![xxiii
© 2010 Taylor & Francis Group, LLC
Preface
It is 20 years since the only other book exclusively concerned with defoaming was
published [1]. However, during the intervening years, much new work has been pub-
lished and many new insights revealed. It therefore seems appropriate to revisit the
topic with a new book.
The earlier book was concerned not only with the scientific aspects of the mode
of action of antifoams but also with a number of different industrial applications.
Indeed the latter formed the greater part of that book. By contrast, this book is more
concerned with the basic science of defoaming. In this it replicates the emphasis
revealed by published literature over the past two decades. The industrial application
of antifoams does in fact produce challenging scientific issues specific to particu-
lar applications. Where appropriate, such issues are described here in the relevant
chapters. However, they are apparently rarely addressed, at least in published works.
Much application still appears to rest on simple empiricism augmented by naive
and sometimes completely incorrect views of the mode of action of antifoams. It is
hoped that this book will help stimulate a more enlightened approach to application,
particularly by those antifoam manufacturers who supply products for specific appli-
cations to many different customers.
The main part of the book then concerns the science of defoaming. The general
properties of foams are presented in an introductory chapter, which serves as a back-
ground to much that follows. Another chapter describes the experimental techniques
involved in measurement of the foam properties relevant for study of antifoam action
and the various techniques used in establishing the mode of action of antifoams.
Various aspects of antifoam behavior and relevant theory are considered in later
chapters.
Most commercially effective antifoams are oil based. The relevance of the entry
and spreading behavior of such oils at the surfaces of foaming fluids to their anti-
foam function has been the subject of much debate over the past half century. A
chapter is therefore specifically devoted to the entry and spreading behavior of oils
and the role of thin film forces in determining that behavior.
The role of bridging foam films by particles and oil drops in the mode of action of
antifoams was emphasized in the earlier book. Also included was the first evidence
that the role of particles in synergistic oil–particle antifoam mixtures for defoam-
ing aqueous foams concerns rupture of air–water–oil pseudoemulsion films. The
past two decades have seen a general acceptance and further development of these
aspects of the theory of antifoam action. Particularly noteworthy have been the
insights provided by Professor Nikki Denkov and his coworkers at the University of
Sofia. All of this is reviewed in a chapter specifically concerned with the mode of
action of antifoams.
The volume of foam generated in the presence of antifoam is a function of anti-
foam concentration. Both the relevant phenomenology and a statistical theory can
be shown to be consistent with experiment provided the effectiveness of individual](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-29-320.jpg)
![1
1 Some General
Properties of Foams
1.1 INTRODUCTION
Before considering the behavior of antifoams, we review the relevant properties of
foams. Only a brief summary is given here. It is for the most part only concerned
with those aspects that may have relevance for the understanding of antifoam action.
For more complete accounts, the reader is referred to the many books [1–9] and
reviews on the subject [10–27].
This brief review includes definition of the structural features of foams. A sum-
mary of the processes occurring in foam films follows with particular emphasis on
the factors that determine the stability of those films. Finally, we include an outline
of the processes of drainage and diffusion-driven coarsening, which concern the
entire body of a foam and not just the constituent parts.
1.2
STRUCTURE OF FOAMS
We first consider the structure of a polydisperse foam. That structure is exemplified
by the photograph reproduced in Figure 1.1. This image depicts a foam that has been
aged, where both drainage and diffusion of gas from small bubbles to large bubbles,
as a result of differences in capillary pressure, have occurred. In the lower part of
the foam, bubbles are spherical (so-called kugelschaum) and of small size with a
relatively low overall gas volume fraction. Collections of spherical bubbles, without
the distortions associated with film formation, form at gas volume fractions, ΦG
foam
,
of ≤~0.74 in the case of monodisperse bubbles and ≤~0.72 in the case of polydisperse
bubbles [28]. As the liquid drains out of, for example, a polydisperse foam so that
ΦG
foam
becomes 0.72, the bubbles distort to form polyhedra. This polyhedral foam
(polyederschaum), with a relatively high gas volume fraction, consists of thin foam
films joined by Plateau border channels.
In the case of the foam depicted in Figure 1.1, there is clear segregation of bubble
sizes, with larger bubbles being present at the top of the foam column. The extent
of such vertical segregation in polydisperse foams varies according to the method
of generation. It probably depends on the extent of mixing during foam generation,
gravity segregation [5], and even the so-called brazil nut effect [29]. Segregation of
such a polydisperse foam can in fact apparently be facilitated by rapid continuous
wetting of a foam column from above so that high liquid volume fractions prevail.
Bubble movement can then occur without requiring the distortion of bubbles so that
gravitational segregation is in turn specifically facilitated [5].](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-33-320.jpg)
![2 The Science of Defoaming: Theory, Experiment and Applications
It is worth noting that the Plateau borders between bubbles of different sizes in
Figure 1.1 are in fact curved, being concave with respect to the larger bubbles. This
implies that the adjacent foam films are also curved so that the capillary pressure is
larger inside the smaller bubbles. Such differences drive the process of gas diffusion
leading to coarsening of the foam where average bubble sizes increase.
We illustrate the salient structural features of a foam in Figure 1.2, by relating
computer-generated simulations reported by Weaire and Hutzler [5] to two actual
images of polydisperse foams, each of different gas volume fraction, made by Hartland
and Barber [30]. The comparison is, however, intended to be only qualitative, particu-
larly with respect to gas phase volume fractions in the respective experimental images
and simulations. The latter concern assemblies of the so-called Kelvin cells, tetrakaid-
ecahedra, with six flat quadrilateral faces and eight curved hexagonal faces. However,
polydispersity means that such polyhedra are not at all present in the foams imaged by
Hartland and Barber [30].
As shown in Figure 1.2b, Plateau borders join together to form a network of chan-
nels, containing almost all the liquid in the foam and through which drainage occurs
in the gravity field. The junctions or nodes of the Plateau borders in the interior of
a dry foam (where ΦG
foam
1
→ ) invariably involve four borders meeting at a regular
tetrahedral angle of 109.5°. The Plateau border cross-sections are seen to be con-
cave triangular in shape, with each of the vertices terminating in a foam film where
0.5 cm
FIGURE 1.1 Aged polydisperse foam with dry polyhedral foam (polyederschaum) at top of
column and spherical bubbles (kugelschaum) at bottom of column.](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-34-320.jpg)
![3
Some General Properties of Foams
Plateau border
node
109.5°
120°
Plateau border
cross section
Kelvin structure with Φg ~ 0.99
Kelvin structure with Φg ~ 0.9
(a)
Plateau
border
network
h
Foam film; h rf
0.5 cm
0.5 cm
(b)
(c)
(e)
(d)
(h)
(g)
(f)
2rf
FIGURE 1.2 Structural elements of foam. (a) Image of polyhedral foam against wall of
vessel. (b) Simulated three-dimensional Plateau border network for monodisperse Kelvin
cell foam. This network contains almost all of the liquid in foam. Liquid drains through
network under influence of gravity. (c) Plateau border node where four borders meet at tet-
rahedral angle. (d) Cross section of plateau border where vertices terminate in foam films.
(e) Simulated Kelvin cell “dry” foam with gas volume fraction, ΦG
foam
, of ~0.99. (f) Image of
“wet” foam or froth with gas volume fraction close to limit of 0.72 (for a polydisperse foam,
reference [28]) where Plateau borders and foam films are absent. (g) Simulated Kelvin cell
“wet” foam with gas volume fraction, ΦG
foam
, of ~0.9. (h) Foam films are present between
bubbles in simulated Kelvin cell foams shown in (e) and (g). Such films are characterized
by thicknesses several orders of magnitude less than their width. (Images (a) and (f) from
Hartland, S., Barber, A.D., Trans. Inst. Chem. Eng., 52, 43, 1974. With permission from
Institution of Chemical Engineers. Images (b), (c), (e), and (g) from Weaire, D., Hutzler, S.,
The Physics of Foams, 1999, by permission of Oxford University Press.)](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-35-320.jpg)
![4 The Science of Defoaming: Theory, Experiment and Applications
the angle between the films as they intersect the borders is 120°. These structural
features represent the essentials of Plateau’s rules [31], which describe a condition
of mechanical equilibrium and apply equally to all both monodisperse and polydis-
perse foams in the limit where ΦG
foam
1 [5]
→ .
Images, such as those presented in Figures 1.1 and 1.2, are of actual foams, gener-
ated in various containers. These images usually represent only the layer of bubbles
adjacent to the walls of those vessels, a limitation often imposed by the depth of
field of the relevant photographic equipment together with the relative opacity of the
foam. Such images exaggerate the proportion of large bubbles in a polydisperse foam
as a result of a statistical sampling bias [10]. Moreover, the surface of the contain-
ing vessel, which represents the plane of observation, means that bubbles adjacent
to that plane are necessarily distorted as indicated by, for example, Steiner et al.
[32] and later by Cheng and Lemlich [33]. In the case of a monodisperse foam, the
Plateau borders of bubbles adjacent to the confining plane form regular hexagons.
The origin of this geometry has been neatly demonstrated by Weaire and Hutzler
[5] in a simulation with Kelvin cells. Tessellation of Kelvin cells in a monodisperse
foam contained in a vessel is in principle possible if the edge bubbles are formed
by slicing a Kelvin cell across the middle as indicated in Figure 1.3a. The resulting
structure with a hexagonal planar surface is shown in Figure 1.3b. Those Plateau
borders directed away from that plane are all oriented at 90°. In this respect, these
structures violate one of Plateau’s rules. However, even in the case of a monodisperse
Plateau borders form
a hexagonal cross
section at plane of
observation
Kelvin bubble is bisected to
form a bubble that can still
tessellate despite touching the
wall of the retaining vessel
(a)
(b)
FIGURE 1.3 In a monodisperse Kelvin foam, even bubbles that touch the surface of retain-
ing vessel must tessellate. Simulation suggests that this is permitted if those bubbles have
structure formed by bisecting a Kelvin cell as indicated in (a). Bubbles with resulting struc-
ture must be extended to restore original volume. As shown in (b), the edge of bubble then
presents a hexagonal cross section to plane of observation at wall of vessel. (From Weaire, D.,
Hutzler, S., The Physics of Foams, 1999, by permission of Oxford University Press.)](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-36-320.jpg)
![5
Some General Properties of Foams
foam, Kelvin cells only exist one layer into the foam from the surface cells. Beyond
that, the structure becomes disordered and Kelvin cells are not apparently found [5].
The dimensions of these structural features of foam are subject to change as a
result of the action of various processes. Senescence is facilitated by processes of
drainage and diffusional coarsening. However, the ultimate arbitrator of the fate of
foams concerns the stability of films, which arguably represents the most important
factor in determining both foamability (i.e., the tendency to form a foam) and foam
stability. With thicknesses typically at least three orders of magnitude less than the
diameter of the circle circumscribed by the edges of a Plateau border, they represent
the most fragile aspect of foam structure. Indeed, if the films break, then the struc-
tures described here cannot exist if the continuous phase is liquid and consideration
of their putative nature becomes purely academic. Design of effective antifoams
tends therefore to be focused on destruction of foam films.
1.3 FOAM FILMS
1.3.1
Surface Tension Gradients and Foam Film Stability
Films formed by adjacent bubbles in a pure liquid are extremely unstable. Pure
liquids therefore do not form foams. This arises in part because of the response
of the films to any external force such as gravity. Consider, for example, a ver-
tical plane-parallel film of a pure liquid in a gravity field. There is no reason
why any element of that film should move in response to the applied gravitational
force with a velocity different from that of any adjacent element. No velocity gra-
dients in a direction perpendicular to the plane of the film surface against the
air will therefore exist. There will then be no viscous shear forces opposing the
effect of gravity. The film will exhibit plug flow (resisted only by extensional vis-
cous forces) with elements accelerated downward tearing it apart. The process is
depicted in Figure 1.4a.
This behavior can be drastically altered if we arrange for a tangential force to
act in the plane of the liquid–air surface so that the surface is essentially rigid. In
the case of a vertical plane-parallel film of a viscous liquid with such rigid surfaces,
subject to gravity, a parabolic velocity profile will develop as shown in Figure 1.4b.
This means that velocity gradients will exist in a direction perpendicular to the film
surfaces. A viscous stress will therefore be exerted at the air–liquid surface. This
stress must be balanced by the tangential force acting in the plane of the surface.
That force can only be a gradient of surface tension. This balance of viscous forces
and surface tension gradients at the liquid–air surface can be written as
d
d
d
d
AL
L
σ
η
y
u
x
y
x
=
=0
(1.1)
where σAL is the air–liquid surface tension of the foaming liquid, ηL is the viscosity,
uy is the velocity of flow in the y direction, y is the vertical distance, and x is the
horizontal distance in the film.](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-37-320.jpg)
![6 The Science of Defoaming: Theory, Experiment and Applications
Thus, we find that if the force of gravity is to be resisted by the film, then a surface
tension gradient must exist at the air–liquid surface. Combining the relevant Navier–
Stokes equation with Equation 1.1 as a boundary condition, Lucassen [15] shows that
in the case of a vertical film in the gravity field the gradient is
d
d
AL L
σ
y
=
ρ gh
2
(1.2)
where h is the film thickness, ρL is the liquid density, and g is the acceleration due to
gravity. This gradient can only exist where differences of surface composition can
occur. We therefore require the presence of more than one component in the film.
Indeed, it is possible to speculate that in the case of, say, aqueous foams, diffusion
of water through the gas phase may rapidly remove any differences in concentration
between different parts of a foam film if only one solute is present. In this case, at
least two solutes (or three components) would be required.
