely air, oxygen, hydrogen, helium, carbon dioxide, and argon. Tabulations of the density, viscosity, thermal conductivity, specific heat, solubility, are given in Appendix 4.1. The data has been extracted from the Inter- national Critical Tables and from Stephen and Stephen "Solubilities of

Inorganic and Organic Compounds".

Section 4.1. A description of the freezing histograms

Figs. 4.1. and 4.2. are the histograms of freezing temperature plotted in steps of 10 microns diameter in air and oxygen respectively.

The temperature intervals are 0.1°C, and the scale of counts per unit temperature interval is shown on the diagram too. Several features of these histograms are immediately apparent:

(i) they are similarly shaped and almost symmetrical within the respective size groups.

(ii) the extreme width of the freezing spectrum is about 1.5°C. The temperature interval between the 25% and 75% probability points can be calculated from these histograms. It is 0.4°C.

(iii) In the case of oxygen there is a discernible elevation of

FREEZING OF WATER IN AIR

- 10 10 - 15 15 - 20 20 - 25 25 - 30 30 -35 35 - 40 40 - 45

MICRONS RADIUS FIG.41

FREEZING OF WATER IN OXYGEN

10 10 - 15 15 - 20 20 - 25 25 - 30 30 - 35 35 - 40 40 - 45 MICRONS RADIUS FIG.4.2 110

mean freezing temperature with increase of droplet size from 10 to

50 microns diameter, followed by a sharp tailing off to lower freezing

temperatures. This elevation is not discernible for freezing in air,

but the sharp lowering above 50 4 diameter is readily seen.

(iv) the mean freezing temperature for a given droplet size

depends on the nature of the cooling gas to a small extent. This can

be seen in Fig.4.3. where the histograms for the freezing of 30-40 mic-

rons diameter droplets are shown for the six gases for direct comparison.

Taking helium for the time being as 'standard', the largest depart-

ure is exhibited by carbon dioxide, -0.6°C, and argon +0.7°C. In trying

to compare any real differences in freezing temperatures of small drop-

lets in the different gases, several effects have to be taken into account:

(a) a relative 'cooling rate effect', mentioned in Ch.II., which

depends on the relative viscosities of the gases. In the neighbourhood

of -370C, the effect is kkx,h, to be a depression of 0.5°C for an increase in cooling rate by a factor 10(see Sec.4.2.)

(b) a thermal lag of the droplets (Sec.3.1.) due to the differing

thermal conductivity of the gases. This thermal lag of temperature is

given by the product

Thermal relaxation time x temperature gradient x terminal velocity,

and, as far as the thermal conductivity is concerned (alia aequalis) will

be proportional to 1< , , where k is the thermal conductivity (see

also the tabulation in Sec.4.3.)

carbon dioxide, there may be a depression in the freezing temperature in

the same way that soluble salts depress the freezing temperature of bulk water by an amount depending on the solute concentration. An expression 111 FIG.. 4.3

15 - 20/A..t. -25 cm-1 d

HELIUM HYDROGEN

r- -36 -37 -36 -3'7 -38 TEMP °C

-25

AIR OXYGEN

n -36 -38

-25

CARBON DIOXIDE ARGON

n 11 1-1 -36 -37 -38 -36 -38 112 for this freezing point depression may be obtained from Raoult's Law and the Clausius-Clapeyron equation using standard thermodynamics.

This raises the question, 'how much gas is dissolved in saturated supercooled water at -40°C? To answer this question the temperature coefficient of solubility must be known. The calculation giving the final depression of freezing point is quite lengthy - for this reason it is presented (for the case of carbon dioxide) in Appendix 4.2.

The table below shows these calculated corrections for droplets of 30-40 microns diameter so that the real differences in freezing temperature are brought out more clearly. THERMAL scAlvAwly coo,w, `CORRECTED' MEAN GAS LAG .DEPp,Ess,on/ RATE FREE214Q TEMP.

HELIUM + 0.02°C 37.2°C

HYDROGEN + 0.05 - + 0.15 37.2

AIR + 0.1 - - 36.9

OXYGEN + 0.1 - - 36.9

ARGON + 0.2 - - 36.4

CARBON + 0.2 DIOXIDE +0.71 - 36.9

CHLORINE - - - (?)

* It must be emphasised that these 'corrected' mean freezing temperatures -1 are not corrected to a cooling rate of 1°C sec 1 but to the cooling rate of 30-40 micron diameter droplets falling at terminal velocity in helium gas ( 4°C sec-1).

Pruppacher (loc.cit.) has shown that the eventual freezing temperature of a solution is the additive amount of the natural supercooling and the straight depression by solute concentration. Making use of his 113 observation and the calculated 001ute depression by carbon dioxide of 0.71°C, one can say that the experimentally observed depressioh after allowance has been made for a relaxation lag is almost certainly a solute-depression effect; carbon dioxide, therefore, shows no una- ccountable abnormalities.

On the other hand argon, which has the same physical characteristics as air except for a slightly smaller solubility, shows a marked elevation of mean freezing temperature relative to air (0.5°C). The solubilities of air and argon are much too low to account for this effect by solute depression. One is tempted, therefore, to invoke the 'clathrate compound' hypothesis which could explain this elevation in terms of the 'ordering' of the water molecules into 'Gages' around the gas molecule: in this way an enhancement might be expected in the ordering of molecules by fluctuations prior to a phase change. Xenon and chlorine are known to form crystallised hydrates in which the gas atoms or molecules are trapped in 'cages' of hydrogen-bond linked water molecules. It was therefore tempting to repeat the measurements in xenon or chlorine gas to look for any possible marked elevation. However, the cost of xenon on the scale required for this research is prohibitive. Difficulty was experienced with chlorine gas (preventing the use of photography) because the faint scattering of light by the small water droplets was partly obscured by scattering from the dense green-yellow gas. The brightening had to be detected by eye, thus limiting the precision of the temperature measurement. The small droplets appeared to freeze at a temperature between -36 and -37°C, appearing like white 'dots' in the greenish gas. Larger droplets of size above approx. 40 microns diameter left a white trail in their wake which quickly disappeared. The

114 cause of this trail is unknown. Chlorine, in the presence of water vapour,is a very corrosive gas. It would be appropriate to mention here that the cold chamber was completely destroyed.

Summarising, one can say that there is an effect on the mean freezing temperature depending on the type of cooling gas used. These effects are small but almost certainly outside the limits of experimental error. At this stage it would be unwise to try to draw any concrete conclusions about the mechanism of these effects.

Section 4.2. A more detailed analysis of the shapeof the freezing histograms

and an estimate of the'cooling rate effect' in the vicinity

of -37 C. From Figs. 4.1. and 4.2. it appears that the mean freezing temperature decreases with increasing droplet size. This is contrary to the behaviour shown for homogeneous nucleation in Fig. 2.4. (a) or (b). However, the results are not directly comparable for two reasons. Firstly, it will be remembered that the homogeneous nucleation curve in Figs. 2.4. (a) and (b) was drawn for t: = 0.6 seconds, i.e. in each temperature interval in which

is determined, the water sample should be kept for 0.6 seconds. In this technique, the ' time ' for which a water droplet remains in a 0.1% temperature interval is determined by the relation

d TdzdT = v d t dz dt Then,

d t = 0.1 seconds ( 4.1. )

0( V

This time factor varies as the droplet radius varies. Before the measured mean

115 freezing temperatures can be transposed onto Fig. 2.4. (a) or (b) an adjustment will have to be made to allow for this effect.

Secondly, no adjustment has been made to allow for the temperature lag of the larger droplets behind the environment temperature, i.e. the temperatures recorded by the thermocouples situated in the gas environment are not the same as those which would be recorded by an ' imaginary thermocouple situated inside the droplet. A further adjustment will have to be made to allow for this effect. The cooling rate adjustment will be dealt with first.

Equation ( 2. 18. ), Ts

logo ( 1 - P ) V T ) dT s s Jo shows that for constant probability and volume , if the cooling rate is increased by a factor 10 then the integral must be increased by the same factor, i.e. T must be increased. From the shape of the freezing histo- s grams, T ) can be evaluated over a small temperature range and the 5*( s integral can be evaluated numerically. This method can be used to estim- P ate the temperature increment over which 7*( Ts) dT increases by s a factor 10 . Before doing so, a few qualifications ought to be made concerning the shape of the histograms.

For tho. histograms of 30 - 40 micron diameter droplets shown in Fig. 4.37 the systematic corrections have been given in tabulated form in section 4.1. Any errors in shape can be due to. one of the following causes:

(a)insufficient temperature resolution

(b)insufficient size resolution, where the difference in temperature lag becomes appreciable with small changes in radius. The

116

tabulation at the end of Sec. 4.3. shows that if the histograms are plotted

in 5/, radius steps, this problem becomes serious above 20/4- radius in air

and 30/... radius in helium. No allowance has been made for this effect when

plotting the freezing histograms: above these sizes they should not be inter-

preted as being ' true ' freezing histograms.

The first reservation is more difficult to answer, for although

the thermocouples were calibrated to read to - 0.1 C, the effect of entrain-

ment of air by the droplets is difficult to estimate. Since, however, the

droplets were photographed a few at a time, and their natural fall speeds

separated their arrival at the freezing level, the effect of entrainment

is probably small too.

