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A Critical Review of the Growth, Drainage and Collapse of Foams
Jianlong Wang, Anh V. Nguyen, Saeed Farrokhpay
PII: S0001-8686(15)00217-1DOI: doi: 10.1016/j.cis.2015.11.009Reference: CIS 1600
To appear in: Advances in Colloid and Interface Science
Please cite this article as: Wang Jianlong, Nguyen Anh V., Farrokhpay Saeed, A CriticalReview of the Growth, Drainage and Collapse of Foams, Advances in Colloid and InterfaceScience (2015), doi: 10.1016/j.cis.2015.11.009
This is a PDF file of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting, typesetting, and review of the resulting proofbefore it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers thatapply to the journal pertain.
http://dx.doi.org/10.1016/j.cis.2015.11.009http://dx.doi.org/10.1016/j.cis.2015.11.009
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A Critical Review of the Growth, Drainage and Collapse
of Foams
Jianlong Wang1, Anh V Nguyen
1* and Saeed Farrokhpay
2
1School of Chemical Engineering, University of Queensland, Brisbane, Queensland 4072,
Australia 2JKMRC, University of Queensland, Brisbane, Queensland 4072, Australia
*Correspondence: [email protected]
Highlights
• Mechanisms governing foamability and foam stability are reviewed.
• Studies on interfacial properties and foam films are inadequate for foam properties.
• Solid particles significantly affect film stability and foam drainage.
• Foam growth and collapse are poorly quantified in the literature.
Graphic abstract
Abstract
This review focuses on the current knowledge regarding (i) the mechanisms
governing foamability and foam stability, and (ii) models for the foam column kinetics.
Although different length scales of foam structure, such as air-water interface and liquid film,
have been studied to elucidate the mechanisms that control the foamability and foam stability,
many questions remain unanswered. It is due to the collective effects of different mechanisms
involved and the complicated structures of foam sub-structures such as foam films, Plateau
borders and nodes, and foam networks like soft porous materials. The current knowledge of
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the effects of solid particles on liquid film stability and foam drainage is also discussed to
highlight gaps in our present level of understanding foam systems with solid particles. We
also critically review and summarize the models that describe macroscopic foam behaviors,
such as equilibrium foam height, foam growth and collapse, within the context of the
mechanisms involved.
Keywords: foam, foam stability, foam drainage, foamability, froth flotation
Contents
1. INTRODUCTION ............................................................................................................................................ 3
2. MECHANISMS GOVERNING FOAMABILITY AND FOAM STABILITY ................................................................ 4
2.1 EFFECTS OF INTERFACIAL PROPERTIES ON FOAM PROPERTIES ...................................................................................... 4
2.1.1 Surface tension and foamability ........................................................................................................... 4
2.1.2 Surface viscoelasticity and foam properties .......................................................................................... 7
2.2 EFFECTS OF LIQUID FILM PROPERTIES ON FOAM PROPERTIES ..................................................................................... 11
2.2.1 Intermolecular forces between two interfaces ................................................................................... 11
2.2.2 Effects of antifoam behaviors on liquid film stability .......................................................................... 12 2.2.2.1 Entry barrier .................................................................................................................................................. 13 2.2.2.2 Role of oil spreading in antifoam performance ............................................................................................. 14 2.2.2.3 Effect of adsorption kinetics of surfactants on the antifoam performance .................................................. 15
2.2.3 Effects of solid particles on liquid film stability ................................................................................... 16 2.2.3.1 Minimum energy to remove particle at interface,
removeG and maximum capillary pressure for the onset
of coalescence, maxcP ................................................................................................................................................. 17
2.2.3.2 Bridging-Dewetting mechanism of liquid film rupture by solid particle ........................................................ 18 2.3 FOAM DRAINAGE FOR AQUEOUS FOAMS ............................................................................................................... 19
2.4 FOAM DRAINAGE IN THE PRESENCE OF SOLID PARTICLES ........................................................................................... 20
3. FOAM COLUMN KINETICS .......................................................................................................................... 25
3.1 MODELS OF EQUILIBRIUM FOAM HEIGHT .............................................................................................................. 25
3.1.1 Hrma et al.’s model ............................................................................................................................. 25
3.1.2 Hartland et al.’s model ........................................................................................................................ 26
3.1.3 Pilon et al.’s models ............................................................................................................................ 28
3.1.4 Limitations of the models for equilibrium foam height ....................................................................... 32
3.2 MODELS OF FOAM GROWTH AND COLLAPSE .......................................................................................................... 32
3.2.1 Models without consideration of foam collapse ................................................................................. 33
3.2.2 Empirical equations for foam growth and collapse ............................................................................ 35
3.2.3 Foam growth model relating to critical Plateau border size ............................................................... 37
3.3 SUMMARY ..................................................................................................................................................... 38
4. CONCLUDING REMARKS ............................................................................................................................. 39
ACKNOWLEDGEMENTS .................................................................................................................................. 40
REFERENCES ................................................................................................................................................... 41
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1. INTRODUCTION
Foams are highly concentrated dispersions of gas (dispersed phase) in a liquid
(continuous phase) [1]. Due to their lightness and large specific surface areas, foams are
widely applied in our daily lives and industry. Examples of applications in which foams are
used [2] include food, cosmetics, cleaning, surface treatment, building materials, reducing
pollution (e.g. wastewater treatment [3] and heavy metal removal [4] by foam fractionation)
and extraction of nature resources, i.e., froth flotation.
When producing a column of foam, different mechanisms either to produce and
stabilize the foam column, such as the formation of liquid films and foams or to destroy it,
such as coarsening of foams and drainage and rupture of liquid film and foams, get involved
[2]. The combined effects of these mechanisms determine the lifetime of a foam or foam
stability. The foamability of a solution is a measure of its capacity to produce a foam. During
the foam formation, the surface energy E A increases with the creation of gas-liquid
interfaces with the surface tension, and interfacial area A. The increased surface energy
means the foam formation is not a spontaneous process, and the input of energy is
indispensable to generate a column of foam. Based on the different ways to put energy into
the liquid phase to generate foam, foaming techniques are categorized into physical, chemical
and biological foaming [5]. We only focus on the mechanical foaming (the rotor-stator mixer
method in Section 2.1.1 and the bubbling method in Section 3) that belongs to the physical
foaming techniques in this review for the sake of its common application. Surfactants must
also be present in the solution to promote the production of foam. The addition of surfactants
stabilizes the liquid film and foams by altering the static surface tension (Section 2.1.1),
adsorption kinetics (Section 2.1.1), surface viscoelasticity (Section 2.1.2) and interactions
between two interfaces of a liquid film (Section 2.2.1). On the contrary, antifoams prevent the
formation of foam by destroying the liquid film (Section 2.2.2). Solid particles also play an
important role in the film stability (Section 2.2.3) and foam drainage (Section 2.4). Foam
drainage for the aqueous foams (Section 2.3) is also discussed due to its importance for the
foam growth and collapse. Finally, the models for the foam growth and collapse are reviewed
(Section 3).
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2. MECHANISMS GOVERNING FOAMABILITY AND FOAM STABILITY
Foamability and foam stability are two main foam properties of surfactant solutions.
Foamability is an overall capacity of surfactant solution to produce foams, whereas foam
stability refers to the lifetime of a foam column. These two terms are interrelated. For
example, the foamability of a transient foam is believed to depend on its stability. Although
these two terms are commonly used in the literature, there are no simple universal physical
parameters to quantify them. Foam height and foam lifetime have been applied to
characterize foamability and foam stability [6-8]. However, these two criteria are not
satisfactory because they are not only dependent on the chemical composition of foaming
solutions but also on the method of foam generation. Moreover, a general theory to explain
the mechanisms of foam formation and stability for all types of foam system does not exist
[9-11] because the magnitude and mutual importance of the different types of effects can vary
significantly, depending on the stage of foam life and the conditions of its existence [10].
