Scaled Models for Nucleation ∗

Barbara N. Hale

Department of Physics and Graduate Center for Cloud Physics

Research,

University of Missouri-Rolla, Rolla, MO 65401

(*Published in”Atmmospheric Aerosols and Nucleation”, Ed. by

P.E.Wagner and G. Vali, Lecture Notes in Physics, 309, 323 (1988))

I. INTRODUCTION

I.1 Motivation for the Scaled Models

Scaled models for nucleation can provide considerable convenience in plotting and analyz-

ing experimental data. In some applications, the scaled models can provide quick estimates

of critical supersaturation ratios, Scr, or supercoolings required for onset of nucleation; the

latter estimates are particularly useful when numerical substance data are unavailable. But

the models offer something more far-reaching: they allow one to isolate the universal temper-

ature dependences and to focus on the substance parameters which dominate the nucleation

process. Finally, for the experimentalist and theorist alike, the scaled models offer a much

appreciated opportunity to analyze data for a spectrum of materials simultaneously.

In this review, the term ”scaled nucleation model” refers to a formalism in which the

classical nucleation rate, J , (and all expressions derived from J ) are expressed in terms of

T/Tc, P/Pc and ρ / ρc. [1] The latter are the reduced temperature, pressure, and number

density, respectively. The subscript c denotes quantities evaluated at the critical point — the

PV T equilibrium point where the distinction between vapor and liquid vanishes. From the

critical point quantities one can form factors (such as Pc/[ρckTc] ∼ 3/8 ) having numericalvalues nearly substance independent. These factors result as one multiplies and divides

J by Pc, Tc and ρc — in such a way as to convert all P, T and ρ to reduced quantities.

Finally, scaled functional forms (generally available in the literature [2]) for equilibrium

1

vapor pressure, surface tension, and number density are substituted into the formalism

[3,1]. The final result is an expression for J which explicitly displays the ”corresponding

states” properties of nucleating substances.

The scaling of J is not a new idea. Near the critical point, (at T ∼ Tc ) such scaling

of the nucleation rate has been considered extensively [4-10]. In particular, Binder [4]

presented a scaled form for the (slightly modified) classical nucleation rate valid near Tc.

The major difference between Binder’s form and the scaled models described in this review

is the applicable temperature range: in the models of this review the expressions for J and

S are not intended for use near Tc. Rather, the intent of these models is to provide a

scaled model for J valid far below Tc — in a range of temperatures relevant to atmospheric

nucleation and freezing phenomena. (Recall that for water Tc = 647.26K. ) A more detailed

comparison of the formalisms is given in Section I.2, below.

Some time ago, Wu, Wegener and Stein [11] demonstrated an approximately linear re-

lationship between lnScr and T 3/2 using experimental SF6 vapor-to-liquid homogeneous

nucleation data. However, the exploitation of a scaled or explicit temperature dependence

of ln Scr for T < Tc was apparently not further pursued until McGraw [12] examined a cor-

responding states formalism and demonstrated that the data for ln Scr fell into identifiable

groups of substances when plotted versus the reduced temperature, TTc. Motivated primarily

by McGraw’s results, we presented a universal temperature dependence, [3]

lnScr = C[TcT− 1]3/2 (1)

where C = 0.05A3/2o , and demonstrated that the experimental data for a range of materials

agreed with this scaling law. (Note that Ao[TcT− 1]n2/3 = 4πr2 σ

kTin the classical energy of

formation of an n molecule cluster). It was also pointed out that the data thus plotted fell

roughly into two groups with slopes in the ratio of 3 to 2. [3] Rasmussen and Babu [13,14]

made use of Eq.(1) to illustrate a crucial correlation between C and the Eotvos constant

[15]. However, the explicit relationship between the C and the Eotvos constant was not

given. The first scaled homogeneous nucleation model far below Tc which explained this

2

relationship and the reason for two groups of substances was presented in [1]. Before giving

details of [1] a discussion of the standard model near Tc is in order.

Similarities With Critical Point Phenomena and an Introduction to Scaling

There is considerable similarity between the standard scaling of J near the critical point

and the scaling in the models of this review. First, however, we point out that [TcT− 1] is

different from the = ±[1− TTc] dependence generally employed in critical point formalisms.

Both forms are (for all practical purposes) equivalent in the analysis near Tc. But a peculiar

property of the [TcT−1] function appears to be that its substitution for in some critical point

formalisms dramatically extends the range of applicability.[16] This result is not widely

used (although apparently recognized) by those working with critical point phenomena.

An important quantity used in the present review, as well as in critical point phenomena,

is called the scaled supersaturation:

x ≡ ln[S

A3/2] (2)

The scaled supersaturation was introduced by Binder and Stauffer [5]. However, in the

models of this review A = Ao[TcT− 1] , whereas Binder uses A = b , where b is a constant

proportional to the surface tension. In particular, the scaled supersaturation used by Binder

and Stauffer [5], has the form lnS/(b ) βδ, where β and δ are the standard critical point

exponents [2]. Near the critical point ∼ [TcT− 1], and βδ ∼ 1.54 [5] — very close to the

classical three dimensional fluid value of βδ = 3/2.