Surface tension gradients due to differences in the surface excess of soluble
surface-active components may exist only when either the surface is not in equilibrium
with the bulk composition or there are concomitant differences in bulk composition
parallel to the surface. In the case of the former, the magnitudes of the gradients are
of course determined by the rate of transport of surfactant to the relevant surfaces.
With concentrated surfactant solutions, transport rates by diffusion will be rapid and
surface tension gradients will be diminished so that shear rates are only balanced at
y
x
(duy/dx)x=0
0
uy (x =0) ≠ 0
uy (x =0) = 0
(duy/dx)all x
= 0
(a) (b)
FIGURE 1.4 Limiting range of velocity profiles in draining foam film. (a) Plug flow.
(b) Flow with parabolic velocity profile when film surfaces are immobile. Arrows represent
magnitude of fluid velocities uy in the y direction.](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-38-320.jpg)
![7
Some General Properties of Foams
the surface of films if the air–liquid surfaces become mobile (i.e., so that the velocity
uy(x = 0) ≫ 0). This would result in enhanced rates of drainage from films. In the
extreme, the surface tension gradients will tend to be eliminated altogether. Thus,
it has occasionally been reported that foamabilities decline at extremely high con-
centrations of surfactant in aqueous solution. Conversely, however, if foam films are
denuded of surfactant because of extremely slow transport rates, then the maximum
surface tension gradients that can be achieved will be small. Such films will there-
fore be susceptible to rupture when exposed to external stress. However, the com-
plex problem of assessing both the effect of rate of transport on the surface tension
gradients in foam films and the overall resultant impact on foam film stability, when
subject to an external stress, has not, apparently, been fully addressed.
Differences in bulk composition are possible in a thin foam film as a result of
stretching of the film. If the film is sufficiently thin, then any stretching causes a
depletion of the bulk phase surfactant solution between the air–liquid surfaces of
the foam film as more surfactant adsorbs on those surfaces. Distances perpendicular
to the film are small so that, provided the stretching occurs reasonably slowly, the
equilibrium inside the film element may be always maintained. Depletion of bulk
phase the surfactant concentration will therefore necessarily mean an increase of
the surface tension of the film as it is stretched. This will, however, only occur if
reduction of the surfactant concentration causes a concomitant increase in surface
tension. In the case of a pure surfactant at concentrations above the critical micelle
concentration (CMC), this may not always happen.
We find then that it is possible to generate a surface tension gradient in a foam
film by stretching various elements of the film to different extents. The increase in
surface tension due to stretching imparts an elasticity to the film. This property of
foam films was first recognized by Gibbs [34] and is usually referred to as the Gibbs
elasticity εG. It is defined as
ε
σ σ
G
AL AL
dln dln
=
2d 2d
A h
= − (1.3)
where A is the film area and the factor 2 arises because of the two surfaces.
A plot of εG against concentration for a submicellar aqueous solution of sodium
dodecyl sulfate (SDS) is shown in Figure 1.5 by way of example. Here, we see that,
except at very low concentrations, decreases in film thickness at constant concentra-
tion produce increases in Gibbs elasticity so that (dεG/dh)c ≤ 0. Thus, as the film
becomes thinner, stretching will cause a relatively greater depletion of surfactant
in the intralamellar liquid and the surface tension will increase to a greater extent.
The plot of Gibbs elasticity against concentration shown in Figure 1.5 clearly
reveals a maximum at concentration cmax. At extremely low concentrations of surfac-
tant, we find that upon stretching of the film, there is essentially no contribution from
the intralamellar liquid, and the surfactant behaves as an insoluble monolayer. Here,
with an increase in surfactant concentration, both the surface excess and the elastic-
ity of the monolayer increase. However, further increases in the surfactant concen-
tration will eventually mean that it significantly exceeds that required to compensate
for stretching of the air–liquid surface, so εG → 0. These two opposing consequences](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-39-320.jpg)
![8 The Science of Defoaming: Theory, Experiment and Applications
of increasing concentration conspire to produce the maximum in a plot of Gibbs
elasticity.
Lucassen [15] considers the effect of Gibbs elasticity on the development of sta-
bilizing surface tension gradients in a foam film in the gravity field. He points out
that if εG decreases as the film is stretched, it will tend to be dynamically unstable.
Under these circumstances, any stretching force will tend to increase the area of
the thinnest part of the film. Such situations will tend to prevail for films formed at
concentrations on the low side of cmax. Thus, we can write
d
d
d
d
G G G
ε ε ε
h h c
c
h
c h
=
∂
∂
+
∂
∂
(1.4)
where c is the concentration. We can therefore have dεG/dh 0 if c cmax because
then (∂εG/∂c)h 0, dc/dh 0, and (∂εG/∂h)c→ 0 as c → 0.
For vertical films prepared from concentrated solutions, the requirement that the
surface tension gradient satisfy Equation 1.2 implies that rapid stretching will occur. In
the case of micellar solutions of certain pure surfactants, this may require achievement
of submicellar concentrations in order that εG 0 and therefore dσAL/dx 0. For verti-
cal films prepared from extremely dilute surfactant solutions where c cmax, Lucassen
[15] shows that the magnitude of εG for thin elements at the top of the film may be less
than that of thinner elements lower down. Any force acting on the film, such as an
increase in weight as it grows, could mean catastrophic extension of the thinnest ele-
ments because of their lower Gibbs elasticities.
In summary, then, we find that surface tension gradients are necessary if freshly
formed foam films are to survive. These gradients may occur if surface tensions
h = 1.0 microns
h = 2.0 microns
5 10
Concentration of SDS (mM)
cmc
10
20
30
40
50
εG (mN m−1)
FIGURE 1.5 Gibbs elasticities of submicellar SDS solutions at two different film thick-
nesses. (After Lucassen, J. Dynamic properties of free liquid films and foams, in Anionic
Surfactants, Physical Chemistry of Surfactant Action (Lucassen-Reynders, E.H., ed.), Marcel
Dekker, New York, Surfactant Sci. Series, Vol. 11, Chapter 6, p. 217, 1981.)](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-40-320.jpg)
![9
Some General Properties of Foams
depart from equilibrium values. This will happen when foam film air–liquid sur-
faces are expanded at rates that are fast so that equilibrium with the bulk surfactant
concentration cannot be maintained. They may also occur when films are thin so
that stretching may deplete intralamellar bulk phase to give rise to a Gibbs elastic-
ity. Unfortunately, there are few experimental observations that clearly reveal the
importance of surface tension gradients in determining foam behavior. Perhaps the
best examples are reported by Malysa et al. [35] and Prins [36].
1.3.2 Drainage Processes in Foam Films
Any freshly generated foam film that survives will now be subject to a capillary
pressure exerted by the curved surfaces of the adjacent Plateau borders. That pres-
sure will tend to suck liquid out of the foam film. The resultant process of film drain-
age is surprisingly complex.
The simplest description of foam film drainage is obtained if the film is supposed
cylindrical with immobile plane-parallel surfaces. Such behavior is represented by
the Reynolds equation [37]
− =
d
d L
h
t
h P
r
2
3
3
2
Δ
f
η
(1.5)
where h is the film thickness, rf is the film radius, and ΔP is equal to the capil-
lary pressure jump, pc
PB
, at the air–liquid surface of the Plateau border. pc
PB
is in
turn equal to σAL∣κPB∣, where ∣κPB∣ is the modulus of the air–liquid curvature of the
Plateau border. If the film is sufficiently thin (~100 nm), then the applied capillary
pressure will be modified by the air–liquid–air (ALA) disjoining pressure, ΠALA(h),
so that the total applied pressure becomes p h
c
PB
ALA
− Π ( ). Drainage will therefore
cease if p h
c
PB
ALA
− Π ( ), where a metastable equilibrium can prevail. We consider the
effects of disjoining pressure on film stability in Section 1.3.3.
The behavior of real foam films is rarely represented by the Reynolds equation.
Films of radius ~100 microns are, for example, generally not plane parallel [6]. The
surface of a draining foam film is characterized by a balance of the surface tension
gradient and the relevant viscous stress as indicated by Equation 1.1. In turn, this sur-
face tension gradient implies a gradient in surfactant adsorption that will be subject
to relaxation by diffusion from the bulk phase and along the surface. Maintenance of
the balance of the surface tension gradient and viscous stress will therefore always
require a countervailing flow in the surface in the direction of the bulk phase flow.
Complete immobility of the air–liquid surface (so that the velocity uy(x = 0) = 0) is
therefore never achieved [6].
In the case of relatively large foam films ~100 microns, where the air–liquid
surface is close to a condition of immobility, drainage occurs in an axisymmetric
manner but with a non-uniform thickness. A thick region develops in the center of
the film, the so-called dimple, while a thinner region surrounds this dimple—the so-
called barrier ring [38]. A schematic cross-section of a film with a dimple is depicted
in Figure 1.6a. The behavior of such a film, formed inside a cylindrical glass cell (the](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-41-320.jpg)
![10 The Science of Defoaming: Theory, Experiment and Applications
Scheludko cell [13, 39]; see Chapter 2, Section 2.3.1), has been analyzed numerically
by Joye et al. [38]. They deduce that the formation of a dimple requires that the capil-
lary pressure inside the maximum putative dimple that can be formed be less than
the capillary pressure in the Plateau border meniscus. This means that r R
d PB
− −
( )
1 1
1
/ ,
where rd is the radius of curvature of the maximum dimple and RPB the radius of cur-
vature of the Plateau border as depicted in Figure 1.6b. Suction from the latter will
not then be matched by sufficient pressure from the dimple so that liquid is selec-
tively removed from the region adjacent to the meniscus, forming a thin barrier ring.
If, on the other hand, the film is small and the capillary pressure in the maximum
dimple is greater than the capillary pressure in the Plateau border, then liquid flow
from any putative dimple readily matches that sucked into the border and neither a
barrier ring nor a dimple are formed. If a dimple is formed, then the concomitant
restricted liquid flow in the barrier ring contributes to a diminished rate of drainage
Dimple in center of film
Plateau border
capillary
pressure
Thin barrier ring
forms that inhibits
drainage from dimple
(a)
(b)
Rc
rd
RPB ≈ Rc
Dimple Plateau
border
Cell wall
FIGURE 1.6 Axisymmetric drainage of foam film with immobile air–water surfaces.
(a) Dimple and barrier ring that form. (b) Defining maximum dimple size where barrier ring is
absent and where film is formed in a cylindrical Scheludko cell (references [13, 39]; see Section
2.3.1). Radius of curvature, RPB, of Plateau border air–water surface is approximated by radius,
Rc, of cell—rd is radius of curvature of air–water surface of the dimple. (Reprinted with per-
mission from Joye, J. et al., Langmuir, 8, 3083. Copyright 1992 American Chemical Society.)](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-42-320.jpg)
![11
Some General Properties of Foams
relative to that predicted by the Reynolds equation. As the barrier ring becomes thin-
ner, then eventually repulsive disjoining forces are predicted to become important in
determining the final rates of film thinning. At high electrolyte concentrations, the
barrier will become extremely thin so that the dimple drains extremely slowly [38].
In practice, axisymmetric foam film drainage is not usually observed in the case
of solutions of simple surfactants. Although dimples are usually formed with such
systems, the resulting films exhibit a hydrodynamic instability where dimples dis-
gorge directly into one side of the Plateau borders [40]. This asymmetric drain-
age is always significantly more rapid than the axisymmetric drainage. The latter
appears to be observed only when asymmetric drainage is suppressed by extreme
surface rheological behavior of the adsorbed surfactant. We illustrate this in Figure
1.7 where the drainage rates of horizontal foam films formed in a Scheludko cell [13,
39], and prepared from solutions of SDS and SDS–dodecanol mixtures, are com-
pared with the surface shear viscosities of those solutions. Rapid asymmetric drain-
age is associated with the low values of the surface shear viscosity typical of aqueous
solutions of simple soluble surfactants.
Joye et al. [40] offer an explanation, based on the earlier work of Stein [42, 43], for
the hydrodynamic instability that produces asymmetric drainage. We illustrate this
in Figure 1.8. Here it is supposed that thickness fluctuations occur in the barrier ring
(linearized in the figure). In the thicker regions the flow of liquid from the dimple
to the Plateau border meniscus will increase relative to the unperturbed flow rate.
This will in turn produce an increased surface tension gradient in the direction of
the flow. Conversely in the case of the thinner regions, the flow rate in the direction
of the meniscus will be slow relative to the unperturbed flow rate. In turn this will
produce a decreased surface tension gradient. As a consequence of these two oppos-
ing trends in surface tension gradient in the direction of flow from the dimple to the
meniscus, another surface tension gradient orthogonal to the unperturbed gradient
0
50
100
150
200
250
300
350
400
−5 −4 −3 −2 −1 0
log10 (Surface shear viscosity/g s−1)
Film drainage time (s)
Asymmetric
drainage
Axisymmetric
drainage
FIGURE 1.7 Film drainage time as function of surface shear viscosity for SDS and SDS +
dodecanol (100:1) solutions using a Scheludko cell and film radius of 100 microns. (Reprinted
with permission from Joye, J. et al., Langmuir, 8, 3083. Copyright 1994 American Chemical
Society, using data from Djabbarah, N.F. et al., Colloid Polym. Sci., 256, 1002, 1978 [41].)](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-43-320.jpg)
![12 The Science of Defoaming: Theory, Experiment and Applications
will exist between the thin and thick regions where the surface tension of the former
region will be slightly lower than the latter. That orthogonal surface tension gradient
will therefore drive liquid from the thin regions to the thick regions, reinforcing the
perturbation. The latter will therefore grow until discharge of the whole dimple is
facilitated. Joye et al. [40] have subjected this putative process to a linear stability
analysis and have shown that the hydrodynamic instability is suppressed, so that
axisymmetric drainage occurs, if either the dimension of the film is small enough
or if the surface shear viscosity is high enough. The latter is of course consistent
with the experimental findings shown in Figure 1.7. However, we must emphasize
that the instability is ubiquitous in draining foam films formed from aqueous solu-
tions of common surfactants. Such films therefore exhibit rapid asymmetric drain-
age by means of this hydrodynamic instability for which no analytical theoretical
expression is available for the prediction of the rate of drainage.
