Bubble Nucleation and Detachment

9
Bubble Nucleation and Detachment

STEVEN D. LUBETKIN Eli Lilly & Company, Greenfield, Indiana, U.S.A.
I. INTRODUCTION
Bubbles are a rather spectacular embodiment of the forces produced by
surface tension. Their spherical shape is a testimony to the isotropic nature of
the gas/liquid interfacial tension. Of all the forces involved in shaping and
making bubbles, surface tension is preeminent, and this dominant position is
reinforced by the appearance of the surface tension to the third power in the
exponential in the rate expression for bubble nucleation as shown in Eq. (6).
In this review, the author has chosen to emphasize a property of gas bubbles
that has been noted in the past, but whose implications have not been fully
worked out—that the surface tension is a function of the bubble size. This is
not true for cavitation or boiling in unary liquids.
Bubbles play a bigger part in our lives than is often realized. An example is
that most fundamental of processes: boiling. Boiling is nothing more than the
formation of bubbles in a liquid where the vapor pressure is above ambient
pressure. When the liquid is water, boiling is the way in which steam is formed,
and because that in turn is the first step in most electricity production, the
efficiency of the process is important. Gasoline separation from crude oil also
depends on bubble generation (distillation), so two of the key sources of en-
ergy, the pillars of modern technological society, rest firmly on bubble
generation. Bubbles play a crucial role in determining the efficiency of these
processes, and so ultimately in the cost of energy. That is the most obvious
example, but it by no means exhausts the list, because boiling or gas bubble
evolution also occur in many other situations, almost if not equally impor-
tant: Heat pumps, heat exchangers, refrigerators, and electrolysis cells are
industrially important examples. The formation of bubbles is an essential step
in the formation of many foamed plastics, and these materials play a key part
in our lives—principally as insulation foams and structural foams. If we open
Copyright n 2004 by Marcel Dekker, Inc. All Rights Reserved.

the discussion, and consider processes where bubbles play a significant part,
then a representative list becomes more extensive:
1. Efficiency of boilers (steam, refrigerators, heat exchangers).
2. ‘‘The bends’’ or bubble formation in the tissues of divers.
3. The eruption of volcanoes and geysers.
4. Steel making.
5. Bubble chambers for detection of subatomic particles.
6. How does transpiration occur in plants more than 10 m tall?
7. Ships screws and other cavitation effects, including spillways, valves, and
other hydraulic equipment.
8. Electrochemical cells and gas evolution, and the sound of electrolysis.
9. The long-range attraction between hydrophobic surfaces.
10. Bubble jet printers.
11. Beer gushing and oil well gushing.
12. Cell damage in fermentation processes.
13. Ecology of the ocean.
14. Insulation, structural, and other foams.
15. Electrostatic (spark) ignition of volatile vapors.
16. Sonoluminescence.
17. Disaster at Lake Nyos and the release of subterranean gases.
18. Cloud formation, and ultimately the global pattern of rainfall.
19. Measurement of surface tension/contact angle.
20. Strength of liquids.
21. Champagne, beer, and carbonated drinks manufacture.
22. Flotation for ore benefication.
23. ‘‘Cracking’’ knuckles.
24. Theories of cosmology.
25. Mechanical failure in nuclear fuel rods.
26. Antibumping granules.
27. Holes in Swiss cheese, bread, and other food foams.
This review is primarily directed at bubbles of gas or vapor surrounded by a
bulk liquid phase, and in what follows, the word ‘‘bubble’’ will be taken to
have that meaning unless the context implies otherwise. Bubbles can be of
three essentially different types, although only two of these are commonly
met. The two common forms are: First, the soap bubble, which consists of an
approximately spherical surfactant bilayer bounded on each side by a gas
phase. The second is a volume of gas, also approximately spherical, sur-
rounded by a liquid. The third, much less usual is the ‘‘negative bubble,’’
which occasionally appears when a drop of liquid falls through the interface
of a liquid in the presence of a surfactant species. The drop can acquire a
‘‘coat’’ of the gas phase through which it fell, and this thin gas coat is in effect a
bubble, filled with liquid. All three types can be brought into a single

definition, if instead of specifying a liquid or gas phase, we can accept as a
definition: ‘‘An approximately spherical or part spherical gas-filled cavity
partially or fully surrounded by a fluid phase.’’
This definition could also encompass various other possibilities, some of
which will be addressed here in more detail than others. These include the fact
that bubbles in water greater than a certain size (a radius r>f1 mm) will not
be spherical, and as the size gets bigger, so too does the deviation from
sphericity. Also included in this definition are bubbles attached to an
interface, be it liquid or solid or indeed that of another bubble. In each case,
the presence of a second interface will generally cause distortion from the
roughly spherical shape, or at least truncate the sphere.
The size of the bubble is significant at the small end of the range as well as at
the large. There is a theoretical absolute lower limit on the size of bubbles. For
a given set of thermodynamic conditions, the critical radius, r*, is the smallest
size of bubble that is in an unstable equilibrium with the supersaturated
solution. Bubbles of smaller size than this radius will spontaneously dissolve
or collapse. In this context, smaller means even one molecule smaller. The
gain of one molecule increases the radius past the critical value, and the larger
bubble is now more likely to grow than collapse. The loss of one molecule
from the critical bubble leaves the radius in the subcritical range where the
Laplace pressure is greater than the vapor pressure in the bubble, which thus
tends to collapse. At the critical size, the probability of collapse or growth is
approximately equal. It is a corollary of the fact that there is an excess
pressure in the bubble, that any gas- (or vapor-) filled bubble cannot
indefinitely persist unless the solution surrounding it is supersaturated. Thus
all bubbles in a saturated solution are unstable. Another way to look at this is
that r* = l for a saturated solution, so that all bubbles are below the critical
size in a saturated solution. Yet another way to view this is that the presence of
a single bubble is sufficient to supersaturate an otherwise saturated solution.
These may be important considerations when the role of bubbles [1] in the
long-range interaction between hydrophobic surfaces is discussed.
II. HOMOGENEOUS NUCLEATION

There are many treatments of nucleation processes in the literature, although
those addressing bubble nucleation are far less common than those dealing
with crystal or droplet nucleation. The so-called classical nucleation theories
(which can trace their lineage directly back to Volmer and Weber [92] and
Becker and Döring [21]) are conceptually simpler than the more modern
theories. They have also been outstandingly successful at predicting some
nucleation phenomena. For these reasons, the present review only encom-
passes this type of theory. There are a number of reviews available on bubble

nucleation, and the treatment here draws on those in Refs. 2–8, 26 and Hirth
(1969).
For a phase change to occur in a homogeneous system, a necessary con-
dition is that the system must be unstable, in the sense that the new (daughter)
phase must have a lower chemical potential l than the old (mother) phase, lV.
Referring to Fig. 1, the full line labeled ‘‘binodal’’ represents the phase
boundary, and is the vapor-pressure vs. temperature curve. In raising the
temperature, so passing from B to A, the system starts at B with a relatively
stable liquid phase, and a relatively unstable vapor phase. As it reaches the
binodal, the system is at equilibrium, with equality of chemical potential for
FIG. 1 At point ‘‘A,’’ bubbles begin to appear. Two pathways to point ‘‘A’’ are
shown. BA represents raising the temperature at constant pressure, whereas CA causes
boiling by reducing the pressure at a constant temperature. The portions of the line BA
above the full curve, or of CA to the left of the full curve until they reach ‘‘A’’ are
sometimes referred to as the ‘‘widths’’ of the metastable zone. Ostwald’s metastable
limit is the kinetic limit of stability, and is shown here as the dotted line. The spinodal
represent the thermodynamic limit of instability.

each phase: There is no drive to change state. As soon as it moves above the
full line, toward A, the vapor phase is more stable than the liquid, and bubbles
may form. The probability of this happening is a strong function of the
chemical potential difference, Dl (Dl = l lV) as will be discussed below.
Displaced from the full line by some distance, and crudely parallel to it
(Ref. 70), is the dotted curve on which the point A lies, representing the
‘‘metastable limit.’’ This line has no absolute theoretical significance. It is an
empirical limit set by experiment, and much beyond which (further toward the
line labeled ‘‘spinodal’’) is a region that is usually experimentally inaccessible,
because of the effectively instantaneous bubble nucleation that then takes
place. In practice, increasing the drive by moving further above the point A,
has little effect. At the spinodal line, there ceases to be any barrier to
nucleation—the new phase will very rapidly appear; the limiting rate of
bubble formation is set by transport of matter (diffusion) or of heat, or both,
with these terms appearing in the preexponential of the nucleation equation.
In this region, the large value of Dl ensures that the exponential term is very
close to 1. Under these conditions, the kinetics are determined by the
preexponential alone. Strictly speaking, at the spinodal, nucleation ceases,
and spontaneous phase change takes place—it is impossible for the super-
heated state to persist.
The further above or to the left of the binodal curve one goes, thus
increasing Dl, the greater the rate at which the phase change will occur. At
the binodal, the rate is zero, but as regions further above or to the left of the
coexistence curve are reached, this rate rises to a high value as the departure
from equilibrium increases. The magnitude of the departure is measured by
the chemical potential difference Dl, between the liquid and vapor states
under the prevailing conditions of temperature, T and pressure, P.
Of course, the ‘‘width’’ of the metastable zone is not a precisely defined
quantity. The concept of having ‘‘no nucleation’’ is reasonably clear (to be-
come completely clear, a time for which the observer is willing to wait to see a
nucleation event becomes important as noted by Fisher [9]). In contrast, what
constitutes an observable or worse yet, a ‘‘rapid’’ rate of nucleation is open to
discussion. Conventionally, an observable rate of homogeneous nucleation
(taking place in the bulk of the mother phase, well away from the influence of
any surface) is taken to be 1 nucleation event in 1 cubic centimeter per second
( J = 1 cm3 sec1, which corresponds to 106 m3 sec1). A rapid rate would
then perhaps be 3–4 orders of magnitude greater. For heterogeneous nucle-
ation (taking place at or on a surface), the rate is defined in terms of events per
unit area per unit time, and J has units of cm2 sec1; for this case, defining the
area is not straightforward because usually the surface on which nucleation
takes place is itself heterogeneous, chemically or topographically or both, and
the nucleation event thus takes place at preferred or active sites. The presence

of the surface has a number of potential effects, which are discussed in more
detail below.
However, the key point is that the change from essentially zero rate to a
very large rate occurs sharply, within a very small range of Dl. This is
illustrated in Fig. 2.
The sharpness experimentally appears as a limiting superheat before which
nothing happens, but beyond which very rapid boiling takes place. Calculat-
ing this rate is the province of nucleation theory, and the so-called ‘‘classical’’
theory is outlined below.
A. The Rate of Homogeneous Bubble Nucleation

The free energy required for the formation of a bubble in a single component
system is a function of the bubble size. To see why this is so, it is sufficient to
note that the surface area of the bubble is proportional to r2, and thus for each
unit of interfacial area created by the new bubble, energy has to be expended
in proportion to that surface area, and in proportion to the surface tension, c:
DGarea ¼ 4kr2 c
FIG. 2 The nucleation rate is indistinguishable from the X axis for most of the
range. In a narrow interval of supersaturation, r, it dramatically rises to very large
values. It is this sudden change from essentially zero to a large rate that gives rise to the
appearance of a kinetic ‘‘metastable limit.’’ The metastable limit is crossed somewhere
between the two arrows.

If the system is superheated and is thus is above the full line (the binodal) in
Fig. 1, then the vapor phase is of lower chemical potential than the liquid.
Consider a bubble exposed to an applied, external (hydrostatic) pressure P V,
and having an internal pressure, p. The decrease in free energy is proportional
to the volume of the new phase produced, and is thus proportional to r3:
4
DGvolume ¼ kr3 ð p P VÞ
3
The total free energy change for the formation of this bubble is then given by:
4
DGtotal ¼ DGarea þ DGvolume ¼ 4kr2 c kr3 ð p P VÞ ð1Þ
3
Regardless of the details of the functional form of the DG terms, it can be seen
that the total free energy must go through a maximum as r increases, because
the positive term DGarea grows as r2, and thus dominates the total at low r,
while the negative term DGvolume grows as r3, and thus dominates as r gets
larger. The maximum occurs at some characteristic value of the size, r*, which
strongly depends on the distance (Dl) the system has moved to the left or
above the equilibrium full curve. The position of the maximum can be
evaluated as shown below. The graph of DGtotal as a function of the bubble
size is shown in Fig. 3.
The maximum in the curve is identified as the height of the kinetic barrier,
DGtotal
* , which has to be surmounted for the phase change to take place. The
free energy increase on the path to the summit is in effect the ‘‘activation
energy’’ for the ‘‘reaction’’ leading to the appearance of the new bubble. By
analogy with the conventional Arrhenius expression for the rate of a reaction,
we might write for J, the rate of nucleation of bubbles:

DGtotal
*
J ¼ C exp ð2Þ
kT
The formal similarity to the conventional Arrhenius equation is deceptive;

in the Arrhenius equation, Ea is essentially a constant, or at worst a weak
function of the chief experimental variable, T:

Ea
R ¼ A exp
kT
but the situation with DGtotal

* is different. It depends very strongly on the
experimental variable, ( p PV), as we will see below. The preexponential, C in
common with the A in the Arrhenius rate expression, represents an encounter
frequency. The details will be discussed below, but for the present discussion,
it can be treated as approximately constant.