Bearing in mind the complicated interplay of various mechanisms, this review focuses on the
existing theories that describe the mechanisms governing the foamability and foam stability.
2.1 Effects of interfacial properties on foam properties
The adsorption of surfactant molecules on the air-water interface alters the interfacial
properties, and the foam properties change accordingly.
2.1.1 Surface tension and foamability
If the external energy applied to generate the foam is constant, then the foam surface
area is inversely proportional to the surface tension. Therefore, a lower surface tension will
increase the foamability of a solution from the perspective of surface energy. The other factor
that influences the foamability is the bubble breakup. Foam generation involves creation and
deformation of gas-liquid interface. When the mechanical method applies to foam generation,
bubbles deform due to the external forces (e.g., shear stress or pressure of turbulence) that are
subjected to the bubble surface. If the deformation is significantly large, bubbles can be split
into smaller ones, and foam generation is possible. The critical Weber number, which is a
dimensionless ratio between the inertial force (causing the deformation) and the surface
tension (restoring the bubble sphericity), has been used as the criterion of the bubble breakup
[12-18]. The bubble breakup occurs if the external forces applied to the bubbles make the
Weber number exceed its critical value, which is of the order of unity. A lower surface
tension ends up a large Weber number, therefore, promotes the bubble breakup process.
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Although the role of surface tension in the foamability has been recognized regarding
surface energy and bubble breakup process, the equilibrium surface tension is not reached
instantaneously, and the dynamics of adsorption of surfactant molecules must be considered
when the adsorption time is longer than the time scale of foam generation [10]. The evolution
of the surface tension value is controlled by two processes: (i) the diffusion of surfactant
molecules to the surface and (ii) the adsorption of surfactant molecules on the interface,
which must overcome an associated energy barrier. Either of these processes may become the
rate-determining step. The average lifetime of the bubbles at the solution/air interface during
foam generation has been recommended as a rational choice of an adsorption reference time
[19].
Research efforts of using different designs have focused on the effect of dynamic
surface tension on the foamability. For example, the relationship between foam formation and
dynamic surface tension of non-ionic and anionic surfactants has been studied using a rotor
test, in which air is introduced to the surfactant solutions by a special stirring device [19]. A
good correlation between foam formation and dynamic surface tension values at t = 100 ms
has been found (Figure 1). The relationship between foam formation and dynamic surface
tension has also been investigated using the Ross-Miles test [20-22]. Recently, the foam
formation by using a sparger was also related to the dynamic surface tension [23]. Figure 2
shows a good correlation of foam height with the surface tension reduction rate, which is
defined as follows [24]:
01/2 *2
mRt
(1)
where 0 is the surface tension of the solvent, m is the quasi-equilibrium surface tension at
which the decrease rate of the surface tension is smaller than 1 mN/m per 30s, and *t is the
time when the surface tension is equal to 0 / 2m . Although the correlation between
foam height and surface tension reduction rate has been found, we note that the surface
tension reduction rate in Eq. (1) [22, 24] is only an empirical equation and lacks a
fundamental basis.
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Figure 1. The relative dynamic pressure at t = 100 ms and the rate of foam formation for the
non-ionic surfactant C12(EO)6 (a, b) and the anionic surfactant C12SO3Na (a’, b’). The rate of
foam formation refers to a rotor speed of 900 min-1
. The graphs are reproduced from [19].
Figure 2. Overview of the results from the paper by Rosen et al. [22], showing the relation
between R1/2 and the foam height in a Ross-Miles test, reprinted from [23]. The solid line is
obtained by fitting a logarithmic equation (R2 = 0.9).
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2.1.2 Surface viscoelasticity and foam properties
A century ago the effect of surface tension gradients on foam and film stability was
discovered and described by Gibbs and Marangoni [25]. In 1941, Levich developed a theory
to relate surface tension gradient with surface dilational elasticity [26]. However, only until
1970 [27] a new methodology for measuring the dilational rheology via a harmonically
oscillating bubble has paved the way for the first commercial instrument for routine
experimental determination of the dilational surface elasticity [28], based on oscillating drops
and bubbles. Various aspects of surface viscoelasticity have been reviewed [29-33] and a
book devoted to surface viscoelasticity has been published recently [34].
The dilational elasticity modulus, E , is defined as the ratio of the surface tension
change, , to the relative increase in surface area, /A A , i.e., / lnE d d A . Surface
dilational elasticity and viscosity are expressed as 0' / cosE A A and
0'' / / sinE A A , respectively, where , 0A , , are the phase shift between the
surface tension and area oscillations, the mean surface area and the angular frequency (Figure
3, bottom panel). For the film elasticity, its value should be two times the value of E because
of the presence of two film interfaces. It is noted that an analogy between the Gibbs elasticity
and the surface dilational modulus has been made [35]. The Gibbs elasticity refers to the
increase in the film surface tension resulting from a decrease in the surfactant concentration
within the interlamellar solution caused by the small extension of the film relative to the film
size. In the Gibbs theory, the film elasticity originates from the deterioration of the interstitial
surfactant solution with the assumption that the thickness of the lamellae is very small. The
Marangoni elasticity, as related to the Marangoni effect, originates from the transport of
surfactant molecules from the adjacent bulk phase to the interface for the case of the non-
equilibrium state of thick foam films [36]. For soluble surfactants, the magnitude of the
elasticity modulus depends on the frequency of external disturbances or the oscillation
frequency. The adsorption layer will behave as an insoluble monolayer when the frequency is
sufficiently high, and the Marangoni dilational modulus reaches its limiting value [37, 38]. At
low frequencies, the adsorption layer behaves as a viscoelastic surface because of relaxation
processes, i.e., diffusional exchange that occurs in and near the interface (The phase shift
between surface tension and area changes in the bottom panel of Figure 3). Therefore, the
modulus E has both elastic and viscous components. The surface viscoelasticity of
adsorption layers and the origin of the dilational viscosity are analyzed in a recent review
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[39]. The dilational surface viscosity is categorized as 1) “true” dilational surface viscosity,
which originates only from the transport of surfactant molecules at the interface, and 2)
“apparent” dilational surface viscosity caused by the surfactant diffusion from the bulk
solution.
Surface viscoelasticity is related to the non-equilibrium state of the adsorption layer
[40]. A distortion of its existing equilibrium state and the absence of adsorption equilibrium
at a freshly created interface are the two main reasons underlying the non-equilibrium state of
the adsorption layer. Because the interface has not yet attained its equilibrium adsorption
coverage during foam generation, surface viscoelasticity plays a significant role in
foamability. Assuming that the surface area of the film elements increases while its volume
and the total amount of surfactants remain constant, and the diffusion of surfactant molecules
can be ignored, the surface concentration of surfactant molecules decreases during liquid film
thinning (Figure 3, top). The reduction in surface concentration induces a local increase in
surface tension (Figure 3, bottom), which in turn resists the stretching of the liquid film.
Consequently, bubble coalescence rate decreases. In practice, the film thinning process is
simulated by oscillating a droplet using drop profile analysis tensiometer (Figure 3, middle).
The surface tension gradients induced at the air-water interface can stabilize the foam films
by retarding drainage [41].