For the case of droplets of 15 - 20/, radius freezing in air,

the freezing histogram and_7*( Ts) over this temperature range are shown in Figs. 4.5. and 4.6. respectively. Fig. 4.6. was derived as follows:

(i) By definition, the number of critical aggregates formed in a

droplet of volume v cm3 within t seconds at a temperature Ts below 0°C is,

on average,

m = 7*(T s) .v.t ( 4.2. ) In view of the stastical nature of the freezing process, m = 1 does not

imply a definite freezing event. If the mean rate of formation of aggregates

° at Ts C below 0°C is m, the probability of freezing is 00 -m mr e = ( 1 - e-m ) ( 4.3. )

rj00 Probability of freezing 0.01 1.0 0.01 0.63 0.1 2.0 0.1 0.86 0.3 3.0 0.18 0.95, and so on. o.4 0.33 0.7 0.50 0.9 0.59 U)

L.1-1

0 ou C.< Co 0 —CO

C.) Csl 0001

U) CS3 003 0 .ov J f/ 09 f: 09 0Z (N 7 C) i 59 I > '21/1 LU

0 1), S?:

...c) 0 0 . CJIA 0.3

ii 0.

0 0 )

118

(7, 01= U.77T.L.: •

10 10

109

_to

8 10

io7

c or- 10 6 -3 .6 •7 •;? •9 37 •1 .2 .3 .5 .6

I- I 9

(ii) The probability of freezing is calculated as the ratio

No. of droplets observed to freeze for each temperature Total no. of droplets taken in the interval

increment. This probability is then used to find m as a function of Ts, from

which 7*(T s) can be found through equation ( 4.2. ).

From Fig. 4.6. one can derive that near -37°C an increase of

* ( Ts ) dT s by a factor of 10 occurs over ^-0.8°C. However not too

much importance should be attached to this estimate since it depends rather

critically on the shape of the tail ends of the histograms, where a difference

of a few counts will alter the calculated value of i7* strongly. The pur-

pose of this analysis is to show that, in principle, the cooling rate

correction can be derived from the shape of the histograms; but in view of

the uncertainties involved it would be better to assume a cooling rate

which has actually been measured ( Carte, 1956, Sec. 2.7.b.) as 0.5°C

per factor 10 in cooling rate.

Returning to the cooling rate adjustment, the ratio of 0.6 sec.

to the time spent by a falling droplet ( at terminal velocity v in a temp-

erature gradient °C cm-1 ) in a 0.1°C temperature interval is o -1 0.6 tzs v 6.v for a gradient 1 C cm . 0.1 TERMINAL VELOCITY COOLING RATE ADJUSTMENT

1.0 Chn .S-ec 0.4 °c 5.0 0.7 10.0 0.9 15.0 1.0 25.0 1.1 In this manner one can adjust the mean freezing temperatures whereby one

' imagines ' that the different sized water droplets have been kept at

each temperature interval for a period of 0.6 seconds.

120

Section 4.3. An assessment of the thermal relaxation effect.

When the thermal inertia of the droplet becomes large, the rate of transfer of heat from the droplet surface is insufficient to keep its temperature in step with the temperature change of the surroundings. This is obviously the reason for the sharp fall of the environment temperature at the level of freezing of the larger droplets. If the temperature lag can be calculated precisely, the true mean freezing temperature can be obtained for a greater size range, which is one of the objectives of this research. The steady state lag is given by the solutions to equations (3.6.) as = `r: ncr.C.1._ v. az This result demands that the relaxation time be known.

If a droplet radius r cm. is suddenly plunged into an environment of temperature different from its initial environment, it will come to 0 - 1/e) of this temperature difference within a time "Ngiven by the solution to the 2 heat conduction equation) where ir ri6w cw (4.4.)

3 Kgas and 4 are the density and specific heat of water respectively, and / WI Cw K is the thermal conductivity of the environment gas. gas In general, if the surrounding gas is unsaturated, evaporation will assist the transfer of heat away from the droplet by the absorption of latent heat.

A combined solution of the heat transfer equation and the evaporation equation shows that ( 4.4. ) must be modified to give the relaxation time as

2 r ,O c ( 4.5. ) w w 3 L'K L D gas V where L is the latent heat of vapourisation and D is the diffusion coeff- v

121

icient of water vapour in the gas, and Cy$ is the slope of the ct.T saturation curve at the temperature concerned (Kinzer and Gunn 1951).

Lastly, if the terminal velocity of a droplet is large compared

with the velocity of propagation of the heat diffusion field, one

further term is added to the increase in cooling rate owing to the

'ventilation effect', a factor fi F The thermal relaxation time r//5 . now becomes

4'1(0,1 3[J(,0 4- I— vi s lr f lt-F (4.6.) J

where s is a characteristic heat diffusion thickness identified with Iji,d (2r) I e.,,„ , and F is a numerical factor which tends to unity -5 -2 -1 as Re- C) . Now for air, 1/ . has magnitude 5x10 cal.cm sec , and -6 -2 -1 o 1-v-b drs has magnitude 2x10 cal.cm sec at -40 C; the evapor- cur ation term, therefore, is a factor 20 less than the main heat transport

term by conduction. The relaxation time can therefore be simplified to

(4.7.) 3 {14. Anl {It F' '2

where F = 0.24 F, 0.22 F, for air and helium at -40°C respectively. There is some controversy over the behaviour of the factor F(see for example

Fuchs' "Evaporation and droplet growth in gaseous media" p.59), but as I the term 0.22 F Reg is of order 0.09 for an 80 micron diameter droplet,

this term cannot dominate the relaxation time for these droplet sizes.

Therefore, accepting F as unity is quite adequate for the purpose of the

following calculations. The calculated relaxation times as a function

122

° of radius for air and helium at -40 C have beenA Fig.3.4. and are shown again here in Fig.4.7. together with some experimentally measured points

which will be described in Appendix 4.3. At this stage the equation for

the temperature lag i9 = 1011-D.17.1r and the relaxation times enable the dz. temperature lag to be calculated as a function of droplet size, o -1 \ (a) in Helium ( dTo 1 C cm- 1) dz

radius (microns) Equilibrium temperature lag

10 1.4x10-3 °C

20 22x10-3 30 0.1 40 0.28 50 0.6

6o 1.1 70 1.8

(b) in Air

10 0.01

20 0.16 3o 0.7 4o 1.9 50 4.o 6o 7.4

Section 4.4. Derivation of the true mean freezing temperature at a

standard cooling rate. FIGURE 4.7

THERMAL RELAXATION TIMES OF WATER DROPLETS IN AIR -200 AND HELIUM AT -40°C, ASSUMING NO EVAPORATION.

z 0 ).1 150 L CJ

100

-50

;50. 1 4

The solid lines in Figs. 4.8. and 4.9. marked ' cooling rate and relaxation corrected ' should give the true mean freezing temperature as a function of droplet size for t = 0.6 seconds. In this form it is directly comparable with the predictions of homogeneous nucleation theory.

The line in the case of helium is directly transposed onto Fig. 2.4. (b), with the addition of a few points plotted for droplet diameters less than

10 microns. Earlier in this thesis it was pointed out that these tiny droplets froze, on average, at exactly the same level as the 10 micron diameter droplets. A cooling rate correction taking into account the very slow rate by virtue of their small terminal velocity is sufficient to enable their mean freezing temperature to be plotted also.

The way in which these mean freezing temperatures fall on the predicted line is quite impressive. Just as striking is the excellent match between Longhorn's observations of the mean freezing temperature of 100 micron diameter droplets and the present measured mean freezing temperature ( shown in Figs. 2.4.(a) and (b) ). These two sets of measurements

have been obtained by totally different techniques: this lends to support the fact that the basic property being investigated is the bulk property of water unaffected by its surroundings, be these gaseous media or insoluble organic liquids.

One must not forget that in the case of freezing in helium, up to a diameter of 120 microns the cooling rate adjustment is a factor which strongly controls the shape of the finally adjusted curve. The magnitude of this cooling rate effect has not been determined precisely in this research for reasons stated in Sec. 4.2.; instead a measured cooling rate effect is used of 0.5°C per factor 10 in cooling rate. The slope of the finally adjusted line depends more-or-less on this value being correct. r. 2

FREE/II"! C.: 0:: V\11•:1 /I ou LIA1)11.7-. 0.8 °C CM .3") 0 \''¼J D E. a‘J. Z s>.

COOL I 1.,IG RATE C., LAY,ATION CORRECTF D

COOLING RATE CORRECTED

EXPERIMENTAL CURVE

-40

-42

MICRONS DIA. -

2,0 3,0 4,0 5,0 6,0 7,0 cp _ 9,0 190 110, 120 F IG. 4.8 32 36 28 40 44 10 loom,

o O 0 TEM RE V PERATU 30 I 50 C

MICRONS DIAMETER 70

FREEZING OFWATERINHELIUM TEMPERATURE GRADIENT WATER: DEIONIS 2/77 X • COOLING RATEUNCORRECTED

90 6 21 X

oCOOLING RATECORRECTED COOLING RATEAND RELAXATION CORRECTED Al

110 ° C 13 4 CM

X 130

1 2 7

Also, in making this adjustment, it has been assumed that the cooling rate adjustmentis valid for two orders of magnitude differences in cooling rate.

The points on the theoretical homogeneous nucleation curve in Fig.

2.'4. (b) have been derived from the experimental measurements by a com- plex process. This,in effect, is the price one has to pay in adopting the technique of freezing without a substrate. The two adjustments just described have tended to obscure the direct theme behind this research, and therefore some main points of Ch. 2 Sec. 2 and this Ch. 4 will be linked together here in stepwise form to draw together the experimental work and the theory. To recapitulate:

(i) In Ch. 2 Sec. 2 an expression was derived for the net rate of formation of critical aggregates, u+Acr nkT kT e h For the mean freezing temperature, P

:r *, v . t = 0.7, and therefore

-U + &r- nkT 0.7 e kT ( 4.9. ) vt h

In the case of water, Lu* can be calculated as a function of supercooling in terms of the bulk properties d's, Lf

2 146 1 sl (1;1 2 ) L dT 3 s f

In principle „this a* can be substituted into ( 4.9. ) to predict the mean freezing temperature as a function of volume of pure water and 128 the duration of superpooling or cooling rate. However, the unknown par-

defeats this approach, and therefore a reverse method is ameter e5"sl used to fix cl at one measured supercooling and then resubstitute sl this value into ( 4.9. ) to predict the shape of the curve over an extended volume. The homogeneous nucleation curve of Figs. 2.4. (a) and

(b) has been derived in this way, drawn for t = 0.6 sec. using Mossop's observation that droplets of diameter 0.8r froze in 0.6 sec. at -41.2°C

+ 0.4°C. This method does not take into account any temperature variation

one must not forget that the location of the curve depends of 6sl strongly on the chosen value.