Another (field theoretic)model for near-critical point nucleation by Langer and Turski

[6] uses a quantity closely related to the scaled supersaturation: the scaled supercooling, τ =

δT/( Tc).The scaled supersaturation (or the scaled supercooling) influences the nucleation

rate primarily via the energy of formation (divided by kT ) of the critically sized ( n = n∗

) cluster [9]:

g(n∗) = (xo/x)2 = (τo/τ)2. (3)

The xo and τo are constants dependent on critical point amplitudes. [4-10] In the classical

theory, xo = 2/33/2. However, τo is less well defined for temperatures far below the critical

3

point —and not independent of T . The classical nucleation rate, J , is proportional to exp-

g(n∗):

J = Joexp[−(xo/x)2] (4)

— and, as was pointed out by Binder, in the case that the kinetic prefactor, Jo, has only

slight T and P dependence, the (x/xo) is nearly constant for fixed J . This leads directly

to the approximate scaling law for the ”scaled supersaturation”, x

x =lnS

A3/2∼ constant (5)

The same sort of arguments lead to the scaling law for ln Scr far below the critical

temperature —and to all the scaling laws in this review.Classical nucleation theory and

theories applicable near the critical point differ primarily in the approximation of the kinetic

prefactor which describes the growth of the clusters subsequent to the nucleation event — and

before observation of macroscopic effects. [4-10, 17] In predicting critical point phenomena

the prefactor must account for diffusion controlled growth and the vanishing of the diffusion

constant as T approaches Tc. [5] But for temperatures far below the critical temperature,

diffusion controlled growth is not in general applicable and the rate of formation of the

new phase is primarily dictated by the birth (nucleation) of new phase embryos (that is

by g(n*)).[4,10] In this low temperature region the classical kinetic prefactor for vapor-to-

liquid nucleation is proportional to the equilibrium vapor pressure squared and appears to

be highly temperature and material dependent.

It was this seemingly unwieldy temperature dependence of the classical kinetic prefactor

which prompted Rasmussen and Babu [13] to comment that a theoretical explanation for

the scaling law of Eq. (1) for ln Scr was lacking. The resolution of this difficulty lies in

casting the kinetic prefactor into an approximately material independent (and nearly TTc

independent) form [1]. With this accomplished, the classical theory predicts the correct

scaling law for ln Scr. One can also use this method to develop a modified lnS scaling law

for constant J not corresponding to onset of nucleation. Finally, one can incorporate a term

4

into the energy of formation which takes account of the translation of the center of mass of

the embryonic cluster. [18, 19]

The organization of the review is as follows. The scaled model for vapor-to-liquid ho-

mogeneous nucleation is presented and compared to cloud chamber and diffusion chamber

data in Section II. In Section III the homogeneous nucleation model is modified to treat the

case of liquid-to-solid phase transitions, and applied to homogeneous freezing temperatures

for a range of substances. In Section IV the scaled nucleation models are extended to in-

clude heterogeneous nucleation phenomena; this model is applied to Vonnegut and Baldwin’s

data [20] for ice nucleation in a supercooled water sample containing silver iodide particles.

Comments and conclusions are in Section VI.

II. A SCALED MODEL FOR HOMOGENEOUS

VAPOR-T0-LIQUID NUCLEATION

II.1 Formalism

The classical Becker-Doring theory [22] for the steady-state homogeneous nucleation rate

[23,24] (including the so-called Zeldovitch factor [25]) can be written as follows [1]:

J = Jo exp− [xo/x]2, (6)

where

Jo = JcI [P1/Pc]α[TcT]α (7)

= JcI [ρ1/ρc]α[8/3]α (8)

and α = 2 in the standard classical model. If one includes the translation of the center of

mass of the cluster, one finds that α = 1. See Appendix A. The factors in Jo are defined

as follows:

I = 2[ρc/ρ2]2/3[Ω(1− T

Tc)]1/2 (9)

5

and

Jc =Pc

h[λcρc]2/3

Pc

kTcλc. (10)

The ρ, P , h, k, S, and Γ are the number density, pressure, Planck constant, Boltzmann

constant, supersaturation ratio and inverse thermal wavelength cubed ( [2πmkT/h2]3/2 ),

respectively. Subscripts 1 and 2 indicate quantities in the parent and daughter phase,

respectively. The form for the exponent, (xo/x)2, follows from the classical free energy of

formation (divided by kT ) for the n - atom/molecule cluster:

g(n) = An2/3 − nB (11)

where B ≡ lnS. [26] Classically, A is equal to the surface tension (divided by kT ) times

the area per surface molecule. From the usual condition for the critically sized cluster,

dg(n∗)/dn = 0, one readily obtains the number of molecules in the critical cluster,

n∗ = (2A

3B)3 (12)

g(n∗) = 0.5n∗B = (xo/x)2 (13)

and

xo = 2/33/2. (14)

