Creator:David Turnbull
Date Created:September 21, 1951
Place Created:Schenectady, New York
Keywords:solidification of supercooled Mercury droplets
Context:article from The Journal of Chemical Physics
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RES. LAB. REPRINT 1902
GENERAL 0 ELECTRIC
KINETICS OF SOLIDIFICATION OF SUPERCOOLED LIQUID MERCURY DROPLETS
BY
DAVID TURNBULL
SCHENECTADY, NEW YORK
Reprinted from The Journal of Chemical Physics, Vol. 20, No. 3, 411-424, March, 1952
Printed in V. S. A.
Kinetics of Solidification of Supercooled Liquid Mercury Droplets
David Turnbull General Electric Research Laboratory, Schenectady, New York (Received September 21, 1951)
The solidification rate of supercooled liquid mercury droplets is strongly dependent upon the nature of foreign substances on their surface or suspended in them. Droplets (2-8 microns diameter) coated with mercury laurate solidify at rates that are proportional to droplet volume and satisfactorily described by the theory of homogeneous nucleation of crystals. Droplets coated with mercury acetate solidify with frequencies that are proportional to droplet area and in good agreement with the theory of heterogeneous nucleation.
The results on Hg2l2-coated droplets are interpreted on the hypothesis that nucleation is effected by suspended crystallites of one kind. For some droplet dispersions the kinetic results are apparently best described by a multiplicity of solidification frequencies/area or volume. Various hypotheses for this multiplicity are considered.
BECKER and Doring1 have developed the theory for the rate of homogeneous nucleation2 of liquid droplets in supersaturated one-component vapors. This theory predicts values of the supersaturation ratios required to nucleate liquid droplets rapidly in several one-component systems in excellent agreement with the measured values of Volmer and Flood,3 with the exception of that for methyl alcohol.
By following the basic assumptions of the Becker-Doring theory, Turnbull and Fisher4 have developed a theory for the rate of homogeneous nucleation in condensed systems according to which the frequency of formation, I, of crystal nuclei/volume of supercooled liquid is:
I=KV exp[-aa*/(AF,YkTl	(1)
where
Kv = n*(A/9rMkT/h) exp[-AFA/kT~\, (2)
<r=interfacial energy/area between crystal nuclei and the liquid, AF„ = the free energy change/volume for the transition liquid—^crystal, n*= number of atoms in the surface of a nucleus of critical size,
1	R. Becker and W. Doring, Ann. Physik [5] 24, 719 (1935).
2	Nucleation that occurs without the catalytic acid of foreign bodies is designated "homogeneous." "Heterogeneous" nucleation is catalyzed by such bodies. D. Turnbull, J. Chem. Phys. 18, 198 (1950).
3	M. Volmer and H. Flood, Z. physik. Chem. 170A, 273 (1934).
4	D. Turnbull and J. C. Fisher, J. Chem. Phys. 17, 71 (1949).
n=number of atoms/volume of liquid,6 &.FA = free energy of activation/atom for transport across the liquid-crystal interface, and a = constant determined by the nucleus shape. A is defined by the equation
A F/kT=Ai*+Bi,	(3)
where AF is the free energy of forming crystal embryos containing i atoms. For nuclei that are spherically shaped a= l6n/3 and
cA/9ir)i=(a/kT)K2v/9r)*,	(4)
where » = volume/atom of the crystal.
The free energy of forming crystal embryos and nuclei on the surface of nucleation catalysts can be described* in terms of i, a, AF„, and 0, the equilibrium contact angle between the crystal nucleus and catalyst when surrounded by supercooled liquid. By combining earlier concepts2'4 an approximate expression for the frequency of nucleation/area of catalyst, I„ may be written as follows,
I.=K, exp[-aaV(0)/(AF,)2*r],
(5)
where
K,=n*(A/9ir)*n.(kT/k) exp[-AFjhT\. (6)
6 In earlier publications, n was erroneously defined as the num-i ber of atoms/sample though the calculations were based upon a correct definition of n. The author is indebted to R. S. Bradley for calling his attention to the error made in the definition. A. D. Turnbull, J. Appl. Phys. 21, 1022 (1950). D. Turnbull, J. Chem. Phys. 18, 768 (1950).
412
DAVID TURNBULL
ns= number of liquid atoms in contact with catalyst/ area. For nuclei having the form of spherical sectors,
/(0)=(2+cos0)(l-cos0)2/4
(A/9t)*= {ff/kT)\2v/9ir) *[(2 - cos0+ cos30) /4]1'6.
The factors that, according to Ecjs. (1) and (5), determine I and Is are at least approximately known excepting <r6 and 6. Therefore, the validity of the nucleation theory can be tested by measuring the variation of I or Js with temperature and comparing the values of (-K^)exp or (i£s)exp computed from these experimental results with the theoretical values predicted by Eqs. (1) or (5).
Crystal nucleation frequencies of large continuous masses of supercooled liquid are generally unsuitable for testing the predictions of nucleation theory because these masses almost inevitably contain nucleation catalysts of unknown composition and variable potency.2'7'8 Let the number of these accidental catalytic centers in a given mass of liquid be mo. It has been postulated2,8,9 that if the mass is broken into a number of isolated droplets m5>rrh crystal nuclei must form in the major portion of the droplets homogeneously or with the aid of a known crystalline film sometimes used to isolate the droplets. The evidence supporting this postulate has been fully summarized elsewhere.6,10 If a mass of liquid is dispersed to a number of droplets m<mo, it may be found that the rate of nucleation in the majority of the droplets, though greater than the rate of homogeneous nucleation, is very much less than the rate of nucleation of the original mass. This result is explained if, following Levine,11 the additional hypothesis is made that the concentration of accidental catalytic centers with a given catalytic potency, £=l//(0), increases as p decreases. Levine has given a formal treatment of the problem that is analogous to the statistical theory of brittle fracture by stress raisers of variable potency developed by Weibull12 and Fisher and Hollomon.13 Levine's theory successfully describes the experimental results of Dorsch and Hacker14 on the freezing of water droplets.
It appears that the most promising method of testing the predictions of the theory of crystal nucleation in
supercooled liquids is the measurement of the nucleation frequency in a collection of small droplets sufficiently dispersed to make »2»Wo.2,9 Then crystal nucleation in droplets isolated by films that are not nucleation catalysts should be homogeneous and describable by Eq. (1). In droplets coated with crystalline films that catalyze nucleation, the nucleation frequency should be proportional to the surface area of the films and describable by Eq. (5).
It has been demonstrated2'16 that the rate determining step in the solidification of very small metal droplets is nucleation. Thus the nucleation frequency in small droplet dispersions can be computed from direct measurement of the solidification rate on the basis that the period of crystal growth is negligible in comparison with the nucleation period.
Consider the kinetics of isothermal solidification of a dispersion of droplets having a wide distribution of diameters. We make the following definitions: D = diameter of the droplet, vd = volume of droplets of diameter D, as = surface area of droplets of diameter D, and Fc = total volume of all droplets of diameter D that remain liquid at time, t. A function y(t) is defined by the equation
y(0= if Vd/D£ VD=(l/V) f VndD. (7) £>=0 D= 0	J n
At t= 0
y{t) = f={ 1/7
•'o
VD°dD.