(ii) The experimental curves of Figs. 4.8 and 4.9 are not directly comparable with this theoretical line for two reasons:

(a) Because the droplets fall through a fixed temperature gradient at terminal velocity, the time factor for which a droplet remains within a specified temperature interval varies with droplet size. To make the

mean freezing temperatures comparable, an adjustment is made whereby one

' imagines ' that all the different sized droplets are held within each

temperature increment for 0.6 sec.

(b) When the temperature lag of the larger droplets behind the en-

vironment becomes appreciable, the thermocouples measuring the environment

temperature no longer record the true droplet temperature. To offset this

effect, and to extend the useful size range of the measurements, a

second correction is made for this temperature lag. For droplets of 120

microns diameter, this lag is calculated to be 1.1°C and 7.4°C in o -1 helium and air respectively when the temperature gradient is 1 C cm .

A more extensive idea of the magnitude of this effect can be gained 1_29 from the table at the end of section 4.3.

The points plotted in Fig. (2.4. (b) ) are those 'corrected' measurements in helium gas. These points were chosen because the large

thermal conductivity of helium allows the greatest size range to be utilised, and secondly its solubility is very low ( see Appendix 4.1. ).

Over the size range 7 - 130 microns diameter there is good agreement

between the derived points and the theory. The range utilised is still rather small, but by this technique it is impossible to extend the

precision measurements above 130 microns diameter since the 'corrections'

then become extremely large. For example, the temperature lag varies 4 as r and is already 1.1°C for 120 micron diameter droplets freezing in helium.

The full experimental verification of the homogeneous nucleation

curve is by no means complete, but it is gratifying to know that when

all conceivable precautions over accidental nucleation are taken, deeper

supercooling than that shown by the theoretical curve has not been

observed. There now seems no doubt that this semi-empirical curve does

indeed describe the homogeneous nucleation of water with the empir-

ically determined parameter e5 . Further evidence for this concl- sl usion will be discussed in the next section, and a summary of the

conclusions that can be drawn from this research will be given in

Sec. 4.6.

Before beginning the next section,some values for &I , r*

n* , calculated from the experimental measurements will be tabulated

here.

130

EQUIV. RADIUS MEAN T *c.i-3 r* n* s c1 5 micron -37-s 9.6 x108 15.0)(10 1.0703 160 10 -37.2 1.2 15.7 1.10 170 25 -36.2 7.7 106 16.6 1.12 180 50 -35.4 9.6 x105 17.3 1.16 200 70 -35.0 3.5 17.7 1.17 210

Section 4.5. A discussion of the physical model describing

the phenomenon of supercooling, and its relation to the

description of the other physical properties of liquid

water.

The model adopted and described in Chs. 1 and 2 explained the phenomenon of supercooling as the inability of thermal fluctuations to produce ice-like aggigates of sufficient size to nucleate the water until a temperature of -37°C is achieved. The number of molecules in an ice-like aggregate existing in equilibrium with supercooled water can be estimated through the relation for the equilibrium radius == .2 6s` i—forr T ir with sl as an adjustable parameter. Fig. 4.11. shows Y1*(T ) in the s range - 60°C to 0°C calculated in this manner with the integration per- formed numerically in 2°C steps. At -37°C the critical aggregates are built

5000 '

no. of molecules in an equilib. size cluster calculated from r ..2 o_•:„ with aa, a) 22 e. rs c;:.12, 10, r1.--5.dT b)20

„P 1 c)18 • 1000

/12 500

100

ay. no. of molecules per cluster, 'flickering cluster' model.

onset of horn. nucl.

Tern1 —40 —2,0 20 40 60 80 's I 1 132 of between one and two hundred molecules.

No specific structure of the liquid water is assumed other than that it has some 'irregular' structure, and that local thermal fluctuations cause these aggregates to continuously appear and disappear. The model requires that at any instant in time some fraction of the water must exist in hydrogen bonded form according to the magnitude and frequency of these local energy fluctuations. An expression has been derived for the rate of formation of these critical sized aggregates, which, in the steady state must equal their rate of disappearance, viz: 1)t, 4- Ar kT

If the experiments described in this thesis are representative of homogeneous nucleation, the experimental mean freezing temperatures can be used to calculate g* ( 15x10-13erg) and n* ( -,150-200 molecules) at -37°C.

Now the formation of a critical-size aggregate occurs when a suitably

'cold' region is created in the liquid by the local energy fluctuation, -15 which has a magnitude of order 15x10 erg. Relaxation of this 'cold' region can occur by the formation of additional hydrogen bonds in this region, the energy of formation of the bonds being taken up in icreased • thermal motion of the localised region. Since the magnitude of the hydro- -15 gen bond energy is of order 3.4x10 erg, at most about 5 additional bonds can be formed in the region which has experienced this fluctuation. With the addition of these few molecules, this region becomes sufficiently large to act as a critical-sized aggregate, which, if it has an ice I-like struc- o ture, consists of 150 to 200 molecules near -37 C. The inference to be gained from these figures is that in the supercooled liquid there already exist transient hydrogen-bond-linked molecular aggregates or 'clusters't

133 the largest of which (near -37°C) have a size only slightly smaller than that of the equilibrium aggregate.

Frank and Wen (1957) have described how these clusters can

'instantaneously' form and arappear by introducing an element of

'co-operativeness' into the hydrogen bonding scheme: this is itself due to the partially covalent character of the hydrogen bond. The hydrogen bond can be described in terms of the 'resonance' scheme shown below

• • • •

• N "' . • • C 0. The formation of a hydrogen bond between a and b introduces a partial charge separation which is inducive to further bonding along a chain in a two dimensional scheme. Consequently, the formation of a single bond causes further rapid bonding according to the progress of local thermal fluctuations, whereas the breaking of a single bond can cause the rapid disinte- gration of the whole aggregate. It is possible that these aggregates can haVe a large number of structural arrangements, but the tridymite-like arrangement in ordinary ice is likely to occur often, especially at low temperatures, because it involves a relatively large number of hydrogen bonds for a given aggregate size. This, in fact, is just a more detailed description of the origin and disappearance of the aggregates invoked in the model of Chs. I and II.

By postulating the existence of a 'chemical reaction' between mono- meric water molecules and the hydrogen bonded aggregates of the form

H 0 2 (4.1o.) 1_34

(similar to Rontgen's approach (1892) ), Frank and Wen were able to account for several of the anomalous properties of water, viz:

(i) the temperature of maximum density, and its variation with pressure

(ii) the decrease of the viscosity of water by the application of pressure

(iii) the tentative explanation for the existence of a single die- lectric relaxation time, determined as the mean life time of the aggreg- -11 ates, approx. 10 sec.

(iv) the structural changes of water to an increased "ice-likeness" when hydrocarbons and other non-polar substances are dissolved. This tendency towards increased "ice-likeness", i.e. the formation of excess hydrogen bonds, can explain the anomalous phenomena such as the large heats of solution, volume changes on solution, and the large heat capacity of solutions.

Litovitz and Carnevale (1955) have made use of the same model to successfully account for the anomalous acoustic absorption of water above that by its sheaf viscosity. The pressure waves shift the equilibrium of the reaction (4.10.) with a phase lag, causing the excess absorption of sound energy. Smith and Lawson (1954) have used essentially the same model to calculate the variation in the velocity of sound waves in water with temperature and pressure.

The fact that the model can even qualitatively explain these anomalous properties is itself good evidence for its correctness, but direct experimental evidence for the existence of these aggregates or clusters has not come forward so far. Frank (1964) has made a survey of the ex- 1 5 perimental evidence and concludes that some 'evidence' comes from infra- red Raman absorption (the 175 cm-1 band in liquid water seems to be the -1 counterpart of the 220 cm band in ice which has been ascribed to a normal mode of vibration in the ice lattice), but this must be regarded as presumptive rather than proof. Koefoed (1957) argues that the difficulty of forming crystallisation nuclei spontaneously in supercooled water is a forceful argument against the existence of 'small fragments' of ice-like structure in liquid water. In this section it will be shown that this argument is fallacious, and, to the contrary, the supercooling phenomenon strongly supports the model and is (so far) the only study which gives some direct evidence in support.

Nemethy and Scheraga (1962) have adopted this model and have been able to derive the thermodynamic parameters of liquid water by the methods of statistical thermodynamics. Their method is essentially as follows:

(i) For a given number of molecules (N) in an aggregate the fraction of water molecules 4,3,2, hydrogen-bonded, y4,y31y2, can be expressed in terms of the fraction of singly bonded molecules y and N by counting from 1 constructed ice models (tridymite-like).

(ii) If the mole fraction of monomerx water is X K , the mole fraction of water i times bonded is

i= 9E (1 - 1c-) I- 1,7,3,1+

On an energy level diagram the 4-bonded water molecule is taken to be ground state, and the 3,2,1,0 bonded molecules are assumed to have energy levels equally spaced above the ground state. The partition function for

1 mole of water molecules distributed over these energy levels is constructed of the form

136 (.2c, , N )

( X . - E. .--... ° tiK == NI O . l'/W KT .-.....