FIG. 3 In this diagram, the Y axis represents DGtotal, and the X axis is the radius of
the bubble, in units of r*, the critical bubble size. The height of the curve at the point
where x = 1, or r = r* is the height of the kinetic barrier the system must overcome to
undergo the phase change, DG*. The curve crosses the X axis at the point r = 1.5r*,
and this is where the surface term exactly equals the volume term, and DGtotal = 0.
The position of the maximum in the free energy curve can be obtained by
differentiating expression (1) with respect to the radius
BDGtotal 12
¼ 0 ¼ 8krc kr2 ð p P VÞ
Br 3
or:
BDGtotal
¼ 0 ¼ 2c rð p P VÞ
Br
therefore
2c
r* ¼ ð3Þ
ð p* P VÞ

* 2 r
DGtotal ¼ 4kr c 1
2
3 r*
when r = r*: DG*total = 4pr*2c[1 (2/3)] = (4/3)pr*2c

4kr*2 c
J ¼ C exp ð4Þ
3kT

Now
2c
r*c ðFig: 4Þ ð5Þ
rP V
Because
2c 2c 2c 2c
r* ¼ c ¼ ¼
ð p* P VÞ p P V P Vða 1Þ rP V
So
4 4c2
DG*total ¼ kc
3 ðrP VÞ2
or
16kc3
DG*total ¼
3ðrP VÞ2
Because

DG*total
J ¼ C exp
kT
Substituting for DGtotal
* :
" #
16kc3
J ¼ C exp ð6Þ
3kTðrP VÞ2
FIG. 4 Log plot of the radius of the critical bubble as a function of the super-
saturation, r. The size increases very rapidly as the supersaturation decreases.

Note that (rP V) is an approximation for the full expression accounting for
nonidealities, etc.

gpl cP V
þ PV
rV cl r S
This is more fully discussed later.
B. Bubble Nucleation by Raising the Temperature

(Boiling)
The Clausius–Clapeyron equation gives the slope of the ( P,T) curve at any
point:
dP Hf Hf qV qL
¼ ¼
dT TðrV rL Þ TðqL qV Þ
Given that qV = MP/RT (assuming ideal gas behavior), and that qV/qL << 1,
integrating gives:
Pl Hf M T L Tsat
ln ¼
PL R T L Tsat
Using the Kelvin equation at the critical size, r*:
2cTsat
T L Tsat ¼ V
q DHf r*

4kr*2 c
J ¼ CVexp
3
Taking the conventional detectable rate of nucleation to be J = 1 cm3 sec1,
and substituting in the expression for J
4kr*2 c
lnð1Þ ¼ 0 ¼ lnðCVÞ þ
3
4kr*2 c
lnðCVÞ ¼
3
3lnðCVÞ
¼ r*2
4kc
1
3lnðCVÞ 2
r* ¼
4kc
1
2cTsat 4kc 2
T Tsat ¼ r
L
q DHf 3lnðCVÞ
12
T L Tsat 1 16kc3
¼ r ð7Þ
Tsat q DHf 3kTL lnðCVÞ

The calculated superheat using Eq. (7) is compared with the measured
values of Kenrick et al. [10] for various simple liquids in Table 1A. Tsat, TL are
the temperatures corresponding to the saturation pressure, and the actual
temperature in the liquid, respectively. A typical early experiment of Kenrick
et al. [10] involved immersing a U tube, open to the atmosphere, and partially
filled with the freshly distilled liquid to be studied, in a thermostatic bath that
had been set to the test temperature. The U tube was immersed and given
sufficient time (5 sec) for the tube and liquid to thermally equilibrate. If boiling
did not take place within that time, the thermostatic bath temperature was
raised, and the experiment was repeated, until a temperature (TL) was
established at which boiling did take place within the allotted time. The
general agreement of the theory with the data is remarkably good, given the
simplicity of the experimental methods.
More sophisticated and more accurate methods for examining the boiling
behavior of pure liquids have been developed. Subdividing the liquid phase
into many small droplets reduces the probability of any given drop containing
a nucleation catalyzing surface. As a procedure, it also has the great ad-
vantage of removing the surface of the container as a potential site for hete-
ronucleation. Trefethen [11] appears to have been the first to use this method,
although Turnbull [12] had earlier used the same technique to look at nuc-
TABLE 1A Calculated and Early Experimentally Measured Values of the

Superheat Required for Boiling
Experimental boiling Calculated Theory/measured

Liquid temperature (K) from Eq. (7) ratio
Water 543 539 0.99

Methanol 453 435 0.90
Ethanol 474 444 0.85
Diethyl ether 416 400 0.89
Benzene 480 476 0.98
Chlorobenzene 523 534 1.04
n-Pentane 419 419.5 1.00
n-Hexane 455 453.3 0.99
n-Heptane 484 485.8 1.01
i-Pentane 411 410.1 0.99
Cyclopentane 453 451 0.99
Methylcyclopentane 473 470 0.99
Cyclohexane 489 492.3 1.02
Water 543 539 0.99
Mean F SD 0.97 F 0.05

TABLE 1B Calculated and More Recently Measured Values of the Superheat
Required for Boiling
Measured Calculated
Boiling point superheat superheat Theory/measured
Substance (1 bar) limit limit ratio
Methane 161.5 107.5

Ethane 88.6 4 3.5 0.88
Flourethene 72.2 16.9
Sulfur dioxide 10. 50.
Propene 47.7 52.4 50.3 0.96
Propane 42.1 53.0 55.3 1.04
1,1-Difluoroethane 24.7 70.4
Propadiene 34.5 73.
Cyclopropane 32.9 77.5
Propyne 23.2 83.6 88.2 1.06
2-Methylpropane 11.8 87.8 87.7 1.00
Chloromethane 24.2 93.0
2-Methylpropene 6.9 96.4 99.3 1.03
1-Butene 6.3 97.8 100.2 1.02
Chloroethene 13.9 100.9
1,3-Butadiene 4.4 104.1
Butane 0.5 105. 105.2 1.00
Trans 2-butene 0.9 106.5
Perfluoropentane 27.0 108.3 108.9 1.01
Cis 2-butene 3.7 112.2
2,2-Dimethylpropane 9.5 113.4
Ethyl chloride 12.3 126.
Perfluorohexane 50.9 136.6 137.4 1.01
2-Methylbutane 27.9 139
1-Pentene 30.0 144. 141.9 0.99
Diethyl ether 34.5 147. 145. 0.99
Pentane 36.0 147.8 148.3 1.00
Perfluoroheptane 70.9 161.6 161.2 1.00
Carbon disulfide 46.3 168.
Chloroform 61.7 173.
2,3-Dimethylbutane 58.0 173.2 175.9 1.02
Acetone 56.2 174.
Cyclopentene 44.2 173.2
Perfluorooctane 94.8 183.8 185.2 1.01
Cyclopentane 49.3 183.8 173.2 0.94
Hexane 68.7 184. 184.3 1.00
Methanol 65.0 186.0 186.5 1.00
Ethanol 78.5 189.5 191.8 1.01
1-Hexyne 71.3 192.

TABLE 1B Continued
Measured Calculated
Boiling point superheat superheat Theory/measured
Substance (1 bar) limit limit ratio
Hexafluorobenzene 74.5 194.7 195.4 1.00

Methylcyclopentane 71.8 202.9
Perfluorononane 114.5 205.3 205.7 1.00
Heptane 98.0 214. 214.5 1.00
2,2,4-Trimethylpentane 99.2 215.3 214.9 1.00
Cyclohexane 80.7 219.6 216.3 0.98
Perfluorodecane 133.0 223.9 223.1 1.00
Benzene 80.1 225.3
1,3-Dimethylbenzene 139.1 235.
1-Octene 121.3 237.1
Methylcyclohexane 100.9 237.2 232.0 0.98
Octane 125.7 239.8 242.7 1.01
Chlorobenzene 132 250.
Bromobenzene 156. 261.
Aniline 184.1 262.
Nonane 150.8 265.3 262.0 0.99
Decane 174.1 285.1 282.8 0.99
Cyclooctane 148.5 287.5
Mean F SD 0.997 F 0.03
Source: Refs. 3 and 29.
leation of the solid phase from liquid mercury. Trefethen’s methods were
improved by Wakeshima and Takata [13]. Skripov and Sinitsyn [14] essen-
tially used the same method to look at boiling in the presence or absence of
ionizing radiation. Apfel [15] adapted the method by using a standing acoustic
wave to halt the rise of the drop at a chosen level, and thus to more accurately
control the degree of superheating. Finally, Skripov and Pavlov [16] used
pulse heating methods, where very rapidly heated surfaces, typically platinum
wires (with heating rates in excess of 106K sec1) result in explosive boiling.
Using resistance thermometry allows the temperature at which boiling takes
place to be accurately defined.
C. Bubble Nucleation by Lowering the Pressure

(Cavitation)
Essentially the same theory as above can be applied with no modification to
the case of cavitation. The experimental variable is now the applied (hydro-

static) pressure, not the temperature. Therefore a more convenient form of
Eq. (7) is sought, expressed in terms of ( p* P). With very minor mod-
ification in the preexponential factor, [C in Eq. (6) is changed to C VV] the same
equation can be used:
" #
16kc3
J ¼ C VVexp ð8Þ
3kTðrP VÞ2
Consulting Fig. 1, it can be seen that in traversing the path CA, the destination
(A) is the same as before, while the route has changed. It is of course well
known that reducing the applied pressure (e.g., atmospheric pressure) results
in the lowering of the boiling point of water. At sufficient low applied pres-
sure, the water would boil at room temperature, thus reaching the line
representing the kinetic superheat limit, though not now at the point A, but
further toward the Y axis. Thus boiling and cavitation are part of a continuum
of behavior, and it is not surprising that they share closely similar mathe-
matical descriptions.
Experimentally, applying negative pressures to liquids to cause cavity for-
mation (cavitation) is not simple. Reducing the pressure in the vapor space
above a liquid using a vacuum pump is ineffectual in causing cavitation in
low vapor pressure liquids. Means of applying substantial negative pressures
are needed. Early experiments, e.g., those of Meyer [17] in this area were
performed by filling and sealing glass tubes at high temperatures, and then
cooling them in a controlled fashion. The differential shrinkage rate on
cooling of the glass and the liquid resulted in negative pressure (tension) be-
ing applied. Knowing the temperature at which cavitation occurred, knowl-
edge of the coefficients of expansion of the glass and the liquid allowed a
calculation of the tension. Other methods for putting liquids under known
tension include the use of metal bellows [18], centrifugal force applied to
tubes containing the experimental liquid [19], and acoustic methods (see, e.g.,
Ref. 20).
Using the centrifugal method, Briggs obtained the data in Table 2,
consisting of experimental fracture tensions for various liquids. These values
are compared with the theoretical calculation based on the full Eq. (10).
The predictions are generally in reasonable agreement with experiment,
although mercury appears to be an exception, and was not used to calculate
the ratio, average, or standard deviation.
The good agreement of theory and experiment in the case of boiling, as seen
by inspection of Tables 1A and 1B, contrasts with the somewhat worse
agreement for cavitation at room temperature (Table 2) and the generally very
poor agreement for gas bubble formation in liquids, the data for which are
given in Tables 6–8, and are discussed below. This may be a reflection of the

TABLE 2 Calculated and Theoretical Fracture Tensions for Various Liquids at 20jC
Theoretical Measured
fracture fracture Theory/measured
Liquid tension/atmosphere tension/atmosphere ratio
Water 1380 270 5.11

Chloroform 318 290 1.10
Benzene 352 150 2.35
Acetic acid 325 288 1.13
Aniline 625 280 2.23
Carbon tetrachloride 315 275 1.15
Mercury 23100 425 54.35
Mean F SD 2.18 F 1.55
Source: Ref. 57.
fact that boiling is an effective means of reducing the dissolved gas content of
liquids, while cavitation is not equally so. However, most cavitation experi-
ments start with careful degassing and distillation (which of course involves
boiling) of the test liquid. It may also be relevant that many cavitation
experiments are focused on the first nucleation event, rather than on
generating massive numbers of bubble nuclei. The effects of gas on bubble
nucleation are discussed below. However, it is unlikely that dissolved gas
accounts for the very poor concordance for mercury’s experimental and
theoretical fracture tensions.
D. A Closer Look at the Preexponential

Up to this point, the preexponential has been taken to be a constant,
independent of the temperature and pressure. An empirical justification for
this is that the nucleation rate depends on the exponential of the free energy of
formation of the critical nucleus, and that this strong dependence effectively
swamps any variation in the preexponential. An illustration of the relative
importance of the two terms is shown in Fig. 5.
Here values typical of boiling have been used, and a change of 5% in the
exponent gives the same order of magnitude change in J as does a 1000%
change in the preexponential. In terms of the sudden onset of nucleation
experimentally observed, the exponential is dominant. Generally speaking, it
is safe to ignore the variability of C. This does not mean that it must be so;
where the surface tension is very low, or very close to the critical point, or
where the spinodal is closely approached, then the kinetics may be dominated
by the preexponential.