(8)
Also from (7) and (8) dy(t)/dD=VD/V and dy°/dD = Vd°/V°. Let the nucleation frequency per droplet = &z>; it follows from the assumptions that the volume rate of solidification of droplets of diameter D conforms to the radioactive decay law and is given by,2
-dVD/dl=kDVD. The total volume rate of solidification is,
d=oo
-dV/dt= I) kDVD= I kDVDdD. d=o	J o
s There is a good possibility (particularly in view of the quanti-
tative success of the Becker-Doring theory applied to the nuclea-
tion of droplets) that a may be approximately equal to the ap-
propriate liquid-crystal interfacial tension <r' measured in macro-
scopic systems. As yet a' has not been measured with sufficient accuracy to test this possibility.
7	M. Avrami (Now Melvin) J. Chem. Phys. 7, 1103 (1939).
8	D. Turnbull, J. Appl. Phys. 20, 817 (1949).
9	D. Turnbull, Trans. Am. Inst. Mining Met. Engrs. 188, 1144 (1950).
10	D. Turnbull, Thermodynamics in Physical Metallurgy (Am. Soc. Metals, Cleveland, Ohio, 1950), pp. 282-306.
11	J. Levine, Nat. Advisory Comm. Aeronaut. Repts., Tech. Mem. Notes 2234 (1950).
12	W. Weibull, Roy. Swed. Inst. Eng. Research No. 151, (1939).
13	J. C. Fisher and J. H. Hollomon, Metals Technol. 14, No. 5 (1947).
14	R. G. Dorsch and P. T. Hacker, Nat. Advisory Comm.
Aeronaut. Repts. Tech. Mem. Notes 2142 (1950).
Integration of (9) and (10) gives V D=V D° txV[-kDt],
(9)
(10)
(11)
7= Z°°7fl» exp[-M]
= f 7fl°[exp{-kDt)2dD. (12) •mi
For homogeneous nucleation
kD = Il>D,
(13)
5 D. Turnbull and R. E. Cech, J. Appl. Phys. 21, 804 (1950).
KINETICS OF SOLIDIFICATION Table I. Summary of methods used to prepare dispersions of mercury droplets.
413
Coating on droplets	Designation"	Method of dispersion	Composition of dispersing mixture Fluid Oxidizing agent			Acid	Droplet diameter range (microns)
Mercury sulfide (black)	Hg(S)	Manual shaking	CS2 (30 cc)	Sulfur 0.05 g			50-200
Sodium oleate (adsorbed)	Na Oleate	Manual shaking	EtOH saturated with Na Oleate				100-400
Mercury iodide (yellow)	Hg2I2b	Manual shaking	EtOH (30 cc)	0.3 g iodine			15-70
Mercury acetate	Hg(Ac)	Waring blendor	Silicone oil (150 cc)	0.2 g acetyl peroxide			4-12
Mercury stearate	Hg(St)	Waring blendor	Mineral white oil (150 cc)	Air	0.5 g stearic acid		2-7
Mercury laurate	Hg(Lau)	Waring blendor	Silicone oil (150 cc)	0.5 g lauroyl peroxide	0.1 g acid	lauric	2-8
Mercury bcnzoate	Hg(Bz)	Waring blendor	Silicone oil (150 cc)	0.7 g benzoyl peroxide	0.3 g acid	stearic	2-8
Dilatometer fluid
Ethyl alcohol (EtOH)
EtOH
EtOH
Methyl cyclopentane
Methyl cyclopentane
Methyl cyclopentane
Methyl cyclopentane
" These designations are not intended to indicate the precise chemical formula of the film but only the method of its formation. The author did not find the data necessary for calculating whether the Hg2++ or Hg++ salts are thermodynamically more stable under the experimental conditions. Also, it is possible that some of the carboxylates are oxy salts.
b According to thermodynamic calculations, Hgil2 is the more stable phase (relative to Hgli) in equilibrium with mercury under the experimental conditions.
and Eq. (12) may be rewritten F= if VD° exp(-IvDt)
d- 0
= f VD°[exp {-IvDt)~]dD. (14) Jo
In case nucleation is catalyzed by the film on the droplet surface,
kD = I.dD	(IS)
and Eq. (12) becomes
.D=oo
F= Z FB°exp(-W)
d=o
= f VD°[exp(-I.aD()JdD. (16) o
Experimentally, we obtain y°=f(D) and V—f(t) at various constant temperatures. By using the relation y°=f(D), Eqs. (14) and (16) may be compared with the experimental relation V=f(t) by a numerical or analytical procedure. From this comparison it may be decided whether the data are fit best by assuming the nucleation frequency proportional to the volume of the droplet [Eq. (14)] or to its surface area [Eq. (16)].
It follows that d2X/dtdT\x =xu where X= (F0- V)/ F0) is proportional to dl/dT or dlJdT for a given dispersion. Therefore d InI/dT or d InI,/dT can be evaluated from V=J{l) for a given dispersion at various constant temperatures even though y°=f(D) is not known.6b
Vonnegut16 has measured the isothermal rate of solidification of tin droplets (1 — 10 micron diameter)
16 B. Vonnegut, J. Colloid Sci. 3, 563 (1948).
isolated from each other by an oxide film as a function of temperature. Analysis of his data5b indicated that the magnitude of d2X/dtdT\x=x^ is in approximate agreement with the prediction of Eq. (14). However, ya—f{D) was not determined. Later measurements of Pound and LaMer17 of V=fit), in conjunction with measurements of y°=f(D), on oxide-coated tin droplets showed that (dX/dl)r decreased much too rapidly with time to be compatible with a single value of I or I,.
We have measured V=/(/) at various temperatures for dispersions of mercury droplets coated with various surface films. Preliminary reports of some of the results have been published elsewhere.6b'19 The purpose of this paper is to describe and analyze these and additional results more completely.
EXPERIMENTAL
A. Materials and Preparation of Dispersions
Purified triple-distilled mercury was used in all the preparations to be described. In forming one group of dispersions 2 cc of mercury and the forming solution were shaken up in a test tube. To form the second group, 5 cc of mercury and the forming solution were agitated in a Waring Blendor for net times of three to eight minutes. The agitation was interrupted at intervals in order to prevent excessive heating of the blendor contents.
The method of forming the dispersions, the composition of the forming solutions, and the range of droplet diameters contributing significantly to the volume of the dispersed mercury are indicated in Table I. All of
17	G. M. Pound and V. K. LaMer, J. Am. Chem. Soc. (to be published).
18	D. Turnbull, Science 112, 448 (1950).
18 D. Turnbull, Phys. Rev. 83, 880 (1951).
414
DAVID TURNBULL
TO POTENT.
DILATOMETER
TO ICE JUNCTION
5 JUNCTION THERMOPILE
COOLING TUBE
OUTER DEWAR
GLASS WOOL-
Fig. 1. Schematic drawing of bath for maintaining constant temperature.
the organic compounds used in the preparations were Eastman cp chemicals, excepting that purified lauroyl peroxide20 was furnished by W. E. Cass and acetyl peroxide was obtained in the form of 25 percent dimethyl phthallate solution from Buffalo Electrochemical Company. The iodine'and sulfur used were reagent grade materials.