I. (Ni oX L. ) 1• 1 i.o..... CW where the f. describe the vibration, rotational, and translational free- 2. dom allowed to each molecular species.

These quantities are evaluated as 5 — f, )2 s • —1 i, .... II ( i — €_ ki-) 3.1 A it& ' i+roM • ilk ' i from experimentally determined infra-red absorption bands.

(iii) The partition function is evaluated in the 'usual' manner by

equating it to the maximum term for any given temperature. This is deter-

mined by solving simultaneously the equations

0 ) —

This method enables NI ), T ) , and x ( T) to be found, once

the energy level spacing is fixed. Nemethy and Scheraga use this energy

separation as an adjustable parameter to find the best fit between calcu-

lated and measured thermodynamic quantities, through the relations

F CT) = - ki ApZ E(T) = Crz ) )7/

50) - E F Cd = 3 QT)) 7 T

Their curve obtained for N(T) is shown in Fig.k. ., for temperatures

between 00C and 100°C. 137

Unfortunately the calculations were terminated at 0°C; consequently the values of the most probable cluster size, N(T), have been extrapolated below 0°C in the form of the dashed line shown. Except for Cv- (which falls off too rapidly with temperature) the calculated and experimental points agree remarkably well over the temperature interval 0-100°C.

The X-ray diffraction data for the radial distribution curve can be reproduced quite well, as can the P-V-T data of the anomalous properties of water by choosing the molecular 'free volume' of the unbonded molecules as an adjustable parameter. Miller (1963) has shown that an excellent empirical correlation exists between the temperature dependence of viscosity and the 'free volume', f, introduced into an equation for the vis- q cosity 11 = Nemethy and Scheraga have used the results of similar calculations to explain quantitatively the anomalous properties of the solutions of hydrocarbons in water.

One phenomenon that has not been treated quantitatively by this statistical method is the supercooling of water. It is not difficult to show that the data obtained from this model would indeed predict the phenomenon of supercooling and also give a fairly precise estimate of the temperature region of homogeneous nucleation. It can be clearly seen in Fig.4.1i. that between 0°C and -30°C the most probable aggregate size produced by thermal fluctuation is much smaller than the equilibrium sized ice-like aggregate.

In this temperature region the model predicts that pure water would exist supercooled in metastable equilibrium. Below -30°C the most probable aggregate size approaches the size of the critical nucleus described in Chs.I and II. One would therefore expect the temperature region of homogeneous nucleation to lie below -30°C. A more precise estimate of the temperature region of homogeneous nucleation is limited by X38

(a) the uncertainty of the numerical value of 6L

(b) the unknown size distribution of the aggregates at any specified temperature

(c) uncertainties introduced into N(T) through the assumption of five equally spaced energy levels (see Nemethy and Scheraga 1962).

The conclusion to be drawn from this section is that the original model adopted to interpret the supercooling experiments also accounts very well, both qualitatively and quantitatively, for a considerable number of the regular and anomalous properties of liquid water. Of the more recent models developed to describe the structure of liquid water, e.g. the

'vacant lattice point' model (Forslind, 1952,1953), the 'water hydrate' model (Pauling 1959,1960), the 'distorted bond' model (Pople 1951), none

describe so well such a range of properties as the model just described

(sometimes called the 'flickering cluster' model).

The fact that the predicted temperature region of homogeneous nuclea-

tion almost co-incides with the temperature region measured experimentally in this research leaves no doubt that the freezing behaviour described in

this thesis is characteristic of homogeneous nucleation. This phenomenon

supplies the first piece of direct evidence for the verification of the

existence of these aggregates and their size over a limited temperature region in the neighbourhood of -35 to -38°C. 139

Sec. 4.6. Conclusions drawn from the observations and measurements described in this chapter, The establishment of the theoretical homogeneous nucleation curve of Figs.

2.4.a and b , in the manner described in Ch. 2 has led to an effort to verify its prediction of a variation of the attainable supercooling depending on the volume of the water sample and the cooling rate. Longhorn and Mason( 1958 ) have extracted from the literature the reports of supercooling which seem to verify this curve, or at least its general features which are quite distinct from the type of line shown in Fig. 2.5. ( The latter characterise the supercooling of water droplets infected by tiny ice nuclei ).

With the appearance in the literature of aircraft observations of liquid water below -40°C, the question arose as to whether the laboratory measurements were representative of true homogeneous nucleation. ( One must remember that the curve was located by experiments which utilised condensation of water vapour on artificial nuclei, and that the manner of freezing of supercooled water is quite different from that of many other liquids ). One of the purposes of this research has been to simulate as closely as possible the conditions under which natural cloud droplets freeze ( prolonged supercooling and the effect of pressure remain outside the scope of this research ).

Roturning to the nucleation curve, at larger volumes the theoretical curve may not be fixed precisely, for it will be remembered that the 'location' of the curve depends strongly on the chosen value for 6s1 , and no temperature dependence of this parameter has been incorporated into the thermodynamic expression for,N5*. Those deep supercoolings described in Sec. 2.5 seem to come very close to this curve, though how much the decrease in attainable supercooling (over the small samples) is due to a real volume effect and how much is due to residual nucleation is difficult to decide, bearing in mind the extreme difficulty of removing tiny particulates from volumes of 140 this size.

In the search for this volume effect, what is required is a single technique covering as wide a size range as possible, combined with precision temperature measurement and prevention of accidental nucleation. Of the methods described in the literature survey, most have suffered from the disadvantage that as the volume becomes larger, heterogeneous nucleation plays an increasing role in determining the supercooling behaviour. In this category come the methods of freezing in glass capillaries or on glass, freezing on metals or coated substrates, and suspension between two liquids. It occasionally happ-

ens that large volumes can be strongly supercooled by these techniques (e.g.

Meyer and Pfaff 1935) but this is for the most part rare. Surmounting this

difficulty is accidental nucleation by the insoluble particulates in the

water itself. The diffusion chamber does not suffer from these disadvantages

but the small droplet size range and the strong temperature gradient prevent

the detection of a volume effect.

The new technique described in this thesis was derived mainly by mod-

ification of the diffusion chamber principle, but in the conversion one of the

main conveniences was lost,i.e. the ability to clean the coolant gas by rain-

ing out the particulates, and the growth of droplets from the vapour in a

clean environment. The manner in which this has been overcome, and the ad-

vantages gained in the conversion have been described in Sec. 3.1.

Fig. 2.4. b. shows that the research has been successful to some

extent in showing for the first time a distinct volume effect which can not

be ascribed to accidental nucleation by a substrate. That the useful size

range only extended to 140,/^diameter was a disappointment, but this was not

altogether unexpected. Serious difficulty arose in the interpretation of

the correct freezing temperature for the large droplet sizes. The other ex- 141 perimental difficulty had begun to limit this approach, i.e. precision temperature measurement. This necessitated the use of 'adjustments' which varied as the square and fourth power of the droplet radius: their use above 1401/1/ v diameter in helium gas is unsatisfactory. At present the extension of this method does not seem a practical feasibility. However, the technique was successful in that droplets of size much larger than 140/, could be consistently supercooled to temperatures near -37°C, the heterogeneous nucleation of these larger droplets up to 200/Adiameter being extremely rare. One is led to conclude that in the other methods of supercooling the substrate is the main agency which causes nucleation as the volume becomes larger.

After all conceivable precautions have been taken over cleanliness, this research has pointed out that

(i)Supercooled airborne water droplets will freeze at a temperature near -37°C depending on the volume of the sample and the cooling rate.

( There may be a slight pressure effect but this is not expected to alter the supercooling behaviour strongly).

(ii)The measurements show a distinct volume effect which cannot be ascribed to accidental nucleation by the substrate. The measured volume effect, and that predicted by theory agree very well. The fact that the shape of the measured line is identical to the semi-theoretical line is evidence that the freezing is true homogeneous freezing.

(iii) The supercooling behaviour of water, when highly purifiedlis found to be similar when studied under grossly different experimental conditions. A very pleasing aspect of this work is the way in which these measurements align with those of Langham, Carte, and Mossop (see Figs. 2.4.a and b ). This is further evidence that the nucleation is a property of pure water. (iv) Over a limited range amenable to analysis, Ad) * and Ill.* have

been calculated and shown to be consistent with the 'flickering cluster' model for water structure. It has been demonstrated that this model predicts the phenomenon of supercooling, and determines the region of homogeneous nucleation to lie in the vicinity of -30°C.

(v)kenall effect has been found depending on the nature of the dissolved gas which must in some way affect the water structure. ( For lack of more information, it would be inappropriate to speculate here ).

Items (ii), (iii), and (iv) together point towards the conclusion that what has been observed in the course of this research has been the true homogeneous freezing of water. 142

CHAPTER V.

The Splintering and Shattering of Freezing Water Droplets and

their Relation to the Rapid Glaciation of Natural Clouds.

Section 5.1. The Evidence for a Rapid Glaciation Mechanism

The Bergeron-Findeisen process which describes one method for the onset of precipitation in middle and northern latitude clouds was briefly described in Ch. I. It demands the appearance of ice particles in the upper regions of a cloud which will grow preferentially from the vapour until they are large enough to fall through the updraught, collecting supercooled water droplets as they do so,

If water droplets supercool, at what temperature (below 0°C) do the first ice particles appear in natural cloud tops? Aircraft

penetration flights (e.g. Peppier 1940, Findeisen 1942, Koenig 1963)

show that this temperature is very variable: it has been reported in

greatest detail for layer type clouds as distinct from the scanty information available about cumulus clouds. Peppler's detailed in-

vestigation of the summit temperatures of layer clouds shows that the

ice phase may be completely absent at -30°C, yet snow crystals have

been reported in other clouds whose summit temperatures extended only

to the -6°C to -7°C level. These two cases are extremes mentioned to

point out the variability of the temperature of the onset of the ice

phase: in general, though,Ippler's many hundreds of observations over

the North German Plain show that when the summit temperature of natural

clouds falls below -12°C, the ice phase is more likely to be present 143

than absent.