If one assumes the scaled form for the surface tension,

σ = σ0o(Tc − T ) (15)

where σ0o is a material dependent constant [27], the A takes a simple form:

A = (36π)1/3Ω[TcT− 1] (16)

where

Ω ≡ σ0okρ2/3

. (17)

6

In this scaled surface tension model Ω is minus the partial derivative with respect to T

of the surface tension per molecule. Hence, the Ω is the effective excess surface entropy per

molecule (in units of k) in the embryonic cluster. The bulk liquid value for Ω ( the Eotvos

constant [15] ) is approximately 2 for most liquids. For associated liquids Ω is smaller ( 1.5 )

and reflects the reduced excess entropy for surface molecules as dipole moments align at the

interface. The grouping of liquids into these two general categories gives the two slopes for ln

Scr noted in the introduction. [1] The corresponding values of Ao are about 10 for ordinary

substances and 7 for associated liquids. This approximate material independence of Ao

was noted when calculating thermodynamic properties of microscopic clusters using Monte

Carlo methods and effective pair potentials. [29] In some preliminary work it was found

that Ao ∼ 10 for Lennard-Jones argon clusters [29] and Ao ∼ 7.5 for Rahman-Stillinger [30]central force (rigid molecule) water clusters [31]. These values of Ao correspond to Ω = 2.1

and Ω = 1.7 for (Lennard-Jones) argon and (rigid molecule central force) water, respectively.

Using Eqs. (6) - (10), the scaled supersaturation, lnS/A3/2, becomes,

lnS

A3/2=

xoδo√ln(Jc/J)

(18)

where

δo =

·1 + [−α ln(Pc/P

1o ) + α lnS + lnI + 2ln(

TcT.)] / ln(Jc/J)

¸−1/2(19)

For a range of temperatures satisfying 0.3 < TTc

< 0.5,

δo ∼ 1 + 0.7Wo[

TcT− 1]v

2ln[Jc/J ], (20)

and for lnJ = 0,

δo ∼ 1.13± 0.04[TcT− 1]v. (21)

In obtaining this approximation, the following are used:

ρc/ρ ∼ 1/3 (22)

7

and,

ln[Pc/P1o ] ∼Wo[

TcT− 1]υ. (23)

For most substances Wo can be roughly represented by L/kTc ∼ 7±2, where L is the latentheat of vaporization near the boiling point. The v ∼ 1 and in subsequent approximations,v = 1 will be used.

While there is some cancellation of the ln( Pc/P1o ) term by lnS and lnI + 2ln(Tc

T) in

Eq. (19), the latter term contributes less than 1.5% to δo. The major contribution to the

approximation in Eq. (21) comes from ln( Pc / P 1o ). For substances (such as toluene)

which have relatively small values of Wo and for which Ω ∼ 2 (non-associated liquids) thereis a considerable cancellation of ln( Pc/P

1o ) by lnS and the temperature dependence of δo

is weak. Finally, one can show that

lnJc ∼ 72± 3 (24)

for most substances. For example, the values are 72.8, 74.7, 71.8, 70.8, 73.7 for the sub-

stances ethanol, water, toluene, nonane and argon, respectively. Since the square root of

ln Jc enters into the expression for ln Scr, a 4% error in ln Jc produces 2% error in ln Scr.

Using lnJc = 72 the following approximate scaling laws result for J ∼ 1cm−3sec−1 :

lnScr/Ω3/2 ∼ 0.53[Tc

T− 1]3/2; (25)

and for larger J of physical interest,

lnS ∼ lnScr[1 + lnJ/(2lnJc)]. (26)

The major deviations from these approximate scaling laws occur at low temperatures

where [ TcT-1] is large (> 1.5). One can show also that the critical cluster size (for onset of

nucleation) takes the form:

n∗ = (3/4π)1/2[lnJc/Ω]3/2[TcT− 1]−3/2. (27)

= 106[2/Ω]3/2[TcT− 1]−3/2 (28)

8

In expansion chamber experiments it is often more convenient to use the supercooling. Eq.

(23) (with the approximation v = 1 ) and Eq.(25) yield

δT 0Wo/Ω3/2 ∼ 0.53[Tc

T− 1]3/2 (29)

where,

δT 0 ≡ TcTfinal

− TcTinitial

. (30)

The modification of this formalism for the case which includes the free energy associated

with the translation of the center of mass is treated in Appendix A.

II.2 Comparison with Experimental Data

The approximations in Eqs.(25) and (26) serve as good predictors for lnS over a range

of nucleation rates. Figure 1 shows experimental homogeneous vapor-to-liquid data for

lnScr/Ω3/2 for a number of substances [32-38] using bulk values [28] for Ω. The data for ln

Scr conform to the approximate scaling law in Eq.(25) rather well in spite of the scatter in

data and the approximation of Ω by the bulk value. In fact, the ln Scr data appear to be

more linear in [TcT− 1]3/2 than the corrections to Eq.(25) (via δo, Eq. (19)) would indicate.