It is believed that the film on the droplet surface consisted of a mercury compound in all cases excepting for the dispersion formed in the alcoholic sodium oleate solution. When the air pressure in the blendor containing the stearic acid solution was reduced to about 0.001 atmosphere, the mercury did not break up into droplets. Therefore, it is believed that the Hg(St) film formed as a result of reaction of the stearic acid with an oxide film formed by the interaction of air and mercury. The chemical film formed in peroxide solution is believed to result from the acid catalyzed direct addition of mercury to peroxide
In all cases the amount of the film forming agent used was less than that required to form by complete reaction an Hg2++ salt constituting a volume fraction of 0.02 of the dispersed mercury.
Measurements of the droplet size distribution L(y°—f(D)~\ in the dispersions where time, temperature, and medium of dispersion were varied to the extent used in the experiments established that y°=f(D) was not measurably affected by these variations, with the possible exception of Hg(S) dispersions (on which such observations were not made) and the Na Oleate dispersions. However, Hg2I2 dispersions did coarsen with time when one-half the indicated amount of iodine was used in forming them.
B. Temperature Measurement and Control
The extent of solidification of mercury was measured dilatometrically. In these measurements two schemes were used on occasion to maintain the temperature con-
20 The source and purity is described elsewhere. W. E. Cass, J. Am. Chem. Soc. 72, 4915 (1950).
stant. The first of these schemes was used in the experiments on Hg(St) and Hg2I2 coated aggregates. The constant temperature chamber consisted of a well-insulated can containing freon 11 entirely surrounded, excepting for outlets, by a bath of the same fluid. Further details on the construction of this bath have been published.9 To maintain the temperature of the outer bath constant to within ±0.2 °C the potential developed by a five junction thermopile, having one set of junctions in the outer bath and the other set in a Dewar flask containing a constant temperature mixture, was used to activate a General Electric Recording potentiometer. The recorder was part of a relay circuit so that as the bath reached the control temperature, the recorder needle closed a relay circuit thus causing the bath heater circuit to open. Liquid nitrogen was circulated in cooling coils in the outer bath. This method of temperature control requires a compound in the Dewar flask at a phase transition temperature within 5 to 10° of the bath temperature to be maintained. With the indicated degree of control of the outer bath, the inner bath temperature remained constant to within ±0.01 °C. The inner bath was agitated by bubbling air through it and its temperature was checked every three to five minutes.
In the other isothermal experiments, a much simpler and equally effective method was used to control temperature. In this method the outer bath was a 4-liter Dewar filled to a level well above the fluid level in the inner bath with a partly melted compound at a temperature 20 to 30° lower than the temperature to be maintained in the inner bath. The container for the inner bath was a 1-liter unsilvered Dewar flask maintained in position by glass wool support and Teflon spacers so that its top was 2 in. to 3 in. below the top of the outer Dewar. The space between the top of the outer and inner Dewars was filled with a cork insulator. Figure 1 shows the arrangement schematically. Heat leaked slowly from the inner to the outer bath. To compensate for this loss, air originally at the ambient temperature was passed through the inner bath. This rate of air flow was regulated by a flowmeter and valve arrangement sufficiently sensitive so that a given pressure differential could be maintained to within a millimeter of water. After the system reached thermal equilibrium with a given rate of air flow, the temperature in the inner bath was held constant to within 0.01°C by small manual adjustments of the rate at intervals ranging from three to fifteen minutes. To determine when these adjustments would be necessary, the temperature of the inner bath was measured every two to five minutes. In the isothermal measurements on Hg(Lau) coated droplets (—117 to —120°) freon 12 was used in the inner bath and an equilibrium mixture of liquid and solid freon 12 in the outer bath.
The temperature was calculated from the potential developed by a five junction copper-constantan thermopile. This potential was measured on a Type K poten-
KINETICS OF SOLIDIFICATION
415
tiometer. One set of thermocouple junctions was placed in the bath a small distance from the middle of the dilatometer bulb. The other set was immersed in a Dewar containing an equilibrium mixture of distilled water and ice made from distilled water. After each set of measurements, the thermocouple calibration was checked against the melting temperature of pure mercury. In all cases, this calibration agreed with the standard calibration charts for copper-constantan thermocouples to within 0.05°C.
C. Dilatometry
A general description of the dilatometer has been published.9 The dilatometer bulbs were about 6- to 7-cm long and ~0.5-cm internal diameter. Five to seven g of mercury (ca £cc) droplets from a given dispersion were introduced into the dilatometer bulb and the dispersing medium changed to the fluid indicated in Table I by several decantations and additions. The capillary bores of the various dilatometers were from 0.055- to 0.065-cm diameter. The drop in fluid level in the dilatometer capillaries caused by the complete isothermal solidification of mercury ranged from 4.5 to 6.5 cm. The fluid level in the capillary was measured, relative to a fixed point on the outer surface of the capillary, to within 0.02 cm by means of a cathetometer.
Preliminary experiments were made on each of the dispersions to determine the temperature range in which solidification occurs at a reasonable rate. The experimental procedure used in these "continuous" cooling experiments has been described.9 Figure 2 shows the dilatometer readings obtained from a cooling-heating cycle on Hg(St)-coated droplets. A linear section of the cooling curve AB was established corresponding to the thermal contraction curve for the dilatometer fluid and the liquid mercury. It was confirmed that the curve B—>A was identical to A-+B and it was therefore concluded that no appreciable solidification had taken place at T> Tb. The curve CD is the thermal expansion curve of dilatometer fluid and solid mercury.
xy, the vertical displacement of the fines AB and CD at Txv is the dilatometer contraction, hha, proportional to the change in volume of the dispersion if all the mercury contained in it solidified isothermally at Txy. The variation of A/z0 with temperature in the range of rapid solidification is negligible. Aha was measured to within one percent uncertainty.
Temperatures of beginning solidification 7\- and of final solidification Tf measured in the cooling-heating experiments are defined as the temperatures between which the points on the cooling curves do not fall, within experimental error, on the extrapolation of either of the lines AB or CD. Actually, Tf and 7\ are weak functions of the rate of cooling and in this investigation are specified for a cooling rate of l°/minute.
No values of Tf and 7\ were accepted as characteristic of a given dispersion unless two additional conditions were established as follows: (1) The volume contraction
presumed to be associated with solidification was registered as an expansion during the heating cycle at the accepted melting temperature of mercury. (2) The sample evolved a substantial quantity of heat (heat of solidification) when cooled through the temperature range Ti to Tf.
The temperature range in which the heat of solidification was evolved was determined by measuring as a function of temperature the deflection, 8, of a sensitive galvanometer activated by a differential thermocouple having one junction in a well embedded in the sample and the other junction embedded in a well in the temperature bath. When the bath was cooled at a constant rate of l°/min, S exhibited a pronounced maximum at a temperature between Ti and Tf. Values of Ti and Tf measured by this method were in excellent agreement with values measured dilatometrically in all cases, excepting for some experiments where a dispersing fluid was used that gelled at T<Ti.
In the isothermal dilatometric experiments the sample bulb was always thoroughly dispersed by shaking immediately before the experiment in order to insure that the rate of volume contraction would not be limited by the flow of fluid into the mercury. The dilatometer bulb and part of the capillary, which had been precooled to a temperature about 10° above that of the inner bath, were immersed in the inner bath after a constant temperature Tc had been established in it, and dilatometer readings h(t) were taken as a function of time. After measuring hit) over a three- or four-hour period, the sample was either (a) quickly quenched into an auxiliary bath maintained at a temperature about 10° below Tf, held in this bath for about 15 minutes then returned to the original bath at Tc, or (b) the bath at Tc was slowly cooled to about 10° below Tf, held for 15 minutes at the lower temperature, and then warmed back up to Tc- In either event after about 15 minutes more of thermal equilibration at Tc a final dilatometer reading hf was taken. The dilatometer reading hi corresponding to the fraction of sample
Fig. 2. Cooling-heating cycle for dispersion of Hg(St)-coated Hg droplets.
