scaled models for nucleationweb.mst.edu/~hale/papers/vienna6.pdfhence, the Ωis the effective...
TRANSCRIPT
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
REFERENCES
[1] B. N. Hale, Phys. Rev. A 33, 4156 (1986)
[2] See for example H. E. Stanley, Introduction to Phase Transitions and Critical Phenom-
ena, Oxford, New York(1971)
[3] B. N. Hale, Nucleation Symposium, 56th Colloid and Surface Science Symposium, Vir-
ginia Polytechnic Institute, Blacksburg, VA, (1982).
[4] K. Binder, J. Phys. C 4, 51 (1980).
[5] K. Binder and D. Stauffer, Advanc. Phys. 25, 343 (1976).
[6] J. S. Langer and L. A. Turski, Phys. Rev. A 8, 3230 (1973),Phys. Rev. A 22, 2189
(1980).
[7] J. S. Langer and A. J. Schwartz, Phys. Rev. A 21, 948 (1980). 8. H. Furukawa and K.
Binder, Phys. Rev. A 26, 556 (1982).
[8] K. Binder, Phys. Rev. A 25, 1699 (1982).
[9] J. D. Gunton, ”The Dynamics of First Order Phase Transitions”, in Phase Transitions
and Critical Phenomena, Ed. by C. Domb and J. L. Lebowitz, Vol. 8, Academic Press,
New York, (1983).
[10] B.J.C. Wu, P. P. Wegener, and G.D. Stein, J. Chem. Phys. 68,308 (1978).
[11] R. McGraw, J. Chem. Phys. 75, 5514 (1981).
[12] D. H. Rasmussen and S. V. Babu, Chem. Phys. Letters 108, 449(1984).
[13] D.H. Rasmussen, M. R. Appleby, G. L. Leedom, S. V. Babu, and R. J. Naumann, J.
Crystal Growth 64, 229 (1983).
[14] In 1886 Eotvos found that the quantity (M/d)2/3, whereM is the molecular weight and
d is the density, was a linear function of temperature. In cgs units, the proportionality
24
constant for the ideal liquid is 2.12. For associated liquids, the constant is reduced. F.
H. MacDougall, Physical Chemistry, Macmillan, New York, 1936, p. 96. The Eotvos
constant defined in this way is actually times kN2/3a , where Na is Avogadro’s number.
This slight correction factor is 0.984.
[15] M. F. Collins and A. M. Collins, Phys. Rev. B 35, 394 (1987); seealso references 1-6 of
this Phys. Rev. B article.
[16] R.G. Howland, N.C. Wong and C.M. Knobler, J. Chem. Phys. 73, 522 (1980).
[17] J. Lothe and G. M. Pound, J. Chem. Phys. 36, 2080 (1962).
[18] H. Reiss, J. L. Katz and E. R. Cohen, J. Chem. Phys. 48, 5553(1968). Nucleation
Phenomena, Ed. by A. C. Zettlemoyer, Elsevier, New York 1977.
[19] B. Vonnegut and M. Baldwin, J. Climate and Appl. Meteor. 23, 486(1984), See also
Ref. 21 which investigated repeated nucleation in a supercooled water sample which is
believed to contain no heterogeneous sites.
[20] M. K. Wang and B. Vonnegut, J. Rech. Atmos, 18, 23 (1984.
[21] R. Becker and W. Doring, Ann. Phys. 24, 719 (1935).
[22] N. H. Fletcher, The Physics of Rainclouds, Cambridge University, Cambridge (1969),
Chapt. 3.
[23] F. F. Abraham, Homogeneous Nucleation Theory, Academic Press,New York (1974).
[24] Y. B. Zeldovi Tc h, Acta. Physicochim. (URSS) 18, 11 (1943).
[25] The Ai2/3 -iB notation appears to have been first used by D.Turnbull and J. C. Fisher,
J. Chem. Phys. 17, 71 (1948).
[26] A linear fit to the surface tension data for liquids is quite successful over a wide range of
temperatures, including a region not too far from the critical temperature (but outside
25
the region of interest for critical phenomena). Some liquids (such as water) show anom-
alies in the slope near the freezing point. These anomalies alies are apparently related
to the properties of associated liquids.
[27] Eotvos constants for a variety of materials are given in [13]. In comparing the results
of the scaling laws in this paper with the data, the ideal value of 2.12 is used for the
majority of sub stances. Exceptions are 1.5 for water and ethanol and 2.35 for nonane,
octane and n-butylbenzene. For SF6 the value 2.0 from [13] is used. The latter values
agree approximately with σ/[[TcT− 1]kTρ2/3] for these substances for T/Tc of interest.
For example this classical formula gives 2.44 ± 0.01 for nonane, 2.35± 0.02 for octaneand n-butylbenzene, 2.17 ± 0.03 for toluene and carbon tetrachloride, 2.10 ± 0.01 forchloroform, 1.45± 0.05 for water and 1.40± 0.20 for ethanol for 0.35T/Tc0.50.