FIG. 5 Composite plot of the values of the exponential and preexponential terms
needed to cause a change of 1000% (from 0.2 to 2) in the rate of nucleation, J, which
is plotted along the X axis. The left-hand Y axis is the preexponential term, which
changes from 11025 to 11026—a change of 1000%. The right hand Y axis (the
exponential term) changes from about 56.9 to 59.2—about 5%.
C can be broken down into two components: a concentration term relating

to the total number, n, of possible sites at which a monomer (atom, molecule)
can join the growing cluster, and a frequency factor, f, relating to the prob-
ability of a successful incorporation:
C ¼ nf
The number of molecules per unit volume in the liquid state is denoted by n,
and the frequency factor, f, can be estimated from the theory of absolute rates.

kT ED
f¼ exp
h kT
Thus we obtain for the complete expression for the rate of nucleation:
0 1
4
E þ kr* 2
c
kT B D
3 C
J¼n exp@ A
h kT

by rewriting this equation, we can emphasize the structure of the preexpo-
nential

kT ED 4kr*2 c
J¼n exp exp ð9Þ
h kT 3kT
The quantity exp (ED/kT), where ED is the activation energy for diffusion
through the liquid, emphasizes the importance of the diffusion in the liquid
state. Volmer [2] derived a very similar rate expression, where the activation
energy was related to the evaporation of a molecule rather than its diffusion in
the liquid state:
0 1
1=2 4
H r þ kr* 2
c
6c B 3 C
J¼n exp@ A
m ð 3 bÞ kT
or:
1=2
6c Hr 4kr*2 c
J¼n exp exp ð10Þ
m ð 3 bÞ kT 3kT
here b is ( p* P)/p*. For the situation where b=3, Eq. (10) does not hold.
The discontinuity arises as a result of an approximation used in the deriva-
tion. The interested reader is referred to the original reference for details.
E. Zeldovich Factor
The theories represented by the Eqs. (9) and (10) are based on the assumption
that critical size nuclei are built up by a sequence of additions of monomers,
with each step in the process being in a quasi equilibrium with the step before
(and after):
A þ A f A2
A þ A2 f A3
A þ A3 f A4
ð11Þ
::::::::::::::::::::
A þ Ai1 f A*
A þ A* ! Aiþ1
The critical nucleus A* contains i* monomer units. At this critical size, the
capture of a single monomer unit causes the critical nucleus to become free
growing, descending the curve to the right of the maximum in Fig. 3. To be
able to treat this distribution as quasi steady state, a conceptual procedure has
to be imagined where any nucleus that gets larger than the critical size is
returned to the mother phase as monomer units. The distribution can then be

approximated as being at a steady state. It was the avoidance of this
approximation that prompted Becker and Döring [21] and Zeldovich [22]
to solve the set of equations making allowance for the reduction below the
steady state concentration due to both promotion above the critical size, and
by the rate of the reverse reaction leading to the critical nucleus. The solution
introduces a ‘‘nonequilibrium factor,’’ Z, usually having an order of magni-
tude of about 102, and given by:

DG*total
Z¼
3kkTi*2
A further correction makes allowance for thermal nonaccommodation.
The condensation coefficient, b, is generally close to unity for bulk phases, but
there are exceptions, as noted by Hirth and Pound [5]. For the case of
monomer units impinging on an isolated subcritical nucleus entirely in the
vapor phase, and consisting of relatively few molecules, disposing of the ex-
cess thermal energy becomes an issue. This accounts for a reduction in the
value of b significantly below one. However, for the case of boiling, thermal
accommodation is most unlikely to be an issue, bearing in mind that the
mother phase is here the condensed phase, and all subcritical bubbles are
expected to be fully thermally accommodated with their surrounding mother
phase. Taking these factors into account, the full expression for the rate of
nucleation becomes:
1=2
6c Hr 4kr*2 c
J ¼ bZn exp exp ð12Þ
m ð 3 bÞ kT 3kT
F. Non-Steady-State Nucleation
Imagine a uniform liquid system at equilibrium. Suddenly, the conditions are
changed so that now the system is superheated. The system does not instant-
aneously arrive at the steady state represented by Eq. (11). The equilibria take
a finite amount of time to develop, and during this time, the nucleation rate is
below the calculated steady state rate.
This delay or time lag, s, can be very considerable. Dunning and Shipman
[23] found a time lag of about 100 hr for concentrated sucrose solutions. For
typical gaseous mother phases, s is usually of the order of microseconds. For
boiling liquids, s is likely to be of the order of tens of milliseconds, but this
depends on the viscosity. For foam formation in molten polymers (this is the
basis of the manufacture of insulation foams), these time lags can become
comparable with or exceed the processing time, and hence may become the
dominant kinetic step in bubble nucleation.

Kantrowitz [24] showed that the rate of nucleation as a function of time,
J(t) was related to the steady state rate, J0 by:
s
JðtÞ ¼ J0 exp
t
with
i*2
s¼
g
Here g is the rate of growth of the critical nucleus, of i* monomer units.
Typical plots are shown in Fig. 6.
G. Two Component (Gas/Liquid) Systems

The discussion so far has centered on the behavior of pure liquids when the
liquid phase was destabilized by changing the external conditions. In this
section, in addition to the liquid, a second component is considered. One of
the commonest and also most interesting cases is where the second component
is a gas. A number of possibilities arise, including the case where the gas forms
an essentially ideal but rather weak solution. In this case, the gas provides a
FIG. 6 Plots of the non-steady-state nucleation rate for increasing sizes of critical
nucleus. The time lag (s) before any appreciable nucleation takes place is indicated on
the plot for the case of the n3 nucleus. On occasion, this lag can be a substantial
proportion of the overall delay before nucleation—for concentrated sucrose crystal-
lization, it has been estimated at 100 hr. Even when the time lag has elapsed, there may
be further significant delays as the rate of time-dependent nucleation, J(t), gradually
builds up to the steady state value, J0.

contribution to the pressure in the critical bubble, thus allowing it to be
smaller than would otherwise be the case. We have already seen that it is easier
to produce a small critical nucleus than a large one, so the conclusion here is
that the presence of the gas acts to reduce the thermodynamic drive needed to
produce the phase change. If the gas/liquid solution is not ideal in either the
vapor phase or in the liquid phase, then even at low levels, the effects of the
dissolved gas can be enhanced. We will discuss the theory in more detail
below.
The whole new field that opens up is where the gas is present in relatively
large amounts, and now the nucleation and phase change represent not the
ebullition of the liquid, which in this case mainly acts as a carrier solvent for
the dissolved gas, but the appearance of bubbles of the previously dissolved
gas itself, with a lesser contribution from the vapor pressure of the solvent. Of
course, there is a continuum of behavior from pure boiling in a single com-
ponent system consisting of the liquid, through the bubble evolution of es-
sentially the pure gas from solution—and this continuity is reflected in the
mathematical description of the process, with essentially one equation des-
cribing both extremes.
H. Dilute Gas Solution

The point of departure for this discussion is a dilute solution of a gas in a
liquid, where the solution and vapor phases are ideal. In this case, the gas
provides some additional pressure in the growing bubble, and this in turn has
the effect of reducing the size of the critical nucleus. This can be seen by
examining the denominator in Eq. (6)
" # " #
16kc3 16kc3
J ¼ C exp fC exp
3kTð p* P VÞ2 3kTðrP VÞ2
(rPV) is an approximation to the pressure in the critical bubble. We will see
that the approximation has four parts: (1) that the vapor and the dissolved gas
are ideal in both vapor phase in the bubble and in the solution, (2) that the
vapor pressure in the critical nucleus is unaffected by the applied pressure,
p* + P ,V (3) that the surface tension is independent of the pressure, and (4) that
p*faP V. We now deal with these approximations. The presence of gas
dissolved in the solvent, and hence having some characteristic partial pressure
in the nucleus, supports a part of the compression due to the surface tension.
Therefore the simplest possible model [25] gives the nucleation rate in the
presence of a dissolved gas as:
" #
16kc3
J ¼ CVVVexp ð13Þ
3kTð p* þ p2 PVÞ2

Here the pressure of the dissolved gas in the nucleus is p2. This extra pressure
results in the critical nucleus being smaller than without the gas being present,
and this in turn implies that the nucleation is easier. For the single component
system, we have seen that the radius of the critical nucleus is given by Eq. (3):
2c
r* ¼
ð p* P VÞ
The addition of the gas pressure in the nucleus gives the expression:
2c
r* ¼
ð p* þ p2 P VÞ
Because p2 is always positive, r* for the second case is always smaller than for
the first. Our assumption of a weak solution equates to the statement that p*
and p2 are of comparable magnitude.
Take now the case where allowance is made for the effect of applied
pressure on the vapor pressure of the solvent. The calculation depends on the
Kelvin equation, and expresses the relationship between the saturation vapor
pressure, pl and the actual vapor pressure under the applied hydrostatic
pressure, PV. The factor g is given by Ward et al. [4]:

VðPV pl Þ cV2
g ¼ exp
kT cV1
p2 ¼ gpl
This gives an expression for the radius of the critical nucleus, r* in terms of the
concentration of the dissolved gas, c, and the equilibrium concentration, cl:
2c
r* ¼ ð14Þ
cP
gpl þ P
cl
The definition of the saturation ratio, a, is that a = c/cl and because kHp = c,
combining these two definitions and using the expression for the supersatu-
ration, r = a 1 allows for some simplification:
2c 2c
r* ¼ ¼
ðgpl þ aP V P VÞ ðgpl þ rP VÞ
Comparing with Eq. (5), it is immediately apparent that r* is smaller than
before, and that nucleation will therefore be easier. For relatively involatile
solvents, with gases dissolved in them, g is close to 1, or somewhat less than 1;
also, pl is relatively small. It is on this basis that the approximation used in
Eq. (6) is justified.

The nonideality of the gas mixture in the nucleus, and of the solution, can
be accounted for by modifying Eq. (14):
2c
r* ¼ ð15Þ
gpl cP V
þ PV
mV cl mS
Here mV and mS are the activity coefficients in the vapor and in the solution,
respectively. For a full derivation, the interested reader is referred to Ref. 4.
Finally, the surface tension is generally a function of the dissolved gas
pressure, in accordance with an isotherm such as the Gibbs adsorption
isotherm. This was pointed out by Hirth and Pound [5] and Hirth et al. [26].
It is a known, but somewhat obscure, fact that gases can act as surfactants
(Refs. 27, 28, 79–87, 89–91). The more soluble gases appear to be more
effective than the less soluble. The mechanism of action of gases as surfactants
is not clear. The classical picture of a surfactant as a molecule with hydrophilic
and hydrophobic parts is clearly inapplicable to most gases, and completely
inappropriate for gaseous elements, whether atomic or molecular. Nonethe-
less, the positive adsorption of these species is a well-established experimental
fact, as is the resulting lowering of the interfacial tension, as reported for
example by Lubetkin and Akhtar [28]. Thus dissolved gases act to lower the
surface tension, and the extent of lowering depends upon the adsorbed
amount.
I. Higher Gas Concentrations

The reduction in surface tension is highly significant for the rate of nucleation
of bubbles, because the surface tension appears to the third power in the
exponent in the expression for the nucleation rate, Eq. (6). With dilute gas
solutions, although this effect is important, it may not be dominant. However,
as the gas concentration increases, so the importance of the surface tension
term is increased.
Based on the information in Table 3, the size of the first coefficient, b, of the
reduction in surface tension with pressure strongly depends on the nature of
the gas, ranging from helium at one extreme, to n-C4H10 at the other. The
common atmospheric gases N2 and O2 have relatively modest coefficients,
with the surface tension falling about 0.1 mN m1 atm1, whereas for n-
C4H10, the corresponding reduction is nearly 3 mN m1 atm1. Similar data
has been reported for ethene in cyclohexane by Lubetkin and Akhtar [28],
where the coefficient b was about 1.7 mN m1 atm1.
At low pressures, the surface tension decreases in a roughly linear fashion,
so that to a reasonable approximation, the surface tension can be represented
by the equation c = c0 + (dc/dp)p, and (dc/dp) = b, where b is the first

coefficient shown in Table 3, and c0 is the surface tension against air at
atmospheric pressure. For helium, the available data suggests that b is very
close to zero, but probably a negative quantity. The small size of the effect for
helium makes it difficult statistically to distinguish b from zero. For all other
gases so far examined, b is a negative quantity, showing that the gases are
positively adsorbed, and reduce the surface tension. At the very least, this
reduction in surface tension will result in a significant reduction in the size of
the critical nucleus, as seen from Eq. (3) and its variants. That equation may
be written:

2 c0 þ dp
dc
p* 2ðc0 þ bp*Þ 2c 2ba
r* ¼ ¼ ¼ þ
ðp* P VÞ ðp* P VÞ rP V r
The coefficient b is negative, thus the second term represents a reduction in the
size of the classical critical nucleus (which is given by the first term). Note that
because the dissolved gas is the main contributor to the pressure in both
subcritical and critical size bubbles, the necessary precondition for the
lowering of the surface tension (the presence of a substantial pressure of
surface active gas) is fulfilled. The assumption that the gas concentration is
large is equivalent to the statement that the denominator in the nucleation
equation, while retaining the same form as before, now represents the
dominance of the dissolved gas in the critical nucleus. Neglecting the term
TABLE 3 Variation in Surface Tension of Water with Dissolved Gas

Content, Expressed as c = c0 + bp + cp2 + dp3, with p in Bar
Gas b c d
He 0.0000 — —
H2 0.0250 — —
O2 0.0779 +0.000104 —
N2 0.0835 +0.000194 —
Ar 0.0840 +0.000194 —
CO 0.1041 +0.000239 —
CH4 0.1547 +0.000456 —
C2H4 0.6353 +0.00316 —
C2H6 0.4376 0.00157 —
C3H8 0.9681 0.0589 —
N2O 0.6231 +0.00287 0.000040
CO2 0.7789 +0.00543 0.000042
n-C4H10 2.335 0.591 —
Source: Ref. 64.