416
D A V I D TURNBULL
Fig. 3. Comparison of duplicate isotherms on a dispersion of Hgala-coated Hg droplets.
solidified X = 0 at Tc is calculated from the equation hi=hf+ Ah0. X=f(t) is given by X=[hi—h{t)~\/ Aho. The time origin was taken as the ordinary time when h(f) = hi.
Additional experiments were carried out on the Hg(Lau)-coated droplets to find out how closely the temperature of the sample approached the temperature of the bath. For this purpose, a dilatometer was used containing an embedded thermocouple well. The potential difference developed in a thermocouple with one junction in the well and the other in the bath was measured as a function of time at constant bath temperatures. In the same experiments X=j{t) was measured. It was found that for transformation rates R^ than 0.05 g of mercury/min the temperature difference registered by the differential couple was <0.02°C. After quenching the precooled dilatometer into the bath at Tc, the temperature inside the dilatometer generally approached Tc within a period of about five minutes. In most of the isothermal data to be reported R was well below 0.05 g/min.
The reproducibility of the isotherms was checked by repeating measurements on several for each dispersion. For all, X=f(t) was reproduced to within the accuracy of the h(t) measurements. Figure 3 illustrates the reproducibility of two isotherms on the Hg2l2 coated mercury droplets. The second experiment at — 87.10°C was made immediately after the first one, but one week's time elapsed between the two experiments at —87.36°.
Fig. 4. Hg droplet size distribution in Hg(Lau) dispersions.
D. Droplet Diameter Measurement
To measure droplet' diameter, a Spencer microscope with an apochromat lens and Aplanat eyepiece was used. The lens had a 4-mm objective and was fitted with a correction collar. Illumination was provided by transmitted light from a zircon arc source. A prism was used to project the field on the wall and the diameter of the image measured, after bringing it into focus, to within J mm. For droplets having diameters in the range 2-12 microns, the magnification factor was 1160. Samples were taken directly from the dilatometer after the isothermal measurements and dispersed in mineral white oil. Due precautions were taken to obtain representative samples and to insure that the droplets would not be flattened by the microscope cover glass.
Droplet size determinations were made on two or more withdrawals from each dilatometer sample. The droplet size distributions for these withdrawals checked satisfactorily. The diameter of a total of about 2000 droplets in each dilatometer sample was measured. The volume of sample having droplet diameters between
D and D+^D (where ^[£>=500/1160 microns for the magnification factor of 1160) was plotted against D. From smoothed curves [(Vdy°/dD)=f(D)] through these points curves were constructed of the function y" =f(D) (see Figs. 4 and 5). D' is defined by
0.5= (1/F°) f VD°dD=( 1/F°) f VdHD. (17) ■A)	•'c
RESULTS
A. Continuous Cooling Experiments
Figure 6 is a diagram that compares the results of the continuous cooling (rate ~l°/minute) experiments for the various dispersions. The ordinate is the supercooling, AT {AT=Tm—T where Tm is the melting temperature), and the abscissa is the logarithm of the droplet volume. Tm—Ti and Tm—Tf calculated from the continuous cooling experiments are the AT coordinates of the corners of each rectangle. The logw coordinates of the corners define the range of volume of droplets that contribute significantly to the total volume of the mercury. A change of a factor of 10 in the cooling
KINETICS OF SOLIDIFICATION
417
rate shifts the AT coordinate of the rectangles 1° or less. These experiments prove conclusively that a very small droplet size in itself is not a sufficient condition for marked supercooling of liquids, for in most instances the size distribution in the various dispersions overlap considerably or even completely yet their nucleation frequencies differ by many orders of magnitude.
It must be concluded that the most important factor determining the position of a dispersion in the solidification diagram (Fig. 6) is the potency of the most effective nucleation catalyst on the surface or in the volume of the constituent droplets. Compared to this factor, the effects of droplet volume and cooling rate are second order. Nucleus formation must be initiated by catalysts in all of the dispersions with the possible exception of Hg(Lau) and Hg(Bz).
The chemical composition and structure of the effective catalytic substance may not correspond to that of the compound covering the major part of the droplet surface. Minute amounts of impurity in the forming compound could have formed a catalyst more potent than that formed by the compound itself, (though different samples of the forming substance gave the same result). However, the fact that the supercooling of the dispersions varies so widely even though their droplet size distributions overlap proves that, with the exception of Hg(Lau), Hg(Bz), and Na Oleate dispersions, the effective catalysts could not have been bodies suspended in the mercury prior to the formation of the dispersion. Further, the fact that the solidification range of a given dispersion is very narrow indicates that the effective catalytic substances in the dispersion must be chemically identical or closely related.
It has been established9 that the solidification of Na Oleate dispersions is coincident with a breakdown in the protective action of the absorbed film. Therefore, the position on the solidification diagram may not indicate the actual AT—log v relationship for Na Oleate dispersions. The main interest attaching to the Na Oleate results is that droplets so large (0.4-mm diameter) can be supercooled so much (35°).
After long holding periods at room temperature, the solidification behavior of Hg(Lau) and Hg(St) dispersions changed discontinuously from that defined by the Hg(St) and Hg(Lau) rectangles in Fig. 6 to that defined by the HgX rectangle with no accompanying change in droplet size distribution. Before this transition is complete, two discrete ranges of solidification [HgX and Hg(St) or Hg(Lau)] are found in the same dispersion. No measurable part of dispersions coated with Hg(Bz) reverted to the Hg(X) class over a two-week period. Holding Hg(Lau) dispersions at — 40°C or lower completely suppressed the transition for at least several weeks.
Addition of small amounts of glacial acetic acid to HgX dispersions in methyl cyclopentane caused a marked increase (by a factor of 10 to 20) in the mean droplet diameter and a discontinuous transition in
LOG V -10.0
-5.0
I Hg (X)
~r
I HgtS)
AT
| |NoOLEATE ] Hg(Ac) Hg^
60 -
Hg(St)
iHg(Lou)
Hg(Bl) _1_
I	10	100 500
DROPLET DIAMETER (MICRONS)
Fig. 6. Solidification diagram for mercury—curves in certain rectangles are isokinetic relations.
solidification behavior to that characteristic of Hg(Ac) dispersions.
These facts seem consistent with the following hypotheses: (a) Hg(Lau) and Hg(St) films hydrolyze to form patches of mercury oxide on the surfaces of the droplets, (b) Mercury oxide is an effective catalyst for the nucleation of mercury crystals, (c) Acetic acid converts the oxide to mercury acetate. In further support of these ideas, it was observed that the transition to Hg(X) was made markedly slower by increasing the viscosity of the dispersing medium and in some instances by greatly increasing the shortest available diffusion path between the droplets and atmospheric components.
B. Homogeneous Nucleation
Isothermal Solidification of Ilg(Lati) Coated Droplets
From our interpretation of the continuous cooling experiments, it follows that the temperature range of rapid homogeneous nucleation should either correspond to or fall below T<> T> T, for Hg(Lau) or [Hg(Bz)] coated droplets. Therefore, it is important to find whether or not the isothermal solidification behavior of these dispersions follows the predictions of the homogeneous nucleation theory.