The linearity of the data of Katz , et al., [37] for toluene is particularly striking, and it

is noteworthy that almost all of the Katz data [35-38] fit this linear dependence extremely

well. In Figure 2 is plotted the J = 104 cm-3/sec expansion cloud chamber data of Miller,

Anderson and Kassner for water [39,40] and of Schmitt, Adams and Zalabsky for toluene [41]

and nonane [42]. The expansion chamber data appear to be consistent with the [TcT− 1]3/2

temperature dependence for lnS/Ω3/2 as predicted by Eqs.(25) and (26).

It is interesting to compare the experimental nonane data for J ∼ 1 (Katz et al., [36]),J ∼ 104 (Schmitt et al., [42]) and J ∼ 108 (Wagner and Strey [43]) in a way which emphasizesthe role of prefactor and exponent for J . The exponent,

(xo/x)2 =

16π

3Ω3[

TcT− 1]

3

/lnS2, (31)

and if one uses Ω = σ/[kTρ2/32 (Tc

T− 1)] and literature derived values for liquid surface

tension, σ, [44] the standard classical model obtains. For most non-associated liquids the

9

Figure 1. Natural logarithm of the threshold (J = 1cm⁻³sec⁻¹) supersaturation ratio, Scr, divided by Ω3/2 from diffusion chamber and nozzle beam experimental data. The data points are for toluene [37] ( ∆ ), nonane [36] (x ), water [32] ( ), n-butylbenzene [37] ( ), sulfur hexafluoride [11] ( + ) carbon tetrachloride [38] ( ), chloroform [38] ( ⊗ ), ethanol [35] ( ), octane [37] ( * ), argon [34] [taken from McGraw [12], Fig. 1] () and acetic acid [33] ( ). The dashed line is 0.53[Tc/T-1]3/2 from Eq.(25). The values used for Ω are [13]: 2.35 for nonane, octane and n-butylbenzene, 1.5 for water and ethanol, and 2.0 for SF₆. For the remaining substances the ideal gas value 2.12 is used [15].

Figure 2. The lnS/Ω3/2 for J∼10⁴cm⁻³s⁻¹ from the expansion chamber data for water [39,40] ( ), nonane [42] ( x ) and toluene [41] ( ) The dashed line is the prediction from Eq. (26).

Ω so calculated is stable with respect to T (in the range of TTcof interest) to about 0.5%.

[28] The stability of these values will depend somewhat on the choice of extrapolation for

σ and ρ2 at low temperatures. In Fig. 3 is plotted the ln[Jo/J ] (using literature values for

P 1o [45]) versus (xo/x)

2 for these data in the classical model (no scaling of σ ). The three

sets of data fall not too far from the straight line —which is the prediction of the classical

model. The dashed lines indicate errors in J of 10±3. While the Wagner and Strey data

and the Katz et al. data appear to be closer in magnitude to the classical model prediction,

the expansion chamber data of Schmitt et al., show a more nearly linear relationship. The

high temperature data correspond in general to larger values of (xo/x)2. Thus in Fig. 3 the

high temperature (expansion chamber) data lie furthest from the solid line and the classical

model prediction.

Both Schmitt et al [42] and Wagner and Strey [43] found major discrepancies in com-

paring their data with the classical model at low temperatures. For example, Schmitt et

al.(using an equilibrium vapor pressure [46] different from the vapor pressure [45] used by

Katz and Wagner and Strey) found that their data disagreed with the classical model by

factors of 109 in J at low temperatures. On the other hand Wagner and Strey (using an

expression for the surface tension [45] different from that used by Schmitt et al and Katz

[44]) found that their data disagreed with the classical model by factors of as much as 106

at low temperatures. As can be seen in Fig 3, the vapor pressure [45] and surface tension

[44] formulae used by Katz bring all the data into approximate mutual agreement with the

classical model. We note that this does not imply that these particular formulae are without

problems.

This dilemma emphasizes the need to sort out competing temperature dependences of

terms in (xo/x)2, and to assume a valid equilibrium vapor pressure at low temperatures.

Finally, it seems appropriate to note that the expansion chamber data of Kassner, Miller

and Anderson [39,40] and of Schmitt et al., [47,41] offer stringent tests for the temperature

dependence of the theory, and it is unfortunate that this data has been so long overlooked.

10

Figure 3. The ln[Jo/J] versus the classical energy of formation, (xo/x)² for: the nonane diffusion chamber data of Katz et al. [36] ( x ), the nonane expansion chamber data of Schmitt et al., [42] where J≈10⁴ ( · ) and the nonane expansion chamber data of Wagner and Strey [43] ( ) where J≈10⁸. The x=A3/2/lnS is the scaled supersaturation. The solid line is the prediction of the classical theory, i.e., ln[Jo/J]= [xo/x]² with J=10⁸; the dashed lines indicates the range of data corresponding to 10±3 errors in J. The literature values of σ [44] and of Po¹ [45] for nonane are used in plotting all the data.