in gpl in comparison with rPV becomes a better approximation as the
proportion of gas increases.
At the critical size, the bubble is in (unstable) equilibrium, with the pressure
in the critical bubble given by p*. Therefore the reduction in the size of the
critical nucleus could be very significant and approximately may be given by:
2ðc0 þ baP VÞ 2c0 2bðr þ 1Þ
r* c ¼ þ
rP V rP V r
Below the critical size, the pressure in the incipient bubble is p, which is greater
than p* or rPV. Thus below the critical size, the adsorption of the gas and the
corresponding reduction in surface tension (and presumably, the increased
growth rate to the supercritical size) could be even greater than for the case of
the critical nucleus. The extent of the reduction in surface tension is dependent
on the kinetics of adsorption as balanced by the rate of collapse or growth of
the subcritical bubbles. These kinetic details are not known at present. It
seems very likely that this reduction in surface tension is implicated in the very
poor agreement between theory and experiment in the case of gas bubble
nucleation. The b coefficient for He gas is very close to zero. If He is used in a
bubble nucleation experiment, is it anticipated that the predictions of classical
nucleation would be verified? Not entirely, because it is not only the
adsorption of the gas reducing the surface tension that matters. The pressure
of gas in the critical bubble will reduce its size, and will also affect the vapor
pressure of the liquid, and corrections for this can be applied. These aspects
are discussed in more detail below. Nonetheless, it is not surprising to see He
at the top of the list of supersaturations experimentally measured (see Table
7), with a ratio theory/measured close to that measured for water cavitation,
Table 2, and noticeably smaller than for all other gases. In other words, He is
not acting as a surfactant, so that the nucleation process is similar to that
found in unary bubble nucleation systems.
III. HETERONUCLEATION
Up to this point, it has been assumed that the nucleation, whether caused by
cavitation, boiling, or dissolved gas evolution, was taking place in the bulk of
the mother phase, without the influence of any heterosurface. Generally
speaking, the presence of such a surface makes nucleation easier, so that
unless special precautions are taken, nucleation will preferentially take place
at surfaces. However, the number of sites upon which such heteronucleation
may take place is limited, whereas the number of sites for homonucleation is
essentially unlimited. Thus as the supersaturation increases, homogeneous
nucleation may come to dominate the rate of phase change, although
heteronucleation is simultaneously taking place. This does not contradict

the very useful generalization that most nucleation is indeed heteronuclea-
tion. Even when careful precautions have been taken, it is nearly impossible to
eliminate all possibility of hetero effects. When a surface is involved in the
nucleation step, the texture of the surface itself may be relevant. With very few
exceptions, solid surfaces have sites of differing adsorption energies or them,
very often associated with chemical inhomogeneities. In addition, topograph-
ical features including steps, pits, and scratches are of significance. For this
reason, our discussion starts with idealized smooth solid surfaces and liquid
surfaces.
A. Heteronucleation at a Smooth Solid Surface

Different surfaces have different catalytic potencies for nucleation, and of
course, the catalytic efficacy depends on the nature of the liquid, the gas, and
the surface. Generally speaking, and considering only atomically smooth,
clean solid surfaces, the greater the contact angle of the liquid on the solid, the
better the surface is in catalyzing the bubble nucleation. This is the opposite of
the case for the formation of liquid droplets from a supersaturated vapor,
where a well-wetted solid surface can encourage condensation at or at least
very close to the binodal, or put another way, at very low supersaturations. In
pores of suitable size, condensation can take place below a (= p/p0)<1. This
is called capillary condensation. A counterpart would be bubbles with an
overall negative curvature, and such bubbles may have longer lifetimes. This
may well be relevant to the experimental observation of apparently kinetically
stable bubbles attached to hydrophobic surfaces, and which are implicated in
long-range attractive forces between such surfaces, as discussed by Christen-
son and Claesson [1].
Recall that the contact angle is conventionally measured through the liquid
(condensed) phase. Fig. 7 shows the important conceptual point that the cap-
shaped bubble is part spherical, and that / is the fraction that the cap
represents of the volume of the whole sphere. Taking the sphere to represent
the volume of the critical nucleus under these conditions, if the nucleation is
homogeneous (or if the contact angle is zero, which amounts to the same
thing), then of course the critical nucleus is the whole sphere. As the surface
becomes a better and better catalyst, or in other words, as the contact angle
gets larger, the fraction, /, represented by the cap, of the volume of the whole
sphere shrinks. We have already seen that a decrease in size of the critical
nucleus corresponds to an increase in the rate of nucleation. Thus the factor /
is a multiplier that appears in the expression for the rate of nucleation, in the
exponential, as seen by inspection of Eq. (17).
Geometrically, it is rather easy to see that / is the ratio of the volumes of
the cap-shaped nucleus to that of the whole sphere (Fig. 8).

FIG. 7 The relative volume / of the spherical bubble cap and the total sphere
changes as the contact angle changes, and becomes rapidly smaller as the contact
angle increases above 90j. The work of formation of the nucleus correspondingly
decreases, so that bubble nucleation becomes easier as the contact angle gets larger.
The volume of the whole sphere is given by:

4 3
Vtotal ¼ kr
3
The volume of the spherical cap ABDC is:
kh2
Vcap ¼ ð3r hÞ
3
Vcap h2
¼ 3 ð3r hÞ
Vtotal 4r
r
sinðh 90Þ ¼
rh
Eliminating h gives

Vcap 2 þ 3 cos h cos3 h
¼
Vtotal 4

Vcap 2 þ 3 cos h cos3 h
/ðhÞ ¼ ¼ ð16Þ
Vtotal 4

FIG. 8 The geometry of the spherical cap ACDB, showing the cap height, h. It
is easy to show that h = R(1+cos h), and from this to deduce that Vcap/Vtotal = / =
(2 + 3 cos h cos3 h)/4. For liquid drops nucleating on a surface, Vcap/Vtotal = / =
(2 3 cos h + cos3 h)/4.
The simplified expression for the nucleation rate for a bubble on a flat, smooth
solid surface, upon which the contact angle is h, is given by:
" #
16kc3 /ðhÞ
J ¼ CVVVVexp ð17Þ
3kTðrP VÞ2
For more details, see Ref. 29. The preexponential, CVVVV is modified from the
previous values [see, e.g., Eq. (8)] in two significant respects. First, the value of
n (the number of molecular positions in the liquid per unit volume) is mod-
ified. For homogeneous nucleation, n is approximately (1/X), where X is the
molecular volume. For heterogeneous nucleation, the number of possible
sites is greatly reduced, and because a surface rather than a volume is
involved, the dimensionality has to be reduced. A useful approximation is
to take n2/3, which is approximately the number of molecular positions per
unit area of heterosurface. This change is reflected in the fact that the unit of J
in this case is cm2 sec1, rather than the cm3 sec1 for Eq. (8). The second is
the appearance of the factors 2s = [1 + cos(h)], and /, defined by Eq. (16),
both of which involve the contact angle of the bubble on the nucleating

surface. Allowance can also be made for the Zeldovich nonequilibrium factor,
as before.
1=2
6c Hr 4kr*2 c
J ¼ n2=3 s exp exp
m/ð3 bÞ kT 3kT
Because Fig. 7 could equally be used to describe a cap-shaped liquid drop
nucleus on a solid surface, the same type of analysis will apply to the case of
heterogeneous nucleation of a liquid from a supersaturated vapor phase. The
only difference is that now the contact angle (as always, measured through the
liquid) is equal to the complement (= 180 h) of that for a bubble, and thus
the expression for the fraction of the whole sphere has its signs reversed:

2 3 cos h þ cos3 h
Vcap ¼ Vtotal
4
The plots for /(h) for both drops (circles) and bubbles (squares) are shown in
Fig. 9.
Because, in principle, /(h) can be arbitrarily small, so too, can be the ex-
ponential. Thus it is theoretically possible for a solid surface to be sufficiently
catalytic for bubble nucleation to cause bubbles to form at any positive
supersaturation. To sufficiently lower the exponential term to explain some of
FIG. 9 The two functions / (for bubbles, shown as squares, and for drops, shown
as circles). The nucleation of drops gets easier as the liquid becomes better wetting
(contact angle f0), while the nucleation of bubbles gets easier as the liquid becomes
increasingly poorly wetting. Contact angles are measured through the liquid phase
(or more generally, through the denser phase).

the experimental observations (see Section V) the contact angle would have to
be of the order of 98% of perfect nonwetting (h f 176j). This is not likely to
be a widespread occurrence. In fact, perfect nonwetting is probably more of
an idealization than is perfect wetting (h = 0j).
B. Heteronucleation at a Liquid/Liquid Interface

Apfel [30] and Jarvis [31] have analyzed the thermodynamics of the bubble
formed at an interface between two immiscible liquids. The situation is shown
in Fig. 10:
Following Blander [3] in defining mA and mB as
c2A þ c2AB c2B
mA ¼ cosðk hÞ ¼
2cA cAB
c2B þ c2AB c2A
mB ¼ cosðk /Þ ¼
2cB cAB
1 3
U ¼ 3 cA 2 3mA þ m3A þ c3B 2 3mB þ m3B
4cA
" #
16kc3 U
J ¼ CVVVVVexp ð18Þ
3kTðrP VÞ2
FIG. 10 A bubble at the interface between two immiscible liquids, showing the pa-
rameters needed to calculate the free energy of formation of the critical bubble.

In addition to the modifications to Eq. (16), the preexponential, CVVVVV includes
a factor of /, as it did for the case of the smooth solid surface. Equation (18) is
clearly predicated on the assumption of thermodynamic equilibrium when
calculating the work of formation of the critical nucleus. This is equally true
for other treatments that involve the contact angle as a variable. Given that all
nucleation takes place well away from equilibrium, it is important to keep in
mind the possibility of misleading interpretations from this cause. Specifical-
ly, this stricture applies to the use of Eq. (18) for liquid drops that are super-
heated while rising through a second immiscible liquid.
Finally, Eq. (18) specifically applies to the case of mercury as the second
liquid phase, and this may be important when attempting to identify the locus
of the nucleation event for electrolytic bubbles produced at liquid mercury
electrodes. Some generalizations can be made based on the relative values of
the various interfacial tensions: (1) If cB z cA + cAB, then it requires less
energy to form a bubble homogeneously in A than at the interface; thus the
nucleation will be homogeneous. (2) If cA z cB + cAB, the bubbles are most
stable in liquid B, and tend to detach from the interface. (3) If both conditions
cA< cB + cAB and cB< cA + cAB hold, then bubbles are most stable at the
interface, and the nucleation will be heterogeneous.
C. Heteronucleation at Rough Solid Surfaces

Rough surfaces introduce two new complications. First, the topography now
directly impacts the work of formation of the critical nucleus and second, the
possibility arises of trapping gas or vapor in sites of suitable shape and size.
This is important because the presence of any supercritical amount of gas
phase anywhere in the system effectively eliminates the need for nucleation.
Included in this category would be small gas bubbles either free or surface
resident, and any so-called Harvey nuclei consisting of trapped gas in a
crevice, depression, or pit. Such nuclei are believed to be widespread, and a
number of authors (Ref. 55) have claimed that they are responsible for bubble
‘‘nucleation’’ phenomena at low supersaturations. The case for this is far from
proven, although there is good reason for expecting this to be an important
contributor to the phase change at low driving forces. Here we address the
question of how various shaped steps, pits, cracks, scratches, depressions, and
prominences affect the nucleation barrier. We will also see that as bubbles
detach from a nucleation site, they frequently leave a significant amount of
gas or vapor attached to the solid. This remaining gas may act as a center for
the production of further bubbles, without the need for nucleation. In these
circumstances, these so-called nucleation sites are essentially ‘‘one-off,’’ and
having each produced a single bubble by a nucleation mechanism, subse-
quently act as repetitive sites for the evolution of further bubbles. They may

become deactivated, in which case another nucleation event would be
required to reactivate them. Often, sites will behave in this way, but under
some circumstances (where the contact angle is either zero, or close to it),
detachment may not leave any gas at the site, and this may be common. The
question is considered in more detail elsewhere in this review.
D. Surface Topography
As examples, consider the conical projection shown on the left in Fig. 11a and
the conical pit shown on the right, Fig. 11b. Other geometries have been
examined in several other publications (e.g., Ref. 3 or 32) but here these will be
taken as representative. Take first the conical pit of half-angle b illustrated in
Fig. 11b.
The reversible work of formation, W, of the bubble in this geometry is
evaluated as:
Z PS þPV
W ¼ cLV ALV þ ðcSV cSL ÞASV ðPV PL ÞVV þ VdP
PV
FIG. 11 A bubble forming at a conical projection (a, on the left) or in a conical

cavity (b on the right). The half angle of the cone is b in each case.