SAMPLE NO.I
19.00°
TIME (MINUTES)
Fig. 7. Solidification isotherms of Hg(Lau)-coated droplets—Sample No. 1.
418
DAVID TURNBULL
200 400 600 800 1000 1200 TIME (MINUTES AT-117.75°)
Fig. 8. Comparison of isotherms calculated on the basis of volume or surface nucleation hypotheses with master isotherm for solidification of Hg(Lau)-coated Hg droplets—Sample No. 1. The "uniform size" curve was calculated on the assumption of a uniform droplet diameter.
For the isothermal experiments, two Hg(Lau) dispersions having different droplet size distributions were prepared. y°=f(D) for these dispersions is given in Fig. 4. To prepare the dispersion (Sample No. 2) having the smaller D' value, the blendor was run for a longer period of time than for Sample No. 1.
X=f{t) curves for Sample No. 1, calculated from dilatometric data at six temperatures in the range -117.75 to -119.00°, are plotted in Fig. 7. In order to test the volume and surface nucleation hypotheses, it is convenient to shrink all the data to a "master isotherm" by letting the magnitude of the time unit vary with temperature. Figure 8 shows the master isotherm and the best multiplying factors t to apply to the ordinary time units to obtain the best superposition of the isotherms.
In constructing this isotherm, the data obtained at the shortest times at the lowest temperatures were disregarded because the transformation rate under these conditions substantially exceeded 0.05-g Hg/min. In order to test Eq. (14) derived from the homogeneous (or volume) nucleation, hypothesis X=f(t'), where t' = at and t is expressed in time units at —117.75° (r= 1.0), was calculated from (14) with the use of the relation y°=f{D). Then a is evaluated by making X=f(t') coincide with the experimental X=f(t) relation at two values of X, in this instance 0.10 and 0.30. Making the substitution t'=at in X=f(f) gives the relation X=fv(t) predicted by Eq. (14) (volume nucleation). An analogous procedure where the calculated and experimental isotherms are made coincident at the same values of X as in testing the volume nucleation hypothesis, was used to calculate a relation X=fa(t) predicted by Eq. (16) (surface nucleation).
The comparison of X=fv(i) and X=fa(t) with the master isotherm X=f(t) determined experimentally is shown in Fig. 8. The data are described by X=fv(t) to
well within the experimental uncertainty but X—fa{t) falls somewhat outside the range of uncertainty in the experimental points for X> 0.5.
The same order of agreement of X=fv(t) and X=fa(t) with the data was obtained when the unsmoothed droplet diameter distribution was used in place of the smoothed y°=}{D). From this result, it is concluded that the errors that may have been made in determining the droplet size distribution are not of sufficient magnitude to affect the validity of the conclusion that the data are best fitted on the basis of the volume nucleation hypothesis.
From the described analysis, values of kn and [from Eq. (13)] I are calculated for —117.75°. I for the other temperatures is found by multiplying I at —117.75° by the appropriate r-factor listed in Fig. 8.
The sources and probable magnitude of the principal errors that affect the intercomparison of I values measured on the same sample are as follows:
1.	±0.02° uncertainty in the control and measurement of temperature may cause ±3 percent error in I.
2.	±0.03 cm uncertainty in final cathetometer reading may lead to ± 1 percent error in I.
Thus the probable deviation of I values measured on the same dispersion at the same temperature is of the order of 4 percent.
The greatest part of the probable deviation of I values measured on different samples, prepared in the same way, is contributed by the uncertainty in the measured droplet size distribution, for any error in the measurement of D is magnified three times when I is calculated. It is estimated that (considering errors in sampling, microscope magnification, measurement, and curve smoothing) D' was measured with an accuracy of ±5 percent, therefore, the corresponding probable error in 7 is ±15 percent. An additional term, affecting the intercomparison of different samples is the ±2 percent probable error in I caused by ±0.05 cm, uncertainty in A ho. By taking all these terms into account, the maximum probable error in the absolute magnitude
Fig. 9. Comparison of isotherms calculated on the basis of volume or surface nucleation hypotheses with master isotherm for solidification of Hg(Lau)-coated Hg droplets—Sample No. 2. The "uniform size" curve was calculated on the assumption of a uniform droplet diameter.
KINETICS OF SOLIDIFICATION
419
of I and the maximum probable deviation when samples prepared in the same way are intercompared is ±21 percent.
Three isotherms were measured on Sample No. 2. The master isotherm and r-values calculated from these data are given in Fig. 9. Analysis of the results by the same method used on Sample No. 1 established that they are best described (and satisfactorily so) on the basis of the volume rather than the surface nucleation hypothesis. / values were calculated as before.
Equation (1) gives I=f(T) based on the homogeneous nucleation theory. From thermodynamic principles,
d(AFv)/dT=-ASv,	(18a)
where ASv is the entropy of solidification/volume. According to the existing thermodynamic data,21 ASv is virtually independent of temperature;22 therefore, to a good approximation,
AF„ = —(ASV)AT	(18b)
and Eq. (1) may be rewritten
I=KV exp[- ao*/(ASv)\ATykTl. (19)
If <j is assumed to be independent of temperature, it follows that In/ is a linear function of \/T{AT)2 having the slope,
d \nI/d[\/T{ATf~]= -a^/{ASvYk- (20)
log/ is plotted against 1/T(AT)2 for both samples in Fig. 10, and an excellent straight line relation obtains for each sample as required by Eq. (1). The average deviation from the mean value of / between Samples 1 and 2 in the overlapping temperature range of the two sets of measurements is 14.5 percent and within the calculated maximum deviation of 21 percent.
Table II. Summary of results on kinetics of homogeneous and heterogeneous nucleation in mercury.
Surface coating	Type of nucleation	a erg/cm1	e	logK. logX. Exptl. Calc Exptl. Calc
Hg(Lau)	Volume (Homogeneous)	31.2		42., 35., ......
Hg(Ac)	Surface (Heterogeneous)	31.2	72°	...... 26.8 27.,
Hg..I2	Volume (Heterogeneous)	a		69.6 35.6 ......
Hg(St)	Heterogeneous	31.2	87°	31. 35. 27.„ 27.3
Fig. 10. Variation of nucleation frequency/volume, I, with supercooling, AT, in the solidification of Hg(Lau)-coated Hg droplets.
» K. K. Kelley, Bureau of Mines Bulletins No. 371 (1934) and No. 393 (1936).
22 The author is indebted to R. A. Oriani for establishing this fact.
a The nucleation frequency in iodide coated droplets can be satisfactorily described by an apparent value of a —29.2 erg/cm2.
A further check on the validity of the volume nucleation hypothesis is obtained from these data by comparing the measured value of ko' for Sample No. 2 with the values calculated from the results on Sample No. 1 on the basis of the surface (ko')a and volume (ko')v hypothesis. At the temperature —118.81° in Sample No. 2, the experimental value of &d'=0.0052 min-1 in fair agreement with (&//)„=0.0065 min-1 and in marked disagreement with (&!/)„= 0.0095 min-1. Though it is recognized that the deviation between X=f(t) and X=fa(t) for any one set of data is hardly large enough to be convincing, it is believed that the result just cited and the fact that X=f(t) is in best agreement with X=fv(t) for two dispersions fully substantiates the conclusion that kw in these experiments is proportional to droplet volume.