Peppler (1940) also studied the characteristics of cumulus clouds arising in the same regions. A total of 609 clouds were studied by aircraft traverses, but unfortunately the temperature onset of the ice

phase was not dealt with in detail. He writes that, in general, the ice phase is absent until summit temperatures reach -10°C, and that a larger fraction of cloud tops contain ice particles as the temperature

progressively falls below -10°C. Koenig (1963) has written of a field

study programme in Missouri in which repeated aircraft flights were made through the tops of simmer time cumulus clouds. The 0°C isotherm at this time of year is typically located at ca. 15,000 ft., and the level to which the cumulus clouds penetrated (ca. 20,000 ft.) enabled

them to achieve summit temperatures of -5°C to -S°C. These flights

showed that cumulus tops may glaciate within a period of ten minutes

or less, and that there was seemingly a dependence of this rapid glacia-

tion on the appearance of water droplets of about 1mm. diameter in a

concentration of 50 m.-3 or more. This observation is probably sub-

stantiated since rain was often seen to fall from the base of the cloud

before the appearance of ice particles, i.e. before the commencement

of the Bergeron-Findeisen process. Towards the end of this chapter

the significance of these observations will become more apparent.

Another piece of evidence for this rapid glaciation of cumulus

clouds is the phenomenon seen clearly from the ground when a cumulus

top loses its vigorous 'bubbling' activity to become a fibrous, diffuse,

silky-white. Such a transition, usually accompanied with an increase FIGS. 5.1. AND 5.2.

THE ONSET OF THE APPEARANCE OF A DIFFUSE TOP TO CUMULUS CONGESTUS. 1 I_5 of upward growth rate and penetration height, takes place within a time scale of order ten minutes. The phenomenon is interpreted as the transition of supercooled water droplets to ice crystals, the release of latent heat in the process being responsible for the increased buoyancy of the cloud and its further upward growth.

On the other hand, after the appearance of the diffuse white top there can sometimes be not only a slowing down of the vertical growth rate but a separation of the main body of the cloud from its diffuse top.

This phenomenon is partly shown in FIGS.5.1. and 5.2. taken at about 1 minute separation. The clouds are classified as 'cumulus congestus'; they developed near Oklahoma City in mid-Nay 1966 to reach a height of nearly 30,000 ft., The middle cell was just in the process of developing a diffuse top and at the same time appeared to be losing its vigorous convective activity. Not long after these photographs were taken the diffuse top seemed to move away from the 'roots' of the main cloud and gradually fragmented apart. Ludlam (1955) has suggested that this diffuseness is caused by the falling out of the larger cloud droplets as the updraught diminishes or disappears, leaving behind the residual of smaller droplets. This phenomenon is known as 'fibrillation', and is to be distinguished from the 'glaciation' process.

Summarising this section, one might say that there is evidence of a rapid glaciation process taking place inside convective cumulus clouds which is generally associated with a further vigorous upward growth. The only direct evidence, though, is from the aircraft penetration flights, and even this method is encumbered with instrumentation 146 difficulties. There is obviously more room for field work in this very important aspect of the rainfall mechanism.

Once a few ice particles have appeared, what mechanism enables the very rapid glaciation of all the other cloud particles to take place?

Some phenomena observed in the course of this research may help to track down the answer to this question.

Section 5.2. Early Ideas on the Glaciation Mechanism.

Whilst conducting some electrification experiments in a laboratory wind tunnel Findeisen (1943) noticed indirectly that frost deposits subjected to an air flow shed small ice splinters, since a charge was being developed on the body. Later observations repeating this experiment showed that this splintering only occurred if the frost deposit had fragile dendrite arms: smooth surfaces shed no splinters in the wind flow. Findeisen also reported some details of the riming of small ice targets by supercooled water droplets. He was of the opinion that the production of splinters by riming was a factor 104 more copious than the breaking of splinters from frost deposits. These riming experiments have been repeated many times since (Kramer 1948, Latham and Mason 1961,

Brownscombe 1966), but the results obtained for the sign of the charge and its magnitude for specified conditions are quite discordant. It still appears that the conditions for electrification by the ejection of splinters from impacting supercooled water droplets are rather special and are not yet fully understood.

It was first thought that the mechanism of the breaking of fragile dendrite ice crystals may have been responsible for the multiplication 147

of ice crystals in natural clouds. These splinters would collide with,

and nucleate, the other supercooled water droplets. However, laboratory

Investigations by Nakaya (1954), Mason and Hallett (1956) on the growth

habit of ice crystals in a supersaturated environment show that this

breaking of dendrite arms cannot be the responsible mechanism for glaci-

ation of cloud summits where these do not reach to heights below the -10°C level. That this is so stems from the fact that fragile dendrite arms

only grow on ice crystals below -10°C: above this temperature the growth

habit is in the form of mechanically stronger hexagonal plates or prisms.

Koenig et.al. (1963) have found that crystal multiplication can be already

well advanced before the cloud summit temperature falls to -10°C. This

is substantiated by his observations of the lack of any dendritic

fragments impacted on collection slides during aircraft traverses through

the cloud summits.

The observation of another possible mechanism, the bursting of

freezing water droplets into several pieces or the ejection of tiny

ice splinters has recently drawn much attention. This mechanism is

invoked to explain not only the rapid glaciation of natural clouds,

but to explain the electrification of thunderclouds also. The electrical

charge of residues left after the ejection of splinters has been the

subject of intensive research. Accounts of this mechanism, e.g. M'son

and Latham(1961), Latham (1965), draw considerable support from the

research of Mason and Maybank (1960) and Langham and Mason (1958). Both

of these researches, which deal with the fragmentation of freezing

water droplets, will be described and discussed later in this chapter. 148

A simplified model for this mechanism goes briefly as follows2

(a)Consider a droplet of water uniformly supercooled to -Ts°C containing a certain amount of surrounding gas dissolved in solution.

With increasing supercooling the solubility of the surrounding gas varies as log(solubility) = A (5,1.) T increasing with increasing supercooling very sharply in the case of carbon dioxide, though less severly with other gases.

(b)Now suppose the droplet nucleates. Ice dendrites will grow / -T / ) of the quickly through the supercooled water causing a fraction k s/Lf supercooled droplet to freeze almost instantaneously, raising the temperature of the mixture of ice and water to 0°C. At the same time C dissolved gas, corresponding to the difference of solubility at -1Is and 0°C comes out of solution and is trapped as tiny bubbles in the ice mesh (The appearance of these tiny bubbles and crystal facets in the ice water mixture is the cause of diffration and reflection leading to the

'brightening' of a droplet on freezing when suitably illuminated).

(c)A fraction (1-T ) of the droplet is left to freeze by the s/Lf conduction of latent heat through the droplet surface to the surrounding gas. Initially, this will form an ice shell around the ice-water mixture, since in the cold environment the droplet will freeze from the outside inwards. The larger the thermal conductivity of the gas the less will be the time (alia aequalis) required for the formation of an ice shell.

(d)Suppose the stage has been reached when an ice shell has formed, yet there is still liquid water to be frozen inside the shell. Since the specific volume of ice is greater than that of water, continued freezing 149 from the outside inwards will cause a pressure build up which can either relax if there are sufficient gas bubbles to go back into solution, or, if there is insufficient gas present the droplet may crack the outer shell or burst apart throwing out the inner material in the form of ice spikes or splinters. The manner in which this pressure relaxes is shown clearly in the photographs of FIGS.5.3., 5.4., 5.5., 5.6. The first three photographs are of water droplets frozen in free fall in air -1 whilst falling through a temperature gradient of order 2°C cm . hey(T were caught on a glass mirror and photographed with oblique 30° illumination from both sides for best contrast and resolution of detail).

The smallest droplets of FIG.5.5., of order 20 - 30 microns diameter show no tendency to produce spicules. FIG.5.6. shows a water droplet freezing in free fall in cold hydrogen gas - the manner of the pressure relaxation is much more vigorous in this case. FIG.5.7. shows both large and small ice splinters being ejected from the surface of a water droplet freezing in carbon dioxide gas. Carbon dioxide is a special case because of its extreme solubility which radically alters the freezing behaviour of a water droplet.

Returning to the simple freezing model, it is apparent that the criterion for splintering or shattering is an un-balance between the amount of gas present in the form of small bubbles in the dendrite mesh and the amount of liquid water inside the shell which has subsequently to freeze. A gas of good thermal conductivity will also assist the splinter or shattering mechanism by being able to form a strong shell completely round the droplet.

FIG. 5.3. 150

WATER DROPLETS CA. 60 MICRONS DIAMETER FROZEN WHILST AIRBORNE. THE SAME SIZE DROPLETS AS IN FIG.5.3. PHOTOGRAPHED AT A SMALLER MAGNIFICATION TO SHOW A MORE GENERAL FIELD OF VIEW. LeJ

• • "INOWI 04' 1 s. 7 1 31. - e r t. 4

Olt

'4 . or W ROM D

V • - 1 4 S N

t) OP RON IC es 20 M . • 53

FIG. 5.6.

film 800 A.S.A f. 2, 1 sec

THE SHATTERING OF A WATER DROPLET IN HYDROGEN GAS WHILST IN

FREE FALL. THE NUCLEATION TEIFERATURE IS ca. -37 C. 154

FIG. 5.7.

film 800 A.S.A. f.2, 1 sec.

ill. Hg discharge lamp

a large splinter was ejected from the r.h. droplet rt this level followed by many smaller ones

1 cm.