A related phenomenon to asymmetric drainage in horizontal cylindrical foam
films is that of the so-called marginal regeneration. This process occurs in vertical
foam films subject to both capillary suction from the Plateau borders and gravity. It
is manifest as an apparent turbulent motion on the margin of foam films where thin
elements of film are drawn out of the Plateau border and thick elements are sucked
Barrier ring
Dimple
Plateau
border
a
b
c
d
Sinusoidal thickness
fluctuation in the
barrier ring
Thick film elements ab
permit faster bulk flow of
fluid than thin film
elements cd
FIGURE 1.8 Origin of hydrodynamic instability driving asymmetric drainage in horizontal
foam films. Thickness fluctuation in barrier ring means that region ab is thicker than con-
tiguous region cd. Fluid flow rate from dimple to Plateau border therefore increases relative
to cd in ab. Both relative shear stress and corresponding surface tension gradient must also
increase in ab according to Equation 1.1. This means that surface tension in ab will be gener-
ally greater than cd, which will mean additional surface tension gradient, orthogonal to that
between dimple and Plateau border, driving fluid from the thin to the thick parts of barrier
ring. This will reinforce instability. However, establishment of differences in surface tension
gradients ab and cd between dimple and Plateau border will be resisted by high values of
surface shear viscosity leading to suppression of instability. (Adapted with permission from
Joye, J. et al., Langmuir, 8, 3083. Copyright 1994 American Chemical Society.)](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-44-320.jpg)
![13
Some General Properties of Foams
into the border causing continuous regeneration of the margin of the film. The thin
elements of film then rise in the gravity field until they reach a height where the film
has the same thickness. The phenomenon is readily observable in reflected white light
where interference colors distinguish film elements of different thickness. Marginal
regeneration can easily be seen in the large vertical foam films (of size ~1 cm) held
in metal or glass frames, which are often used for study of aspects of antifoam action
(see, e.g., references [44–46]). It is also often visible in the large bubbles formed at the
top of foam columns generated by air entrainment (e.g., during hand dishwashing). A
detailed description of the phenomenon is to be found in the monograph of Mysels et
al. [3]. Images of films exhibiting marginal regeneration are given in Figure 1.9.
Joye et al. [40] argue that marginal regeneration is driven by the same hydrody-
namic instability as that which causes asymmetric drainage in horizontal cylindrical
foam films. The main difference concerns the role of gravity in marginal regenera-
tion, which causes thin film elements resulting from the growing perturbations to
rise up the film. The net effect of the process is relatively rapid drainage of the film.
As with asymmetric cylindrical film drainage, no analytical expression is available
for the prediction of the rate of film drainage. The hydrodynamic instability is sup-
pressed if the surface dilatational modulus is sufficiently high [15, 47]. It is probable
that high surface shear viscosities may also lead to suppression of the instability
as is found with small horizontal films. We note, however, that high surface shear
viscosities often apparently accompany high surface dilatational moduli, which
renders selection of the key property difficult. The resultant “rigid” films drain by
a Poiseuille-like flow for which Mysels et al. [3] present an analytical expression.
Vertical films, which exhibit marginal regeneration, typically drain at rates two
orders of magnitude faster than rigid films in which the instability is suppressed [3].
It should be clear then that asymmetric drainage of horizontal cylindrical films
and marginal regeneration in vertical foam films, although often apparent, are
not always exhibited. Liquid flow by capillary suction from the Plateau border is,
(a) (b) (c)
FIGURE 1.9 Marginal regeneration. (a) Film in inverted triangular glass frame. (b) Film in
rectangular glass frame illustrating marginal regeneration at bottom of film (it is absent from
top of film). (c) Large film (~5.5 cm radius) in cylindrical vessel illustrating marginal regen-
eration at angle of only ~10° to horizontal. In all cases, rising film elements move up to film
regions of the same thickness with which they coalesce. (Image (a) after Mysels, K.J. et al.,
Soap Films, Studies of Their Thinning, Pergamon, London, 1959. Image (b) after Nierstrasz,
V.A., Marginal regeneration, PhD Thesis, Delft University of Technology, 1996 [48].)](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-45-320.jpg)
![14 The Science of Defoaming: Theory, Experiment and Applications
however, always present. It is therefore misleading for Weaire and Hutzler [5] to state
that “the traditional but somewhat obscure phrase for this (liquid flow by capillary
suction) process is marginal regeneration.” For the relevant hydrodynamic instabili-
ties to occur and cause marginal regeneration, the air–liquid surfaces should addi-
tionally have the required rheological characteristics (i.e., low surface viscosity, etc.).
1.3.3 Disjoining Forces and Foam Film Stability
By means of these processes, foam films drain until they either form metastable
films or rupture. Which fate awaits a given film is decided largely by the nature of the
forces that exist across a foam film as it thins. These forces are usually termed “dis-
joining” forces and are positive if they resist the thinning of the film and negative if
they assist thinning of the film. They are, however, only significant in comparatively
thin films of ≤100 nm. The metastability of the film requires that resulting disjoining
pressure balances the capillary pressure exerted by the Plateau borders. The disjoin-
ing pressure is a function of the film thickness—plots of disjoining pressure against
film thickness (disjoining pressure isotherms) largely determine the properties of
those metastable films.
The disjoining pressure, ΠALA(h) in air–liquid–air foam films, stabilized by
adsorbed surfactant, is made up of contributions from several components. These
include electrostatic forces (from overlapping electrostatic double layers), van der
Waals forces, structural forces (arising from the presence of close-packed layers of
micelles or nanoparticles in films [49]), steric forces (derived from the overlapping
head group—head group interactions of surfactants or chain–chain interactions of
polymers adsorbed on opposite sides of the film), and oscillatory forces, etc. These var-
ious contributions conspire to produce disjoining pressure isotherms of various forms.
Arguably, the simplest isotherm is one dominated entirely by the van der Waals
forces, which are invariably present. Such an isotherm is shown schematically in
Figure 1.10, where the disjoining pressure is always negative and where it becomes
more negative as the film thins so that
d
d
ALA
Π ( )
.
h
h
0 (1.6)
In consequence, any perturbation of the film thickness that produces thin and
thick regions will tend to grow spontaneously because molecules in the thin regions
will transfer to the thick regions. However, any perturbation of a film to produce
thick and thin regions must also inevitably increase the surface area of the film. This
will increase the number of molecules in the relatively weak attractive force field
close to the air–liquid surface. Such an increase in surface area will therefore be
resisted by an opposing force—the surface tension [51, 52].
Thickness perturbations, of a thermal or mechanical nature, may be consid-
ered to be of a wavelike nature. A symmetrical sinusoidal perturbation is shown in
Figure 1.11. Here the thin part of the film is subject to two opposing forces. Thus, a
capillary pressure due to the surface tension tends to suck liquid back into the thin
part of the film, and a disjoining force tends to push liquid away.](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-46-320.jpg)
![15
Some General Properties of Foams
The magnitude of the capillary pressure is determined by the curvature of the
film surface. For a given amplitude of the perturbation, the curvature is determined
by the wavelength. Thus, the shorter the wavelength, the more marked the curvature
and the stronger the capillary pressure. Vrij and Overbeek [51, 52] deduce a criti-
cal wavelength, λcrit, above which disjoining forces will dominate over the capillary
pressure and the perturbation will spontaneously grow. The critical wavelength is
( )/
λ
π σ
crit
AL
ALA
d d
=
2 2
1
2
Π h h
(1.7)
The rate of growth of the perturbation will increase with increasing wavelength λ for
λ λcrit because the damping effect of the capillary pressure will decrease. However,
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
10 20 30 40 50 60 70
h (nm)
ΠALA(h)
(104 N m−2)
FIGURE 1.10 Plot of disjoining pressure isotherm for a plane-parallel air–water–air film with
only van der Waals interactions. Here disjoining pressure, ΠALA, is given by ΠALA(h) = AH/6πh3
where AH is Hamaker constant of 3.7 × 10–20 J. (From Isrealachvilli, J.N. Intermolecular and
Surface Forces with Applications to Colloidal and Biological Systems, Academic Press,
London, 1985 [50].)
Liquid movement due to
disjoining pressure
Liquid movement due to
capillary pressure
Unperturbed
film surfaces
FIGURE 1.11 Sinusoidal thickness perturbations in thin liquid foam film.](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-47-320.jpg)
![16 The Science of Defoaming: Theory, Experiment and Applications
for sufficiently long wavelengths, the rate of growth will eventually begin to decline
because of the increased distances over which the film liquid has to be moved against
viscous resistance. An optimum wavelength of 2crit for the maximum rate of growth
of a perturbation therefore exists [51, 52]. As we show in Figure 1.10, dΠALA(h)/dh is
strongly dependent on film thickness, which leads to λcrit decreasing in proportion to
the square of the film thickness in the case where only van der Waals forces prevail.
Film collapse is supposed to occur when the amplitude of the fastest-growing per-
turbation equals the thickness of the film. Vrij and Overbeek [51, 52] have produced
the simplest description of this process. They calculate the minimum total time for
film drainage (unfortunately estimated using the Reynolds equation [37]) and subse-
quent growth of the fastest-growing perturbation. The average thickness of the film
at the moment of rupture is the critical thickness hcrit. Experimental measurements
of hcrit for microscopic foam films are in the region of a few tens of nanometers (see,
e.g., reference [53]).
Films that are completely dominated by van der Waals forces and exhibit the type
of instability described by Vrij and Overbeek [51, 52] are likely to be either free of
surfactant or have low levels of surfactant adsorption. The presence of high levels
of adsorbed surfactant means that positive contributions to the disjoining pressure
become significant. In particular, long-range electrostatic and van der Waals forces,
together with extremely short range steric forces, can produce isotherms with two
minima. Such isotherms are often exhibited by aqueous surfactant solutions [6, 54].
This type of isotherm is illustrated schematically in Figure 1.12. It is characterized
by two regions where dΠALA(h)/dh 0 and two regions where dΠALA(h)/dh 0.
ΠALA(h)
h
E
C
D
A
B
h1
h2
h3
Π
max
ALA(h3)
pPB2
c
= ΠALA(h2)
pPB1
c
= ΠALA(h1)
Stable common black films where
dΠALA(h)/dh 0
20–50 nm
Stable Newton black films where
dΠALA(h)/dh 0
~5 nm
FIGURE 1.12 Schematic diagram of disjoining pressure isotherm exhibiting two stable
regions where common black foam films and Newton black films can be formed, respectively
(see text for full explanation). ΠALA is air–liquid–air (ALA) foam film disjoining pressure and
h is film thickness.](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-48-320.jpg)
![17
Some General Properties of Foams
Bergeron [55] has argued that adsorbed surfactant present at the surfaces of the film
can also produce other complications in the analysis of the effect of thickness pertur-
bations. Dilatational effects, for example, due to adsorbed surfactant, tend to dampen
the fluctuations rendering rupture less probable. Moreover, fluctuations due to lateral
adsorption density fluctuations could lead to lateral fluctuations in disjoining pressure
to further complicate the picture. These issues are far from fully resolved.
Despite these complications, disjoining pressure isotherms such as that shown in
Figure 1.12 offer an explanation for much of the behavior of the thin liquid films,
formed by aqueous solutions of simple surfactants, subject to an applied capillary pres-
sure. Consider, for example, the regions in those isotherms where dΠALA(h)/dh 0. This
means that the disjoining pressure increases as the film thins so that any perturbation in
film thickness will be resisted. Such regions are therefore stable. Conversely in regions
where dΠALA(h)/dh 0, any perturbation in film thickness will be enhanced leading to
instability. For the isotherm shown in Figure 1.12, there are therefore thickness regions
where stable films can be formed (regions BC and DE) and thickness regions where
unstable films are formed (regions AB and CD). In consequence, regions AB and CD
are not accessible to experimental measurement. In this type of isotherm, the region
AD is largely determined by electrostatic and van der Waals contributions and can be
readily accounted for by the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory
(see, e.g., reference [6]). However, the region DE, if it is present, is usually largely due
to short range steric interactions between molecules adsorbed on opposite sides of the
film and cannot be accounted for by that theory.
Consider now the likely evolution of a film with a disjoining pressure isotherm
similar to that shown in Figure 1.12, which is subject to a capillary suction, pc
PB1
,
from the adjacent Plateau border. Such a film will drain in the unstable region AB
until the critical thickness is reached whereupon perturbations in thickness will
not grow to produce holes and film rupture but will rather produce stable regions,
or “spots,” which continue to expand and cover the whole film until it reaches an
equilibrium thickness h1 where the capillary pressure equals the disjoining pressure
(i.e., p h
c
PB1
ALA
= Π ( )
1 ). The film thicknesses involved at this stage are in the region
of 20–50 nm. Such films are so thin that destructive interference of light occurs so
that they (and indeed the “spots”) appear black in reflected light. They are usually
termed “common black films.”