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Figure 3. Schematic presentation of the adsorbed surfactant molecules during the thinning of
a liquid film by stretching (top, reproduced from [42]), oscillation of a droplet using drop
profile analysis tensiometer (middle) to simulate liquid film thinning and transient change in
the surface tension and surface area of a droplet with a mixture of 0.5 mM SDS and 5mg/L
dodecanol, oscillated at the frequency of 0.05 Hz (bottom). Amplitudes of surface tension and
area, with a clear phase shift between surface tension and area changes. Reprinted from [43].
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Good correlations between the liquid film or foam stability and the surface
viscoelasticity have been reported [38, 44-60]. However, many questions remain challenging:
Is the good correlation between film stability and surface rheology indicative of a similar
correlation with foam stability? Are measurements at the interfaces (i.e., surface
viscoelasticity) and on a single film (i.e., film stability) sufficient to understand and predict
the foamability and foam stability [2]? What is the appropriate frequency at which the surface
viscoelasticity can correlate with the foamability and foam stability? For the first and second
questions, because the foams cannot be considered as a simple combination of foam films,
the correlation between film stability and surface rheology cannot be directly applied to foam
stability. Moreover, it is challenging to correlate the foam column stability with the liquid
film stability [61, 62]. This discrepancy has been explained by “the conditions of the
existence of the foams and the single film were quite different and, therefore, the different
types of forces could operate there [10].” For the last question, on the one hand, the
frequency of the measurements of the surface viscoelasticity should be relevant to the surface
age during foam generation (0.1–1 s) regarding foamability. On the other hand, the two main
processes that determine the foam lifetime, that is, the Ostwald ripening and bubble
coalescence, have been shown to be controlled by the low- and high-frequency surface
elasticities, respectively [51]. The roles of surface viscoelasticity in the foamability and foam
stability remain poorly understood. The link between surface viscoelasticity and foam
properties are summarized as follows [2]:
For a given applied stress, a large dilational surface elastic modulus results in a small
strain, the foam film will stretch less, and is less likely ruptured, by disturbances.
The film elasticity will restore the interface by bringing back surfactants and by limiting
the stretch of the interface.
The Marangoni effect will draw surfactants back to the interface and liquid into the film,
thereby reducing the possibility of liquid film rupture.
Low surface viscoelasticity facilitates stretching interfaces, resulting in liquid film rupture,
but high surface viscoelasticity causes a solid-like response of the interface, causing
fracture.
Surface elasticity modifies the process of drainage in foam films and Plateau borders and
controls the appearance of bell-shaped liquid drops that have a destabilizing effect on
films.
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2.2 Effects of liquid film properties on foam properties
2.2.1 Intermolecular forces between film interfaces
Foams are made of liquid films and Plateau borders. Therefore, the link between liquid
film properties and foam properties is critical to our understanding of their behaviors. Foam
films inspired many pioneers like Boyle, Hooke [63], Newton [64], Plateau [65] and Gibbs
[25, 66]. The history of foam and foam films is very rich as recently compiled [67]. The
disjoining pressure and DLVO theory developed for describing the liquid film stability are
considered as milestones in the theoretical development of foam films.
When two bubbles meet in foam, forces act between the two interfaces. These forces
determine the stability of the liquid film that separates the foam bubbles. The disjoining
pressure, , was introduced by Derjaguin in 1930’s to describe the force per unit area
between the two interfaces of a liquid film [68, 69]. The DLVO theory is based on the
following attractive and repulsive interactions [17, 42, 70-74]:
The van der Waals interactions originate from the electromagnetic fields between two
dipoles. The attractive London-van der Waals disjoining pressure is considered:
36van
A
h (2)
Where A is the Hamaker constant and h is the film thickness. The van der Waals
attractions always destabilize foam films. Therefore, repulsive forces produced by
surfactant and electrolyte molecules are needed to balance the attractive force to
stabilize foam films and foam dispersions.
The electrostatic interaction becomes evident when the interfaces are electrically
charged in the presence of ionic surfactants. The repulsion between the two charged
interfaces stabilizes foam films. The presence of salts such as NaCl screens this
repulsion. The Debye length defines the effective range of the Coulombic force on
ions against their thermal diffusion at a charged interface:
022
Bk T
nze
(3)
where 0 and are the dielectric constant of vacuum and the relative permittivity of
the solution, Bk is the Boltzmann constant, T is the absolute temperature, n is the
ion concentration in the solution, z is the valence of a symmetric z:z electrolyte and e
is the elementary charge. The repulsion can be clearly observed when the film
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thickness h falls below 2 D . The contribution of electrostatic repulsion to the
disjoining pressure approximately decays exponentially:
exp /dl Dh (4)
In addition to the DLVO disjoining pressure components, non-DLVO disjoining pressures,
non-DLVO, such as the steric repulsion, hydration repulsion and hydrophobic attraction have
also been considered [17], giving.
van dl non DLVO (5)
Figure 4. Schematic of a disjoining pressure isotherm for foam films that includes
contributions from dl , van and steric for the non-DLVO pressure. Reproduced from [75].
Shown in Figure 4 is an example of the disjoining pressure versus film thickness. A
positive disjoining pressure in the film is fundamental to film stability and the existence of a
foam. Therefore, a strong and long-range repulsive interaction in the film is necessary for
good foamability and foam stability.
2.2.2 Effects of antifoam behaviors on liquid film stability
Antifoams are oils, hydrophobic solid particles or a mixture of both that are present in
the solution and prevent the formation of foam [76, 77]. Antifoams have been widely used in
many industrial applications, such as pulp and paper production, food processing, textile
dyeing, fermentation, wastewater treatment, and the oil industry [76, 78-81]. In the froth
flotation of naturally hydrophobic minerals, such as coal, graphite, sulfur and molybdenite,
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nonpolar collectors are used [82, 83]. They are commonly petroleum-based hydrocarbon
liquids, such as diesel oil. Therefore, nonpolar collectors potentially exhibit antifoam
behaviors and affect the froth stability. Moreover, some nonionic surfactants, such as
dodecanol, can also influence the foam properties by exerting antifoam actions. In this
section, the mechanisms of liquid antifoam actions and the effects of surfactants on antifoam
activity are discussed. Systematic reviews discussing antifoams are available in the literature
[76, 77, 79]. It should be noted that the terms “fast antifoams” and “slow antifoams” have
been previously introduced [77]. The former term denotes “the antifoams whose globules can
enter the surfaces of the foam films and to destroy these films in the early stages of film
thinning”, whereas the latter indicates “antifoams whose globules first leave the foam films
and destroy the foam after entering the walls of the Plateau borders.” [77]. Here, only the
mechanisms of “fast antifoams” are discussed here.
2.2.2.1 Entry barrier
Antifoams must enter the liquid film to destroy a liquid film or foam layer. Antifoams
with a low entry barrier completely collapse the foam in seconds, whereas antifoams with a
high entry barrier require hours to destroy the foam. The interaction energy per unit area in an
asymmetric oil-water-air film [84] and the so-called generalized entry coefficient [85-88]
have been introduced to represent the entry barrier. However, the determination of their
values is very difficult [77]. Alternatively, the capillary pressure of the air-water interface at
the moment of oil drop entry, CRCP , has been proposed as a quantitative characteristic of the
entry barrier because it is related to antifoam efficiency [89-97]. The film trapping technique
(FTT) has also been developed to measure precisely the value of CRCP [98, 99].
Because the antifoam activity strongly depends on the magnitude of the entry barrier
[77], it is crucially important to understand the factors that affect the entry barrier. Here, the
two main points discussed in the literature concerning the factors that affect the entry barrier
of an oil droplet are summarized.