From the In/ vs \/T{AT)- lines for the two samples a composite straight line was constructed as shown in Fig. 10. Assuming that mercury crystal nuclei are spherically shaped (a=16ir/3), a was calculated from the slope of the composite line [Eq. (20)] to be 31.2 ergs/cm2. (Also see summary of results—Table II.) The radius of the nucleus23 r*= — 2<r/(AFv) was calculated to be 11.8/1° at —118°; whence it follows that it contains 830 atoms and «*=220.
To calculate Kv from the theory of homogeneous nucleation [Eq. (2)] it was assumed that AFA=(AFA)Via, where (AFA)vi, the free energy of activation for viscous flow is calculated from the viscosity of mercury using the equation for viscosity calculated from absolute rate theory.24 In the temperature range of the measurements exp(—AFA/kT)<^ 10~2. With this assumption logKv calculated from Eq. (2) is 35.1 and is to be compared with the experimental value of 42.1 ± 2 determined from the intercept of the composite line in Fig. 10 on the log/ axis.
To explain this disagreement of a factor of at least 105 between the Kv and (Kv)exvi some of the assumptions made in deriving and applying the theory have to
23 M. Volmer, Kinetik der Phasenbildung (Steinkopff, Leipzig, Germany (1939).
21 Glasstone, Laidler, and Eyring, The Theory of Rate Processes
(McGraw-Hill Book Company, Inc., New York, 1941), pp. 477-
516.
420
DAVID TURNBULL
Fig. 11. Solidification isotherms of dispersion of Hg(Ac)-coated Hg droplets.
140
TIME (MINUTES)
be examined more critically. In the derivation of the theory the frequency factor for the addition of an atom to the nucleus from the liquid was assumed to be (kT/h) from the absolute rate theory. There is no assurance that this assignment is valid for an atomic process as complex as the nucleation reaction. In addition, it is assumed that crystal embryos mix ideally with the supercooled liquid. It is believed that together these factors might lead to an uncertainty as large as 10±2 in the calculated Kv. However, the assumptions that ASv and a are temperature independent are probably of more importance. The term exp[—atr3/(AFvYkT~] is very temperature sensitive (such that I changes by a factor of 5/°C) so that even a very small dependence of <7 or AS„ upon temperature has a potent effect upon the measured slope [d\n[/d(\/T(A.T)2~\ and hence on log(-fC„)es:p. For example, suppose that a depends upon temperature according to the equation: <r= <ro-\-bT and that AFv is given by AF„= -A5„°(Ar)(l+cAr)i. It follows,
d lnI/dll/T(AT)^l-aa3/(ASvyk']
X {1+[T{Tm-T)/(3T- Tm)J_3b/c+c]}. (21)
In Eq. (21) the factor — aff3/(A5„)2^ is the slope that would be measured if a and ASV were temperature independent.
If c=0 a value of 6 — 0.0008/° is sufficient to change the slope 18 percent and the apparent value of Kv by a factor of 106. In order to cause a change of the same order of magnitude when b=0, c must be 0.0016/°.
Fig. 12. Comparison of isotherms calculated from volume and surface nucleation hypotheses with master isotherm for Hg(Ac)-coated Hg droplets.
It seems reasonable to suppose that the combined effect of b, 0, and the relatively large AT values obtaining in the experiments accounts for the major part of the disagreement between Kv and (/C„)eXp-25 On the basis of this analysis, it may be concluded that the experimental results are compatible with the predictions of the homogeneous nucleation theory.
The only other possible interpretation of the data that seem to merit serious consideration is that nucleation is catalyzed by inclusions colloidally suspended in the mercury. In order to account for the results, the minimal concentration of the inclusions would have to be 1013 to 1014/cm3 (corresponding to a volume fraction 10~6 to 10~8) all having equal catalytic effectiveness/cm2. Since the same results are obtained in continuous cooling experiments with Hg(Bz), and two-Hg(Lau) dispersions gave the same I within experimental error, it is concluded that these inclusions could not have had their origin in the preparation of the dispersion but, if present, must have been suspended in the original mercury sample.
The arguments that make the colloidal catalyst interpretation less plausible are:
(1)	The homogeneous nucleation theory describes the results semiquantitatively.
(2)	Hg(Lau) dispersions from different lots of mercury exhibited the same solidification behavior within experimental error in continuous cooling experiments.
Since /=f(T) and kD is given by Eq. (13), it follows that
kD=f(T)vD.	(22)
From (22) we may derive "isokinetic" relations between vd and T at constant ho as follows,
=c
(23)
One such isokinetic curve for Hg(Lau)-coated droplets is drawn in Fig. 6. The ko value of this curve is
25 We do not assert that the signs of b and c are actually as indicated. Intuitively it might be believed that b should be negative rather than positive. However, our knowledge of liquid-crystal nuclei interfaces is not sufficiently advanced to justify such a belief. Our purpose is to show that the neglect of the factors b and c might, so far as our present knowledge goes, account for the disagreement between Kv and (Kv)elp.
TIME (IN MINUTES AT-85.31")
UNIFORM SIZE SURFACE NUCLEATION VOLUME NUCLEATION
t°° T SYMBOL _
KINETICS OF SOLIDIFICATION
421
fixed by the co-ordinates of a corner of the Hg(Lau) rectangle.
C. Heterogeneous Nucleation
1. ku Proportional to Droplet Area: Isothermal Experiments on II g{Ac)-Coaled Droplets
The primary purpose of this investigation was to measure the kinetics of homogeneous nucleation of mercury crystals. Consequently, the isothermal solidification behavior of dispersions in which nucleation is believed to be heterogeneous was studied in less detail than that of the Hg(Lau) dispersions.
Figure 11 shows the X=f(t) isotherms obtained at five different temperatures on a dispersion of Hg(Ac) coated droplets. The isotherm at —85.01° does not_ extrapolate to X=0 at / = 0 because the dispersion was inadvertently precooled to a temperature just below — 85.01° for a short time prior to the running of the isotherm.
The data were analyzed by the same methods used to analyze the data on the Hg(Lau) dispersion. Figure 12
Fic. 14. Solidification isotherms of dispersion of Hg2I2-coated Hg droplets.
shows the master isotherm and Fig. 5 the smoothed curve for y°=f(D). In evaluating a the calculated and experimental curves were made coincident at X=0.3 and 0.5. Comparison of the calculated curves with the master isotherm is made in Fig. 12. Clearly the data are most satisfactorily described on the basis of the surface nucleation hypothesis [Eqs. (15) and (16)]. For 0<X<0.2 the experimental rate is greater than calculated on the basis of either the surface or volume nucleation hypotheses, but this departure is not considered significant in view of the small values of X and t.
Is was calculated at various temperatures from the t values shown in Fig. 12. Log/s is a linear function of 1/T(AT)2 as shown in Fig. 13. From the measured slope and the value of a obtained from the experiments on homogeneous nucleation, the contact angle 9 was calculated to be [see Eq. (5)] 72°. Log(^s)CTP was found to be 26.8 which compares well with the value 27.4 calculated from Eq. (6) on the assumption that all elements of area in the droplet surface are equally effective for nucleation catalysis.