[28] B. N. Hale and R. C. Ward, J. Stat. Phys. 28, 487 (1982).
[29] F. Stillinger and A. Rahman, J. Chem. Phys. 68, 666 (1978).
[30] P. Kemper and B. N. Hale, see ” Monte Carlo Simulations of Small Water Clusters:
Effective Surface Tension” in this volume.
[31] R. H. Heist and H. Reiss, J. Chem. Phys. 59, 665 (1973).
[32] R. H. Heist, K. M. Colling and C.S. Dupuis, J. Chem. Phys. 65,382 (1976).
[33] B.J.C. Wu, P.P. Wegener and G.D. Stein, J. Chem. Phys. 69, 1776(1978).
[34] J. L. Katz and B. J. Ostermier, J. Chem. Phys. 47, 478 (1967).
[35] J. L. Katz, J. Chem. Phys. 52, 4733 (1970).
[36] J. L. Katz, C. J. Scoppa, N. G. Kumar, and P. Mirabel, J. Chem. Phys. 62, 448 (1975).
[37] J. L. Katz, P. Mirabel, C. J. Scoppa, and T. L. Virkler, J. Chem. Phys. 65, 382 (1976).
[38] R. J. Anderson, R. C. Miller, J. L. Kassner and D. E. Hagen, J. Atmos. Sciences 37,
26
2508 (1980).
[39] R. C. Miller, R. J. Anderson, J. L. Kassner and D. E. Hagen, J. Chem. Phys. 78, 3204
(1983).
[40] J. L. Schmitt, R. A. Zalabsky and G. W. Adams, J. Chem. Phys. 79, 4496 (1983).
[41] G. W. Adams, J. L. Schmitt, and R. A. Zalabsky, J. Chem. Phys. 81, 5074 (1984).
[42] P. E. Wagner and R. Strey, J. Chem. Phys. 80, 5266 (1984).
[43] J. L. Jasper, J. Chem. Ref. Data 1, 841 (1972).
[44] Selected Values of Properties of Hydrocarbons and Related Compounds (Thermodynam-
ics Research Center, Texas A & M University College Station, Texas, 1965), American
Petroleum Institute Research Project
[45] G. F. Carruth and R. Kobayashi, J. Chem. Eng. Data 18, 115 (1973).
[46] J. L. Schmitt, G. W. Adams, and R. A. Zalabsky, J. Chem. Phys. 77, 2089 (1981).
[47] N. H. Fletcher, The Chemical Physics of Ice, Cambridge University Press, Cambridge
(1970), Chapter 4.
[48] D. Turnbull and J. C. Fisher, J. Chem. Phys. 17, 71 (1949), J. Chem. Phys. 17, 429
(1949).
[49] Handbook of Chemistry and Physics, 56th Edition, CRC Press, Cleveland, (1975-76), p.
D-179.
[50] In order to derive this form from Fletcher’s Eq. 4.21, page 94 of Reference [48], the
following expression was used: Di* = [kT/h]4π[3/(4πρ1)]2/3n∗2Γ1/3ρ1 , where 393is given
in Section II.1 of this review.
[51] This form for w is estimated from the experimental data of K. T. Gillen, D. C. Douglas
and M. J. R. Hoch, J. Chem. Phys. 57 5117 (1972). See also p. 44 of Reference [53].
27
[52] C. A. Angell, Supercooled Water in Water: A Comprehensive Treatise, Ed. by Felix
Franks, Plenum, New York (1983), Table I, p. 14.
[53] D. Turnbull, J. Appl. Phys. 21, 1022 (1950).
[54] K. A. Jackson, Ind. Eng. Chem. 57, 28 (1965). Jackson estimates that the surface tension
is approximately one half the heat of fusion for metals and about one third the heat of
fusion for semi-metals, organic compounds and alkali halides.
[55] J. Q. Broughton and G. H. Gilmer, J. Chem. Phys. 64, 5759 (1986).
[56] H. J. De Nordwall and L. A. K. Staveley, J. Chem. Soc. 224, (1954); D. G. Thomas and
L. A. K. Staveley, J. Chem. Soc. 5, 4569 (1952).
[57] D. Turnbull and B. Vonnegut, Ind. Eng. Chem. 44, 1292 (1952).
[58] Some substances have anomalous values for Ω. See S. Glasstone, Physical Chemistry,
Van Nostrand, New York (1948), p. 492 for additional discussions of the Eotvos constant.
[59] C. A. Croxton, Statistical Mechanics of the Liquid Surface, Wiley, New York (1980),
pp. 158-163 also discusses the temperature derivative of the surface tension.
[60] P. V. Hobbs, Ice Physics, Clarendon, Oxford (1974), Chapt. 7.
[61] N. H. Fletcher, Aust. J. Phys. 13, 408 (1960). 63. G. Vali and E. J. Stansbury, Can. J.
Phys. 44, 477 (1966).
[62] G. Vali and E. J. Stansbury, Can. J. Phys. 44, 477 (1966)
28