THE SHEDDING OF SMALL ICE SPLINTERS BY 60 MICRON DIA.

WATER DROPLETS FALLING IN COLD CO2 GAS. 155 FIG. , 5.8.

. SOL. Large suspended drops Small airborne drops nucleating 0°C Johnson 1966 near —37°C GAS m I/I. • nucl. near 'nucl. near 0°C —15°C

shatter shatter, onset 40 microns rad. HELIUM 9.7 dT .., 20°C sec-1 dt

shatter shatter, onset 20 microns rad. HYDROG. 21.4 • dT 012 °C sec-' dt

few shatter no shattering, no shattering, no splintering some AIR 29.2 if Tair<20.0 splintering

no shattering, no splintering

OXYGEN 49.6

no shattering, no splintering

ARGON 56

vigorous copious splintering, occasional CARBON splintering, fragmentation of larger droplets. occasional 15/4r droplet .03 splinters DIOXIDE 1713 shatter. 20trt as many as 10 —30. 156

On the basis of this model, nucleation near 0°C favours splintering T / or shattering (lower solubility and larger fraction of water ( 1 - s/Lf) to subsequently freeze), whereas nucleation far below 0°C does not favour splintering or shattering.

One must also bear in mind that not far below 0°C, the freezing will be slow and a good ice shell may not be formed. It appears then, that the conditions for splintering or shattering may be rather critical.

Evidence that this model is essentially correct can be seen from the table of FIG.5.8. which is constructed from Johnson's research (1966) and these observations. The techniques used were totally different yet the phenomena reported are very similar. These are attributed to the action of the coolant gases and not to any anomaly of technique.

Johnson has studied the freezing behaviour of mm. size water droplets

suspended on thin nylon thread and cooled in a cryostat. These larger

droplets were either nucleated near 0°C by innoculation with ice crystals

or allowed to supercool to between -15°C and -20°C. In his type of

experiment there is the added advantage that a droplet can be left super-

cooled to reach solution equilibrium which is of order ten times longer

than the time taken to reach thermal equilibrium. In this research this

factor was not controllable since the droplets were freely falling. In

the case of the larger droplets, the attainment of solution equilibrium

prior to freezing becomes uncertain, but by the simplified model, this

would enhance the shattering process.

Carbon dioxide is an exception to this model. Its extreme solubility'

causes a nucleated droplet to bubble and froth, throwing off many tiny 157

ice splinters when these bubbles burst at the surface.(see FIG.5.7.).

It may be that the occasional 'fragmentation' observed of the larger

droplets is caused by the tremendous quantity of gas coming out of

solution.weakening the dendrite mesh. Small droplets up to 30 kdia.

generally leave a few splinters behind at the freezing level, whereas

larger droplets may eject as many as 10 or more in a single thin line

as if they had been forced out of a single crack in the ice shell.

Returning to FIG.5.8. it can be seen that with increasing solubility

and decreasing thermal conductivity the likelihood of shatter decreases

and, in the case of air, Johnson's work shows that nucleation near o 0 C at this solubility does cause a few drops to shatter if the surround-

ing air temperature is cold enough for a good ice shell to form quickly.

One small anomaly has been found in this table, that droplets

shatter more readily in hydrogen whose solubility is greater than helium

at-40°C. Other than viscosity or solubility, the physical properties

of the two gases are very similar. The answer may lie in the fact that

in hydrogen small droplets fall faster and do not have the same time to

dissolve the surrounding gas as droplets in helium. This non-attainment

of solution equilibrium would aid the shattering mechanism.

Having described the main points of this model of droplet shattering

and demonstrated its essential correctness, it is now possible to assess

the value of earlier research on shattering and splintering, in particular

the relevance of these studies to the explanation of the cause of crystal

multiplication in natural clouds. 158

Section 5.3. Previous research on Droplet Splintering and Shattering.

Schaefer 1952 (loc.cit.) was probably the first to observe the

splintering of airborne droplets on freezing. His apparatus was the diffusion chamber described in Chapter II. He writes, in the wake of some of these fast moving droplets tiny brilliantly illuminated

particles appear which seem to hang in the air motionless before they fall. Observed under a X10 microscope these points of light appear like

stars which rise 1 to 2 mm. before falling. At this moment the star disintegrates and clusters of 4 to 50 shiny ice crystals fall like a miniature display of fireworks."

Hallett 1956 repeated these observations with a similar diffusion chamber and reported the results in greater detail. As was shown earlier in Chapter III, the size range of droplets grown in a diffusion chamber is very small; for this size range of up to 20,/,diameter Hallett estimated

the average number of splinters produced per droplet to be about 5, yet

some produced as many as 10. (These tiny splinters grow very quickly at the freezing level in the diffusion chamber and can by counted by eye).

Bigg. 1957 also observed small clusters of splinters forming indirectly as droplets froze. The apparatus Bigg used was a shallow dish containing supercooled sucrose solution at -20°C(which served as an ice nucleus detector) underneath a block of cold copper at -30°C.

The cold copper block caused spontaneous condenstaion and occasional freezing in the thin layer of air above the sucrose solution. Bigg

counted as many as twelve crystals growing in the sucrose solution in

'clusters', although the average number per droplet was about five. 159

Magono 1955 also produced photographs similar to FIG.5.6. of the tracks of several particles emanating from the point at which a droplet froze in free fall above dry ice. The cooling gas must obviously have been carbon dioxide.

Langham 1958 reports the violent shattering of airborne droplets

50 - 100 microns diameter on freezing with the ejection of up to five splinters. (A re-evaluation of the droplet size from the photographs shown in his Thesis, natural size, indicate that the terminal velocity -1 was 50 cm.sec corresponding to droplets of diameter -v150/6%). These photographs are very similar to the one shown in FIG.5.6. except that some small particles a few microus in size remained at the level where the droplets shattered. These were identified as splinters.

The experimental apparatus in this case was a long column of cold nitrogen gas cooled from the botton upwards by the slow evaporation of liquid nitrogen. The droplets were produced by dispersing, unfortunately, silver iodide colloid to cause the droplets to freeze at -40C before completely falling into the liquid nitrogen. Langham quotes the freezing temperature to be about -5°C in an environment temperature of -65°C. Since air is composed of 79% nitrogen one would not expect a gross difference in the freezing behaviour in air or nitrogen. Maybank 1960, following the observations of Langham reports a more detailed investigation of the shattering and splintering of droplets in the size range 60,/v.to 2mm. diameter. Of these, the larger ones were supported on a shellac tipped thermocouple, the smaller ones on the underside of fine amyl acetate fibres inside a small cryostat. The 160 droplets were either nucleated at 0°C or -5°C by immersing them in a cloud of ice crystals produced by nucleating a water fog with tiny pieces of solid carbon dioxide or by silver iodide colloid suspended within the water drops. Splinters wore detected by the growth of ice crystals in a tray of supercooled sugar solution inside a transfer chamber. Splintering and/or shattering were observed over the entire droplet size range.

Summarising this section, one might conclude that there are many well documented cases in the literature of the shattering and splintering of supercooled water droplets after nucleation. This mechanism has been invoked to explain the rapid glaciation of supercooled cumulus tops

( Koenig 1963 ), and the thunderstorm electrification problem with the associated ice crystal economy of the thundercloud ( Latham and Mason 1961,

Latham 1965, Browning and Mason 1962 ).

In this research several thousand droplets were photographed freezing at -37°C in free fall in air, yet surprisingly not one shattered or produced a visible splinter. What, then, was the difference in the freezing mechanism in these rasmarches? The answer was not hard to find once a common feature of the earlier research had been spotted.

Sec. 5.4. A criticism of the previous research on splintering and shatter-

ing, and an assessment of the relevance of these experiments

to the glaciation and electrification problem.

A common feature of the diffusion chambers used by Schaefer and Hallet was that solid carbon dioxide was the base coolant. Biggs coolant 161

-for the sucrose solution is not mentioned in his paper). Mhgono and Maybank

both utilised solid carbon dioxide too. Did the chambers in fact leak

letting in carbon dioxide gas which might change the freezing behaviour

by virtue of its extreme solubility? To test this possibility the

chamber used in this research was filled with carbon dioxide gas and

careful observation made of the manner in which small droplets froze.

Indeed, this splintering and fragmentation was now very obvious appearing

exactly as described by Schaefer and Hallett. The small droplets left

behind at the freezing level clusters of 2-5 visible splinters, whereas

some of the larger droplets (80 - 150p diameter) often threw out ten or

more splinters which remained glittering suspended in a single line as

if they had been ejected from a single opening in the droplet surface.

In Maybank's case, the evaporation of the pellets of solid carbon

dioxide in the nucleation chamber must have caused the drops to absorb

carbon dioxide gas into solution, just as happened in the diffusion

chamber experiments. Secondly, Maybank's method of transferring small

droplets 60 microns diameter from one cold chamber to another affords

very poor temperature control since the thermal relaxation time is of

order 0.05 seconds, enabling a small droplet to change its temperature

easily in the transfer process. Johnson(1966)has repeated Maybank's

experiments in the absence of carbon dioxide and has been unable to agree

with Maybank's findings.

Magono, of course, froze his droplets in pure carbon dioxide gas.

Clearly then, as far as the relevance to atmospheric crystal

multiplication is concerned: these experiments have been incorrectly cited 162

for evidence of such a mechanism.