If now the capillary pressure increases to pc
PB2
so that it exceeds the maximum at C
in the isotherm then the film will drain to another unstable region and will jump from
C to form another stable film at D. Equilibrium will again be established when the film
thins to thickness h2 where p h
c
PB1
ALA
= Π ( )
2 . Such films are extremely thin, being little
more than bilayer leaflets of thickness in the region of 5 nm. They are usually termed
“Newton black films.” Further increases in capillary pressure to p h
c
PB3
ALA
max
Π ( )
3
should simply lead to film rupture. However, thickness fluctuations cannot exist in such
thin films. It has therefore been argued by Kashchiev and Exerowa [56] that rupture
occurs by nucleation of holes caused by thermal fluctuations in these films.
The transition from a disjoining pressure isotherm dominated by van der Waals
forces, such as that shown in Figure 1.10, to one with at least a region where dΠALA(h)/
dh 0 is present can be observed at a certain surfactant concentration. That is
of course represented by a transition from rupture to black spot formation. The](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-49-320.jpg)
![18 The Science of Defoaming: Theory, Experiment and Applications
concentration at which this transition occurs is designated cblack. It occurs because of
the effects of changes in the adsorption layer of surfactant on the thickness depen-
dence of ΠALA(h) and dΠALA(h)/dh. Foam film stability tends to increase markedly at
concentrations above cblack.
Values for cblack in microscopic films have been measured for a number of surfac-
tants [57]. In the case of aqueous solutions of surfactants, cblack is generally signifi-
cantly lower than the CMC. Thus, for example, for SDS, cblack = 1.6 × 10–6 M [57] and
the CMC = 8.4 mM [58], and for dodecyl hexaethyleneglycol (C12EO6), cblack = 4.9 ×
10–6 M [57] and the CMC = 8.7 × 10–6 M [59] at 25°C.
We have seen then that film rupture may occur because surface tension gradients
are not sufficiently high to enable the film to withstand stress, because dΠALA/dh is
always positive so that rupture is inevitable at a certain critical thickness, or because
the Plateau border capillary pressure exceeds any maximum in the relevant disjoining
pressure isotherm. However these phenomena are associated with low concentrations
of surfactant (at least if we consider films formed slowly so that equilibrium between
the air–liquid surface and the intralamellar liquid is maintained). Thus for example,
we have cmax ≪ CMC for dεG/dh 0 and cblack ≪ CMC. Elimination of both causes
of rupture should therefore be readily achieved at sufficiently high concentrations of
surfactant. The poor discrimination in foamability often found with relatively con-
centrated aqueous micellar solutions of surfactants may well be attributable to that
cause. Interesting differences in foamability are, however, often revealed when films
are either formed rapidly so that equilibrium adsorption is not obtained and conditions
for stability are thereby violated or if antifoam is added to the solution.
1.4
PROCESSES ACCOMPANYING AGING OF FOAM
1.4.1 Capillary Pressure Gradients
Foam film stability is, as we have seen, determined in part by the lack of balance
between the disjoining pressure and the capillary pressure applied to the films by the
Plateau borders. The capillary pressure also drives the process of film thinning, which
precedes film rupture. This in turn influences the frequency of foam film rupture. The
relative magnitudes of the capillary pressure and the hydrostatic head in the foam also
determine the bulk drainage behavior of the foam. If the capillary pressure at the top
of the foam balances the hydrostatic head, then bulk drainage will not occur. As we
show in later chapters, the stability of the films between antifoam entities and the gas
liquid surface—the so-called pseudoemulsion films [60]—may also be determined by
the lack of balance between the disjoining pressure in the pseudoemulsion film and the
Plateau border capillary pressure. It is therefore important to clearly define the nature of
the pressure distribution in the continuous phase of a foam as represented by the system
of Plateau border channels. In this, we follow closely the arguments of Princen [61].
We first consider a polydisperse foam under mechanical equilibrium where drain-
age is absent. The upper region of such a foam is shown schematically in Figure 1.13.
The continuous gas–liquid surface at y = 0 covers the dome-shaped tops of bubbles
and the upper Plateau borders. The Laplace pressure jump at the gas–liquid surface
of the Plateau borders is the difference between the atmospheric pressure, Patm, and](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-50-320.jpg)
![19
Some General Properties of Foams
the pressure PPB(y = 0) in the Plateau borders at y = 0. We therefore define the Plateau
border capillary pressure, p y
c
PB
( )
= 0 , by
p y P y P
c
PB
PB atm AL t
( ) ( )
= = = − = −
0 0 σ κ (1.8)
where κt is the curvature of the air–liquid Plateau border surface at the top of the
foam column where y = 0 and where we note that κt is negative with respect to the
fluid in the border. As stated by Princen [61], κt is a constant everywhere at the top
surface of the foam and is independent of the detailed shape of that surface.
For the total Plateau border pressure PPB(y) at some distance y into the bulk of the
foam, we must allow for the hydrostatic head and must therefore write
P y P gy p y P
PB atm L c
PB
atm
( ) ( )
= + + = = +
0 0
ρ ρL
L AL t
gy − σ κ (1.9)
All this is illustrated in Figure 1.13.
At equilibrium the pressure in the liquid phase at the bottom of a foam column
must equal the atmospheric pressure plus that due to the weight of the foam. That
pressure must also equal the Plateau border pressure at the bottom of the foam.
Therefore, from Equation 1.9, we can write
P y H P gH P g
PB 0
e
atm L 0
e
AL t atm L
( )
= = + − = +
ρ σ κ ρ
0
0
1
H
y
0
e
G
foam
d
∫ −
( )
Φ (1.10)
where H0
e
is the height of the equilibrium foam. The integral accounts for the weight
of the foam column where ΦG
foam
is the gas phase volume fraction (and where the den-
sity of the air is neglected). In the case of an equilibrium foam in the dry limit, the
average gas volume fraction ΦG
foam
→ 1. We can therefore often neglect the integral
Plateau border pressure
PPB(y = 0) = Patm – σAL |κt|
κt κf
Plateau border pressure
PPB(y) = PPB(y = 0) + ρLgy
Patm
y = 0
y
κfi
κi
Top of
foam
column
FIGURE 1.13 Schematic illustration of factors contributing to Plateau border pressure
in foam. (Adapted with permission from Princen, H.M. Langmuir, 4, 164. Copyright 1988
American Chemical Society.)](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-51-320.jpg)
![20 The Science of Defoaming: Theory, Experiment and Applications
in Equation 1.10 and deduce that the capillary pressure jump at the top of the foam
σAL ∣κt∣ approximately matches the hydrostatic head ρLgH0 at the bottom of the foam.
However in the case of a draining foam, the capillary pressure jump at the top of the
foam, p y gH
c
PB
AL t L
( )
= =
0 σ κ ρ so that the actual height H of the foam column
exceeds the corresponding equilibrium height H0. If this foam is stable (in that foam
films at the top of the foam do not rupture), it will continue to drain until H = H0
e
as
a result of increases in (the modulus of the) curvature, ∣κt∣, following increases in the
gas phase volume fraction ΦG
foam
. An equilibrium condition is eventually achieved
where σ κ ρ ρ
AL t L L 0
e
= =
gH gH and where we must have
κ κ
t t
where
κt is the
modulus of the curvature at the top of the foam when drainage has ceased.
At equilibrium, the total capillary pressure applied to a foam film should be bal-
anced by the disjoining pressure ΠALA(h) regardless of the position of the film in the
foam. At the top of the foam column, the film will be subject to two capillary pres-
sures—that due to the curvature of the film and that due to the Plateau border, both
of which will conspire to force liquid out of the film, which will be resisted by a posi-
tive disjoining pressure. Therefore, at equilibrium, the disjoining pressure is given by
ΠALA c
PB
c
film
AL t f
( ) ( )
h p y p
= − = +
( ) = +
( )
0 σ κ κ (1.11)
where κf is the curvature of the dome-shaped surface of the bubble films at the top
of the foam column and pc
film
is the capillary pressure jump in those films. In a dry
foam where ΦG
foam
→ 1, then we have ∣κt∣ ≫ ∣κf∣.
The capillary pressure in the interior of the foam where y 0 is given by
p y
c
PB
AL i
( )
= −
0 σ κ (1.12)
where κi is the curvature of the relevant Plateauborders. At equilibrium, the disjoin-
ing pressure in the films in the interior of the foam must therefore be given
ΠALA (h) = σAL (∣κi∣ ± ∣κfi∣) (1.13)
where κfi is the curvature of the film (the sign of which may be either positive or
negative), and where, according to Princen [61], κfi is zero in the case of a monodis-
perse Kelvin foam.
We can use these arguments to describe the evolution of a foam in a gravity field.
Immediately after generation, the foam is wet and Plateau borders are thick with
relatively low curvatures. Drainage of the foam can therefore occur if Equation 1.10
is not satisfied. That process continues until the capillary pressure at the top of the
foam column equals the hydrostatic head whereupon drainage ceases. During this
process, foam films also drain until the disjoining pressure equals the capillary pres-
sure everywhere in the foam and a condition of mechanical equilibrium is attained.
However, this condition represents an unstable equilibrium because gas diffusion
between bubbles will occur in response to differences in capillary pressure. This
will in turn reduce the number of bubbles and therefore cause the Plateau borders to
begin to swell so that drainage again commences. Moreover, the attainment of the](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-52-320.jpg)
![21
Some General Properties of Foams
equilibrium requires that the disjoining pressure isotherm exhibits a maximum suf-
ficiently high to match the total capillary pressure at the top of the foam column so
that Equation 1.10 is satisfied. If that equation cannot be satisfied, foam collapse will
commence as the capillary pressure exceeds that maximum in the disjoining pres-
sure isotherm. This process of foam collapse will start at the top of the foam column
where, as indicated by Equations 1.11 through 1.13, the capillary pressure is highest.
1.4.2 Foam Drainage
The drainage of the continuous phase liquid out of a foam is a surprisingly complex
phenomenon. It is driven by a combination of gravitational, capillary, and viscous
forces. The rate at which it occurs in the case of so-called “free drainage” is easily
observed by monitoring the rate of accumulation of liquid at the bottom of a foam
column. If the foam is generated from a gas of low liquid solubility, then, as we shall
discuss in Section 1.4.3, changes in the structure of the foam due to gas diffusion are
likely to be slow so that they may be neglected. General treatments must, however,
take account of the coupling of gas diffusion with foam drainage (see, e.g., the treat-
ment of Hilgenfeldt et al. [62]).
The observed drainage behavior of foams has often been successfully described
[63] by the classic foam drainage equation of Goldfarb et al. [64]. Derivation of this
equation therefore provides a useful introduction to the nature of that process. This
approach neglects any contribution to the process from structural changes due to gas
diffusion or film rupture and makes the assumption that the liquid content of the foam
is entirely contained within the Plateau borders. Drainage is therefore assumed to
occur by flow down the interconnected system of Plateau borders (illustrated by a sim-
ulation for the case of a foam made up of Kelvin cells in Figure 1.2b). Any contribution
to drainage from foam films is neglected. The gas–liquid surfaces are assumed to be
immobile. Essentially, the approach assumes that Poiseuille flow occurs in the Plateau
borders of a monodisperse foam and that the liquid volume fraction of that foam, (y,t)
(where ΦL
foam
(y,t) =1− ΦG
foam
(y,t)), at any position y and time t, is simply
ΦL(y,t) = APB(y,t)LPB (1.14)
where y is measured from the top of the foam column and APB(y,t) is the cross-
sectional area of a Plateau border. LPB is the total length of Plateau borders per unit
foam volume. The absence of gas diffusion and film rupture means that LPB is not
a function of y or time t. The task of the theory then is to determine APB(y,t), and
therefore ΦL(y,t), as a function of y and t.