It is noted, however, that the nonane and toluene expansion chamber data appear to give

values of lnS which are slightly smaller than the classical model would predict at high

temperatures. An approximate value of =2.35 for nonane can be extracted from the Schmitt

data by plotting the ln(J/S2) versus 16pi3Ω3[Tc

T− 1]3 / lnS2. It is interesting that this is

the value of the Eotvos constant for nonane cited by Rasmussen [13] and that if one uses

Ω = 2.35 the data of Schmitt et al. agrees with the classical model to less than a factor of

ten at all temperatures.

Some comments on the scaling law in Appendix A are relevant. The α = 1 softened

temperature dependence of the kinetic prefactor for J 0 (which predicts a more linear depen-

dence of ln Scr on [TcT− 1]3/2 ) and the more nearly material independence of lnJc ’ merit

some consideration. However,it is well known that J 0/J (without replacement factors) is

1017 and unless the corresponding could be increased by 15% the scaling laws in Appendix

A do not agree with experiment.

III. A SCALED MODEL FOR LIQUID-TO-SOLID

HOMOGENEOUS NUCLEATION

III.1 Formalism for Liquid-to-Solid Nucleation.

In rewriting the classical liquid-to-solid nucleation rate formalism [48] the same general

expression for the energy of formation, g(n), of an embryonic solid cluster containing n

molecules (or atoms) is used: [49]

g(n) = An2/3 −B0n. (32)

The B’ is a quantity analogous to the lnS used in the vapor-to-liquid formalism:

B0 ≡ ln[P 1o /P

2o ], (33)

where as before, the subscripts 1 and 2 denote parent and daughter phase and the o subscript

implies coexisting (equilibrium) vapor pressure. Equation (32) explicitly incorporates the

necessary dimensional features of surface terms ( ∼ n2/3 ), and bulk terms, ( ∼ n ). Devia-

tions in cluster structure or shape can be absorbed into A. The critical cluster size, n∗, and

11

g(n∗) are given by Eqs. (12) and (13) with B replaced by B0, and the scaled supersaturation

is x ≡ B0/A3/2.

In liquid-to-solid nucleation B0 is more conveniently related to the supercooling of the

liquid. In particular, when the melting temperature, Tm, is close to the triple point tem-

perature [50]:

B0 ∼ Bo[Tm/T − 1] (34)

where Bo ∼ Lf/(kTm), and Lf is the entropy of fusion per molecule at Tm. For example,

for water-to-ice nucleation (at water saturation) Bo ∼ 2.6.The conventional steady state

nucleation rate, J , (see for example Fletcher [38]) can be written in the same form as Eq.

(6): [51]

J = Jos exp[−(xo/x)2] (35)

where,

Jos ≡ JcI [ρ1/ρc]2[8/3]2β0 (36)

The ρ1 is the parent liquid phase number density, and I is given by Eq. (9). The is given

by [52]:

β0 = exp[−w] ∼ exp[−13Tm/T ] (37)

where w is the diffusion barrier in the liquid (divided by kT ). The form for Jos is identical

to the form given by Eq. (8) for vapor-to-liquid nucleation except for the factor.

The liquid-to-solid homogeneous nucleation rate is difficult to measure directly. Of more

interest is B0 or, equivalently, the supercooling which produces the onset of nucleation. It

is assumed in this treatment that onset corresponds to JV ∼ 1 s−1, where V is the parent

phase volume in cm3. For liquid-to-solid nucleation the lnI is small ( ∼ −1 ) and onlyslightly dependent on T

Tc. We also note that temperature variations in ln β0 and ρ1 / ρc ≈ 3

are small compared to the large value of lnJc ≈ 72. In this case Jos is nearly constant and

12

Eq. (5) results in the same way as noted in Section II. Thus, for the onset of heterogeneous

freezing:

B0/A3/2 = x = xoδ /√ln Jc (38)

where,

δ ≡ [1 + (−w + lnV + 2ln8− 1)/lnJc]−1/2. (39)

For application of this model to small, micron sized liquid drops ( lnV ∼ −26 )and a bulk liquid sample ( V ∼ 1 cm3 ) using -ln β0 = 16:

δ ∼ 1.5 (micron-sized drops);

δ ∼ 1.1 (bulk liquid,V ∼ 1 cm3.)

As in the scaled homogeneous vapor-to-liquid nucleation formalism [1], a scaled interfacial

tension between the parent and daughter phase, σ12, is introduced where

σ12

ρ2/32 kT

≡ Ω12[TcT− 1]. (40)

If one uses a spherical droplet model for the embryonic cluster A is given by Eq. (16) and

Eq. (35-40) give:

ln[P 1o /P2o ] ∼ 0.48 δ Ω

3/212 [

TcT− 1]3/2. (41)

For V ≈ 1 cm3 the constant [0.48 δ ] = 0.53 and Eq. (41) is surprisingly similar to Eq. (25)

for the vapor-to-liquid case. For the small micron sized drops 0.48δ ≈ 0.72. Finally, usingEq. (34) the supercooling required for onset of nucleation is:

Bo[Tm/T − 1] ∼ 0.48 δ [Ω12[TcT− 1]]3/2. (42)

In view of uncertainties in experimental data for supersaturations and supercoolings and

the difficulties with surface contamination, and measurement of interfacial surface tensions,

13

it appears that Eq.(42) is quite reasonable. In the next section this formalism is applied to

experimental data for homogeneous freezing.