Using Young’s equation, cLV cos h = cSV cSL, and geometrically evaluating
the areas, AA and AB, and volumes, VA and VB, together with the condition
for equality of chemical potential at the critical size:
4
W ¼ 4kr2 c/pit kr3 ðPV PL Þ/pit : here c ¼ cLV
3

1 cos h cos2 ðh bÞ
/pit ¼ 2 2 sinðh bÞ þ
4 sin b
At the critical size, r = r*, the terms can be expanded in a Taylor series.
Taking the leading two terms in the Taylor series leads to the full equation for
the nucleation rate, J, in a conical pit of half angle, b:
!1=2 " #
2=3 1 sinðh bÞ 2c 4kc/pit r*2
J¼n exp ð19Þ
2 km/pit 3kT
It will be seen that both preexponential and exponential terms are affected,
but as noted earlier, changes in the preexponential are relatively unimportant.
The main result is that the function /pit appears as a multiplier in the
exponential, and that /pit in turn depends on both the contact angle, h and
the cone half angle, b. Wilt [32] showed that with a contact angle of 94j, a cone
of half angle 4.7j would produce nucleation in a carbonated beverage with a
saturation ratio of 5. None of these values is physically unrealistic, although
one could argue about how common such sites might be. However, fizziness in
Coca Cola appears to be universal.
For a conical projection, shown in Fig. 11a, the same analysis gives two
new functions.

1 cos h cos2 ðh þ bÞ
/cone ¼ 2 þ 2 sinðh þ bÞ þ
4 sin b
and the term in the preexponential also has its signs reversed:
1=2 " #
2=3 1 þ sinðh þ bÞ 2c 4kc/cone r*2
J¼n exp
2 km/cone 3kT
Wilt showed that such projections are stable toward nucleation, and were not
preferred sites for bubbles to form.
Of the geometries he investigated, Wilt [32] showed that the conical pit,
with high contact angle, was theoretically the only one able to catalyze nucle-
ation to the point that bubbles would be formed at saturation ratios close to 5.
This was important because this is typical of carbonated drinks, and the fact is
that such drinks do indeed produce massive numbers of bubbles when the

pressure is released (i.e., when the supersaturation is established). Simple
theoretical calculations show that without catalytic assistance for homoge-
neous nucleation of CO2 in water, the supersaturation required approaches
1400. Thus there was a very significant difference of more than 2 orders of
magnitude between theory and experiment. The geometry of the pit is one
aspect necessary for the Wilt theory to be successful. The other is the contact
angle. Experiments have been undertaken in an attempt to test this theory.
E. Importance of the Contact Angle

Pyrex glass usually has a high density of silanol groups on its surface exposed
to atmospheric (moist) air. These groups will react with chloromethyl silanes,
thus replacing some of the silanol groups with methyl silane. These groups,
have their -(CH3) groups exposed to the solution, and thus render the surface
hydrophobic [33]. The extent of the hydrophobicity can be controlled by the
concentration of the chloromethyl silane solution exposed to the Pyrex glass.
The parameter describing the hydrophobicity is the contact angle. Some
typical data [34] are shown in Fig. 12.
Both receding and advancing contact angles can be adjusted in the region
from about 0j to about 90j, and the advancing angles are generally about
FIG. 12 The contact angle for a Pyrex glass surface treated with various concen-
trations of dichlorodimethyl silane. The advancing and receding angles are shown.
The Pyrex surface is close to complete coverage at 1000 ppm.

between 5j and 10j greater than the receding angles. The measured angles
decline with the passage of time, by about 0.15j hr1. This is due to the
gradual hydrolysis of the Si–O–Si bond.
Nucleation rates of distilled water/CO2 solutions were measured on two
surfaces. First, a fully silanized surface, where the contact angle was close to
90j, and second, a carefully cleaned Pyrex glass surface, where the contact
angle was close to zero.
The procedure was to equilibrate the solvent with the CO2 gas at the chosen
pressure overnight. The pressure vessel was opened, thus rapidly allowing the
pressure to drop to ambient, whereupon the vessel was quickly closed. As
bubbles formed and burst, the pressure rose in the sealed pressure vessel. The
nucleation rate was measured as the rate of pressure rise, allowing for
adiabatic cooling, and the data shown in Fig. 13 were plotted according to
Eq. (6). The methods and apparatus have been fully described in Ref. 35. The
plots should be straight lines if the theory embodied in Eq. (6) holds. Clearly,
they are not straight lines. The hydrophobic nucleating heterosurface gives a
more rapid nucleation rate than the hydrophilic surface at a given supersat-
uration. The hydrophobicity of silanized Pyrex is greater than that of clean
Pyrex. Thus there is a qualitative agreement with expectations regarding the
FIG. 13 The rate of CO2 bubble nucleation measured as a function of (rP)2 for
two conditions. The upper curve (triangles) is for silanized Pyrex, with a contact
angle of close to 80j. The lower curve (squares) is for clean Pyrex, with a contact
angle close to 0j.

ease of nucleation on heterosurfaces. No significance attaches to the absolute
values of the ordinates—the units are arbitrary.
F. The Geometry of the Nucleation Site

Attempts to exploit the geometry of the sites to control bubble nucleation
have not been particularly successful. An early attempt was that of Griffith
and Wallis [36]. The authors used a copper metal surface carefully finished
with fine emery paper, and then deliberately pricked with a gramophone
needle, which itself had been sharpened. The opening of the approximately
conical cavities was about 25 Am radius. The experimental results showed that
the presence of the pits did indeed reduce the superheat necessary for boiling
at relatively low superheats. Unexpectedly, the pits did not reduce the
superheat needed to cause nucleation to the extent predicted: experimentally
20jF vs. the prediction of 3jF. The radius of curvature of the tip of the
gramophone needle was about 2.5 Am, and the opening was about 50 Am; the
depth is not mentioned, but the cone half angle was about 9j. The authors
attributed the larger than expected superheating results to the fact that the
mean wall temperature did not coincide with the temperature in an active
conical pit.
A different approach was adopted by Lubetkin [37], who made conical pits
in a polymer, CR39. This was performed by first exposing the polymer to high
energy alpha particles, and then etching the polymer along the tracks of those
alpha particles. According to the conditions used, and the energy of the
particles, the tracks can be designed to have a range of shapes and sizes. A
computer simulation corresponding to the conditions used in the study is
shown in Fig. 14. By controlling both the chemical conditions and the etching
time (shown on the figure) pits with various profiles could be achieved,
including various conical pits of known (calculated) diameters at their
mouths, and known half angles, through curved sided approximately conical
pits, with arbitrarily small half angles, through round-bottomed pits. All are
shown in Fig. 14.
Once these pits had been formed, they were exposed to chlorotrimethyl
silane, and contact angles as high as 118j were achieved [38]. The resulting
surfaces with their pits were examined as nucleating surfaces for supersat-
urated CO2 solutions. The predictions of the theory embodied in Eq. (19) are
that this combination of very high contact angle, and small cone half angles
should be an effective catalyst for bubble nucleation at a supersaturation of
about 4. The results were unexpected: The rate of nucleation as measured by
the pressure rise method (used above) was much lower than for glass or
stainless steel. It was surmised that the high contact angle caused bubble
detachment to be much more difficult than on glass or stainless steel, and

FIG. 14 The calculated etch trajectories for irradiated CR39 polymer. The sim-
ulation was performed using conditions designed to represent those actually used in
the study (see text). It can be seen that by stopping the etch at various times (9–81 m are
shown) a range of half angles can be produced. It is possible to produce essentially
straight-sided conical pits, steeply curved pits or round-ended pits, but these geo-
metries cannot be achieved independently of the half angle. By using a controlled,
collimated radiation source, it is possible in principle to produce pits where the axis
intersects the polymer surface at different angles, giving openings of different sizes and
geometries (e.g., ellipses of various eccentricities), but this was not performed in the
study referred to in the text.

that while the bubbles remained attached to the surface, no further nucle-
ation was possible there. In other words, detachment had become the rate-
limiting step in bubble evolution. This question is dealt with in more detail
later.
Volanschi et al. [39] have performed elegant experiments with cavities of
well-defined geometry. An inverted square pyramidal cavity of accurately
known size (depth and half-angle) was made by etching silicon. Electrochem-
ical bubbles were generated in this site, and were observed with an optical
system giving a magnification of about X400. Overpotential and capacitance
measurements were made, and this allowed studies of individual bubble
events. The results led the authors to postulate what they call the ‘‘concen-
trator’’ effect: essentially, that the concentration of liberated electrolytic gas
species is greater at the base of the cavity than elsewhere. Given that such sites
are already favored for nucleation, this added effect could go some way to
explaining the much lower than expected supersaturation needed to cause
electrolytic bubble nucleation.
Marto and Sowersby [40] used glass cavities of known geometry, in a
variety of aqueous solutions, and photographically studied the motion of the
gas/liquid interface during bubble formation and release during boiling, from
the cavities. The authors observed a liquid wave traveling down the walls of
the cavity after bubble departure. If this wave reached the bottom of the
cavity, the cavity often became deactivated as a nucleation site. On the other
hand, if the liquid evaporated before reaching the base of the cavity, it
remained active as a nucleation site.
When bubbles detach from a surface under gravity, a common state of
affairs is that a proportion of the bubble remains attached to the surface, or in
the cavity. The amount remaining is predictable from the physical properties
of the system. Bearing in mind that from point of view of the bubble shape,
the detachment of a vapor or gas bubble is the same physical problem as the
detachment of a liquid drop in the same vapor or gas (see, e.g., Ref. 41), the
treatment of detachment, at least under quasi equilibrium conditions is well
understood. The relevant point is that the detachment of one bubble (however
formed) very often leaves enough gas phase behind to avoid the need for a
fresh nucleation event.
IV. PREEXISTING NUCLEI

There are three types of preexisting nuclei we need to consider: (1) free
approximately spherical bubbles, which generally will be coated to a greater
or lesser extent with surface active molecules, (2) similar bubbles, attached to
a surface, and (3) portions of gas or vapor trapped in crevices, cracks, pits, and
other geometrically nonspherical cavities. Such bubbles may be filled with air

(or other gases) or with the vapor of the liquid (usually water), and we will see
that the composition of the vapor phase can be important.
A. Free Bubbles
An important distinction needs to be made between free and attached
bubbles. Ignoring gravity, free spherical bubbles have a single radius of cur-
vature. Recall that the curvature of an interface is defined as:

1 1
n¼ þ ð20Þ
r1 r2
where r1 and r2 are the principal radii of curvature. For a sphere, r1 = r2 = r,
and the curvature is then 2/r. The Laplace equation can be written as
DP ¼ nc ð21Þ
so for a sphere, DP = 2c/r, as before. When a bubble is attached to a surface,
not only can r1 and r2 be different, but the possibility of a negative curvature
exists, and this may have important consequences. For this reason, we start
with the simplest case, where the bubble is free, and assumed to be spherical.
Bubbles over about 1 Am radius in water will rise under gravity, eventually
intersect the free liquid surface, and burst. The 1 Am limit is not completely
arbitrary. The demarcation between kinetic stability and instability is defined
by the Péclet number, Pe, which is the ratio of the gravitational force to the
Brownian force:
DqgVr
Pe ¼
kT
Dq is the density difference between the liquid and gas phases, g is the
gravitational acceleration. The bubble has a volume V, a radius r, while k is
the Boltzman constant, and T is the temperature. If Pe >>1, then gravity will
overcome the randomizing tendency of Brownian collisions, and the bubble
will rise. If Pe<< 1, then the bubble may remain suspended for an indefinite
period, assuming no other instabilities supervene. The dividing size is that for
which Pe f 1, which for water at room temperature occurs close to r = 1 Am.
A lower limit on size is set by the critical radius under the conditions imposed
on the system. Between these two sizes, free bubbles might in principle survive.
What factors make survival unlikely? Clearly, the most serious is Ostwald
ripening (see, e.g., Ref. 42). Gases have enough solubility in most liquids to
make Ostwald ripening a significant mechanism for bubble size redistribution
to occur on reasonable time scales. For the case of gas bubbles in water, this
mechanism will be quite rapid. The original theory for crystalline solids was
outlined by Lifshitz and Slyozov [43], and applies equally to gas bubbles. In an

assembly of bubbles, where the assembly can consist of any number greater
than one, and where there is more than one size represented, the larger
bubble(s) will grow at the expense of the smaller. In principle, it is easy to see
why the pressure inside the smaller bubble is greater than that in the bigger by
inspection of Eq. (21). Given that solubility is directly proportional to
pressure (Henry’s law), the gas in the smaller bubble is more soluble than
that in the larger, and the latter will grow at the expense of the former. This is
true regardless of absolute size, although the kinetics will be faster for smaller
bubbles. A similar situation holds for assemblies of more than two bubbles.
Eventually, any assembly of bubbles will be reduced to a single bubble; if this
single bubble has a radius greater than about 1 Am, the Péclet condition, Pe >
1, ensures that it will rise and burst. This is simply another way of expressing
the underlying truth that suspensions (of crystals, droplets, or bubbles) are
thermodynamically unstable, and will phase separate given time. As a lower
bound, the rate at which a single bubble in a saturated (but no supersaturated)
solution will dissolve, has been evaluated by Epstein and Plesset [44]. Their
expression for the time, t, required for disappearance from an initial size r0 in a
medium where the surface tension is c,

r2 kH r0 ðPL PV Þ
t¼ 0 þ1
3DRT 2c
The diffusivity is D, and kH is the Henry law constant. PL and PV are the
pressure in the liquid and in the vapor, respectively. Typical calculations are
shown in Table 4.
A subsidiary question is what difference does the composition of the vapor
in the bubble make? The answer, by analogy with the situation with emulsions
[45], is that this may be significant in terms of the kinetics of ripening. With
either gaseous component present on its own, the Ostwald ripening will
proceed as described by Lifshitz and Slyozov [43]. However, when two gases
TABLE 4 Time Required for Disappearance of a Bubble in a Saturated

Solution
r0 Dissolution time Rise rate (U mm sec1) Reynolds number, Re
1 Am 8.msec 0.0022 4106

10 Am 6..6 sec 0.22 4103
100 Am 5.,880 sec 22 f1
1 mm 5.,800,000 sec 330 >>1
The rise rates quoted in the third column are all estimated from Stokes law, and thus will be
inaccurate for sizes above about 100 Am. Since the condition Re = DqUd/l <1 for Stokes Law
to be reliable, the rise rate for the 1 mm bubble was estimated from Ref. 77.