Following Weaire and coworkers [5, 23, 63], we initially suppose the border to
be vertically orientated. The cross-sectional area of the border, APB(y,t), will change
with time as it drains subject to gravitational, capillary, and viscous forces. Consider
then a segment of Plateau border of length Δy, which is shown schematically in
Figure 1.14. The volume of the segment is approximated by [APB(y,t) + (∂APB (y,t)/∂y)t
Δy/2] Δy, which is the limit as Δy → 0 becomes simply APB(y,t)Δy. The total gravi-
tational force driving drainage in the segment is ρg APB(y,t)Δy or simply ρg per unit
volume. The gravitational force is resisted in part by a capillary pressure gradient,](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-53-320.jpg)
![22 The Science of Defoaming: Theory, Experiment and Applications
( / )
∂ ∂
p y
c
PB
t which varies as a result of narrowing of the Plateau border with increasing
y. The segment is therefore subject to a net capillary force of ( / ) ( , )
∂ ∂
p y A y t y
c
PB
t PB
. Δ
or simply ( / )
∂ ∂
p y
c
PB
t per unit volume. The gravitational force is also resisted in part
by viscous friction. By analogy with Poiseuille flow in a cylinder (or even dimen-
sional analysis), the relevant shear force per unit volume is k u y t A y t
geom PB
m
L PB
( , ) / ( , )
η ,
where kgeom is a dimensionless geometrical constant dependent on the shape of the
Plateau border (where kgeom ~ 50), uPB
m
(y,t) is the mean liquid velocity over the cross-
section of the Plateau border [5, 23, 63]. We can therefore write for the net force/unit
volume of the segment of Plateau border that
ρ η
L c
PB
t geom PB
m
L PB
g p y k u y t A y t
( / ) ( , ) / ( , )
− ∂ ∂ − = 0 (1.15)
where the capillary pressure is given by
p A y t
k
c
PB
AL AL geom PB
i
= − = − ( , )
/
σ κ σ (1.16)
RPB = ĸi
−1
pPB
c
= −σAL|ĸi|
pc
PB+ Δpc
PB = pc
PB +
∂pc
PB
∂y t
Δy
Drainage driven by gravity
resisted by capillary and
viscous forces
Cross-sectional area
= APB(y,t) ∝ ĸi
−2
(y,t)
Cross-sectional area =
∆y
RPB ( )
+
∂RPB
∂y t
Δy
APB ( ( )
)
(y,t) +
∂APB
∂y t
Δy
FIGURE 1.14 Plateau border segment subject to increase in cross-sectional area and
decrease in capillary pressure along length Δy and where RPB and APB are radius of curvature
and cross-sectional area of Plateau border, respectively.](https://image.slidesharecdn.com/2169771-250519100200-a91f93d2/85/The-Science-Of-Defoaming-Theory-Experiment-And-Applications-Peter-R-Garrett-54-320.jpg)
The Science Of Defoaming Theory Experiment And Applications Peter R Garrett
- 1. The Science Of Defoaming Theory Experiment And Applications Peter R Garrett download https://ebookbell.com/product/the-science-of-defoaming-theory- experiment-and-applications-peter-r-garrett-4339542 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. The Vaccine Race Science Politics And The Human Costs Of Defeating Disease Meredith Wadman https://ebookbell.com/product/the-vaccine-race-science-politics-and- the-human-costs-of-defeating-disease-meredith-wadman-232145532 The Science Of Ghosts Searching For Spirits Of The Dead Joe Nickell https://ebookbell.com/product/the-science-of-ghosts-searching-for- spirits-of-the-dead-joe-nickell-44880742 The Science Of Family Systems Theory Jacob B Priest https://ebookbell.com/product/the-science-of-family-systems-theory- jacob-b-priest-44945722 The Science Of Stuck Breaking Through Inertia To Find Your Path Forward Britt Frank https://ebookbell.com/product/the-science-of-stuck-breaking-through- inertia-to-find-your-path-forward-britt-frank-44975388
- 3. The Science Of Thai Cuisine Chemical Properties And Sensory Attributes Valeeratana K Sinsawasdi https://ebookbell.com/product/the-science-of-thai-cuisine-chemical- properties-and-sensory-attributes-valeeratana-k-sinsawasdi-46149538 The Science Of Proof Forensic Medicine In Modern France E Claire Cage https://ebookbell.com/product/the-science-of-proof-forensic-medicine- in-modern-france-e-claire-cage-46188460 The Science Of Right In Leibnizs Moral And Political Philosophy The Science Of Right Christopher Johns https://ebookbell.com/product/the-science-of-right-in-leibnizs-moral- and-political-philosophy-the-science-of-right-christopher- johns-46304154 The Science Of Human Diversity A History Of The Pioneer Fund Richard Lynn https://ebookbell.com/product/the-science-of-human-diversity-a- history-of-the-pioneer-fund-richard-lynn-46511354 The Science Of Judo Mike Callan https://ebookbell.com/product/the-science-of-judo-mike-callan-46705260
- 5. Peter R. Garrett THE SCIENCE OF DEFOAMING THE SCIENCE OF DEFOAMING surfactant science series volume 155 Theory, Experiment and Applications
- 7. THE SCIENCE OF DEFOAMING Theory, Experiment and Applications
- 8. SURFACTANT SCIENCE SERIES FOUNDING EDITOR MARTIN J. SCHICK 1918–1998 SERIES EDITOR ARTHUR T. HUBBARD Santa Barbara Science Project Santa Barbara, California 1. Nonionic Surfactants, edited by Martin J. Schick (see also Volumes 19, 23, and 60) 2. Solvent Properties of Surfactant Solutions, edited by Kozo Shinoda (see Volume 55) 3. Surfactant Biodegradation, R. D. Swisher (see Volume 18) 4. Cationic Surfactants, edited by Eric Jungermann (see also Volumes 34, 37, and 53) 5. Detergency: Theory and Test Methods (in three parts), edited by W. G. Cutler and R. C. Davis (see also Volume 20) 6. Emulsions and Emulsion Technology (in three parts), edited by Kenneth J. Lissant 7. Anionic Surfactants (in two parts), edited by Warner M. Linfield (see Volume 56) 8. Anionic Surfactants: Chemical Analysis, edited by John Cross 9. Stabilization of Colloidal Dispersions by Polymer Adsorption, Tatsuo Sato and Richard Ruch 10. Anionic Surfactants: Biochemistry, Toxicology, Dermatology, edited by Christian Gloxhuber (see Volume 43) 11. Anionic Surfactants: Physical Chemistry of Surfactant Action, edited by E. H. Lucassen-Reynders 12. Amphoteric Surfactants, edited by B. R. Bluestein and Clifford L. Hilton (see Volume 59) 13. Demulsification: Industrial Applications, Kenneth J. Lissant 14. Surfactants in Textile Processing, Arved Datyner 15. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applications, edited by Ayao Kitahara and Akira Watanabe 16. Surfactants in Cosmetics, edited by Martin M. Rieger (see Volume 68) 17. Interfacial Phenomena: Equilibrium and Dynamic Effects, Clarence A. Miller and P. Neogi 18. Surfactant Biodegradation: Second Edition, Revised and Expanded, R. D. Swisher 19. Nonionic Surfactants: Chemical Analysis, edited by John Cross 20. Detergency: Theory and Technology, edited by W. Gale Cutler and Erik Kissa 21. Interfacial Phenomena in Apolar Media, edited by Hans-Friedrich Eicke and Geoffrey D. Parfitt 22. Surfactant Solutions: New Methods of Investigation, edited by Raoul Zana 23. Nonionic Surfactants: Physical Chemistry, edited by Martin J. Schick 24. Microemulsion Systems, edited by Henri L. Rosano and Marc Clausse 25. Biosurfactants and Biotechnology, edited by Naim Kosaric, W. L. Cairns, and Neil C. C. Gray 26. Surfactants in Emerging Technologies, edited by Milton J. Rosen
- 9. 27. Reagents in Mineral Technology, edited by P. Somasundaran and Brij M. Moudgil 28. Surfactants in Chemical/Process Engineering, edited by Darsh T. Wasan, Martin E. Ginn, and Dinesh O. Shah 29. Thin Liquid Films, edited by I. B. Ivanov 30. Microemulsions and Related Systems: Formulation, Solvency, and Physical Properties, edited by Maurice Bourrel and Robert S. Schechter 31. Crystallization and Polymorphism of Fats and Fatty Acids, edited by Nissim Garti and Kiyotaka Sato 32. Interfacial Phenomena in Coal Technology, edited by Gregory D. Botsaris and Yuli M. Glazman 33. Surfactant-Based Separation Processes, edited by John F. Scamehorn and Jeffrey H. Harwell 34. Cationic Surfactants: Organic Chemistry, edited by James M. Richmond 35. Alkylene Oxides and Their Polymers, F. E. Bailey, Jr., and Joseph V. Koleske 36. Interfacial Phenomena in Petroleum Recovery, edited by Norman R. Morrow 37. Cationic Surfactants: Physical Chemistry, edited by Donn N. Rubingh and Paul M. Holland 38. Kinetics and Catalysis in Microheterogeneous Systems, edited by M. Grätzel and K. Kalyanasundaram 39. Interfacial Phenomena in Biological Systems, edited by Max Bender 40. Analysis of Surfactants, Thomas M. Schmitt (see Volume 96) 41. Light Scattering by Liquid Surfaces and Complementary Techniques, edited by Dominique Langevin 42. Polymeric Surfactants, Irja Piirma 43. Anionic Surfactants: Biochemistry, Toxicology, Dermatology, Second Edition, Revised and Expanded, edited by Christian Gloxhuber and Klaus Künstler 44. Organized Solutions: Surfactants in Science and Technology, edited by Stig E. Friberg and Björn Lindman 45. Defoaming: Theory and Industrial Applications, edited by P. R. Garrett 46. Mixed Surfactant Systems, edited by Keizo Ogino and Masahiko Abe 47. Coagulation and Flocculation: Theory and Applications, edited by Bohuslav Dobiás 48. Biosurfactants: Production Properties Applications, edited by Naim Kosaric 49. Wettability, edited by John C. Berg 50. Fluorinated Surfactants: Synthesis Properties Applications, Erik Kissa 51. Surface and Colloid Chemistry in Advanced Ceramics Processing, edited by Robert J. Pugh and Lennart Bergström 52. Technological Applications of Dispersions, edited by Robert B. McKay 53. Cationic Surfactants: Analytical and Biological Evaluation, edited by John Cross and Edward J. Singer 54. Surfactants in Agrochemicals, Tharwat F. Tadros 55. Solubilization in Surfactant Aggregates, edited by Sherril D. Christian and John F. Scamehorn 56. Anionic Surfactants: Organic Chemistry, edited by Helmut W. Stache 57. Foams: Theory, Measurements, and Applications, edited by Robert K. Prud’homme and Saad A. Khan
- 10. 58. The Preparation of Dispersions in Liquids, H. N. Stein 59. Amphoteric Surfactants: Second Edition, edited by Eric G. Lomax 60. Nonionic Surfactants: Polyoxyalkylene Block Copolymers, edited by Vaughn M. Nace 61. Emulsions and Emulsion Stability, edited by Johan Sjöblom 62. Vesicles, edited by Morton Rosoff 63. Applied Surface Thermodynamics, edited by A. W. Neumann and Jan K. Spelt 64. Surfactants in Solution, edited by Arun K. Chattopadhyay and K. L. Mittal 65. Detergents in the Environment, edited by Milan Johann Schwuger 66. Industrial Applications of Microemulsions, edited by Conxita Solans and Hironobu Kunieda 67. Liquid Detergents, edited by Kuo-Yann Lai 68. Surfactants in Cosmetics: Second Edition, Revised and Expanded, edited by Martin M. Rieger and Linda D. Rhein 69. Enzymes in Detergency, edited by Jan H. van Ee, Onno Misset, and Erik J. Baas 70. Structure-Performance Relationships in Surfactants, edited by Kunio Esumi and Minoru Ueno 71. Powdered Detergents, edited by Michael S. Showell 72. Nonionic Surfactants: Organic Chemistry, edited by Nico M. van Os 73. Anionic Surfactants: Analytical Chemistry, Second Edition, Revised and Expanded, edited by John Cross 74. Novel Surfactants: Preparation, Applications, and Biodegradability, edited by Krister Holmberg 75. Biopolymers at Interfaces, edited by Martin Malmsten 76. Electrical Phenomena at Interfaces: Fundamentals, Measurements, and Applications, Second Edition, Revised and Expanded, edited by Hiroyuki Ohshima and Kunio Furusawa 77. Polymer-Surfactant Systems, edited by Jan C. T. Kwak 78. Surfaces of Nanoparticles and Porous Materials, edited by James A. Schwarz and Cristian I. Contescu 79. Surface Chemistry and Electrochemistry of Membranes, edited by Torben Smith Sørensen 80. Interfacial Phenomena in Chromatography, edited by Emile Pefferkorn 81. Solid–Liquid Dispersions, Bohuslav Dobiás, Xueping Qiu, and Wolfgang von Rybinski 82. Handbook of Detergents, editor in chief: Uri Zoller Part A: Properties, edited by Guy Broze 83. Modern Characterization Methods of Surfactant Systems, edited by Bernard P. Binks 84. Dispersions: Characterization, Testing, and Measurement, Erik Kissa 85. Interfacial Forces and Fields: Theory and Applications, edited by Jyh-Ping Hsu 86. Silicone Surfactants, edited by Randal M. Hill
- 11. 87. Surface Characterization Methods: Principles, Techniques, and Applications, edited by Andrew J. Milling 88. Interfacial Dynamics, edited by Nikola Kallay 89. Computational Methods in Surface and Colloid Science, edited by Malgorzata Borówko 90. Adsorption on Silica Surfaces, edited by Eugène Papirer 91. Nonionic Surfactants: Alkyl Polyglucosides, edited by Dieter Balzer and Harald Lüders 92. Fine Particles: Synthesis, Characterization, and Mechanisms of Growth, edited by Tadao Sugimoto 93. Thermal Behavior of Dispersed Systems, edited by Nissim Garti 94. Surface Characteristics of Fibers and Textiles, edited by Christopher M. Pastore and Paul Kiekens 95. Liquid Interfaces in Chemical, Biological, and Pharmaceutical Applications, edited by Alexander G. Volkov 96. Analysis of Surfactants: Second Edition, Revised and Expanded, Thomas M. Schmitt 97. Fluorinated Surfactants and Repellents: Second Edition, Revised and Expanded, Erik Kissa 98. Detergency of Specialty Surfactants, edited by Floyd E. Friedli 99. Physical Chemistry of Polyelectrolytes, edited by Tsetska Radeva 100. Reactions and Synthesis in Surfactant Systems, edited by John Texter 101. Protein-Based Surfactants: Synthesis, Physicochemical Properties, and Applications, edited by Ifendu A. Nnanna and Jiding Xia 102. Chemical Properties of Material Surfaces, Marek Kosmulski 103. Oxide Surfaces, edited by James A. Wingrave 104. Polymers in Particulate Systems: Properties and Applications, edited by Vincent A. Hackley, P. Somasundaran, and Jennifer A. Lewis 105. Colloid and Surface Properties of Clays and Related Minerals, Rossman F. Giese and Carel J. van Oss 106. Interfacial Electrokinetics and Electrophoresis, edited by Ángel V. Delgado 107. Adsorption: Theory, Modeling, and Analysis, edited by József Tóth 108. Interfacial Applications in Environmental Engineering, edited by Mark A. Keane 109. Adsorption and Aggregation of Surfactants in Solution, edited by K. L. Mittal and Dinesh O. Shah 110. Biopolymers at Interfaces: Second Edition, Revised and Expanded, edited by Martin Malmsten 111. Biomolecular Films: Design, Function, and Applications, edited by James F. Rusling 112. Structure–Performance Relationships in Surfactants: Second Edition, Revised and Expanded, edited by Kunio Esumi and Minoru Ueno 113. Liquid Interfacial Systems: Oscillations and Instability, Rudolph V. Birikh, Vladimir A. Briskman, Manuel G. Velarde, and Jean-Claude Legros 114. Novel Surfactants: Preparation, Applications, and Biodegradability: Second Edition, Revised and Expanded, edited by Krister Holmberg