Entry coefficient
Antifoam activity has been correlated with the entry coefficient [76, 79]:
AW OW OAE (6)
where AW , OW and OA are the surface tensions of an air-water interface, oil-water
interface, and oil-air interface, respectively. From the thermodynamic perspective, the
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condition for the emergence of an oil droplet at the air-water interface is 0E . However, it
has been correctly stated that a positive value of E does not guarantee high antifoam
performance because, from the perspective of kinetics, the entry barrier also plays a crucial
role [76-80, 100-102]. Regarding their contributions to antifoam performance, the
relationship between the entry coefficient E and the entry barrier is analogous to chemical
thermodynamics and kinetics [77].
Surfactant concentration
The effect of surfactant concentration on the entry barrier has been studied previously
[93]. Figure 5 shows that the entry barrier increases as the surfactant concentration increases.
The oscillatory (structural) forces shown in Figure 5 is non-DLVO forces and appears in thin
liquid films containing colloidal particles (surfactant micelles for this case). The origin of this
structural forces has been discussed in [103]. It is noted that the effect of the surfactant
concentration has not yet to be fully understood [77].
Figure 5. Entry barrier of hexadecane drops, CRCP , as a function of the SDDBS concentration,
SC [93]. All solutions contain 12 mM NaCl. Reproduced from [77].
2.2.2.2 Role of oil spreading in antifoam performance
When an oil droplet emerges at the air-water interface, depending on the sign of the
spreading coefficient, AW OW OAS [104], it either spreads out ( 0S ) or bridges the
two interfaces ( 0S ) [100-102, 105-112]. Both outcomes result in rupture of the liquid film.
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Similar to the entry coefficient, E , S is also a thermodynamic property. A positive initial
spreading coefficient INS (defined using AW without the spreading oil) has been shown to
contribute to antifoam performance [110, 113]. Because the calculation of S requires the
value of the oil-water surface tension, OW , which is difficult to measure, the spreading
pressure, i fAW , which is defined as the reduction in the equilibrium surface
tension of the air-water interface caused by the addition of an antifoam to the aqueous surface,
was introduced [112]. f and i are the equilibrium surface tensions of surfactant solutions
with and without antifoams, respectively. A positive value of AW indicates that it is
thermodynamically favorable for antifoams to spread on the surface of the surfactant solution.
2.2.2.3 Effect of adsorption kinetics of surfactants on the antifoam performance
Some antifoams have been found only to affect the foamability and exert no influence
on foam stability [91, 92, 114]. For example, Figure 6 shows that in the presence of antifoams,
the initial foam volume (foamability) generated by shaking AOT solutions is several times
larger than that of APG solutions. However, the foam stability of APG foams is much higher
than that of AOT foams. Solutions without antifoams exhibit both good foamability and foam
stability. The different effects of antifoams on the foam stability of AOT and APG are
explained by the different entry barrier values ( 125CRCP Pa for APG solutions, and
3CRCP Pa for AOT solutions). The much higher entry barrier for APG solutions makes the
entry of antifoams into the liquid film more difficult and results in more stable foams relative
to AOT solutions. However, during foam generation, the antifoams become more active for
the APG solutions because of the slow adsorption kinetics of APG molecules (Figure 7). The
unsaturated adsorption layers make the entry of antifoams much easier relative to the fully
saturated surfaces. It is noted that the possible higher surface viscoelasticity deduced from the
slow adsorption kinetics of APG molecules may also contribute to the higher foam stability
of APG foams relative to AOT foams. Additionally, the fast adsorption of AOT molecules on
the air-water interface may facilitate foam formation and result in a higher foamability than
the APG solutions (see Section 2.1.1).
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Figure 6. Foam volume vs. time for two surfactant solutions -10 mM AOT and 0.6 mM APG
- containing 0.01 wt% PDMS-silica compound (Bartsch test). For comparison, in the absence
of an antifoam, the initial foam volume was 180 ± 10 mL for AOT and 100 ± 10 mL for
APG, and the foam was stable for the duration of this experiment [91]. Reproduced from [77].
Figure 7. Dynamic surface tension of AOT and APG solutions measured by MBPM: 10 mM
AOT and 0.6 mM APG were used in the foam tests (Figure 6). For comparison, the results for
equal surfactant concentrations (i.e., 2.5 mM) are also shown [91], reproduced from [77].
2.2.3 Effects of solid particles on liquid film stability
Similar to surfactants, solid particles can adsorb to air-water interface and alter the
interfacial properties [115-117]. Although effects of the hydrophobicity [118-122], size [123]
and shape [118, 124] of solid particles on liquid film stability have been observed, the
mechanisms that solid particles stabilize or destabilize liquid films are still not well
quantified. It has also been found that solid particles influence the interfacial properties of
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liquid films and bubbles not only under equilibrium conditions but also under dynamic
conditions (such as the bubble coalescence process) [116]. Despite the complicated effects of
solid particles on the stability of liquid films, here we review the theory of maximum
capillary pressure induced by solid particles to stabilize the liquid film and the bridging-
dewetting mechanism of liquid film rupture by solid particles, respectively, to explain how
solid particles can stabilize and destabilize the liquid film. We also discuss the effects of solid
particles on foam drainage in Section 2.4 from a macroscopic perspective.
2.2.3.1 Minimum energy to remove particle from interface, removeG and maximum capillary
pressure for the onset of coalescence, maxcP
To stabilize the liquid film, solid particles are required to stay at the air-water interface.
The stability of a solid particle at the air-liquid interface is quantified by the minimum energy
required to remove the particle from the air-water interface, removeG as follows [116, 125,
126]:
22 1 cosremoveG R (7)
where R is the radius of the solid particle, is the surface tension of the air-water interface
and is the contact angle of the liquid phase on a solid particle (the particle hydrophobicity).
Sign “+” refers to particle removal into the gas phase while sign “-” refers to the removal of
the particle into the liquid phase. Eq. (7) indicates that the highest stability of solid particle at
the air-water interface occurs when the contact angle equals 90°.
Bubble coalescence occurs only if the two interfaces of the liquid film contact. Suppose
the solid particles form a single layer between the two bubbles, Kaptay [126] derived
theoretical equations for the maximum capillary pressure for the onset of coalescence, maxcP ,
inspired from the work of [127] as follows (Figure 8):
max 2 coscP pR
(8)
where p is a parameter describes the effect of the packing and interfacial coverage by the
particles on the capillary pressure. Eq. (8) indicates that the liquid film will be stabilized by
the single layer of particles if the contact angle is lower than 90° ( maxcP is positive). On the
contrary, the liquid film will be destabilized if the contact angle is larger than 90° ( maxcP is
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negative). Both Eqs (7) and (8) should be considered when the effect of solid particles on
film stability is analyzed. Based on the combined analysis of Eqs (7) and (8), the optimum
contact angle for the stabilization of liquid film by a single layer of particles is around 70°
[126].
Figure 8. Schematic of the packed single layer of spherical particles at the air-water-air
interfaces, reprinted from [126].
It is noted that the experimentally measured maximum capillary pressure for the solid-
stabilized emulsions is just 0.25 of the calculated value based on Eq. (8) [128]. This
discrepancy was explained by a sharp decrease of the capillary component of the film
elasticity in the emulsions and by the defects of the particle layer packing.