(or more) are mixed, then interesting behaviors including the development of
bimodal distributions become possible. As far as the author is aware, these
behaviors have not been reported nor does there appear to be any data on
Ostwald ripening of bubbles in general.
When the discussion focuses on the detachment of a succession of bubbles
from a source, the composition of the remaining gas in the reservoir will
change with time. This point is taken up later. Finally, as was the case for
nucleation, the presence of the gas increases the internal pressure sustainable
in the bubble, and this may be important to the stability of trapped bubbles.
At low bubble densities in the bulk liquid, the probability of collisions and
hence of coalescence is rather small, but the underlying theory of coalescence
of emulsions, or aggregation of suspensions, would be expected to equally
apply to bubbles. It seems likely that coatings of surfactant would exercise
their main influence on stability by prevention of such coalescence. There is a
second possibility and that is that the adsorbed surfactant layer becomes
compressed as the (partially soluble) gas contents of the bubble diffuse out, as
pointed out by Yount [46]. In doing so, it becomes less permeable to the gas,
and at the same time, more rigid (see, e.g., Ref. 47). For such compression to
become a reality, the pressure in the bubble would have to be less than would
normally be accommodated in a bubble of that radius. For it to then function
as a nucleation center, gas or vapor would have to diffuse into the interior,
thus relieving the compression of the surfactant ‘‘skin.’’ There would clearly
be an activation energy barrier to the onward growth.
In summary, then, free bubble suspensions are thermodynamically unsta-
ble, and phase separation will occur, but the time scale for this depends on the
detailed circumstances.
B. Bubbles Attached to a Smooth Surface

The situation changes when the bubble is attached to a smooth surface.
Depending on the contact angle, we have already seen that in the absence of
gravitational distortion, the bubble will adopt a part spherical shape. The
pressure in the cap is still dictated by the curvature, not by the volume, and so
remains essentially the same as for a full spherical bubble of the same radius.
The significant change is that when the bubble grows (as it will in a
supersaturated environment, allowing for the excess pressure in the bubble
owing to the curvature), the buoyancy forces will grow, too. A point will come
where the bubble detaches, as discussed in more detail below. Often when the
bubble detaches, some gas remains at the previous site of attachment of the
now rising bubble. Provided that this new spherical cap has a radius R > r*, it
will act as a site for a new bubble to grow, without the need for nucleation. In
these circumstances, heterogeneous nucleation will be a ‘‘one-off’’ process,
followed by bubble growth and detachment. At sufficiently low contact

angles, or at sites with composite nature, this mechanism may not operate,
and this is discussed below.
Note that attached, part spherical bubbles are subject to the same Ostwald
ripening process as free bubbles, and will therefore disappear in a saturated or
even a mildly supersaturated solution of the gas phase, or when the temper-
ature is sufficiently lowered. It is a matter of experimental observation that
bubbles that detach from surfaces are generally of a size such that Pe > 1, and
bubbles released from surfaces generally rise rapidly through the solution and
burst at the free liquid surface.
C. Sources of Gas Bubbles at a Rough Surface

Next we consider roughness on a scale that allows bubbles to have a three-
dimensional surface of attachment, rather than the two-dimensional circle of
attachment typical of a smooth surface. An appropriate length scale would be
the bubble diameter. The new feature of such attachments is that negative
curvatures become possible, and this has important consequences. We start
by considering the question, where do these bubbles come from?
We will consider only three types of cavities: a crevice (an extended V
shape, such as might be produced by a scratch) with a triangular cross section,
a conical pit, and a reservoir with an arbitrary, but reentrant shape. These
cases are schematically shown in Fig. 15.
For the crevice, this same diagram can also be used to illustrate a conical
depression, a case that will be discussed in a later section.
Taking the crevice first, consider how an advancing thin sheet of water will
enter the crevice. The situation is illustrated in Fig. 16. In A, the contact angle,
h, is less than or equal to 2b, the wedge angle: Strictly speaking, the relevant
contact angle is the advancing angle, ha. This nomenclature derives from the
advance of a liquid over a ‘‘dry’’ surface. In these circumstances, the liquid
will penetrate to the apex of the wedge before meeting the far wall, thus filling
the entire cavity. There will be no air trapped in the cavity by the advancing
water. In B, on the other hand, where h > 2b, the advancing water will trap
some air. As a generalization, one would expect that the greater h is, in
comparison to 2b, the larger the volume of air trapped in a given cavity. A
previously dry surface containing both types of cavity, when exposed to water
or other liquids, might be expected to show different extents of air trapping.
Summarizing the predictions of Bankoff [48] gives Table 5.
The prediction is that steep sided cavities, with large contact angles (i.e.,
poor wetting) are likely to be the sources of bubble production. In well-wetted
conditions, cavities are likely to be unimportant.
In practice, the advancing water will generally not be parallel to the solid
surface nor will it be configured as a thin sheet. The model serves only to point
out the possible importance of the size and shape of such cavities, and the

FIG. 15 Three possible gas reservoirs: On the left (a), a crevice (extended perpen-
dicular to the plane of the page) or conical pit. On the right (b), a small-mouthed re-
entrant cavity. In each case, the opening radius is a.
FIG. 16 A thin sheet of liquid advancing from right to left encounters a crevice.
If h< 2b, then the crevice would fill completely, but if h>2b, then air would be
trapped.

TABLE 5 Correlation of Cavity Type, Contact Angle and Trapped Phase
Type of Cavity Wetting Conditions Phase Trapped
Steep Poor Gas only

Steep Good Liquid only
Shallow Poor Gas and liquid
Shallow Good Neither
possibility that some sites are better at trapping air pockets than others, and to
emphasize the role of the contact angle, as discussed by Bankoff [48], Cole
[49], Atchley and Prosperetti [50], and Carr [51]. Certain conclusions can also
be drawn from this analysis about the behavior of such crevices in supersat-
urated and undersaturated conditions, and this question is considered below.
This includes the possibility of successive cycles of heating and cooling
causing such cavities to produce or eliminate gas/vapor pockets. In this
connection, it should be noted that each successive bubble detaching from a
reservoir takes with it a portion of the gas in the cavity, leaving an increased
proportion of vapor. The composition of the contents of the cavity is thus a
function of time.
Next, let us consider the case of the reservoir cavity illustrated in Fig. 15B.
There is no obvious property relating to contact angle that can be used to
predict the filling of such cavities. The expectation is that the filling would be
governed more by hydrodynamics than thermodynamics, in the absence of
cycling of temperature or pressure, or both. We will examine the effect of such
cycling on the reservoir cavity below. The thermodynamic approach also
hides the importance of kinetics in the filling, emptying, and bubble release
processes. It is this aspect that has been examined by Atchley and Prosperetti.
Of course, both types of cavity can be filled with vapor by a nucleation
event followed by growth. In the case of the conical cavity or wedge, it has
already been established that such reentrant sites are preferred for bubble
nucleation, and with suitable roughness inside the reservoir cavity, this too,
would act as a preferred place for nucleation. The chief difference is that
nucleation can only fill the cavities with vapor not with noncondensable gases
such as N2 or O2.
D. The Effect of Supersaturation or Undersaturation

on a Bubble Trapped in a Conical Pit
For the sake of simplicity, we will consider only the behavior of a conical pit
containing a bubble. This case and others have been the subject of an
extensive literature, which includes Bankoff [48], Griffith and Wallis [36],
Apfel [52], Cole [49], Winterton [53], Trevena [54], and Atchley [50]. We will

first look at a particularly simple case (and one that probably does not exist in
practice) where the contact angle is held constant at 90j, as examined by
Griffith and Wallis [36].
The curvature (= 1/r) of the interface is plotted against the volume of gas
phase for each of the positions of the interface, r1 though r4 (Fig. 17). The case
where r = r# is of particular interest. This value of r may be called the
‘‘threshold’’ radius. Some authors use the term ‘‘critical radius’’ but this
allows confusion with the critical nucleus. The radius is both the radius of the
bubble at the point where the bubble shape is hemispherical (for the fixed
contact angle of 90j) and equal to the radius of the opening of the conical pit.
At this point, the curvature is at a maximum, and the pressure difference, DP,
is also a maximum. If the internal pressure is at least this great, a bubble will
form, grow, and eventually be released from the cavity. Should the pressure be
lower, the meniscus will retreat back into the cavity. For boiling, to get to this
threshold size, the superheat is the minimum needed to cause the growth of a
bubble in this particular cavity. A set of cavities, each with the same opening,
r#, would all become centers for bubble release at the same, sharply defined
value of the superheat:
2c
DP ¼
r#
FIG. 17 The curvature is plotted vs. the volume enclosed for a constant contact
angle of 90j. The local maximum is where the curvature is equal to the radius of the
opening of the cavity.

Now using the Clausius Clapeyron equation,
DP DHV
¼
DT TVV
2cTW VV
r# ¼ ð22Þ
DHV ðTW TS Þ
where (TWTS) is the superheat (the difference between the temperature of

the heated wall, TW, and the saturation temperature, TS). Thus the smaller the
opening of the cavity, the larger the required superheat to generate a bubble.
The following discussion is based on Apfel [52] and Trevena [54]. Denote
the advancing contact angle by ha, and the receding contact angle by hr and
consider the case of cavitation, where negative pressures are applied, and let
the contact angle not be restricted to 90j. Referring to Fig. 18a, a conical
cavity is shown where the applied pressure is greater than the pressure in the
incipient bubble, while in the case B, the applied pressure is lower than that in
the bubble. In the former case, where the meniscus is being forced into the
FIG. 18 Because the advancing contact angle, ha, is generally larger than the re-
ceding angle, hr, the geometry of the meniscus is different when the hydrostatic
pressure is forcing the liquid into the cavity, or when the pressure is reduced so that
the gas expands outward from the cavity.

cavity, the contact angle rises until it becomes equal to the advancing contact
angle, ha, whereupon the meniscus advances down the cavity, maintaining
that angle.
When the applied pressure is reduced, as in the latter case, the meniscus
withdraws from the cavity. The relevant contact angle is now hr, the receding
angle, as shown in Fig. 18b.
The concept of the threshold size is also applicable here. The pressure
inside and outside the bubble are related through the Laplace equation:
2c
DP ¼ ðP PV Þ ¼
r
In the case where the contact angle is exactly ha when the external pressure P is
imposed, the cavity is a threshold cavity, and
2cðcosðha bÞÞ
r ¼ r# ¼
ðP PV Þ
The situation is shown in Fig. 19. The opening of the conical pit has a radius
r0. The conditions (contact angle, radius of contact) are such that the cavity is
‘‘subthreshold,’’ (A), ‘‘threshold,’’ (B) or ‘‘superthreshold.’’
E. Expulsion of Air from a Cavity

Cavities freshly filled with air can expel bubbles (containing mostly air plus
some relatively small amount of the vapor of the solvent) before the system
FIG. 19 The definition of a ‘‘threshold’’ cavity. In the literature, the term ‘‘critical’’
cavity is used. To avoid confusion with the critical bubble the term, ‘‘threshold’’ is
used in the text.

becomes superheated. Cold tap water standing in a glass will often release
bubbles. It is very likely that some of these bubbles are of this type, although
clearly there is a contribution from the supersaturation of the air dissolved in
the water in this case. For a freshly filled reservoir type of cavity, containing
only air, one can compare the expansion (V1/T1 = V2/T2) due to heating from
main water temperature up to room temperature T2. In the example of tap
water left to warm up to room temperature, T1 would be about 10jC, and
room temperature would be say 25jC. To calculate a maximum effect, we can
use say 99jC for T2, with the calculated departure volume of a bubble from
the orifice. This would allow one to predict whether air bubbles would be
emitted from such a cavity before any nucleation could possibly take place:
V2 =V1 ¼ T2 =T1 ¼ 372=273 ¼ 1:36
In other words, a volume equal to 36% of the cavity volume is available to

form a bubble large enough to become detached. Assume that a bubble has to
be about 100 Am in diameter to detach (see below), this would then give an
estimate of the equivalent radius of the cavity needed to expel a bubble of air
from water (just) before boiling as 70 Am. The larger the volume of the cavity
relative to the detachment volume of the bubble, and the narrower the
opening, the lower the temperature required to expel the air bubble(s).
Conversely, if the volume of the cavity is less than about three times greater
than the volume of the detaching bubble, then an air bubble will not be
released before the system becomes superheated with respect to the solvent.
This calculation does not account for the change in solubility of the dissolved
gas with temperature (supersaturation of the solute), which will have the effect
of decreasing the size of the cavity needed.
F. Deactivation of Cavities
As each bubble detaches from a cavity, it carries away a portion of the gas/
vapor mixture from the cavity. This mechanism is also at least in part
responsible for the removal of the dissolved gases from solution by boiling.
Thus the composition of the contents of the cavity changes with time
asymptotically approaching pure vapor of the solvent. To a good approxi-
mation, the cavity will be filled only with vapor at long times. As the
proportion of noncondensable gas in the cavity decreases, the total pressure
for a given curvature of the cavity also decreases. This decrease affects the
ability of the cavity to withstand compression. Not only will the cavity
become more easily deactivated as the external pressure is raised, but as the
temperature is reduced, the water may reenter the cavity. If it fully reenters the
cavity, the site no longer acts as a Harvey nucleus, and a fresh nucleation event
is necessary to reactivate it. The probability of such a nucleation event may be
very much less than the probability of a bubble release from the active Harvey

site. The deactivation process is the main reason for antibumping granules
having to be freshly added to the liquid to be boiled; a cycle of boiling and
cooling will quickly result in the antibumping granules becoming inactive.
G. How Important are Cavities (Harvey Nuclei)?