- 12. 115. Colloidal Polymers: Synthesis and Characterization, edited by Abdelhamid Elaissari 116. Colloidal Biomolecules, Biomaterials, and Biomedical Applications, edited by Abdelhamid Elaissari 117. Gemini Surfactants: Synthesis, Interfacial and Solution-Phase Behavior, and Applications, edited by Raoul Zana and Jiding Xia 118. Colloidal Science of Flotation, Anh V. Nguyen and Hans Joachim Schulze 119. Surface and Interfacial Tension: Measurement, Theory, and Applications, edited by Stanley Hartland 120. Microporous Media: Synthesis, Properties, and Modeling, Freddy Romm 121. Handbook of Detergents, editor in chief: Uri Zoller, Part B: Environmental Impact, edited by Uri Zoller 122. Luminous Chemical Vapor Deposition and Interface Engineering, Hirotsugu Yasuda 123. Handbook of Detergents, editor in chief: Uri Zoller, Part C: Analysis, edited by Heinrich Waldhoff and Rüdiger Spilker 124. Mixed Surfactant Systems: Second Edition, Revised and Expanded, edited by Masahiko Abe and John F. Scamehorn 125. Dynamics of Surfactant Self-Assemblies: Micelles, Microemulsions, Vesicles and Lyotropic Phases, edited by Raoul Zana 126. Coagulation and Flocculation: Second Edition, edited by Hansjoachim Stechemesser and Bohulav Dobiás 127. Bicontinuous Liquid Crystals, edited by Matthew L. Lynch and Patrick T. Spicer 128. Handbook of Detergents, editor in chief: Uri Zoller, Part D: Formulation, edited by Michael S. Showell 129. Liquid Detergents: Second Edition, edited by Kuo-Yann Lai 130. Finely Dispersed Particles: Micro-, Nano-, and Atto-Engineering, edited by Aleksandar M. Spasic and Jyh-Ping Hsu 131. Colloidal Silica: Fundamentals and Applications, edited by Horacio E. Bergna and William O. Roberts 132. Emulsions and Emulsion Stability, Second Edition, edited by Johan Sjöblom 133. Micellar Catalysis, Mohammad Niyaz Khan 134. Molecular and Colloidal Electro-Optics, Stoyl P. Stoylov and Maria V. Stoimenova 135. Surfactants in Personal Care Products and Decorative Cosmetics, Third Edition, edited by Linda D. Rhein, Mitchell Schlossman, Anthony O’Lenick, and P. Somasundaran 136. Rheology of Particulate Dispersions and Composites, Rajinder Pal 137. Powders and Fibers: Interfacial Science and Applications, edited by Michel Nardin and Eugène Papirer 138. Wetting and Spreading Dynamics, edited by Victor Starov, Manuel G. Velarde, and Clayton Radke 139. Interfacial Phenomena: Equilibrium and Dynamic Effects, Second Edition, edited by Clarence A. Miller and P. Neogi
- 13. 140. Giant Micelles: Properties and Applications, edited by Raoul Zana and Eric W. Kaler 141. Handbook of Detergents, editor in chief: Uri Zoller, Part E: Applications, edited by Uri Zoller 142. Handbook of Detergents, editor in chief: Uri Zoller, Part F: Production, edited by Uri Zoller and co-edited by Paul Sosis 143. Sugar-Based Surfactants: Fundamentals and Applications, edited by Cristóbal Carnero Ruiz 144. Microemulsions: Properties and Applications, edited by Monzer Fanun 145. Surface Charging and Points of Zero Charge, Marek Kosmulski 146. Structure and Functional Properties of Colloidal Systems, edited by Roque Hidalgo-Álvarez 147. Nanoscience: Colloidal and Interfacial Aspects, edited by Victor M. Starov 148. Interfacial Chemistry of Rocks and Soils, Noémi M. Nagy and József Kónya 149. Electrocatalysis: Computational, Experimental, and Industrial Aspects, edited by Carlos Fernando Zinola 150. Colloids in Drug Delivery, edited by Monzer Fanun 151. Applied Surface Thermodynamics: Second Edition, edited by A. W. Neumann, Robert David, and Yi Y. Zuo 152. Colloids in Biotechnology, edited by Monzer Fanun 153. Electrokinetic Particle Transport in Micro/Nano-fluidics: Direct Numerical Simulation Analysis, Shizhi Qian and Ye Ai 154. Nuclear Magnetic Resonance Studies of Interfacial Phenomena, Vladimir M. Gun’ko and Vladimir V. Turov 155. The Science of Defoaming: Theory, Experiment and Applications, Peter R. Garrett
- 15. CRC Press is an imprint of the Taylor & Francis Group, an informa business Boca Raton London NewYork Peter R. Garrett THE SCIENCE OF DEFOAMING Theory, Experiment and Applications
- 16. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20130607 International Standard Book Number-13: 978-1-4200-6042-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmit- ted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 17. © 2010 Taylor & Francis Group, LLC Dedication To my wife Carole for her encouragement and forbearance
- 19. xiii © 2010 Taylor & Francis Group, LLC Contents Preface.................................................................................................................. xxiii Author.....................................................................................................................xxv Chapter 1 Some General Properties of Foams.......................................................1 1.1 Introduction................................................................................1 1.2 Structure of Foams.....................................................................1 1.3 Foam Films.................................................................................5 1.3.1 Surface Tension Gradients and Foam Film Stability....5 1.3.2 Drainage Processes in Foam Films...............................9 1.3.3 Disjoining Forces and Foam Film Stability................ 14 1.4 Processes Accompanying Aging of Foam................................ 18 1.4.1 Capillary Pressure Gradients...................................... 18 1.4.2 Foam Drainage............................................................ 21 1.4.3 Bubble Coarsening (Diffusional Disproportionation).....24 1.5 Summarizing Remarks.............................................................28 Acknowledgment.................................................................................29 References...........................................................................................29 Chapter 2 Experimental Methods for Study of Foam and Antifoam Action.......33 2.1 Introduction..............................................................................33 2.2 Measurement of Foam..............................................................33 2.2.1 Bartsch Method: Hand Shaking Measuring Cylinders...................................................33 2.2.2 Automated Shake Tests...............................................35 2.2.3 Ross–Miles Method....................................................35 2.2.4 Tumbling Cylinders.....................................................36 2.2.5 Gas Bubbling...............................................................37 2.2.6 Direct Air Injection.....................................................38 2.2.7 Measurement of Bubble Size Distributions.................39 2.3 Observations with Single Foam Films..................................... 41 2.3.1 Scheludko Cells........................................................... 41 2.3.1.1 Films Containing Antifoam Drops.............. 41 2.3.1.2 Measurement of Disjoining Pressure Isotherms of Air–Liquid–Air Foam Films..................................................43 2.3.2 Dippenaar Cell............................................................44 2.3.3 Large Vertical Films...................................................45
- 20. xiv Contents © 2010 Taylor & Francis Group, LLC 2.4 Air–Water–Oil Pseudoemulsion Films.....................................46 2.4.1 Direct Observation of Pseudoemulsion Films.............46 2.4.2 Measurement of Disjoining Pressure Isotherms for Air–Water–Oil Pseudoemulsion Films..................48 2.4.3 Direct Measurements of Pseudoemulsion Film Rupture Pressures........................................................49 2.5 Spreading Behavior of Oils......................................................52 2.6 Summarizing Remarks.............................................................53 References...........................................................................................54 Chapter 3 Oils at Interfaces: Entry Coefficients, Spreading Coefficients, and Thin Film Forces..........................................................................57 3.1 Introduction..............................................................................57 3.2 Classic Entry and Spreading Coefficients................................58 3.3 Generalized Entry Coefficients, Pseudoemulsion Films, and Thin Film Forces............................................................... 61 3.3.1 Definitions................................................................... 61 3.3.2 Case Where Generalized Entry Coefficient, Eg > 0....65 3.3.3 Case Where Generalized Entry Coefficient, Eg < 0....68 3.3.4 Magnitude of Generalized Entry Coefficients............70 3.4 Mode of Rupture of Pseudoemulsion Films.............................72 3.4.1 Disjoining Pressures and Stability of Pseudoemulsion Films.................................................72 3.4.2 Surface Tension Gradients and Stability of Pseudoemulsion Films.................................................78 3.5 Generalized Spreading Coefficients and Thin Film Forces.....79 3.6 Spreading Behavior of Typical Antifoam Oils on Aqueous Surfaces.....................................................................85 3.6.1 Hydrocarbon Oils........................................................85 3.6.1.1 Spreading and Wetting Behavior of Hydrocarbons on Surface of Pure Water.....85 3.6.1.2 Complete and Pseudo-Partial Wetting Behavior of Hydrocarbons on the Surfaces of Aqueous Surfactant Solutions................................................. 87 3.6.1.3 Non-Spreading (Partial Wetting) by Hydrocarbons on the Surfaces of Aqueous Surfactant Solutions......................94 3.6.2 Polydimethylsiloxane Oils...........................................96 3.6.2.1 Complete and Pseudo-Partial Wetting Behavior of Polydimethylsiloxanes on the Surfaces of Pure Water and Aqueous Surfactant Solutions......................96
- 21. xv Contents © 2010 Taylor & Francis Group, LLC 3.6.2.2 Rates of Spreading of Polydimethylsiloxanes on the Surfaces of Pure Water and Aqueous Surfactant Solutions....................................................104 3.6.3 Effect of Spread Hydrocarbon and Polydimethylsiloxane Oils on the Stability of Pseudoemulsion Films...............................................106 3.7 Non-Equilibrium Effects due to Surfactant Transport...........108 3.8 Summarizing Remarks...........................................................108 References......................................................................................... 110 Chapter 4 Mode of Action of Antifoams........................................................... 115 4.1 Introduction............................................................................ 115 4.2 Antifoam Effects due to Solubilized Oils.............................. 116 4.3 Effect on Foamability of Mesophase Precipitation in Aqueous Surfactant Solutions................................................ 121 4.4 Surface Tension Gradients and Theories of Antifoam Mechanism.............................................................................128 4.4.1 General Considerations.............................................128 4.4.2 Surface Tension Gradients Induced by Spreading Antifoam...................................................................129 4.4.3 Elimination of Surface Tension Gradients................138 4.5 Oil Bridges and Antifoam Mechanism.................................. 141 4.5.1 Oil Bridges in Foam Films........................................ 141 4.5.2 Line Tensions and Antifoam Behavior of Oil Lenses....153 4.5.3 Oil Bridges in Plateau Borders and Stability of Pseudoemulsion Films............................................... 157 4.6 Antifoam Behavior of Emulsified Liquids............................. 165 4.6.1 Antifoam Effects of Neat Oils in Aqueous Foaming Systems...................................................... 165 4.6.1.1 Early Work................................................. 165 4.6.1.2 Short-Chain Alcohols................................ 167 4.6.1.3 n-Alkanes................................................... 175 4.6.1.4 Neat Polydimethylsiloxane Oils................. 180 4.6.2 Antifoam Effects of Neat Oils in Non-Aqueous Foaming Systems...................................................... 183 4.6.3 Partial Miscibility and Antifoam Effects.................. 185 4.6.3.1 Origins of Partial Miscibility in Binary Liquid Mixtures......................................... 185 4.6.3.2 Systems with Lower Critical Temperatures............................................. 188 4.6.3.3 Systems with Higher Critical Temperatures............................................. 198
- 22. xvi Contents © 2010 Taylor & Francis Group, LLC 4.7 Inert Hydrophobic Particles and Capillary Theories of Antifoam Mechanism for Aqueous Systems..........................201 4.7.1 Early Work................................................................201 4.7.2 Experimental Observations Concerning Contact Angles and Particle Bridging Mechanism................203 4.7.3 Theoretical Considerations Concerning Particle Geometry and Contact Angle Conditions for Antifoam Action by Smooth Particles...................... 216 4.7.3.1 Particles with Curved Surfaces and No Edges.................................................... 216 4.7.3.2 Particles with Edges................................... 216 4.7.3.3 Models of Foam Film Rupture by Particles Using Surface Energy Minimization.............................................224 4.7.4 Effect of Rugosities on Antifoam Action of Particles.....................................................................228 4.7.5 Rugosities and Stability of Air–Water–Solid Films.......................................................................233 4.7.6 Particle Size and Kinetics of Foam Film Rupture....239 4.7.7 Antifoam Effects of Calcium Soaps..........................243 4.7.8 Melting of Hydrophobic Particles and Antifoam Behavior....................................................................247 4.8 Mixtures of Hydrophobic Particles and Oils as Antifoams for Aqueous Systems............................................249 4.8.1 Antifoam Synergy.....................................................249 4.8.2 Early Patent Literature.............................................. 251 4.8.3 Role of Oil in Synergistic Oil–Particle Antifoams.....252 4.8.4 Early Hypotheses Concerning Role of Particles in Synergistic Oil–Particle Antifoams......................263 4.8.5 Role of Particles in Synergistic Oil–Particle Antifoams..................................................................267 4.8.5.1 Experimental Observation.........................267 4.8.5.2 Spherical Particles, Spread Oil Layers, and Rupture of Pseudoemulsion Films...............................272 4.8.5.3 Smooth Particles with Edges in Absence of Spread Oil Layers...................277 4.8.5.4 Smooth Particles with Edges in Presence of Spread Oil Layers................... 281 4.8.5.5 Rough Particles with Many Edges............. 281 4.8.6 Antifoam Dimensions and Kinetics..........................290 4.9 Summarizing Remarks...........................................................292 Acknowledgments.............................................................................295 Appendix 4.1.....................................................................................295 References.........................................................................................302