2.2.3.2 Bridging-Dewetting mechanism of liquid film rupture by solid particle
Solid particles can act as antifoam to destroy the liquid films. The bridging-dewetting
mechanism was established to explain the antifoam action of solid particles [76, 77, 79, 107,
109, 118, 129]. First, the solid particle contacts with the two opposite interfaces of the liquid
film during the film thinning and form a solid bridge between them. Then the particle surface
is dewetted, and the three-phase contact lines come in direct contact with each other, which
makes liquid film rupture (Figure 9). Evidently, for solid spheres the film rupture happens
when the contact angle is larger than 90°, which is consistent with the physical meaning of
Eq. (8).
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Figure 9. Schematic presentation of the bridging of liquid film by a solid particle, reprinted
from [129]. For spherical particle, the film rupture happens when the contact angle is larger
than 90°.
2.3 Foam drainage for aqueous foams
Foam drainage is the passage of liquid through a foam. Foam drainage is crucially
important for foam stability. There are three different mechanisms governing the lifetime of a
foam: (i) foam drainage caused by gravity, (ii) coarsening caused by the transfer of gas
between bubbles induced by the capillary pressure differences, and (iii) bubble coalescence
caused by the rupture of liquid films between neighboring bubbles [2]. The coarsening
process cannot directly affect the foam lifetime. However, it has been found that the
coarsening and foam drainage are strongly coupled since the coarsening rate is sensitive to
the liquid fraction that is controlled by foam drainage, and conversely, foam drainage is
strongly dependent on bubble size, which is controlled by coarsening [2, 130-132]. Therefore,
coarsening can influence the foam lifetime via its effects on foam drainage. Among these
three processes, foam drainage determines the liquid fraction of a foam, which is a key
parameter for both coarsening and bubble coalescence [130, 133-145]. Different types of
drainage regime (i.e., forced, free and pulsed drainage) have been observed [2, 9, 146-148]. A
forced drainage experiment consists of pouring the foaming solution at constant flow rate into
a foam that has already been allowed to drain freely [2, 9, 149, 150]. The theories developed
for the foam drainage of aqueous foams are discussed below.
During foam drainage, the liquid is confined in a network of channels or Plateau
borders, which join at a node in four. Therefore, modeling foam drainage primarily focuses
on the liquid flow in the two foam structures. Some researchers have also considered the
contribution of the liquid film to the foam drainage [151, 152]. However, the liquid films
have not been found to significantly contribute to the drainage process because of the small
amount of liquid contained in films relative to that in the Plateau borders and nodes [151].
Studies on foam drainage can be categorized into microscopic and macroscopic
investigations [151]. The former refers to studies at the scale of a single Plateau border,
whereas the latter refers to studies at the scale of at least several bubbles. The first study on
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the microscopic modeling of foam drainage only considers the contribution of Plateau
borders to the foam drainage [153, 154]. The drainage rate is thought to be dependent upon
the mobility of the Plateau border wall or the surface shear viscosity. A Plateau border-
dominated approach to studying foam drainage was developed by subsequent researchers,
who expanded on the initial microscopic modeling attempt [155, 156]. Nguyen improved on
the work of [153] and provided a numerical solution for the liquid flow velocity in a single
Plateau border as a function of surface viscosity [157]. The Plateau border-dominated
approach was challenged when the contribution of nodes to the foam drainage was
recognized, and mobile Plateau border walls were assumed [158]. Since then, the standard
foam drainage equation [153] has been modified [159], and foam drainage models that
consider viscous losses from both the Plateau borders and nodes have been proposed [146,
160].
On the macroscopic level, foam drainage has been analogous to the liquid flow
through a porous medium [2, 161]. However, two key differences between liquid drainage
through a foam and that through a porous medium must always be considered. First, in a
foam, the size of the network (i.e., Plateau borders and nodes) through which the liquid flows
is not fixed but is actually dependent on the flow itself: The bubbles can move apart to allow
liquid to pass and then move back, and the interfaces are not completely rigid but are instead
partially mobile in a foam, depending on the interfacial properties, such as surface shear
viscosity. It has been demonstrated that the Darcy law that describes the fluid flow through a
porous medium at low Reynolds numbers is also applicable to aqueous foams [133, 158, 161-
163]. An alternative way to study foam drainage on the macroscopic level is to adopt
dimensional analysis to compare the existing foam drainage data in a consistent manner and
thus simplify the analysis [162]. One of the applications of foam drainage study is to
calculate the liquid profile in pneumatic foam, which is a key factor to understand the liquid
transport in a foam column. Different theories have been proposed to predict the liquid
profile [164, 165]. However, further experiments are needed to validate these theories.
2.4 Foam drainage in the presence of solid particles
Despite the progress made in understanding aqueous foam drainage [2, 9], foam
drainage in the presence of solid particles remains poorly understood. The study of the
drainage of three-phase foams or froths is crucially important for many industrial applications.
For example, in froth flotation, the wash water is commonly applied to the froth phase to
flush the entrained gangue particles out of the froth and consequently increase the grade of
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hydrophobic particles. The reports on foam drainage in the presence of solid particles are
reviewed here.
Theories of foam drainage for aqueous foams have established benchmarks for
studying three-phase foams in which solid particles are present. Logically, the following
questions have been raised regarding foam drainage in the presence of solid particles: Can the
foam drainage equations for aqueous foams apply to foams with solid particles? How does
the presence of solid particles influence foam drainage?
Not surprisingly, in the past 15 years, many studies on three-phase foams have focused
on the transport or motion of solid particles in the flotation froth [166-168] or surfactant
foams [169-172] because of their paramount importance for industrial applications. The solid
particles in flotation froths have been divided into attached particles (i.e., hydrophobic
particles), which follow the bubbles, and unattached particles (i.e., both hydrophobic and
hydrophilic particles), which mainly follow the liquid [166]. On the one hand, attached
hydrophobic particles adsorb to the air-water interface and act as a barrier to prevent bubble
coalescence and impede the coarsening process [115, 173-175]. However, their effects on
foam drainage remain unknown. On the other hand, studies on foam drainage in the presence
of solid particles have primarily focused on the unattached hydrophilic particles and
nanoparticles [167, 170-172, 176-183]. Nevertheless, these studies contribute to our
understanding of drainage behaviors in the presence of solid particles. For example, by the
scaling behavior (power law) between the drainage velocity and the imposed flow rate in
forced drainage experiments [178], the presence of nanoparticles has been found to induce a
foam drainage transition from a node-dominated regime to a Plateau border-dominated
regime. Moreover, atypical results such as large foam permeability exponents and prefactors
which have not been obtained in aqueous foams, have been recorded in three-phase foams
[176]. It should be noted that these observations in three-phase foams cannot be explained
simply based on the theories developed for aqueous foams. For example, the foam regime
transition in aqueous foams is usually caused by a change in the surface viscosity or interface
mobility [184, 185]. However, it is difficult to make the same claim in foams containing
nanoparticles because the presence of hydrophilic solid particles can change the interfacial
properties to only a small extent [178].
To explain the foam drainage behaviors in the presence of hydrophilic particles, several
mechanisms have been proposed, such as rheology of the powder suspension and clogging in
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the confined regions of the Plateau borders [171, 176, 179, 181-183]. Among them, one of
the important parameters controlling the drainage behaviors of foams with hydrophilic
particles is the confinement parameter, /p cd d , which relates the size of the particle, pd ,
to the maximum diameter of the circle inscribed in the Plateau border cross-section, cd
(Figure 10). The hydrophilic particles are trapped in the network of aqueous foams either by
the mechanism of collective trapping – jamming – of the suspension for 1 [182], or by the
individual capture of the particles by the foam constrictions for 1 [172, 179, 181, 183].