This excellent agreement between theory and experi-
Fig. IS. Comparison of isotherms calculated from volume and surface nucleation hypotheses with master isotherm for Hg2I2-coated Hg droplets.
ment fully supports the concept developed earlier that the kinetics of heterogeneous nucleation can be described with the use of a single parameter, 6, in addition to those required to describe homogeneous nucleation. Deviations between theory and experiment due to any temperature dependence of ASv and a will be smaller in these experiments26 than in the case of homogeneous nucleation because AT is much smaller.
An isokinetic relation between logn and AT for this dispersion is indicated on the solidification diagram in Fig. 6.
2. kd Proportional to Droplet Volume: Isothermal Experiments on HgJz-Coated Mercury Droplets
X=f{t) curves measured at eight temperatures on a Hg2l2-coated dispersion are plotted in Fig. 14. Because of the extraordinary magnitude of the temperature coefficient of the rate, every isotherm was rechecked, with the exception of the one at — 87.55°C. In all cases, the agreement is excellent.
y°=f(D) for this dispersion is given in Fig. 5. The master isotherm and the test of its agreement with the volume and surface nucleation hypotheses is shown in Fig. 15. A much better fit (also an entirely satisfactory one) is obtained on the basis of the volume than on the
TEMPERATURE °c
Fig. 16. Variation of 7 with AT for the solidification of of Hg2I2-coated Hg droplets.
26 Of the order of 102 if it is accepted that the deviation in the homogeneous nucleation results is caused by the temperature dependence of a and ASv.
422
DAVID TURNBULL
Fig. 17. Comparison of isotherm calculated from volume nucleation hypothesis with master isotherm for dispersion of Hg(St)-coated Hg droplets.
surface nucleation hypothesis. Log! is plotted against 1/T{AT)2 in Fig. 16. A good straight line relation is obtained at the five lowest temperatures; (Kv)eKp and the apparent a value derived from the constants of this line are listed in Table II. At temperatures higher than — 86.9° I is much lower (by a factor of 4),than predicted by the low temperature linear relation. When the dispersion was precooled below — 86.9° then warmed to a temperature above —86.9°, the transformation rate at the higher temperature was significantly larger than if the sample had not been precooled.
These results can be explained on the hypothesis that the nucleation is catalyzed by colloidal particles suspended throughout the volume of the mercury droplet. For reasons already given it seems likely that these catalysts must have had their origin in the preparation of the dispersion rather than in some earlier stage of the mercury processing.
The greater total catalytic effect of the suspended particles in comparison with the Hg2I2 film on the droplet surface might be accounted for in one of the following ways:
(1)	The suspended particles are not chemically identifiable as Hg2I2 and their catalytic potency, p, is greater than p for Hg2I2.
(2)	The suspended particles are Hg2I2 having the same p as the surface film. However, the total area of these particles/droplet considerably exceeds the film area/droplet. For example, if the suspended particles were 0.01 micron in diameter and their volume fraction ~0.001, their total area/droplet would exceed the droplet area by a factor of 4.
(3)	The thermodynamic properties and undoubtedly also the lattice parameters of crystals 0.01 micron in diameter differ significantly from those of crystals exceeding one micron size. Therefore, p for 0.01 micron Hg2I2 particles might exceed p for the larger Hg2I2 crystals constituting the surface film.
As a consequence of the exceptionally marked dependence of / on T (a factor of about 200/°) (/f„)exp
~1070 is many orders of magnitude larger than maximum .Kv^lO36 calculable from nucleation theory. On the basis of the known facts, it is not possible to reconcile this disagreement with nucleation theory without an additional hypothesis. It can be verified from the theory that the temperature coefficient of nucleation in heterogeneous processes is larger, the nearer the process is in temperature to the equilibrium temperature. Therefore, the results on the Hg2I2 dispersion suggest that the rate controlling process is the formation of the nucleus of a phase in equilibrium with liquid mercury (or with Hg2I2) at a temperature well below the normal melting point of mercury. Two possible rate-controlling mechanisms to account for the results are suggested:
1.	Formation of a nucleus of a new Hg2I2 phase having a greater catalytic potency for formation of mercury crystals than the parent Hg2I2 phase.
2.	Nucleation of a crystalline mercury phase, /3, having a melting temperature well below —38.87°C. It is supposed that the nucleation of /3 is more strongly promoted by Hg2I2 than is the nucleation of the most stable crystalline phase a. Also it is assumed that the nucleation of a in fl is rapid by comparison with the nucleation of in liquid.
The experimental results can be reconciled with theory on the basis of mechanism (2) if /3 melts around — 70°C.
An isokinetic relation AT=f (logo), for the Hg2I2 dispersion is indicated in Fig. 6.
3.	Heterogeneous Nucleation Showing Multiplicity
of Is at Constant T
(a) Results on Hg(St) Dispersion.—y°=f(D) for this dispersion was practically identical in shape with y°=f(D) for Sample No. 1 shown in Fig. 4 but with D' = 3.8 microns. X=f(t) was measured at four temperatures and the master isotherm constructed from these data is plotted in Fig. 17. After an earlier analysis613 of these results, it was concluded that the conditions for homogeneous nucleation were approached in these experiments. However, it proved impossible to describe the data on the basis of the volume nucleation hypothesis with a single value of I and the discovery was made that Hg(Lau) and Hg(Bz) dispersions supercool to temperatures 18-20° lower than Hg(St) dispersions. Therefore, it is believed that the earlier conclusion was mistaken and that mercury crystals are nucleated heterogeneously in the Hg(St) dispersion. The comparison of the master isotherm with the curve calculated on the basis of volume nucleation is given in Fig. 17. The experimental rate of solidification decreases with increasing X much more sharply than does the calculated rate. These facts can be interpreted on the hypothesis that two or more fractions of the dispersed droplets have different Is values at constant temperature.
KINETICS OF SOLIDIFICATION
423
kD' may be roughly estimated at a specified value of X by assuming the validity of the relation
kD'=dX/dt(\-X).	(24)
Apparent values of I and /» were evaluated from hn' at X=0.3 using Eqs. (13) and (15). The constants 9 and (^«)eiP of Eq. (5) calculated from these data and the t-values in Fig. 17 are listed in Table II. In calculating the log K,= 27.3 given in the table, it was assumed that every mercury atom in the droplet surface was in the catalyst surface. Actually, it seems probable that the fraction of surface mercury atoms also in the catalyst surface was substantially less than unity.
Since Hg(Lau) films apparently do not catalyze the nucleation, it is not considered likely that any one of the effective catalysts in the Hg(St) dispersion is a mercury stearate. It seems more likely that the catalysts are mercury oxystearates and/or substances formed by the reaction of minor components in the stearic and forming solution with mercury.
(b)	Hg(X) Dispersion.—Rates of solidification of Hg(X) dispersions increased.slowly as a result of room temperature aging. Four isotherms measured over a two-day period, during which time aging is hardly perceptible, are plotted in Fig. 18. These isotherms can only be described by several values of I, that differ widely (by factors of at least 1000) at constant temperature. Apparently, the curves approach in time an X asymptote, X~0.2 at -41.72° to X~0.9 at -42.77°. This general behavior is similar to that observed by Vonnegut16 and by Pound and LaMer17 in the solidification of tin.