- 23. xvii Contents © 2010 Taylor & Francis Group, LLC Chapter 5 Effect of Antifoam Concentration on Volumes of Foam Generated by Air Entrainment..........................................................309 5.1 Phenomenology......................................................................309 5.1.1 Introduction...............................................................309 5.1.2 Dispersion of Single Antifoam.................................. 310 5.1.3 Mixed Dispersions of Two Antifoams...................... 313 5.1.4 Relative Effectiveness of Antifoam Entities and Foam Structure.......................................................... 318 5.2 Statistical Theory of Antifoam Action...................................320 5.2.1 Assumptions..............................................................320 5.2.2 Factors Determining Number of Antifoam Entities in Foam Film................................................324 5.2.3 Calculation of Volume of Air in Foam in Presence of Antifoam................................................330 5.2.4 Limitations of Theory...............................................334 5.3 Summarizing Remarks...........................................................336 Appendix 5.1 Effect of Excluded Volume on Antifoam Concentration in a Film Exhibiting Reynolds Drainage........338 A5.1.1 Mean Flow Velocity of Antifoam Entities................338 A5.1.2 Effect of Excluded Volume on Antifoam Concentration in a Draining Film............................. 339 References......................................................................................... 341 Chapter 6 Deactivation of Mixed Oil–Particle Antifoams during Dispersal and Foam Generation in Aqueous Media.........................343 6.1 Introduction............................................................................343 6.2 Deactivation of Antifoam Effect of Polydimethylsiloxane Oils without Particles.............................................................344 6.3 Early Work with Hydrophobed Silica– Polydimethylsiloxane Antifoams............................................346 6.3.1 Separation of Silica from Oil....................................346 6.3.2 Equilibration and Deactivation.................................347 6.3.3 Deactivation, Emulsification, and Drop Sizes...........348 6.4 Deactivation of Hydrophobed Silica– Polydimethylsiloxane Antifoam by Disproportionation......... 351 6.4.1 Experimental Evidence............................................. 351 6.4.2 Theoretical Considerations Concerning Deactivation by Disproportionation.......................... 357 6.5 Effect of Oil Viscosity on Deactivation of Hydrophobed Silica–Polydimethylsiloxane Antifoams................................363 6.6 Deactivation in Other Types of Oil–Particle Antifoams........366 6.7 Theories of Foam Volume Growth in Presence of Deactivating Antifoam...........................................................368
- 24. xviii Contents © 2010 Taylor & Francis Group, LLC 6.7.1 Antifoam–Bubble Heterocoalescence–Kinetic Model of Pelton and Goddard for Foam Generation by Sparging.............................................368 6.7.2 Modified Antifoam–Bubble Heterocoalescence- Kinetic Model Using a Statistical Distribution of Antifoam Drops over Bubbles...................................373 6.7.3 Combination of Kinetic Model of Antifoam Deactivation with Kinetic Model of Antifoam Action........................................................................ 376 6.7.4 Combination of Kinetic Model of Antifoam Deactivation by Disproportionation with Empirical Expression for Antifoam Action..............379 6.8 Summarizing Remarks...........................................................383 Acknowledgment...............................................................................386 References.........................................................................................386 Chapter 7 Mechanical Methods for Defoaming................................................389 7.1 Introduction ...........................................................................389 7.2 Defoaming Using Rotary Devices..........................................389 7.2.1 Designs of Rotary Devices Described in Scientific Literature...................................................389 7.2.2 Commercial Rotary Defoamers................................398 7.2.3 Defoaming Mechanisms of Rotary Devices.............400 7.2.3.1 Role of Centrifugal Force .........................400 7.2.3.2 Role of Shear and Impact Forces on Bubbles in Mechanical Defoaming...........404 7.2.3.3 Defoaming by Inherent Liquid Spray........407 7.3 Defoaming Using Ultrasound.................................................409 7.3.1 Brief History of Defoaming by Ultrasound..............409 7.3.2 Defoaming Mechanism of Ultrasound...................... 415 7.4 Defoaming Using Packed Beds of Appropriate Wettability..... 421 7.5 Summarizing Remarks...........................................................422 Appendix 7.1......................................................................................423 References.........................................................................................428 Chapter 8 Antifoams for Detergent Products.................................................... 431 8.1 Introduction............................................................................ 431 8.2 Powders for Machine Washing of Laundry............................ 433 8.2.1 Front-Loading Drum-Type Textile Washing Machines................................................................... 433 8.2.2 Use of Fatty Acids and Soaps.................................... 435 8.2.3 Use of Non-Soap Particulate Antifoams................... 439 8.2.4 Use of Hydrocarbon–Hydrophobic Particle Mixtures....................................................................441
- 25. xix Contents © 2010 Taylor & Francis Group, LLC 8.2.4.1 Hydrocarbon Mixtures with Alkyl Phosphoric Acid Derivatives.....................441 8.2.4.2 Hydrocarbon Mixtures with Non- Phosphorous-Containing Organic Compounds................................................446 8.2.5 Use of Polydimethylsiloxane-Based Antifoams........450 8.2.5.1 General Properties.....................................450 8.2.5.2 Storage Deactivation and Incorporation in Detergent Powders................................. 453 8.2.5.3 Dispensing.................................................456 8.2.5.4 Enhancement of Antifoam Effectiveness......................................... 457 8.2.6 Hydrocarbon-Based Simulation of Dimethylsiloxane-Based Antifoams.........................460 8.3 Liquids for Machine Washing of Laundry............................. 461 8.3.1 General Properties..................................................... 461 8.3.2 Incorporation of Polyorganosiloxane–Hydrophobic Silica Antifoams in Detergent Liquids......................... 462 8.4 Machine Dishwashing............................................................467 8.5 General Hard-Surface Cleaning Products..............................469 8.6 Summarizing Remarks........................................................... 471 Appendix 8.1..................................................................................... 471 References......................................................................................... 476 Chapter 9 Control of Foam in Waterborne Latex Paints and Varnishes............ 481 9.1 Introduction............................................................................ 481 9.2 Foam and Antifoam Behavior................................................485 9.2.1 General Considerations.............................................485 9.2.2 Effect of Stratified Layers of Polymer Latex Particles on Foam and Pseudoemulsion Film Stability.....................................................................488 9.3 Specific Issues Concerning Oil-Based Antifoams.................489 9.3.1 Incorporation in Paints and Varnishes......................489 9.3.2 Defect Formation in Drying Paint Films..................492 9.3.2.1 General Considerations..............................492 9.3.2.2 Experimental and Theoretical Studies of Cratering Caused by Marangoni Effect Induced by Spreading Oil Drops..................................495 9.3.2.3 Putative Craters Caused by Non- Spreading Oil Drops Bridging Paint Films..........................................................497 9.4 Summarizing Remarks...........................................................499 References.........................................................................................500
- 26. xx Contents © 2010 Taylor & Francis Group, LLC Chapter 10 Antifoams for Gas–Oil Separation in Crude Oil Production...........503 10.1 Introduction............................................................................503 10.2 Surface Activity at Gas–Hydrocarbon and Gas–Crude Oil Interfaces..........................................................................504 10.3 Causes of Foam Formation in Gas–Crude Oil Systems.........509 10.3.1 Disjoining Pressures..................................................509 10.3.2 Origin of Surface Tension Gradients at Gas–Crude Oil Interfaces......................................... 510 10.3.3 Experimental Observations of Foam Behavior......... 511 10.4 Use of Antifoams.................................................................... 515 10.4.1 General Considerations............................................. 515 10.4.2 Polydimethylsiloxanes and Substituted Polydimethylsiloxanes............................................... 517 10.4.2.1 Effect of Solubility of Antifoam Oils........ 517 10.4.2.2 Mode of Antifoam Action in Crude Oils............................................. 520 10.4.3 Other Materials.........................................................524 10.5 Summarizing Remarks...........................................................525 References.........................................................................................526 Chapter 11 Medical Applications of Defoaming................................................. 529 11.1 Introduction............................................................................ 529 11.2 Use of Simethicone Antifoam in Treatment of Gastrointestinal Gas...............................................................530 11.2.1 Therapeutic Application............................................530 11.2.2 Use of Simethicone in Endoscopy............................. 532 11.3 Defoaming of Blood during Cardiopulmonary Bypass Surgery....................................................................... 533 11.3.1 Gas Bubble Oxygenators and Use of Antifoams....... 533 11.3.2 Mechanism of Polydimethylsiloxane– Hydrophobed Silica–Coated Porous Defoamers....... 535 11.3.3 Polydimethylsiloxane–Hydrophobed Silica Antifoam as Source of Emboli.................................. 537 11.3.4 Cardiotomy Defoaming............................................. 539 11.3.5 Defoaming in Cardiopulmonary Bypass Blood Circuits, Which Include Membrane Oxygenators and Cardiotomy/Venous Reservoirs..................................................................540 11.3.6 Defoaming Systems Avoiding Use of Polydimethylsiloxane-Based Antifoam.....................542 11.3.6.1 Potential Replacements for PDMS– Hydrophobed Silica in Cardiotomy Reservoirs..................................................542
- 27. xxi Contents © 2010 Taylor & Francis Group, LLC 11.3.6.2 Potential Use of Defoamer Elements with High Air–Blood Contact Angles.......544 11.3.6.3 Removal of Gaseous Microemboli............545 11.4 Summarizing Remarks...........................................................549 References......................................................................................... 551 Frequently Used Symbols and Abbreviations.................................................... 555
- 29. xxiii © 2010 Taylor & Francis Group, LLC Preface It is 20 years since the only other book exclusively concerned with defoaming was published [1]. However, during the intervening years, much new work has been pub- lished and many new insights revealed. It therefore seems appropriate to revisit the topic with a new book. The earlier book was concerned not only with the scientific aspects of the mode of action of antifoams but also with a number of different industrial applications. Indeed the latter formed the greater part of that book. By contrast, this book is more concerned with the basic science of defoaming. In this it replicates the emphasis revealed by published literature over the past two decades. The industrial application of antifoams does in fact produce challenging scientific issues specific to particu- lar applications. Where appropriate, such issues are described here in the relevant chapters. However, they are apparently rarely addressed, at least in published works. Much application still appears to rest on simple empiricism augmented by naive and sometimes completely incorrect views of the mode of action of antifoams. It is hoped that this book will help stimulate a more enlightened approach to application, particularly by those antifoam manufacturers who supply products for specific appli- cations to many different customers. The main part of the book then concerns the science of defoaming. The general properties of foams are presented in an introductory chapter, which serves as a back- ground to much that follows. Another chapter describes the experimental techniques involved in measurement of the foam properties relevant for study of antifoam action and the various techniques used in establishing the mode of action of antifoams. Various aspects of antifoam behavior and relevant theory are considered in later chapters. Most commercially effective antifoams are oil based. The relevance of the entry and spreading behavior of such oils at the surfaces of foaming fluids to their anti- foam function has been the subject of much debate over the past half century. A chapter is therefore specifically devoted to the entry and spreading behavior of oils and the role of thin film forces in determining that behavior. The role of bridging foam films by particles and oil drops in the mode of action of antifoams was emphasized in the earlier book. Also included was the first evidence that the role of particles in synergistic oil–particle antifoam mixtures for defoam- ing aqueous foams concerns rupture of air–water–oil pseudoemulsion films. The past two decades have seen a general acceptance and further development of these aspects of the theory of antifoam action. Particularly noteworthy have been the insights provided by Professor Nikki Denkov and his coworkers at the University of Sofia. All of this is reviewed in a chapter specifically concerned with the mode of action of antifoams. The volume of foam generated in the presence of antifoam is a function of anti- foam concentration. Both the relevant phenomenology and a statistical theory can be shown to be consistent with experiment provided the effectiveness of individual
- 30. xxiv Preface © 2010 Taylor & Francis Group, LLC antifoam entities remains constant. However, the effect of the concentration of anti- foam can be complicated by the decline in effectiveness of individual entities during foam generation due to disproportionation to form inactive material. Two chapters deal with these issues. In some applications, contamination of a foaming medium with chemical anti- foams is unacceptable. Defoaming can then be achieved, in principle, using mechan- ical means, which include centrifugal or ultrasonic devices. A review of the mode of the defoaming action of such devices is included here. It reveals a subject relatively rich in opportunity for the application of new insight. The author is particularly grateful to various colleagues for stimulating discus- sions on various aspects of the topics covered in this book. Those colleagues included Dr. Paul Grassia (University of Manchester), Dr. Stephen Neethling (Imperial College), Professor Nikki Denkov (University of Sofia), Professor Clarence Miller (Rice University), and Professor Gordon Tiddy (University of Manchester). Peter R. Garrett REFERENCE 1. Garrett, P.R. (ed.). Defoaming, Theory and Industrial Applications, Marcel Dekker, Surfactant Science Series, Vol. 45, 1993.