The other important parameter that accounts for the effect of hydrophilic particles on foam
drainage is the volume fraction of particles in the suspension, p . It has been found that the
foam drainage velocity is reduced when p is sufficiently high even if 1 [182].
Figure 10. Network of aqueous foams containing hydrophilic solid particles. Particles
suspended in interstitial liquid can be either freely transported or trapped, depending on the
confinement parameter, , that compares the particle size with the size of passage through
those constrictions, cd , reprinted from [186].
Very recently, a systematic study on the joint effects of the volume fraction of particles
in the suspension, p and the confinement parameter on the foam drainage in the presence of
hydrophilic particles, was reported [186]. In this study, all the regimes and transitions of
reduced foam drainage velocity induced by the presence of hydrophilic particles are
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identified experimentally based on two parameters, p and . Particularly, the authors
identified the particle inclusion/exclusion transition, which refers to the transition from full
inclusion to full exclusion of solid particles with respect to the foam network. This transition
makes the foam drainage velocity evolves from its minimal value (fully included particles) to
its maximal one (fully excluded particles), due to the decrease in the fraction of particle
surface area in contact with the foam network and consequently the reduction of the drag
experienced by the draining liquid. Different regimes are divided by two particular values of
the confinement parameter, c and * , that correspond to the end of the flowing suspension
regime and the minimal foam drainage velocity, respectively. Specifically, five regimes are
identified for the low and moderate particle volume fractions, i.e., 0.40p (Figure 11a):
(i) Flowing suspension regime ( c ), where the drainage velocity is almost constant (for
0.16p ) or just decreases slightly with ;
(ii) Capture transition regime ( *c ), where the drainage velocity decreases
significantly (for 0.40p );
(iii) Maximum frictional drag regime ( * ), where the drainage velocity becomes minimal;
(iv) Particle exclusion transition regime ( * 10 ), where the drainage velocity increases
to the values that are close to the velocities in the first regime;
(v) Minimal frictional drag regime ( 10 ), where the increase in the drainage velocity is
much less pronounced than that of the previous regime.
For 0.40p , the foam drainage stops in the range of moderate values of , but starts again
for higher values of (Figure 11a). The authors also identified region of the jammed state
by giving the following equations for 1 and * , respectively (Figure 11b):
21 / 3 0.1
0
pack
p
pack
p
( 1 ) (9)
3
6 0.72 11
pack
p
C
C
( * ) (10)
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where is the gas fraction of the foam ( 0.90 in [186]), packp is the critical particle
volume fraction beyond which the particle packing occurs [182], 0packp represents the
value of packp for unconfined conditions and C is expressed as
0.27 2.751 0.57 1 / 0.27 1 3.17 1C by numerical simulations of foam
structures [172].
Figure 11. (a) Reduced viscous drag (inverse of the reduced drainage velocity, 0/V V V ,
defined as ratio of foam drainage velocities with particles and without particles) of foamy
suspensions as a function of the confinement parameter , for several particle volume
fractions. c and * , correspond to the end of the flowing suspension regime and the
minimal foam drainage velocity, respectively. (b) Diagram of the reduced particle volume
fraction as a function of the confinement parameter in the foam network. The jammed state
domain is deduced from experiments. The red and black lines correspond to Eqs. (9) and (10),
respectively. Reprinted from [186].
Recognizing the dependence of foam drainage velocity on p and is significant not
only for understanding the foam drainage with hydrophilic particles but also for the ones with
unattached hydrophobic particles. The works from the Pitois group [186] could help the
understanding of the complex froth behaviors in froth flotation.
The following gaps have been identified by the current knowledge of foam drainage in
the presence of solid particles:
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The effects of attached hydrophobic particles on the foam drainage remain unclear.
The effects of solid particles on the stress state of the gas-liquid interface of foams
remain poorly understood.
It is unclear whether the foam drainage equations for aqueous foams can be applied to
three-phase foams.
3. FOAM COLUMN KINETICS
Foam column kinetics describes the transient behaviors of foams, including the growth,
drainage, and collapse of foams. Foam column kinetics is crucially important for many
industrial applications. For example, the precise control over the froth phase becomes
particularly important in froth flotation because of the strong dependence of its performance
(product grade or selectivity and recovery) on the froth stability [11, 17, 187]. Foam column
kinetics has been used to predict the froth rising velocity, which can be linked to a key
parameter in froth modeling and plant operation, that is, the fraction of bubbles bursting on
the top surface of the froth [188-190]. Foam column kinetics has also been applied in glass-
melting furnaces [191-194]. We critically review the models for equilibrium foam height and
foam growth. Then, we summarize the models for foam column kinetics that are available in
the literature.
3.1 Models of equilibrium foam height
3.1.1 Hrma et al.’s model
Bikerman introduced the concept of “unit of foaminess”, (s) as follows [6, 7]:
max
g
H
j (11)
where max ( )H m is the equilibrium foam height under the superficial gas velocity,
3 2( / ) ( ) / ( )gj m s Q m A m with volumetric gas flow rate Q and column cross-section area A .
Despite its wide application to represent the foam or froth stability [188-190, 192, 195, 196],
this expression is limited by its dependence on the gas flow rate [192, 197]. It is to note that
an expression of Bikerman’s unit of foaminess, , regarding relevant physical quantities and
the average bubble volume has been proposed [198]:
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31/2 1/2
02
1
0.16H
b
f
Vg
(12)
where and are liquid density and viscosity, is the surface tension, bV is the average
bubble volume, g is the gravitational acceleration, f is a numerical factor that characterizes
the boundary conditions (cross-section and surface mobility) for the flow of liquid through a
Plateau border ( 49f for flow through a Plateau border under no-slip boundary conditions,
i.e., the Poiseuille flow, and its value is reduced in the case of slip boundaries [151, 199]),
and 0 and H are the liquid fraction at the bottom and the top of the foam column,
respectively. However, the authors also pointed out that the product of the flow factor f and
the liquid fraction H complicates the interpretation of any measured value for in
Bikerman’s test [198].
An equation has been proposed to express the equilibrium foam height as a function
of the effective average bubble radius, r ; the critical superficial gas velocity, cj , beyond
which the foam will grow without limit; the minimum superficial gas velocity required to
generate foam, mj ; and the superficial gas velocity, j , as follows [192]:
max 2 / /c m c mH r j j j j j j (13)
Equation (13) is subjected to three conditions, namely, (i) m cj j j , (ii) the liquid film
ruptures as soon as the critical film thickness is reached, and (iii) foam height is proportional
to the gas phase flux. Only when cj j and m cj j , Eq. (13) can be expressed as
max / 2 / mH r j j , where 2 / mr j such that a liquid with a low gas velocity threshold is
more “foamable”.
Hrma’s model successfully explains the experimental data qualitatively and gives
useful insights into the mechanism of foam formation and stability [194]. However, the
model cannot predict the equilibrium foam height due to the lack of expressions for cj and
mj . Consequently, it is impossible to validate the model by comparing the model with
experimental data.
3.1.2 Hartland et al.’s model
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The first attempt to predict the equilibrium foam height dates back to 1974 [200]. In
this model, the foam height, maxH , is related to the critical liquid film thickness, , the liquid
density, , viscosity, , surface tension, , gas velocity, v , and bubble diameter, d . It is
expressed by the following equation:
5/4
max
1/4 7/4
0.55 vdH
d g
(14)
The authors also compared their model with experimental results by plotting against
5/7 1/7 4/79/7
max/v d g H as shown in Figure 12 with the following equation as the
theoretical line that comes from Eq. (14):
5/7 9/7
1/7 4/7 4/7
max
0.71 v d
g H
(15)
Because the regression line with a slope 0.62 is close to the theoretical line with a slope 0.71,
it is claimed that little is lost by the assumption of the immobile interface.