III.2 Comparison with Liquid-to-Solid Nucleation Data

In Table I is shown experimental supercooling data for a range of non-metallic

substances. [53] The values of Ω12 are calculated from the nucleation formalism using Eq.

(42) above, Bo = Lf /k Tm , and V corresponding to 50 µ drops. In the last column of

Table I is an estimate of Ωls calculated from the following expression:

Ωls ∼ [TmT]4/3[

Lf

kTm]2/3 (43)

∼ 13[Lf

kTm]2/3

Table I. Comparison of Ωls from Eq. (43) and Ω12 calculated from homogeneous freezing

data [57] using Eq. (42).

Substance [Tm/T− 1] Ω12 Ωls

H2O 0.18 0.42 0.61

CCl4 0.25 0.42 0.47

CHCl3 0.33 0.83 0.84

C6H6 0.34 1.0 1.0

CH3Cl 0.46 0.83 0.80

CH3Br 0.16 0.52 0.71

C3H6 0.14 0.50 0.71

BF3 0.14 0.50 0.64

NH3 0.26 0.88 0.95

SO2 0.20 0.89 1.07

An empirical relationship between the heat of fusion per unit area, L0, and the liquid-

solid interfacial tension, σls, for simple metals was observed by Turnbull [54] ( σls ∼ 0.45L0)and by Jackson [55]. Gilmer’s computer calculation of σls and L0 for a Lennard-Jones

14

system estimates that σls ∼ 0.32L0 [56] The excess entropy can be expected to depend onthe degree of molecular association at the cluster surface — and hence on the substance and

its structure in the solid state under consideration. Since Ωls represents the excess surface

entropy, the entropy of fusion places a maximum value on the degree of supercooling.

It is interesting to consider the application of Eq. (43) to Ωvl for liquid-vapor interfaces

(or Ωvs for solid-vapor interfaces) using the latent heat of vaporization, Lv, (or the latent

heat of sublimation) and the boiling temperature, Tb:

Ωvs ∼ 1/3[Ls/kTb]2/3, (44)

Ωvl ∼ 1/3[Lv/kTb]2/3, (45)

where the subscripts l, s and v denote liquid, solid and vapor, respectively. For example,

for water/ice one obtains Ωvl ∼ 1.9 and Ωvs ∼ 2.65. Also, Ωvs − Ωvl = 2.65− 1.9 = 0.75.That this number is larger than Ω12 is not unexpected, since Eq. (43) appears to give an

anomalously large estimate of Ωls for water. See Table I.

For metals the critical temperatures are generally several thousand degrees — and difficult

to measure. This makes the formalism impractical for metals. The Tc is known for Hg,

however, and in this case one can compare Ωls = 0.08 to Ω12 ≈ 0.09 from the data of

Turnbull [54]. The Ω for metals appear to be an order of magnitude smaller.

IV. SCALED MODEL FOR HETEROGENEOUS NUCLEATION

IV.1. Formalism for Heterogeneous Nucleation

The formalisms of Sections II and III can be modified for heterogeneous nucleation —

following Fletcher [23,48] and Turnbull and Vonnegut [58]. The procedure is to replace Ω

by Ω0:

Ω0 = f(m)1/3; (46)

15

where f(m)1/3 is an effective entropy reduction factor. In Fletcher’s classical spherical cap

model on a plane substrate, the f(m) takes on a simple form [23]:

f(m) =(1−m)2(2 +m)

4(47)

and

m = cos θ,

where θ is the classical contact angle. One can use f(m)1/3 simply as a parameter and we

refer to the corresponding θ as an effective contact angle. Non-spherical cap embryonic

shapes can be explicitly treated by modification of f(m ) [62] . With this Ω0, g(n) is:

g(n) = A0n0 2/3 −Bn0, (48)

where B = lnS for liquid-to-vapor [ or B = B0 as in Eq. (33) in liquid-to-solid] nucleation

and

A0 ≡ [36π]1/3Ω0[TcT− 1] (49)

= Af(m)1/3

and

n0∗ =·2A0

3B

¸3. (50)

= n∗f(m).

Finally, in order to use Eqs. (25) and (42), the δo and δ must be modified to reflect the

heterogeneous site area available in the parent phase. The simplest method is to multiply

Jo by a per unit volume fraction of molecules in contact with the substrate:

Jp = ρ−1/31 a0. (51)

The a0 is the total area (in cm 2 ) of substrate characterized by f(m) per unit volume of

parent phase. The subsequent expressions for δo and δ are given by δo ’and δ ’:

16

δo’ = [1 + [−α ln(Pc/P1o ) + α lnS + lnI + lnJp + 2ln(

TcT)]/ln(Jc/J)]

−1/2; (52)

δ’ = [1 + (−w + lnV + 2ln8 + lnJp − 1)/lnJc]−1/2. (53)

The result for heterogeneous freezing is:

[Tm/T − 1] ∼ 0.48[δ’/Bo] [Ω12 f(m)1/3[TcT− 1]]3/2.