The one remaining observation of droplet shattering or splintering in the absence of carbon dioxide is Longhorn's work. The tiny crystals appearing in his photographs are not a product of the splintering process; they can be produced quite easily by allowing a sufficient number of droplets to freeze in a localised region in a short time. This allows a high supersaturation to build up in locality by the evaporation of water vapour from the droplets at 0°C into the cold surroundings at -40°C.

A stage is reached when condensation into tiny water droplets is achieved which subsequently freeze to form the ice crystals seen, giving the appearance of crystals left behind after the shattering process. If individual droplets are allowed to freeze in this manner the high supersaturation is absent and no crystals appear at the freezing level. It is quite erroneous to attribute these numerous small crystals to the mechanism of splinter production. Yet, without a doubt, shattering did occur in the absence of carbon dioxide. This could either be due to the tremendous temperature difference to which the droplets were suddenly subjected, assisting the shattering process by the non-equilibrium of solution or temperature (this is quite artificial as far as natural cloud processes are concerned) or to the nucleation not far below 0°C by the silver iodide. Longhorn's arrangement, by virtue of its design cannot be used to isolate the responsible mechanism, and again these experiments must be regarded with caution before citing these as evidence for the natural splinter production mechanism. 4) 100

Because of the intense interest in the splinter and shattering mechanism, it was thought to be worthwhile to repeat Langham's work paying special attention to isolate the two factors responsible for the shattering.

The freezing chamber was entirely enclosed with polystyrene foam heat insulation and cooled from the bottom upwards by an ordinary laboratory refrigeration unit. This unit could take the chamber temperature to

-32°C without using any additional coolant, at a measured temperature + 1 gradient of 0.5- 0.1oC cm .

A silver iodide colloid, prepared in monodisperse form after a method by Evans et.al. was strongly diluted and dispersed in the freezing chamber in droplet form from a few microns diameter to about 150/A diameter by ultrasonic and vibrating needle techniques.

May it suffice to report here that no shattering was observed, and no visible splinters were left after the droplets froze near -6°C, with variable air environment temperature between -12°C and -26°C. The entrainment of air by the many larger droplets prevented these observations being recorded to -6°C.

The important conclusion is that Langham's droplets shattered by the tremendous temperature difference to which they were suddenly subjected without having had time to reach either temperature equilibrium or solution equilibrium. This non-attainment of equilibrium prior to nucleation means that an artificially small quantity of nitrogen is dissolved in the supercooled water, and the amount of water to freeze T / after nucleation is greater than the fraction (1 - s/Lf). Both these 164 factors enhance the shattering process.

With the aid of this research, a review of the literature shows that the evidence presented so far for the chain reaction mechanism being splinter production is not valid for the natural glaciation of cloud tops or for thunderstorm electrification. Contamination by carbon dioxide radically changes the freezing behaviour of water by virtue of its extreme solubility. This fact has in some cases been overlooked; in others insufficient care has been taken to ensure its absence. Our knowledge of any chain reaction mechanism is now left very weak. This is exemplified by our inability to answer the following questions:

(i) Are these splinters of sufficient size to be picked out by eye under optimum illumination conditions?

(ii)If small droplets do not splinter or shatter when nucleated in air at -40°C, can they produce splinters when nucleated not far below

0 C?

(iii)If the answer to (ii) is positive, then between these two extremes there must be a supercooling at which the shattering or splinter mechanism ceases. What is this temperature?

(iv)Do larger droplets splinter more readily than droplets of size up to about 100 diameter, i.e. is there an effect as the volume to surface ratio becomes larger?

(v)Does the impacting velocity of a falling hailstone or graupel particle (which, in the case of natural glaciation or electrification determines the temperature at which a supercooled droplet is nucleated) play a part in the splinter production mechanism by preventing or aiding 1 65 the formation of an ice shell?

(vi) That role does the ambient pressure have on the freezing process?

There are three methods currently used in cloud physics to detect splinter production:

(a)Settling out of these splinters into a tray of supercooled sugar solution, where the splinters nucleate the solution causing crystals to grow slowly enough to be counted by eye. Only if the environment is saturated with respect to ice can one be sure that the smallest splinters will grow and settle down to the supercooled solution. Under these conditions there is always the possibility of an ice crystal from some other part of the cold chamber settling into the solution.

(b)Inference from the net chahe left on the residue. This method is indirect and gives no other information about the size of the splinters, the number ejected etc. Johnson(1966), has developed a,technique for simultaneously watching for splinters and charging with a very sensitive electrometer.

(0) direct observation with a well designed optical system.

The optical qualities of the cold chamber used in this research made it tempting to look more closely for the answers to the questions stated.

(±) In the case of carbon dioxide the splinters could readily be seen by eye. They remained in the light beam for up to 20 seconds at

-40°C before slowly evaporating. Their dimensions may have been up to a few microns. A half micron if not smaller ice crystal should be visible under these illumination conditions. 166

One can state, then, that in the case of water droplets up to 100 diameter freezing in undersaturated air at -37°C no splinters are ejected of size larger than, say, )2-- micron. Shattering is not observed either.

(ii) Throughout the course of this research it was found that it is by no means easy to nucleate small airborne droplets at temperatures other than around - 37°C to find an answer to the second question.

Silver iodide colloid will nucleate water droplets below -6°C and is suitable for looking at the splintering and shattering at lesser supercooling. The results indicated an absence of either process as indicated earlier. This method is, of course, artificial from the point of view of natural hailstone-droplet collisions in clouds for reasons which will be stated in answer to question (iv). According to the physical model, in the case of airithe solubility and/or poor thermal conductivity always win over the amount of water left to freeze after the nucleation of small droplets.

In answer to question (iv), Johnson(1966)has seen simultaneous charging and splintering of millimetre sized supercooled water drops, though whether the production rate is sufficient to account for the glaciation process cannot yet be assessed for lack of sufficient quantitative information. It might be opportune here to point out again Koenigts report of the state of cloud particle growth prior to glaciation. These indicate the presence of mm. sized water drops in concentration of order

50 m 3 in cumulus congestus tops.

Lastly, FIG.5.3. shows clearly the spicules formed on water droplets 167

frozen in air at -37°C. These spicules are also seen in photographs in

Koenig's paper on frozen droplets captured by ice crystals in natural clouds. The droplet sizes shown in FIG.5.3. and in Koenig's paper are both around 60-70p diameter. These spicules are evidently firmly attached to the frozen droplet since they do not break off at the terminal velocity. However, it is not known whether collisions with graupel or hail particles inside natural clouds can cause these to become dislodged, or whether the much increased air drag after collision can cause these to break off. Recent work by Brownscombe (1960, who tried to defect these splinters or fragments by blowing supercooled water droplets at an ice target and allowing the air to settle over supercooled sugar solution, showed that this last mechanism does not appear to be operative either, although recently the idea has arisen that these parameters may be very special (e.g. droplet size, wind speed, wind temperature, rate of accretion of supercooled water droplets) and he has by no means exhausted all the possibilities.

It is not possible at this stage to make an assessment of the mechanisms (iv), (v) and (vi) without further careful experimental work.

It is almost certain, though, that small droplets up to 100/^ diameter play no role directly in the splinter chain reaction mechanism. Koenig's observation that large water drops are the controlling glaciating

particles in a natural cloud has some support from this research. 168

Summary and Conclusions drawn from Chapter V

(i)Evidence has been presented that a mechanism is operative which can

cause an entire cloud region to freeze very rapidly once this process

has started. This glaciating mechanism can be well advanced before

natural ice nuclei become active.

(ii)Early ideas that this mechanism may be the breaking of frost deposits

cannot be correct for cloud temperatures above -10°C. The splinter

mechanism is described on the basis of a new simplified model which

correctly predicts the behaviour of freezing droplets in different gases.

(iii)Early research on the splintering mechanism was shown to be invalid

in those cases where carbon dioxide was present. Langham's observations

of droplet shattering in the absence of carbon dioxide have been repeated

and shown to be invalid also.

(iv)Some research has been described which indicates that small droplets

probably play no role directly in the glaciating mechanism. This is

consistent with observations from aircraft flights through Cu congestus

tops, which show that an appreciable number of large supercooled water

drops are present before the glaciating mechanism commences.

(v) Although this research has not been able to point out this mechanism

completely, it would be premature to speculate here because several

possibilities still remain. There is still the added complication of

any effect of ambient pressure (at high altitudes where the ambient

pressure is low the freezing behaviour may alter radically), and no

detailed study has yet been made of droplets larger than 100/LA diameter

freezing whilst falling at terminal velocity. This in itself could turn I 9

out to be a major engineering problem. Evenso, the improvement of the model which describes the freezing behaviour in different gases, together with the recognition of the fact that the nature of the surrounding gas grossly determines the freezing behaviour of a water droplet, constitute two steps forward to the solution of the glaciation problem.

170

APPENDIX 2.1.