Figure 12. Comparison of experimental results with Hartland’s model, replotted from [200].
The model successfully predicts that the foam height increases with the viscosity, gas
velocity and bubble size increase but decrease as the density, surface tension and film
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thickness increase. However, the model assumes that the drainage of the films is represented
by the axisymmetric drainage of liquid from between two flat discs, for which the Reynolds
equation is applied. This assumption can be limited because the gas-liquid interfaces are
mobile or partially mobile [146, 158, 160]. The assumption that the liquid film rupture occurs
when the critical film thickness is reached is also problematic. It has been found that liquid
films do not necessarily burst when their critical thickness is reached. Instead, the rupture of a
liquid film occurs due to two independent and consecutive stages [191]. The first stage is the
thinning of the liquid film due to drainage, and the second stage is the tear of the liquid film
due to random molecular collisions [201] and fluctuations [143]. The characteristic time of
liquid film rupture, f is the sum of two characteristic times: (i) the characteristic time of
drainage, d , and (ii) the lifetime of the critically thin film, c , giving [191, 192]:
f d c (16)
Only when the foam is evanescent, that is, bubbles burst as soon as their films reach the
critical thickness, is 0c . Even if the evanescent foam assumption is correct, in a real foam,
the films are unlikely to burst at a single exerted pressure or critical thickness but will instead
exhibit a distribution of bursting probabilities [202]. Also, the model does not consider the
coupling between liquid film drainage and foam drainage (drainage along Plateau borders),
whereby the pressure exerted on the film varies with position in the foam.
3.1.3 Pilon et al.’s models
Dimensional analysis has been applied to identify two dimensionless parameters, 1
and 2 that include all physical variables relating to the equilibrium foam height [194]:
2
1
Re
m
gr
j j Fr
(17)
max max2
mH j j HCa
r r
(18)
A power-law type relation has been assumed between the two dimensionless parameters, i.e.:
max Ren
HCa K
r Fr
(19)
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where K and n are constant parameters determined from experimental data. Re , Fr , and Ca
are the Reynolds, Froude, and Capillary numbers, respectively, which are defined as:
Re
mj j r
(20)
2
mj jFr
gr
(21)
mj jCa
(22)
The authors found Eq. (19) can fit experimental data over a wide range of thermophysical
properties with K = 2905 and n = -1.80 with a correlation coefficient R2 = 0.95, as shown in
Figure 13.
Figure 13. Correlation of dimensionless parameters 2 vs 1 , reprinted from [194] with
experimental data from [203-208].
Finally, the following relationship between the two dimensionless parameters has been
determined:
1.80
max 2905
Re
H Fr
r Ca
(23)
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It is noted that the relationship between 1 and (liquid volumetric fraction of foam) was
established to compare the published forced drainage data [162, 209]. In these equations,
is the liquid density, r is the average bubble radius, mj is the minimum superficial gas
velocity for the onset of foaming [210], is the liquid viscosity, is the surface tension
and g is the acceleration due to gravity.
Because Eq. (23) neglects both bubble coalescence and inter-bubble gas diffusion and
its application is limited to highly viscous liquids, a third dimensionless number to represent
the Ostwald ripening that occurs in surfactant foams is needed to generalize this model to low
viscosity fluids (e.g., aqueous surfactant solutions) [211]:
3o
m
DS
r j j
(24)
where D is the diffusion coefficient of the gas that diffuses across the liquid film from one
bubble to the other, and oS is the Ostwald solubility coefficient. This dimensionless number
can be interpreted as the ratio of the average contact time between bubbles in the foam,
/ mr j j and the characteristic permeation time, 2 / or DS .
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Figure 14. Correlation between dimensionless numbers 1.82 1 and 3 for aqueous foams
made from different surfactant solutions and gases, reprinted from [211] with experimental
data from [202, 211-217].
The relationship among 1 , 2 and 3 is assumed to follow a power law as:
2 1 3
n mL (25)
where L, n, and m are constants determined from experimental data. It has been found that
both high and low viscosity fluids yield a value of n close to -1.80 via least squares fitting
[194, 211-213]. The authors found that Eq. (25) can fit experimental data over a wide range
of thermophysical properties with the parameters L = 118 and m = -0.96 with a coefficient of
determination R2 = 0.95, as shown in Figure 14.
Finally, the equilibrium foam height can be expressed in a dimensional form based on
Eq. (25) [211]:
1.760.8
max 1.8 0.961.64118
m
o
j jH
r g DS
(26)
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Although dimensional analysis provides a simple and useful tool for the prediction of
equilibrium foam height, there is no physical meaning for these fitting parameters in the
models, such as K, L, m and n. Also, the theoretical model [210] developed for predicting the
superficial gas velocity for the onset of foaming, mj is still needed to be validated by a
consistent set of experimental data [218].
3.1.4 Limitations of the models for equilibrium foam height
All of the above models assume that there is no bubble coalescence on the top of the
foam before the equilibrium foam height is reached. Based on this assumption, the foam will
reach its equilibrium height as soon as the bubbles burst at the top of the foam. However, a
gentler transition from not bursting to bursting is much more common in growing foam or
froth [188-190, 202]. This relatively gentle transition indicates that rupture of liquid films or
bubbles of foams do not occur at the same time but instead exhibit a distribution of bursting
probabilities [202]. Therefore, predicting the change in foam height with time becomes more
important than the equilibrium foam height.
3.2 Models of foam growth and collapse
Most modeling of foam stability has focused on foam collapse rather than growth [1,
202, 219-222]. A mass balance equation for the bubbling gas in a foam column shown in
Figure 15 has been proposed [1, 223]:
2
1
11
z
z
dzdGA A dz A
dt dt (27)
where G is the superficial gas velocity, A is the cross-sectional area, is the the volumetric
liquid fraction of the foam. The first term on the right-hand side of Eq. (27) represents the gas
in the foam, while the second term corresponds to the gas escaped from the collapsed foams.
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Figure 15. A pneumatic foam being generated by bubbling: no foam collapse (left panel) and
collapse has occurred (right panel), reprinted from [1].
Although it is widely accepted that the liquid fraction in the foam is spatially variant
and decreases to an asymptote within a few cm up the foam column length [202, 224], Eq.
(27) can be simplified to obtain the explicit expression for the foam growth rate
/growth
dH dt by introducing the average liquid fraction, , as:
1
g
growth collapse
jdH dH
dt dt
(28)
where gj is the superficial gas velocity, H is the foam height, and / collapsedH dt is the
foam collapse rate, which is dependent on the foam height (volume) [43]. The first term on
the right-hand side of Eq. (28) represents the foam growth rate of stable foams (i.e., no
bubble coalescence). The growth rate of transient foams, therefore, equals the growth rate of
stable foam minus the foam collapse rate. In this section, the models of foam growth are
categorized based on the treatment of the foam collapse rate: (i) the foam collapse rate is not
considered; (ii) the foam collapse rate is expressed empirically; and (iii) the foam collapse
rate is related to the critical film thickness or Plateau border size.
3.2.1 Models without consideration of foam collapse
An equation has been proposed to express the foam height as a function of the
superficial gas velocity and average foam porosity (gas fraction), t [191]:
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gj
H tt
if t (29)
where is the time for the foam height to reach its equilibrium value and
0
1,
H
t z t dzH
(30)
Since ( ) (1 )t , Eq. (29) would be the same of Eq. (28) as long as the foam collapse rate
/ 0collapse
dH dt .