Jones et al. [55] are representative of a significant school of thought that
postulates that under low supersaturations, there is no classical nucleation,
and bubble release is only a matter of growth from a preexisting gas filled
cavity. Depending on the radius of curvature of the interface in the cavity, no
nucleation step is involved, and thus it is irrelevant what the nucleation theory
predicts. This latter approach has been hinted at a number of times in the
literature. The presence of any supercritical amount of any gas phase any-
where in contact with the liquid phase of the system is theoretically enough to
remove the need for nucleation. There are undoubtedly cases where this is the
main mechanism of bubble evolution. The regular emission of strings of
bubbles from beers and champagnes, originating at a single site, are very likely
to be of this type; see, e.g., Liger-Belair et al. [56].
Once again, it is important to separate the cases of a single component
system (boiling, cavitation) from that of the supersaturated gas solution. For
a reservoir cavity, illustrated in Fig. 15b, consider first the case of a super-
saturated gas solution. As bubbles form and leave, the composition of the gas
in the reservoir will approach the composition (gas + vapor) appropriate to
steady state described by the curvature of the opening under the existing
conditions.
Diffusion from the supersaturated bulk gas solution will ensure that the
pressure in the reservoir climbs to a level great enough to cause a next bubble
to detach. As time goes by, and the supersaturation of the solution decreases,
the emission of bubbles becomes slower and slower, and finally stops. The
process has been described by Jones et al. [55]. This is in contrast to the state of
affairs with a single component system. Bubbles gradually detaching rid the
cavity of all noncondensable gas, and the reservoir is then filled with the vapor
of the boiling liquid. In the absence of liquid in the cavity, the only drive to
cause a bubble to leave the cavity is heating of the vapor above ambient
temperatures. It clearly would be expected that the continuous regular strings
of bubble characteristic of beer, wine, and carbonated soft drinks will be less
common in boiling or cavitation.
V. EXPERIMENTAL TESTS OF BUBBLE NUCLEATION

THEORY
Finally, we collect here some representative experimental results, and the
calculations used to interpret them.

A. Boiling
The early experiments of Kenrick et al. [10] involved submerging filled but
open tubes of the study liquid in a thermostatic bath at a given temperature,
waiting for thermal equilibration (5 sec) and noting if there was boiling in that
time. For various reasons, it is probable that there were adventitious nuclei in
these early experiments (Table 1A). However, comparing these data with the
later experiments, summarized by Blander, as shown in Table 1B, there is a
somewhat greater variability in the earlier results, but overall the agreement
with the classical theory is very satisfactory. The later results, mostly obtained
using the rising drop technique, largely avoid the possibility of Harvey nuclei,
but suffer from the use of an equilibrium expression [Eq. (18)], for the
evaluation of the work of formation of the critical nucleus. The agreement
between theory and experiment is nonetheless very satisfactory.
Factors other than Harvey nuclei may affect the results. High energy
radiation is one such influence. It has been shown that radiation does not
greatly affect gas bubble nucleation, but clearly it does affect boiling, this
being the basis of the bubble chamber used to detect subatomic events. The
effect of radiation on the kinetics of boiling nucleation is well illustrated by
the data in Fig. 20 from Ref. 6. The line AB represents the steep decrease
in the time (s) before the droplet boils as the temperature increases. The wait-
ing time decreases from about 550 to 0.5 sec (three orders of magnitude) as the
temperature is raised from A (145.6jC) to B (146.0jC). Measurable rates of
bubble nucleation occur on the branch DC at much lower superheats (the
temperature at D is low as 129.5jC), when the superheated liquid is exposed
to radiation (Fig. 20).
B. Cavitation
A sample of data for the fracture tension of liquids is shown in Table 2. The
data were obtained by Briggs [57] using a centrifugal method to apply tension
to the samples. It can be seen that in all cases, the measured fracture tension is
smaller than the prediction. There is a suspicion that heterogeneous sites at
the walls of the containing tubes may have influenced the results. The ratio
cavitation theory/measured is not as large as for gas bubble nucleation, but is
larger than for the case of boiling (Table 3).
C. Gas Evolution
1. Electrolytic
The pressure in a small, newly formed bubble (at an electrode, for example) is
by definition, rather high, and close to p*. For gas molecules to diffuse into
this bubble and make it grow, the solution surrounding this bubble must be
highly supersaturated, and have r approximately equal to the critical super-

FIG. 20 A plot of experimental data from Skripov [6] for the homogeneous boiling
of pentane. The part ACB is the normal superheat limit. As the temperature is
increased, so the time required for explosive boiling of liquid drops rapidly decreases.
In going from 145.6 to 146.0jC, the waiting time drops from 550 to about 0.5 sec. If
the superheated liquid droplets are exposed to radiation, very much lower superheats
are needed to cause bubble nucleation. Temperatures as low as 129.5jC can cause
bubble formation.
saturation needed to cause nucleation. Conversely, a mature bubble near

departure has only a small excess pressure, and thus needs a correspondingly
small supersaturation to cause growth. Using the rate of growth of bubbles at
or near electrodes as a means of measuring supersaturations in the vicinity is
thus fraught with difficulties. It seems probable that this factor accounts for at
least some of the radically different estimates for the supersaturations shown
in Table 6A,B.
2. Nonelectrolytic Bubbles in Aqueous Solutions
Two main techniques other than electrolytic methods have been used to
obtain data on the nucleation of gas bubbles in aqueous media. These are
pressure release (as used by Lubetkin, and Hemmingsen and coworkers) and
chemical (as used by Noyes, Bowers, and Hey among others). In the former
technique, water is saturated with the study gas at an elevated pressure, then
the pressure is released down to a target pressure ( PV) often atmospheric.
Bubbles form, rise, and burst and the gas is released into the headspace. This

Table 6A Experimental and Theoretical Values for Electrolytic Supersaturation
Author(s) r(ce/cb) or (ce cb)a Notes
Sides [65] f100 A maximum value

Westerheide and 8–24
Westwater [66]
Dapkus and Sides [67] 9–16 On clean, smooth mercury
Shibata [68] 7–70 (1,5... > 100 mA) From transients
Glas and Westwater [69] See Part B of this table From growth
a
The dimensionless version (ce/cb) is preferred in the present review because it directly relates to
a, the saturation ratio, but some authors use the dimensional version ce cb.
Table 6B
H2 O2 CO2 Cl2
1.54 1.36 1.08 1.018

19.9 15.4 1.64 1.324
release is studied by the pressure rise (if the vessel is immediately resealed) or
by measuring the amount of gas evolved, or by acoustically counting bubbles
[38]. In the chemical generation of bubbles, the supersaturation is evaluated
by measuring the total gas released, usually sonicating the solution to ensure
complete gas evolution (Table 7).
3. Nonelectrolytic Bubbles in Organic Liquids

N2 gas bubbles were formed by pressure release from solutions in various
simple organic liquids, by Kwak and Kim [58]. Similar experiments had been
performed on nitrogen bubbles nucleating in ether by Wismer [59], and more
recently, by Forest and Ward [60] (Table 8).
4. Organic Gas Bubbles from an Organic Liquid

Data from Lubetkin and Akhtar [28] are shown for the nucleation of bubbles
of ethene from a supersaturated solution in cyclohexane, plotted according to
Eq. (6). The Y axis is in arbitrary units. The two curves are for nucleation on a
Pyrex glass surface and a stainless-steel surface. The corresponding advancing
contact angles were about 5j and 25j, respectively (Fig. 21).
5. Example Calculation of Bubble Nucleation Rate

A calculation is presented in Table 9 based on the case of a supersaturated gas
nucleating in an aqueous solution at 30jC and atmospheric pressure. The

TABLE 7 Supersaturation Needed to Cause Bubble Nucleation (Mostly
Heterogeneous) in Water or Aqueous Solutions. Supersaturations Generated by
Nonelectrochemical Means
Gas Measured Reference/theoryy Theory/measured ratio
He 230–320 (h)/1400 6–4

Ar 110 (i)/1400 13
N2 100–160 (e)/1400 14–9
O2 100–150 (i)/1400 14–9
O2 95–127 (c)/1400 15–11
H2 80–100 (g)/1400 17–14
CO 80 (f)/1400 18
CH4 80 (i)/1400 18
N2 20–30 (g)/1400 70–47
O2 16–50 (c)/1400 87–28
NO 16 (g)/1400 88
CO2 10–20 (i)/1400 140–70
CO2 5.4–7.6 (b)/1400 260–184
Cl2 5 1400 280
CO2 4.62 (d)/1400 303
CO2 1.3–2.1 (a)/1400 1077–667
Mean F SD 89 F 107
y
Calculation for a gas dissolved in water—see Table 9.
(a) Jones et al. [55]; (b) Lubetkin [37,38]; (c) Bowers et al. [70]*; (d) Hey et al. [71]; (e)
Hemmingsen [72]*; (f) Smith et al. [73]*; (g) Rubin et al. [74]*; (h) Hemmingsen [75]*; (i) Bowers
et al. [76].
References marked with* are claimed to be homogeneous nucleation. The mean and standard
deviation do not include the last entry for CO2.
TABLE 8 Homogeneous N2 Bubble Nucleation in Nonaqueous Liquids
Liquid Theory Measured Theory/measured ratio
Methanol 250 90 3
Ethanol 252 80 3
Chloroform 330 70 5
Carbon tetrachloride 326 53 6
Benzene 361 35 10
n-Hexane 184 56 3
Mean F SD 5.0 F 2.76
Source: Ref. 58. The authors claim that the nucleation is homogeneous.

FIG. 21 Experimental data for the nucleation of ethene from cyclohexane, plotted
according to Eq. (6). The upper data (squares) are for stainless steel, while the lower
curve (circles) is for Pyrex glass. The contact angles are roughly 25j and 5j, respec-
tively.
TABLE 9 Calculation of the Rate of Nucleation Using Eq. (6)

Variable Unit Magnitude Comment
3 1
C m sec 1 10 35
Arbitrary
c N m1 70 103 For water/aira
T K 303
k J K1 1.38 1023
r Dimensionless 1400 The required r for
f1 event cm3 sec1.
This value is used in the
text as theory for gas
bubble nucleation
J m3s1 106 Equivalent to 1 event cm3 sec1
P Pa 101,325 1 atm
16pc3 N3 m3 1.72 102 A
3kT(rP)2 N3 m3 2.52104 B
16pc3/3kT(rP)2 Dimensionless 68 A/B
exp[16pc3/3kT(rP)2] Dimensionless 2.17 1030 exp(A/B)
J = Cexp[16pc3/3kT(rP)2] m3 sec1 2.17 105 f1 event cm3 sec1
a
Note that, in general, the presence of the gas will lower the interfacial tension. This effect has not been
included in the calculation here.

value of the supersaturation, r, was chosen to give a rate of approximately 1
event per cm3 sec1, equivalent to 106 events m3 sec1. The calculation is
based on the simplifying assumptions of Eq. (6), and makes no allowance for
nonideality:
" #
16kc3
J ¼ Cexp
3kTðrPVÞ2
6. Comparison with Calculation of Droplet Nucleation Rate

For comparison purposes, a calculation is performed for the rate of homo-
geneous droplet nucleation from a supersaturated vapor (Table 10). The
equation is that from Dunning and Shipman [23]:
" #
16kc3 M2 NA
J ¼ CVVVVexp
3R3 T3 q2 ðln aÞ2
7. Reconciliation of Theory and Experiment