(c)	Interpretation.—The X=f(t) relations for both the Hg(X) and Hg(St) dispersions can be described by the general analytical relation,
X=Z1°[l-exp (-£,/)]
+X2°[1—exp(— • ■ •Z<°[l-exp(-Jfe<0], (25)
where Xi", _X"2° • • • Xi° are fractions of the total dispersion, each characterized by a single nucleation frequency (kh k2, ■ • - k, per droplet). It is assumed for simplicity, and for the present purpose without a significant loss in rigor, that the droplet size in a given dispersion is uniform. So we have then to interpret directly a multiplicity in k that corresponds to a multiplicity in I,.
Two terms of Eq. (25) with Xi°=0.3 and £^100 k2 are sufficient to describe the Hg(St) isotherms. At least five terms are required for the description of the HgX isotherms.
Two basic interpretations of the multiplicity of k seem worthy of serious consideration. First, it may be supposed that each k value characterizes the catalytic potency of a different chemical substance. In our experiments these were introduced during the dispersion forming process or formed in a subsequent chemical reaction (as was definitely the case for the HgX dispersion). An essential requirement of this interpretation is that the average number, m, of crystallites or surface
patches per droplet of the most potent catalytic substance should not greatly exceed unity.
A second basic interpretation is that catalysis is effected by one chemical substance having different catalytic potencies in different parts of its surface or when formed on droplets of different size. For example, it may be supposed that nucleation takes place preferentially at nonthermally equilibrated defects in the surface of the catalytic crystallite. A specific example of such a defect is the step present in a crystal surface caused by the termination of a screw dislocation in it.27 The specific structure and catalytic potency of these defects may vary considerably and considering the small mass of their containing crystallites, the average number m/droplet of the defects most effective in nucleation catalysis can be of the order unity. It follows that the formal treatment of the problem of the kinetics of nucleation catalyzed by such defects is entirely analogous to the treatment of k multiplicity as a result of catalysis by a variety of chemical substances.
A second possible explanation for a differing catalytic potency in the same chemical substance follows from the fact that the thickness of the droplet surface coating is only of the order of 0.05 micron. Therefore, its thermodynamic properties, lattice parameter, and catalytic potency will be comparatively sensitive functions of its thickness and surface charge density. It is conceivable that the film thickness and its surface charge density may be related to the droplet size in such a way that X=f(t) is described by equations of type (25).
The kinetic treatment of the k multiplicity caused by suspended crystallites, surface patches, and structural defects must take into account the statistical distribution of units of a particular catalyst among the droplets. Kimball and Halford28 have treated this problem formally considering only one kind of catalyst. Their method was to write an equation of type (25) where each term corresponds to a droplet class containing a specified number of catalytic units. Then the Xi°'s and s are
Fig. 18. Solidification isotherms of dispersion of Hg(X)-coated Hg droplets.
27	Burton, Cabrera, and Frank, Nature 163, 398 (1949).
28	Kimball and Halford (private communication to Pound and LaMer, reference 17).
424
D A V I D TURNBULL
related to m and k, the nucleation frequency of a droplet containing one unit, through the Poisson distribution law. The resulting X=f(t) relation is,
X=l-exp{-w[l-exp(-^r)]}	(26)
and X approaches an asymptote [ 1 — exp (—m) ] at t—» , for small values of m (26) reduces to X = m[_ 1 — exp (—kt)~] which corresponds to a single term in Eq. (25) with m=Xi<\
None of the existing data showing multiplicity of k values can be described by the Kimball-Halford theory on the assumption of a single catalyst; to fit the data to this theory requires several more terms of Eq. (26), approximately the same number as would be required with Eq. (25), corresponding to the additional active catalysts. For example, the Hg(St) data require two Kimball-Halford terms.
Pound and LaMer17 have fitted two of their isotherms on tin with the Kimball-Halford formula. Their analysis shows that at least two specific catalysts were active and inspection of their other isotherms indicates that several other catalysts were probably active. The simple Kimball-Halford theory predicts that m is proportional to droplet volume if catalysis is effected by suspended crystallites or to droplet area if effected by chemical or structural heterogeneities in the droplet surface. Pound and LaMer's results are better described on the assumption that m is proportional to droplet area.
Applying the procedure used on the Hg(St) data Pound and LaMer's data gives (ifs)exP~1031, compared with Ks= 1027-6 calculated on the assumption that every atom in the droplet surface is in an active catalytic surface.
It seems probable that the different catalysts responsible for the k multiplicity in the Hg(St) and Hg(X) dispersions are very closely related chemically, and in the case of Hg(X) they may be chemically the same and differ only structurally. The basis of this conclusion is that the solidification diagram (Fig. 4) shows that different crystalline compounds differ widely in their catalytic potency, and it seems very unlikely that two or more wholly unrelated compounds formed in the same dispersion process would exhibit nearly the same catalytic activity. It seems even more unlikely that several unrelated compounds exhibiting catalytic activity characterized by the narrow Hg(X) band in the solidification diagram could all be formed from the Hg(St) dispersion by approximately the same kinetics and all catalytically deactivated by small amounts of acetic acid.
CONCLUSIONS
In this investigation it has been demonstrated unequivocally for the first time, to the best of the author's knowledge, that the rate of crystal nucleation in several dispersions of supercooled mercury droplets follows first-order reaction kinetics described at constant tempera-
ture for a given dispersion by a single value of nucleation frequency/volume or area.
Convincing evidence has been obtained that 2-8 micron diameter mercury droplets coated by a mercury laurate film solidify from nuclei formed homogeneously. It is demonstrated that the data are in satisfactory agreement with the Turnbull-Fisher theory of homogeneous nucleation in condensed systems. This agreement lends credence to the use made of the theory6* to calculate liquid-crystal nucleus interfacial energies of metals where the homogeneous nucleation frequency is known at only a single temperature. (In calculating cr from I and a theoretical value of Kv, it should be remembered that an uncertainty of only one percent is introduced into a for every factor of ten uncertainty in Kv.)
In an earlier publication,63 it was pointed out that "gram-atomic interfacial energies" ag of the strictly metallic elements calculated from presumed homogeneous nucleation frequencies are related to the corresponding gram atomic heats of fusion, AH/, by the equation,
<xg/AHf=0A6±0M.	(27)
When treated according to the earlier procedure, for purposes of comparison, the new data on mercury give <rg/AHf= 0.61. This deviation from the average for the other metallic elements, though significant, is not sufficient to invalidate the generalization that <rg/AH/ ~ constant for the metallic elements.
The first instance has been established, in the Hg(Ac) dispersion data, in which the nucleation frequency is proportional to the surface area of the supercooled droplet. These results are in good agreement with the theory of heterogeneous nucleation kinetics derived from concepts advanced in earlier papers.2'4'6* In this case the nucleation catalytic potency of the surface film can be conveniently characterized by a contact angle 6.
For two mercury dispersions and for all tin dispersions thus far studied a multiplicity of Is values measured at constant temperature is found. It is believed that the catalysts responsible for this multiplicity in mercury are either structural defects of different severity in a common crystalline substance or very closely related chemical substances.
ACKNOWLEDGMENT
The author was ably assisted by W. A. Rocco throughout this investigation. Mr. Rocco is mainly responsible for the design of the simple and effective method used to maintain the temperature constant. E. F. Fullam advised and assisted in the droplet size distribution measurements.
The author has been stimulated by discussions of various aspects of this investigation with J. C. Fisher, J. H. Hollomon, and G. W. Sears. He is also indebted to V. K. LaMer for making the Pound-LaMer manuscript available to him in advance of publication.
SCHENECTADY, N. Y.