The local foam porosity distribution has been assumed as a second-order polynomial:
2
0 1 2,z z
z t a a aH t H t
(31)
where the coefficients 0a , 1a , and 2a are determined based on the boundary conditions at the
top and bottom of the foam layer, as shown in Figure 16:
10, ( )t t (32)
0
,0
z
z t
z
(33)
2,H t t (34)
where 1 t is the porosity at the top of the foam as a function of time only, and 2 is the
porosity at the bottom of the foam with a value of 0.74 for an ordered mono-disperse foam
only [1, 225].
Solving Eq. (31) with the boundary conditions, Eqs. (32)-(34) results in the following
porosity profile of the foam:
2
1 2 1,z
z t t tH t
(35)
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The average foam porosity t is then obtained by replacing ,z t with Eq. (35) in Eq.
(30):
1 22
3
tt
(36)
1 t has been treated as follows [191]: (i) constant porosity at the top, (ii) an exponential
function of time, and (iii) obtained based on an approximate solution of the drainage equation.
A similar model for the build-up and breakdown of foam in a glass melt has also been
derived based on the hydrodynamics of the drainage of liquid lamellae [193].
Figure 16. Schematic of a foam layer generated by bubbling and a coordinate system with
notations where qpb is the volumetric velocity of liquid drainage, reprinted from [191].
Although the authors obtained good agreement between the models predictions and
experimental data for the case of low superficial gas velocity [191], they also pointed out that
the model ignores some key physical processes taking place during foam formation, such as
bubble coalescence on the top of the foam layer and Ostwald ripening between bubbles.
Except for the missing physical processes, the authors did not provide any theoretical or
experimental evidence for the assumption of the second-order polynomial for local foam
porosity distribution in Eq. (31) as well as for the expression of average foam porosity in Eq.
(36).
3.2.2 Empirical equations for foam growth and collapse
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The half decay time ( 1/2t ) at which the foam column height is half of the original height,
was found by expressing the foam collapse rate as [8]:
/dH
k tdt
(37)
Integrating Eq. (36), the expression for the change of foam height with time becomes:
0 1/2/ ln / 0.5H H t t (38)
where 0H is the original foam height, 1/2t is the half foam decay time, and is a constant.
Because the relationship between H and t is known, 1/2t can be obtained by fitting the foam
collapse data [223].
An empirical equation for the growth of the froth in a non-overflowing column placed
into a flotation cell has also been proposed [188-190]:
/max 1 tH t H e (39)
H is the fraction of air remaining in the froth at a given froth height H , which is a
crucial parameter in froth modeling and plant operation involving froth flotation; it is
expressed as:
maxH H tdH t A A
Hdt Q Q
(40)
where maxH is the equilibrium foam height; is the average bubble lifetime, which equals
the unit of foaminess, , defined by Eq. (11); A is the column cross-sectional area; and Q is
the gas flow rate to the flotation cell [189]. Eq. (40) assumes that the liquid fraction of the
froth is zero and H is different from the gas fraction or foam porosity (when
maxH t H , 0H but the gas fraction is not zero). The growth rate of the froth height
can be deduced from Eq. (39) used with max gH j as follows:
1g
dHj H
dt (41)
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Suppose that the liquid content is low, i.e., 0 , by comparing Eq. (41) with Eq. (28), we
can see that the foam collapse rate is proportional to the foam height. Both Eq. (38) for foam
decay and Eq. (39) for foam growth are empirical equations based on experimental results
without any fundamental basis.
3.2.3 Foam growth model relating to critical Plateau border size
Bubble burst rate on the top of the foam layer has been correlated with the critical
Plateau border size (area), critA with assumption that there is no internal bubble coalescence,
as follows [202]:
1 crit
Burst
dHk A
dt
(42)
where
13 PB
gk
C
(43)
is the liquid density, g is the acceleration due to gravity, PBC is the Plateau border drag
coefficient, which can be determined by forced drainage experiments and is the liquid
viscosity.
The foam initially grows at a rate of gj (superficial gas velocity) with zero liquid
fraction assumption ( 0 ). When critical Plateau border size is reached, i.e., a sufficient
force is exerted to overcome the disjoining pressure in liquid film, bubbles (liquid films) at
the top of foam layer starts to burst and the foam either reaches an equilibrium height and
stop growing, or it continues to grow with a reduced growth rate,
1 crit g
Final
dHk A j
dt
(44)
depending on the gas velocity. The critical Plateau border area can be obtained based on the
foam growth curve if the gas flow rate is sufficiently high:
1
1crit
Initial Final
dH dHA
k dt dt
(45)
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The comparison between theoretical and experimental foam growth data for the foam created
from a 4 ml/l Teepol solution is shown in Figure 17.
Figure 17. Comparison between theoretical (lines) and experimental (symbols) foam growth
data for foam created from a 4 ml/l Teepol solution, replotted from [202].
Although main fundamental physics governing the foam collapse process has been
considered, a significant discrepancy between the simulated and experimental results for
growing foams remains (Figure 17). This discrepancy has been explained as follows: “In a
real foam, the films are unlikely to burst at a single exerted pressure, but rather are likely to
exhibit a distribution in the bursting probabilities….This distribution in the film stabilities
probably accounts for the more gentle transition from not bursting to bursting seen in the
experiments” [202]. Also, the assumption that the bubbles start to burst when the critical
Plateau border size is reached requires further experimental validation.
3.3 Summary
The equations to describe the equilibrium foam height and foam growth reported in the
literature are summarized in Table 1.
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Table 1. Summary of equations for the equilibrium foam height and growth and collapse of
foam and froth
Source Equation Note
Equations for equilibrium foam height
[192] max 2 / /c m c mH r j j j j j j -
[200]
5/4
max
1/4 7/4
0.55 vdH
d g
Rigid interface assumption
[194]
1.80
max 2905
Re
H Fr
r Ca
For highly viscous fluids
[211]
1.760.8
max 1.8 0.961.64118
m
o
j jH
r g DS
Applied for surfactant solutions
Equations for foam or froth growth and collapse
[191] gj
H tt
No bubble coalescence
[8] 0 1/2/ ln / 0.5H H t t Empirical equation
[188] /max 1 tH t H e Empirical equation
[202] 1 crit gFinal
dHk A j
dt
After onset of bursting
4. CONCLUDING REMARKS
This review tackles the issue of foamability and foam stability from the perspective of
different length scales of the foam: the air-water interfaces (surface tension and surface
viscoelasticity), the liquid films (intermolecular forces, antifoam actions and solid particles
on its stability), the bubbles (coalescence) and the foam (foam drainage and foam column
kinetics).
As for the foamability, it is widely accepted that foaming solutions with a low
equilibrium surface tension, fast adsorption kinetics, an optimal surface viscoelasticity to
restore the air-water interface after disturbances via the Gibbs-Marangoni effect and a strong
and long-range repulsion between interfaces of a liquid film, have a good foamability.
However, the measurements of the properties of the air-water interface and liquid film are not
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directly relevant to the foam formation because the conditions that we studied are different
from those that occur during foam formation [2]. Many questions remain unanswered,
including: What is the adsorption reference time for the foam formation? What is the
frequency for the surface viscoelasticity that is relevant to the foam formation?
As for the foam stability, it has been found that drainage of liquid films and foams,
coarsening and coalescence are three main processes that determine the foam lifetime.
However, the collective effects and interactions of the three processes make it very difficult
to study the foam stability. Furthermore, the studies on intermediate scales, such as air-water
interface and foam films, are inadequate to explain the mechanism of foam stability due to
the