A derivation of the work of formation and the equilibrium radius of a

droplet in a supersaturated vapour. Consider a closed vessel maintained at constant temperature T

and pressure pr of the vapour. The total thermodynamic potential is

(I) Nvap Nvap Nliq ~liq141-r26.1v ( 2.1.1.) is the number of vapour molecules, Nliq is the number of where Nvap -2 liquid molecules and Cs-ix is the surface energy cm between the phases. a3 In equilibrium, =0, subject to the condition that Nvap + Nliq= c. From these relationships an equation is found giving the thermodynamic potential difference between the vapour and liquid phases in terms of

the equilibrium radius ( 2.1.2) 4:17 vap =26 (Yliq r*

where \rug is the volume per molecule of the liquid phase. Now the difference in thermodynamic potential of the whole

system before and after the appearance of a droplet radius r is ( N + N )90 Ad2= Nliq Nvap Tvap 1+Tr261v liq vap yap 2 - - N ( - ) = 6 lv Nliq yap lig 1 ( 26" v ( 2.1.3.) tor r2 61v - lv liq ) vliq r ( 2.1.4) =4T! 0-iv ( r2 - 2 r3 ) 3 r* which has the form shown in Fig. 1.2. The height of the energy barrier T >k LIT is obtained by substituting r= r* in ( 2.1.4.). This gives -4 2 = 1 6 4 r* . In terms of macroscopically measurable par- 6 lv 3. 1 71 ameters, the quantity no can be obtained in the following limy:

v eTvap( Pr'T ) liqr( P ' T ) = 2e lv liq r* Pr v vliq dp = 2 ( vap dp 6 lv vliq 2.1.5.) frr r* Fl° 11 since at ( poo , T ) cp vap = CP liq

Applying the ideal gas equation to the vapour, and assuming the condition v > v vap liq , equation ( 2.1.5. ) can be reduced to

r* = 2 c5-iv(sj 1 ( 2.1.6.) (liq RT log p e r 3?0,,

Substituting this value for r* in (2.1.4) gives AV' in terms of macroscopic parameters.

APPENDIX 2.2.

A derivation of _ for a crystalline aggregate in a liquid.

Equations ( 2.1.4. ) and ( 2.1.2. ) are valid for this case also, except that in the L.H.S. there appears the quantity 2 Cr v sl sol r* These two equations give 2 A6 = 4 T6s1 vsoi e .1 • 4 ( 2.2.1. ) 3 I'Cr lig(05) - Crsol(r/F55S2 where Ts is the supercooling below the bulk equilibrium temperature. The denominator of ( 2.2.1. ) can readily be evaluated since

d qiici( p,T ) - dT Crsol ( P'T ) = )\ (T 172 where ( T ) is the molecular latent heat of fusion. Then Ts X ( T ) dT liq ( p'T s) (Tsol ( P'T s )

In terms of molar quantities br can be written

M si)2\ = 16 --11 6sl 3 1 3 2 111.511 f dT T L To

APPENDIX 3. Calculation of the droplet excess temperature over the surroundings due

to radiation.

Amount of radiant energy intercepted

by dS2 from dS 1 4 cos t92 dS cos0 T dS2 1 1 1

4 Normal area of receiving surface x solid subtended by radiatorx6 T1

Experimental arrangement. +,2ot

k —25 °c A A1.=',%/4- sq. cm. The temperatures marked are rough

averages over the divided surface C of the chamber. V

-50Z 173

In the main the radiation transfer is in the vertical, since horizontally there is not a great deal of temperature difference between the droplet and the walls. Assume all the enclosure radiates as a black body.

(i) Contribution from the aperture 177-1. 4 • 6' 213 lop (ii)Contribution from the top TrI. 10.10. .6 2.4-8 leo 4 (iii)Contribution from the side walls (upper half) IT), 24 io0 • .4- loZ (iv) Contribution from the side walls (lower half) and base at -50°C

Try:16 223.4 . 21 (v)Amount re-radiated, assuming the droplet radiates as a black body

at -35°C 4-j r26. 238`

-3 2 Net budget ,--z10 r cal. sec-1

Now imagine a steady state set up between the droplet receiving radiant energy and returning this to the environment by straight conduction. The excess temperature of the droplet over the environment is such that these two heat terms balance in the steady state.

2 -3 2 -47( R dT K = 10 r or dR air

d T = r2 dR 2 Kair. R

Integrating this equation and putting in the boundary condition that when R =00 , TR = Tc„.. , 174 (Tr - T D0 ) 1.1+ r

The assumption that the droplet radiates as a black body is not strictly correct since water absorbs preferentially in some infra-red bands, and secondly at the small droplet size range one encounters the Mie scattering region. However, it is unlikely that the final result will be altered strongly.

APPENDIX 4.1.

Physical properties of the gases used in this research. .4 D,-.. litre Jerila ca.I• e..`2 soli 1. e43h1 GAS DENSITY C 4 VISCOSITY THERM. COND. :4.01 .1BI.1 II:P " . P AIR 0 C 1 '293 0.238 170.9 5.33 x to"5 29'2 -40 C 1.51 150.3

HYDROGEN 0 C 0.090 3.36 84.11 38 21 -40 C 0.105 3.26 74.9

ARGON 0 C 1'782 0.121 209 3.77 56 -4o C 2.085 174

HELIUM 0 C 0.179 1.250 186 33.2 9.3 -4o C 0.209 170

OXYGEN 0 C 1.429 192 5.56 49 -4o C 1.67 0.217 166

CARBON 0 C 1.977 0.184 137 3.27 1713 DIOXIDE -40 C 2.31 120 175

APPENDIX 4.2.

-40at C Calculation of the freezing point depression by saturation with CO2 The temperature variation of solubility is given by

d ( log S ) = Q H A H = heat of solution. d T RT2

log S = -6H RT

A T Then log Si - log S2 = A A or, T T 1 2

S log 1 = A ( T - T ) S 2 1 2 T T 1 2 0 Now for carbon dioxide at 0 2O , and at 60 C S = 0.35 °C, Si = 1.7 ml/ml H 2 ml/ml H20, the volume of gas being measured at S.T.P.

These values can be substituted to show that the constant A is 1080 C

Then log o = 1080(273 - 233) s o 273 233 0 This calculation gives the solubility of CO2 at -40 C as 8.2 ml S.T.P./ml H20.

Calculation of the freezing point depression.

This is given by standard thermodynamics as

6T = )1X RT2, where AX = No. mols. solute L f No. mols. solvent

Now the number of gm. mols. of CO2 dissolved in 1 gm. mol. of water saturated at -40°C is 6.66x 10-3, which follows from the previous calculation and the fact that the volume of gas per gm. mol. at S.T.P. is 22.4 litres.

Substituting these values, and Lf as 56 cal. gm.-1 at -40°C gives the

176 2 freezing point depression AT 6.66 x 10-3x 1.99)(233 0-7°C. 56 x 18 This is the result utilised in Sec. 4.1.

APPENDIX 4.3.

A method for measuring the thermal relaxation time of small droplets in free fall.

To calculate the thermal relaxation lag to extend the useful size range of these measurements, the relaxation time of a droplet in free fall had to be calculated in the manner outlined in Sec. 4.3. A check on the magnitude of the calculated values was obtained using a similar technique to that described in Ch.3.

A small heavy brass uniform temperature enclosure could be rapidly moved towards,and away from, a source of different sized droplets

( see the figure of this appendix ). The lighting and photography for the method was the same right angle system described in Ch. 3. These droplets “ entered the chamber through a 4 diameter aperture into a heat filtered light beam entering the base of the chamber through a solid perspex window.

The level of the freezing is determined solely as a function of droplet size and its associated relaxation time as shown below:

The temperature variation is given by the relation which defines the relaxation time as

T - T f = 1 - e Ti - 'T ( 4.3.1. ) f where T, Ti, Tf, are the instantaneous temperature, the initial temperature, and the final equilibrium temperature respectively. The droplet is plunged

r ' 4 - 1 1_77-

C;•74.;-'7 1ON M7ASURING APPARATUS

ATOMISER OF ss c:AMI'ER .:i:-OU3 HATCHED \ TO UND''RNEATH TH7 DROPLET FIG. 3.16

ND DROPLET SOURCE !XSULATION A'RE i\IOT

TO COLD FLUID PUMP

•/".. ••••137'77,1

VIEWING

/ WINDOW

4 277 ,••••• \ "

4 24 RETURN TO COLD PUMP LIGHTING.

1_78

into a temperature difference ( T. - T f ) immediately after entering the aperture. After a time t secs., which can be related to the distance fallen

at terminal velocity,

T - T -z*/v -I f 1 ( 4.3.2. ) T.i 'T f * Now an excellent temperature marker to determine the temperature T after a known time is the brightening at the homogeneous freezing temperature, about

-37°C.Ifz*,v,T.and Tf are known, then'r(r) can be calculated. For example, suppose that a small water droplet at room temperature ( 15°C ) falls into an environment whose temperature is -50°C. Then, ignoring temperature signs, 37 15 - -z*/vq- 1 which gives for any 65 size droplet

z* = log 65 v iC e 13

If z*, v, can be measured, then 'C(r) can be calculated. The technique is extremely valuable since it is a reliable and fairly accurate method for tiny droplets whose temperature could never be followed by any conceivable measuring devices. Some results determined experimentally ( with estimated errors ) are shown in Fig. 4.7. The dashed line on either side of the calculated curve gives a distribution limit which takes into account the statistical variation of freezing temperature about the mean value. Over a limited range, the agreement between the experimental measurements and the theory is good both in the magnitude of the relaxation time and its variation with radius. However, the small size range amenable to analysis was disapp- ointing. At the very small droplet size,below 30A,radius, the effect of the finite thickness of the interface destroys the validity of equation ( 4.3.1. ) 4 I. At the larger sizes the distance z* varies as v`roc r Either an enormous 1 7 9 constant temperature enclosure is needed to extend the range, or else the

i - T ) larger. chamber must be cooled to lower temperatures to make ( T f The first suggestion is impractical, and the second line of attack was soon brought to a halt by the fact that no pumping fluid was available which remains inviscid at these low temperatures. EVen methyl alcohol becomes too viscous for the pump to circulate the coolant. Obviously cold liquid cir- culation is not the best way to cool the cold chamber: a controlled cooling utilising dry ice or evaporating liquid air or nitrogen would enable much lower temperatures to be achieved inside the cold chamber.

The experiment was a success though, in that it gave an experimental check on the calculated relaxation times of those sized droplets utilised in this research, and therefore more confidence can be attached to the thermal lag ' adjustment '. 1 9 0

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