In the case of boiling, the data shows that the theory and experiment agree
remarkably well. The data from Table 1B show an average value of the
superheat calculated from the theory vs. the measured superheat, the ratio
boiling theory/measured = 0.997, with a standard deviation of F 0.03. For
cavitation, the comparable figure for the theoretical and measured fracture
TABLE 10 Calculation of the Homogeneous Water Droplet Nucleation from the Vapor
Phase
Variable Unit Magnitude Comment
3 1
C (or gk) m sec 1 1035
The original equation
uses gk, equivalent
to C here
c N m1 7 102
M kg 1.8 102 Water
NA Dimensionless 6.023 1023
R J K1 mol1 8.314
T K 2.93 102
q kg m3 1 103
r Dimensionless 4
16pc3M2NA N3 m3 kg2 3.36 1018 A
3R3T3q2 J3 kg2 m6 5.23 1016 B
16pc3M2NA/3R3T3q2 Dimensionless 7.76 101 A/B
exp[16pc3M2NA/3R3T3q2ln2r] Dimensionless 1.21 1028 exp(A/B)
J=C exp[16pc3M2NA/3R3T3q2ln2r] m3 sec1 1.21 107 Roughly 1 event cm3 sec1

tensions are cavitation theory/measured = 2.18 F 1.55 (if the values for
mercury are discarded) and 9.63 F 19.77 (if they are included). The compa-
rable figures for the nucleation of gas bubbles in water are somewhat less
easily assembled, the chief difficulties being distinguishing homogeneous from
heterogeneous nucleation, or indeed whether bubble nucleation is involved at
all in a couple of cases, and in the effect of gas adsorption on the surface
tension. Ignoring these potential complications, the theoretical calculation
performed as shown in Table 9 gives a theoretical value of gas in water theory
of about 1400 for gases dissolved in water at ambient temperature of 25jC.
Using this as a basis, the values of the supersaturation calculated as shown
above and measured by various means (pressure release, bubble train timing,
and chemical generation) in situ for various gases give a ratio of gas in water
theory/measured = 89 F 107. For bubbles of N2 in various organic solvents,
a much smaller value of the ratio results: gas in organics theory/measured =
5 F 2.76.
The evidence strongly supports the hypothesis that dissolved gases, parti-
cularly at relatively high concentrations in water, cause substantial lowering
of the interfacial tension in bubbles, and that this lowering is the main reason
for the fact that gas bubbles nucleate at low supersaturations, and thus give
such high values of the ratio theory/measured.
Boiling is an effective means of reducing or eliminating the gas content of a
liquid, and this removal of gas probably accounts for the rather good
agreement of the data with the predictions of the theory in the case of boiling.
The experimentalists who measured the fracture tension of liquids (by
cavitation) were acutely aware that preexisting nuclei (e.g., Harvey nuclei)
would strongly influence their results, and so took pains to attempt to elim-
inate dissolved and suspended gas bubbles before the experiment. The test
liquids were carefully purified by distillation before use, and were degassed.
This care, while not completely effective, probably was enough to reduce the
gas content to the point where the gas could be considered to be a minor
impurity, rather than a major contributor to the results. Thus cavitation data
are relatively reliable, although not providing quite such a good agreement
with theory as in the case of boiling.
By their nature, the experiments on gas bubble nucleation, whether
electrolytic, chemically generated, or by pressure release, all have high
concentrations of gas in solution during the bubble nucleation process, and
thus show the most pronounced effects of surface tension reduction. Two
aspects of this observation are particularly noteworthy: first, that He of all the
gases so far tested shows the smallest ratio of theory/measured—and is also the
only gas with a b coefficient of f0. Second, that organic liquids with N2 gas
dissolved in them (including the case of diethyl ether) show relatively modest
ratios theory/measured. In the case of diethyl ether, the liquid is relatively close

to its critical point (as are all the simple organic liquids, if water is used as the
yardstick), and so the surface tensions are relatively low to start with. Thus
gases thus cannot reduce the surface tension as dramatically for organic
liquids as for water. In brief, the most dramatic effects of dissolved gas are
expected in aqueous gas solutions, and in confirmation of this, the highest
values of the ratio theory/measured are indeed found for these systems.
Furthermore, the more soluble gases produce the largest effects: The upward
trend in the ratio theory/measured as one goes down Table 7 follows the
general trend in increased solubility going down the table. The b coefficient
also follows the same trend, with b increasing with the solubility.
One final observation is that Eq. (6) or any of its variants depends only on
the properties of the liquid, not on those of the gas. Theory predicts that all
gases should nucleate at the same supersaturation, r. Table 7 shows a factor of
about 250:1 in the supersaturation needed to cause nucleation for different
gases. Even allowing for the fact that some of these results may be derived
from homogeneous and others from heterogeneous nucleation experiments
(Table 11), the classical nucleation theory provides no clues to understanding
the systematic variation of the ratio with both the solubility of the gas and the
b coefficient, which together form a most surprising result. However, it has a
simple interpretation in the light of gas adsorption as a mechanism for
reducing the interfacial tension, and thus lowering the supersaturation needed
for nucleation. When a single gas is examined in a variety of liquids as seen in
Table 8, the experimental data show a rather small variability, and much
lower values of the ratio theory/measured. Once again, this is consistent with
the gas adsorption mechanism, but unexpected based on classical nucleation
theory.
TABLE 11 Summary Comparison of Theoretical Prediction and Measured

Drives Required for Bubble Nucleation
System Ratio Comment
Boiling (single 0.997 F 0.03 Heterogeneous

component system) (liquid/liquid interface)
Cavitation (single 2.18 F 1.55 Homogeneous (?)
component system)
Gas evolution (pressure 89 F 107 Heterogeneous (mostly)
release, chemical reaction,
etc.) from water
Gas evolution (N2 from 5 F 2.76 Homogeneous (?)
organic liquids)

VI. THE IMPORTANCE OF DETACHMENT IN
NUCLEATION KINETICS
It is not the intention to thoroughly review the detachment literature here.
The approach will be to mention very briefly the principles of calculating the
detachment volume of a bubble, and to apply these principles to a discussion
of the interaction of nucleation and detachment.
A. Quasi Static Detachment

Detachment must by definition be a nonequilibrium process, but it is a con-
venient fiction to assume that the forces are the same (or at least, close enough
to being the same) as for a genuine equilibrium state. The errors involved
become smaller as the system more closely approximates to equilibrium con-
ditions. This is one reason why detachment should be approached as slowly
as possible during experimental investigations of drop/bubble detachment
volumes.
B. Tate’s Law
The weight, w, of a detaching drop of wetting liquid of surface tension, c,
formed on a circular tube in air with an orifice of radius r is approximately
given by:
w ¼ 2krc
Tate actually expressed his ‘‘law’’ as follows: ‘‘Other things being equal, the
weight of a drop of liquid is proportional to the diameter of the tube in which
it is formed.’’ In practice, a lesser weight, w V, is obtained. The reason is that
only a proportion of the total drop actually detaches—as much as 40% of the
liquid may remain attached to the orifice. Thus the actual weight is given by
the expression 2prc multiplied by a correction factor, f:
wV ¼ 2krcf
The values of f were experimentally derived by Harkins and Brown [61]. For a
given drop weight, it is possible by using the correction factor to calculate
what proportion of the drop weight (and hence volume) is retained on a given
orifice. In calculations involving the interface between two immiscible fluids a
and b, the shape of the interface, and properties such as the enclosed volume
are available, and it only needs an alteration of sign if the two fluid phases are
exchanged. Thus water in air (a in b) is the same mathematical problem as air
in water (b in a), but with a sign reversal. What holds for drops, generally
speaking, holds for bubbles, too. However, as far as the author is aware, there

has been no experimental verification of Tate’s law and the Harkins Brown f
factor for the case of detaching bubbles.
Recourse to empirical factors such as the Harkins Brown f factor can be
avoided by calculations based on the volumes of axisymmetric drops (bub-
bles) as discussed by Boucher [78].
The significance of this for bubble nucleation is that, in principle, it is
possible to calculate the remaining amount of gas at a nucleation site, and
thus to be able to answer unambiguously the question ‘‘Is a fresh nucleation
event needed for the next bubble?’’ It is worth mentioning in this context that
when the inner fluid wets the orifice, then the appropriate radius for
detachment is the outer radius of the orifice, but when the inner fluid does
not wet the orifice, it is the inner radius that is relevant. In the presence of
water as the outer fluid phase, and the generally reasonably good wetting
reported in most nucleation experiments, it seems that often, the nonwetting
bubble will be detaching leaving a minimal amount of gas behind. Composite
nucleation sites, with both hydrophilic and hydrophobic parts, could be an
exception.
C. Axisymmetric Profiles
A useful program was written by Boucher et al. [62] to calculate axisymmetric
profiles (meridian curves) for holms, bridges, pendent, and sessile drops and
bubbles. This program has been rewritten in Quick Basic, and is available
from the author. In combination with the theoretical volume of a bubble
deduced from the known interfacial tension, using the program, and the
theory outlined by Boucher [62] , the remaining volume can be estimated. An
example of the results for the calculated profiles for bubbles with various
shape factors, H (shown alongside the curves), are given in Fig. 22.
D. Detachment vs. Nucleation: The L Number

It has been noted above that heterogeneous nucleation is energetically
favored over homogeneous nucleation, and for this reason, surfaces are
usually implicated in nucleation processes. The kinetics of release of bubbles
in the presence of a surface are different from the kinetics in its absence, the
reason being that the surface introduces a new kinetic step, detachment, into
the overall rate of release. The ratio, L, of the rate of nucleation to that of
detachment is a significant quantity because it determines which is rate de-
termining in the overall release kinetics. For heterogeneous nucleation, the
three most important experimental variables are the saturation ratio, a, the
contact angle, h, and the interfacial tension, c. For detachment, the selection
of important variables depends upon the treatment of the detachment
process. In the case of (quasi) equilibrium detachment, a quantitative

FIG. 22 Plots of bubble profiles for various values of the shape parameter, H,
obtained from the QuickBasic program written by Boucher et al. [62].
treatment exists which allows a prediction of the regions in which detachment

dominates or is dominated by nucleation [37]:

J 3c sinðhÞ ð2 cosðhÞÞ 1=2
L¼
k 2gDq
The fraction J/k represents the ratio of the rate of nucleation J, to the rate of
growth k, of the bubble, while the term in brackets accounts for the effect of
the contact angle. The rate of growth of a bubble has been addressed by many
authors, and adequate theories are available; a recent example is given by
Bisperink and Prins [63].Therefore, in principle, it is possible to assess regions
in which classical nucleation theory will apply, and those where it is not
expected to hold. It emerges that the contact angle plays an important role,
because small contact angles make detachment easy, while inhibiting nucle-
ation, thus making nucleation the rate-determining step. On the other hand,
large contact angles make detachment difficult while promoting heteroge-
neous nucleation, thus making detachment the rate-determining step.
SYMBOLS
A Preexponential
Aj Cluster of j monomer units
Al,m Area of interface between phases l,m

b = ( p* PV)/p*
b, c, d Coefficients in expansion of surface tension
vs. pressure relationship
C Preexponential
cj Concentration
b Condensation coefficient
b Cone half angle
D Diffusion coefficient
Ea Activation energy
ED Activation energy for diffusion h i
g Fractional vapor pressure g ¼ exp VðPVp
kT
lÞ
cV2
cV1
f Harkins Brown adjustment factor
g Gravitational acceleration (f9.81 m sec2)
g Rate of growth of nucleus
h Planck’s constant
Hf Enthalpy of fusion
Hv Enthalpy of vaporization
i* Number of monomer units in critical nucleus
J Rate of nucleation
J(t) Time-dependent nucleation rate
J0 Steady state nucleation rate
k Boltzmann constant
kH Henry’s law constant
k Rate of growth of a bubble
L Ratio of rate of nucleation to detachment rate
M RMM
m Molecular mass
l Viscosity
s Nucleation time lag
P Pressure
PV Imposed hydrostatic pressure, or target pressure
p Pressure in bubble
pl Saturation vapor pressure (over a flat surface)
p* Pressure in critical bubble
p2 Partial pressure of gas in bubble
PL Pressure in liquid (hydrostatic pressure)
Pe Péclet number
R Universal gas constant
r Radius of bubble
r# Threshold radius for cavity
r* Critical radius
r0 Initial radius

Re Reynolds number, = DqUd/l
T Absolute temperature
TL Temperature in the liquid
Tsat Temperature corresponding to the saturated vapor
pressure at standard pressure
TW Temperature of the (heated) wall
U Rise rate of a bubble
V Volume
w Theoretical maximum weight of a detaching drop
wV Actual weight of a detaching drop
Z Zeldovich nonequilibrium factor
DG Change in Gibbs free energy
Dl Change in chemical potential
U Function of the two contact angles at a liquid/liquid
interface
X Molecular volume
a Saturation ratio ( p/PV)
/ Function of the contact angle
/cone A function of the contact angle and cone half-angle
/pit A function of the contact angle and pit half-angle
/(h) A function of contact angle for a spherical cap
c Interfacial tension
c0 Surface tension of the pure liquid in air at a pressure of
1 atm
cLV Liquid/vapor interfacial tension
cSV Solid/vapor interfacial tension
j Curvature ( = 1/r)
l Chemical potential
m Activity coefficient
h Contact angle
ha Advancing contact angle
hr Receding contact angle
q Density
r Supersaturation (= a 1)
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Bubble Nucleation and Detachment

Copyright n 2004 by Marcel Dekker, Inc. All Rights Reserved.

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II. HOMOGENEOUS NUCLEATION

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A. The Rate of Homogeneous Bubble Nucleation

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The formal similarity to the conventional Arrhenius equation is deceptive;

but the situation with DGtotal

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B. Bubble Nucleation by Raising the Temperature

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TABLE 1A Calculated and Early Experimentally Measured Values of the

Experimental boiling Calculated Theory/measured

Water 543 539 0.99

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Methane 161.5 107.5

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Hexaﬂuorobenzene 74.5 194.7 195.4 1.00

C. Bubble Nucleation by Lowering the Pressure

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Water 1380 270 5.11

D. A Closer Look at the Preexponential

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C can be broken down into two components: a concentration term relating

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G. Two Component (Gas/Liquid) Systems

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H. Dilute Gas Solution

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I. Higher Gas Concentrations

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TABLE 3 Variation in Surface Tension of Water with Dissolved Gas

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A. Heteronucleation at a Smooth Solid Surface

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The volume of the whole sphere is given by:

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B. Heteronucleation at a Liquid/Liquid Interface

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C. Heteronucleation at Rough Solid Surfaces

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FIG. 11 A bubble forming at a conical projection (a, on the left) or in a conical

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E. Importance of the Contact Angle

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