The above expression offers an interesting look at the importance of heterogeneous nucle-

ation. The left hand side is always small and approaching zero near the melting temperature.

But for f(m) = 1 the right hand side is finite at T = Tm. The role of f(m) << 1 is to

lower the effective excess surface entropy per molecule and allow the nucleation to reach the

observable (or JV = 1 ) state for small supercoolings. Increased impurity concentrations are

reflected in a decreased δ ’. In the real, contaminated world, the number of heterogeneous

sites is large and the freezing is driven to T = Tm.

IV.2 Comparison with Heterogeneous Freezing of Supercooled Water

Consideration of f(m)1/3 as an entropy reduction factor offers an alternative view of the

role of the substrate in heterogeneous nucleation. For example, if 55% of the ’interfacial’

molecules have bulk solid entropy (via strong attachment to the foreign particulate surface),

one would predict f(m)1/3. ∼ 0.45. In this case a single 0.2µ diameter particulate in a smallwater droplet, would (on the average) produce freezing of the drop within 10 seconds at

−20oC.An interesting application of this idea is to the recent work of Vonnegut and Baldwin [20].

They report results on repeated ice nucleation in a 0.01 gram sample of supercooled water

containing large numbers of 10µ AgI particles. Nucleation events were found to occur in a

wide range of time periods; an average of these time periods, < t >, increased exponentially

with decreased supercooling. In the scaled stochastic model the predicted value of ln < t >

is:

17

ln < t >= −ln(JoJpV ) + f(m)[xo/x]2 (54)

= − ln[JoJpV/Σ]− lnΣ+ 13[Tm/T − 1] + f(m)[[xo/x]2

where Σ = total site area (in A2 ) characterized by f(m). A plot of ln < t > −13[Tm/T −1]versus [xo/x]

2 should give a line for each f(m) site with the slope and intercept predicting

the f(m) and lnΣ values. Such a plot is given in Fig. 4, where (xo/x)2[B2

o/Ω3]/33/1000 is

plotted for convenience.

In this analysis, at least two ’sites’appear to be consistent with the data from [20]. The

solid line predicts f(m)1/3 ∼ 0.21 and Σ ∼ 105 A2 ; the dashed line predicts f(m)1/3 ≈ 0.13and Σ ∼ 40 A2. A third line is possible; however, Vonnegut and Baldwin [20] state that

the high temperature data points correspond to fewer freezing events and as such are less

reliable. The f(m)1/3 = 0.21 corresponds to an effective contact angle of 27o and the f(m)1/3

= 0.13 corresponds to θ ≈ 18 o using Eq. (47). The f(m) obtained from this data via the

scaled formalism does not depend on the (heterogeneous) kinetic prefactor which requires

information about the total site area. Detection of two or more sites in one collection of

particulates presents difficulties from threshold temperature data alone. It is also noted

that the site having the smaller f(m) is suppressed in the data of [20] because of a smaller Σ,

and points out the complications associated with multiple site effects in a stochastic model

for heterogeneous nucleation. Additional data on heterogeneous ice nucleation with time

dependence information exist [61], [63] and analysis via the scaled stochastic models is in

progress.

V. COMMENTS AND CONCLUSIONS

This work on scaled models was motivated primarily by a desire to isolate the substance

independent features of the classical nucleation rate at temperatures far below the critical

temperature, and to identify a universal temperature dependence for J. What emerged are

a scaled energy of formation with [TcT− 1] dependence, a relatively material independent

factor, ln Jc, and the overall weak temperature dependence of the kinetic prefactor for J.

All these results are useful when predicting the lnS scaling laws. The distinct features of

18

Figure 4. The natural logarithm of the average time before nucleation, <t>, (adjusted for diffusion temperature dependence in supercooled water) versus the scaled temperature function for the nucleation in supercooled water samples containing silver iodide particles. The data is taken from Vonnegut and Baldwin [20]. The water samples consisted of approximately 0.01g of distilled water into which large numbers of small silver iodide particles in the size range of 10µ had been added. The slope of the solid (dashed) line gives f(m)^1/3 =0.21(0.13). The larger intercept for the dashed line implies fewer sites or particles associated with the smaller f(m). The data points (left to right) correspond to supercoolings of 9, 8, 7, 6.5, 6, 5.5, 5, 4.4, 4, 3.2and 2.7 degrees C. The authors comment that the last point (at T=-2.7C ) corresponds to 4 freezing events and that those at smaller supercoolings are in general more uncertain.

these scaled models are the use of the scaled surface tension and the effective excess surface

entropy per molecule, Ω.

The scaling law in Eq. (25) appears to describe the experimental ln Scr for onset of

vapor to liquid nucleation rather well, and points out the usefulness of Ω in characterizing

critical supersaturation values. The fact that the bulk value for this quantity is nearly two

for most substances and reduced to about 1.5 for associated liquids provides a convenient

’rule of thumb’ for estimating critical supersaturations for a wide variety of materials.

A comparison of the diffusion chamber and expansion chamber nonane data indicates

that the classical model does a credible job of predicting J for vapor-to-liquid nucleation.

There does appear to be, however, some anomalous temperature dependence — related to

uncertainties in low temperature equilibrium vapor pressure which can generate apparent

discrepancies as large as 106 between data and the classical model. A careful consideration

of the competing temperature factors in (xo/x)2 appears to be in order.

The scaled liquid-to-solid homogeneous nucleation model provides a temperature depen-

dent formalism for analyzing experimental data for a range of non-metallic substances. For

non-metals the Ωls predicted from the experimental data ranges from 0.42 to 1 ; for Hg the

Ωls appears to be about 0.09 — an order of magnitude smaller than for the non-metals stud-

ied. An approximation for Ωls proportional to [Lf/kTm]2/3 is suggested by the formalism

and reflects the surface vs. volume properties of Ω and Lf .

Following Fletcher’s approach, the scaled homogeneous models are extended to include

heterogeneous nucleation. The modifications use Ω0 = Ωf(m)1/3, where the f(m)1/3 is

interpreted as an effective entropy reduction factor characterizing the ability of the foreign

substrate to inhibit molecular motion at the interface. Assuming a stochastic model for

heterogeneous nucleation, a scaled analysis of experimental data on repeated ice nucleation

in an AgI containing supercooled water sample is made. The results indicate the presence

of at least two nucleating sites with effective contact angles of 27o and 18o. An estimate of

the sample total site areas also emerges from the analysis. The latter values are surprisingly

small — corresponding to 105 and 40 A2, respectively. This could reflect difficulties with the

19

theoretical treatment of the fraction of water molecules in contact with the site.

In summary, the scaled models offer a first step toward nucleation formalisms for T << Tc

which are nearly substance independent. It is hoped that future workers will be interested in

improving the models and will find a measure of satisfaction in nature’s simplicity. Perhaps,

too, these discussions will motivate a renewed interest in the Ω parameter — or its near

equivalent, the Eotvos constant. In spite of understanding its physical interpretation, there

appears to be no quantitative explanation as to why Ω (which is the difference between two

much larger numbers of the order of 25) turns out to be about 2. Some materials have

anomalous values of Ωvl and this should be kept in mind. [59, 60] Perhaps the future will

yield a method for calculating f(m)1/3 from related experimental data — or from calculations

on smaller systems of atoms. Finally, it is hoped that the temperature dependences in these

scaled models will prove useful to experimentalists analyzing data.

APPENDIX A

Equation (25) can be modified to include the free energy associated with the translation

of the center of mass of the cluster [18]. For simplicity replacement factors [19] are not

considered. In this case the Jo is multiplied by:·ΓkT

P

¸n0∗3/2exp(−9/4)

and the n∗ of Eq. (12) is multiplied by a small ( 3% ) correction factor:

[1 +0.75

n∗0B

−3∼ [1 + 0.75

·x

xo

¸2]−3. (55)

The resulting J ≡ J 0 is given by Eq. (6) with α = 1, I replaced by I 0:

I 0 =·4

3π1/2

¸1/339/4

·ρcρ2

¸2/3 hxox

i9/2A−7/4

·T

Tc

¸2, (56)

and Jc replaced by J0c ’where:

J 0c ≡ Pc

·Γchρc

¸(57)

20

The δo is replaced by δo(α = 1, I = I 0, Jc = J 0c).

Inclusion of the center of mass translational free energy reduces by one the power ofhPPc

iin the prefactor of J ’. This softens the temperature dependence of the prefactor and predicts

a lnScr which is more nearly linear in [TcT− 1]3/2. In addition, lnJ 0c is almost universally 86

and the prefactor for J 0 becomes less material dependent. For example, for water, ethanol,

toluene, nonane, xenon the values of ln J 0c are 86.2, 86.1, 86.7, 86.7, and 86.5, respectively.

The value for argon (84.4) is notably smaller. The resulting scaling law is:

lnS0crΩ3/2

= 0.44[TcT− 1]3/2 (58)

For J 0 >> 1 lnS ’is related to lnS0cr via Eq. (26) with Jc = Jc ’. There are, in short,

some attractive features of this modification. However, in order for Eq. (58) to agree with

experiment, the effective surface tension (and thus ) must be about 15 % larger than the

bulk values.

ACKNOWLEDGEMENTS

This work is supported in part by the National Science Foundation under Grant No.

ATM83-10854 and Grant No. ATM87-13827. We thank J. Schmitt and G. Adams for

making the details of their data available.

21

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24

constant for the ideal liquid is 2.12. For associated liquids, the constant is reduced. F.

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25

the region of interest for critical phenomena). Some liquids (such as water) show anom-

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26

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27

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28