- 31. xxv © 2010 Taylor & Francis Group, LLC Author After working for Unilever Research at Port Sunlight for more than a quarter of a century, Peter Garrett retired, acquiring visiting professorships at the University of Manchester in the United Kingdom and briefly at Rice University, Houston, Texas. He has worked on various aspects of interfacial science in a career lasting 40 years. These included the equilibrium and dynamic adsorption behavior of surfactant solu- tions, solubilization in microemulsions, kinetics of surfactant dissolution and meso- phase growth, and of course both the basic science of defoaming and its application especially in the detergents industry. He is author or coauthor of more than 30 journal articles and is named as inventor on 14 patents, the majority of which concerned the application of antifoam technology. He is also editor of the book Defoaming, Theory and Applications published in 1993. A member of the Royal Society of Chemistry, he received his PhD degree from the University of Leicester in the United Kingdom in 1971.
- 33. 1 1 Some General Properties of Foams 1.1 INTRODUCTION Before considering the behavior of antifoams, we review the relevant properties of foams. Only a brief summary is given here. It is for the most part only concerned with those aspects that may have relevance for the understanding of antifoam action. For more complete accounts, the reader is referred to the many books [1–9] and reviews on the subject [10–27]. This brief review includes definition of the structural features of foams. A sum- mary of the processes occurring in foam films follows with particular emphasis on the factors that determine the stability of those films. Finally, we include an outline of the processes of drainage and diffusion-driven coarsening, which concern the entire body of a foam and not just the constituent parts. 1.2 STRUCTURE OF FOAMS We first consider the structure of a polydisperse foam. That structure is exemplified by the photograph reproduced in Figure 1.1. This image depicts a foam that has been aged, where both drainage and diffusion of gas from small bubbles to large bubbles, as a result of differences in capillary pressure, have occurred. In the lower part of the foam, bubbles are spherical (so-called kugelschaum) and of small size with a relatively low overall gas volume fraction. Collections of spherical bubbles, without the distortions associated with film formation, form at gas volume fractions, ΦG foam , of ≤~0.74 in the case of monodisperse bubbles and ≤~0.72 in the case of polydisperse bubbles [28]. As the liquid drains out of, for example, a polydisperse foam so that ΦG foam becomes 0.72, the bubbles distort to form polyhedra. This polyhedral foam (polyederschaum), with a relatively high gas volume fraction, consists of thin foam films joined by Plateau border channels. In the case of the foam depicted in Figure 1.1, there is clear segregation of bubble sizes, with larger bubbles being present at the top of the foam column. The extent of such vertical segregation in polydisperse foams varies according to the method of generation. It probably depends on the extent of mixing during foam generation, gravity segregation [5], and even the so-called brazil nut effect [29]. Segregation of such a polydisperse foam can in fact apparently be facilitated by rapid continuous wetting of a foam column from above so that high liquid volume fractions prevail. Bubble movement can then occur without requiring the distortion of bubbles so that gravitational segregation is in turn specifically facilitated [5].
- 34. 2 The Science of Defoaming: Theory, Experiment and Applications It is worth noting that the Plateau borders between bubbles of different sizes in Figure 1.1 are in fact curved, being concave with respect to the larger bubbles. This implies that the adjacent foam films are also curved so that the capillary pressure is larger inside the smaller bubbles. Such differences drive the process of gas diffusion leading to coarsening of the foam where average bubble sizes increase. We illustrate the salient structural features of a foam in Figure 1.2, by relating computer-generated simulations reported by Weaire and Hutzler [5] to two actual images of polydisperse foams, each of different gas volume fraction, made by Hartland and Barber [30]. The comparison is, however, intended to be only qualitative, particu- larly with respect to gas phase volume fractions in the respective experimental images and simulations. The latter concern assemblies of the so-called Kelvin cells, tetrakaid- ecahedra, with six flat quadrilateral faces and eight curved hexagonal faces. However, polydispersity means that such polyhedra are not at all present in the foams imaged by Hartland and Barber [30]. As shown in Figure 1.2b, Plateau borders join together to form a network of chan- nels, containing almost all the liquid in the foam and through which drainage occurs in the gravity field. The junctions or nodes of the Plateau borders in the interior of a dry foam (where ΦG foam 1 → ) invariably involve four borders meeting at a regular tetrahedral angle of 109.5°. The Plateau border cross-sections are seen to be con- cave triangular in shape, with each of the vertices terminating in a foam film where 0.5 cm FIGURE 1.1 Aged polydisperse foam with dry polyhedral foam (polyederschaum) at top of column and spherical bubbles (kugelschaum) at bottom of column.
- 35. 3 Some General Properties of Foams Plateau border node 109.5° 120° Plateau border cross section Kelvin structure with Φg ~ 0.99 Kelvin structure with Φg ~ 0.9 (a) Plateau border network h Foam film; h rf 0.5 cm 0.5 cm (b) (c) (e) (d) (h) (g) (f) 2rf FIGURE 1.2 Structural elements of foam. (a) Image of polyhedral foam against wall of vessel. (b) Simulated three-dimensional Plateau border network for monodisperse Kelvin cell foam. This network contains almost all of the liquid in foam. Liquid drains through network under influence of gravity. (c) Plateau border node where four borders meet at tet- rahedral angle. (d) Cross section of plateau border where vertices terminate in foam films. (e) Simulated Kelvin cell “dry” foam with gas volume fraction, ΦG foam , of ~0.99. (f) Image of “wet” foam or froth with gas volume fraction close to limit of 0.72 (for a polydisperse foam, reference [28]) where Plateau borders and foam films are absent. (g) Simulated Kelvin cell “wet” foam with gas volume fraction, ΦG foam , of ~0.9. (h) Foam films are present between bubbles in simulated Kelvin cell foams shown in (e) and (g). Such films are characterized by thicknesses several orders of magnitude less than their width. (Images (a) and (f) from Hartland, S., Barber, A.D., Trans. Inst. Chem. Eng., 52, 43, 1974. With permission from Institution of Chemical Engineers. Images (b), (c), (e), and (g) from Weaire, D., Hutzler, S., The Physics of Foams, 1999, by permission of Oxford University Press.)
- 36. 4 The Science of Defoaming: Theory, Experiment and Applications the angle between the films as they intersect the borders is 120°. These structural features represent the essentials of Plateau’s rules [31], which describe a condition of mechanical equilibrium and apply equally to all both monodisperse and polydis- perse foams in the limit where ΦG foam 1 [5] → . Images, such as those presented in Figures 1.1 and 1.2, are of actual foams, gener- ated in various containers. These images usually represent only the layer of bubbles adjacent to the walls of those vessels, a limitation often imposed by the depth of field of the relevant photographic equipment together with the relative opacity of the foam. Such images exaggerate the proportion of large bubbles in a polydisperse foam as a result of a statistical sampling bias [10]. Moreover, the surface of the contain- ing vessel, which represents the plane of observation, means that bubbles adjacent to that plane are necessarily distorted as indicated by, for example, Steiner et al. [32] and later by Cheng and Lemlich [33]. In the case of a monodisperse foam, the Plateau borders of bubbles adjacent to the confining plane form regular hexagons. The origin of this geometry has been neatly demonstrated by Weaire and Hutzler [5] in a simulation with Kelvin cells. Tessellation of Kelvin cells in a monodisperse foam contained in a vessel is in principle possible if the edge bubbles are formed by slicing a Kelvin cell across the middle as indicated in Figure 1.3a. The resulting structure with a hexagonal planar surface is shown in Figure 1.3b. Those Plateau borders directed away from that plane are all oriented at 90°. In this respect, these structures violate one of Plateau’s rules. However, even in the case of a monodisperse Plateau borders form a hexagonal cross section at plane of observation Kelvin bubble is bisected to form a bubble that can still tessellate despite touching the wall of the retaining vessel (a) (b) FIGURE 1.3 In a monodisperse Kelvin foam, even bubbles that touch the surface of retain- ing vessel must tessellate. Simulation suggests that this is permitted if those bubbles have structure formed by bisecting a Kelvin cell as indicated in (a). Bubbles with resulting struc- ture must be extended to restore original volume. As shown in (b), the edge of bubble then presents a hexagonal cross section to plane of observation at wall of vessel. (From Weaire, D., Hutzler, S., The Physics of Foams, 1999, by permission of Oxford University Press.)
- 37. 5 Some General Properties of Foams foam, Kelvin cells only exist one layer into the foam from the surface cells. Beyond that, the structure becomes disordered and Kelvin cells are not apparently found [5]. The dimensions of these structural features of foam are subject to change as a result of the action of various processes. Senescence is facilitated by processes of drainage and diffusional coarsening. However, the ultimate arbitrator of the fate of foams concerns the stability of films, which arguably represents the most important factor in determining both foamability (i.e., the tendency to form a foam) and foam stability. With thicknesses typically at least three orders of magnitude less than the diameter of the circle circumscribed by the edges of a Plateau border, they represent the most fragile aspect of foam structure. Indeed, if the films break, then the struc- tures described here cannot exist if the continuous phase is liquid and consideration of their putative nature becomes purely academic. Design of effective antifoams tends therefore to be focused on destruction of foam films. 1.3 FOAM FILMS 1.3.1 Surface Tension Gradients and Foam Film Stability Films formed by adjacent bubbles in a pure liquid are extremely unstable. Pure liquids therefore do not form foams. This arises in part because of the response of the films to any external force such as gravity. Consider, for example, a ver- tical plane-parallel film of a pure liquid in a gravity field. There is no reason why any element of that film should move in response to the applied gravitational force with a velocity different from that of any adjacent element. No velocity gra- dients in a direction perpendicular to the plane of the film surface against the air will therefore exist. There will then be no viscous shear forces opposing the effect of gravity. The film will exhibit plug flow (resisted only by extensional vis- cous forces) with elements accelerated downward tearing it apart. The process is depicted in Figure 1.4a. This behavior can be drastically altered if we arrange for a tangential force to act in the plane of the liquid–air surface so that the surface is essentially rigid. In the case of a vertical plane-parallel film of a viscous liquid with such rigid surfaces, subject to gravity, a parabolic velocity profile will develop as shown in Figure 1.4b. This means that velocity gradients will exist in a direction perpendicular to the film surfaces. A viscous stress will therefore be exerted at the air–liquid surface. This stress must be balanced by the tangential force acting in the plane of the surface. That force can only be a gradient of surface tension. This balance of viscous forces and surface tension gradients at the liquid–air surface can be written as d d d d AL L σ η y u x y x = =0 (1.1) where σAL is the air–liquid surface tension of the foaming liquid, ηL is the viscosity, uy is the velocity of flow in the y direction, y is the vertical distance, and x is the horizontal distance in the film.
- 38. 6 The Science of Defoaming: Theory, Experiment and Applications Thus, we find that if the force of gravity is to be resisted by the film, then a surface tension gradient must exist at the air–liquid surface. Combining the relevant Navier– Stokes equation with Equation 1.1 as a boundary condition, Lucassen [15] shows that in the case of a vertical film in the gravity field the gradient is d d AL L σ y = ρ gh 2 (1.2) where h is the film thickness, ρL is the liquid density, and g is the acceleration due to gravity. This gradient can only exist where differences of surface composition can occur. We therefore require the presence of more than one component in the film. Indeed, it is possible to speculate that in the case of, say, aqueous foams, diffusion of water through the gas phase may rapidly remove any differences in concentration between different parts of a foam film if only one solute is present. In this case, at least two solutes (or three components) would be required. Surface tension gradients due to differences in the surface excess of soluble surface-active components may exist only when either the surface is not in equilibrium with the bulk composition or there are concomitant differences in bulk composition parallel to the surface. In the case of the former, the magnitudes of the gradients are of course determined by the rate of transport of surfactant to the relevant surfaces. With concentrated surfactant solutions, transport rates by diffusion will be rapid and surface tension gradients will be diminished so that shear rates are only balanced at y x (duy/dx)x=0 0 uy (x =0) ≠ 0 uy (x =0) = 0 (duy/dx)all x = 0 (a) (b) FIGURE 1.4 Limiting range of velocity profiles in draining foam film. (a) Plug flow. (b) Flow with parabolic velocity profile when film surfaces are immobile. Arrows represent magnitude of fluid velocities uy in the y direction.
- 39. 7 Some General Properties of Foams the surface of films if the air–liquid surfaces become mobile (i.e., so that the velocity uy(x = 0) ≫ 0). This would result in enhanced rates of drainage from films. In the extreme, the surface tension gradients will tend to be eliminated altogether. Thus, it has occasionally been reported that foamabilities decline at extremely high con- centrations of surfactant in aqueous solution. Conversely, however, if foam films are denuded of surfactant because of extremely slow transport rates, then the maximum surface tension gradients that can be achieved will be small. Such films will there- fore be susceptible to rupture when exposed to external stress. However, the com- plex problem of assessing both the effect of rate of transport on the surface tension gradients in foam films and the overall resultant impact on foam film stability, when subject to an external stress, has not, apparently, been fully addressed. Differences in bulk composition are possible in a thin foam film as a result of stretching of the film. If the film is sufficiently thin, then any stretching causes a depletion of the bulk phase surfactant solution between the air–liquid surfaces of the foam film as more surfactant adsorbs on those surfaces. Distances perpendicular to the film are small so that, provided the stretching occurs reasonably slowly, the equilibrium inside the film element may be always maintained. Depletion of bulk phase the surfactant concentration will therefore necessarily mean an increase of the surface tension of the film as it is stretched. This will, however, only occur if reduction of the surfactant concentration causes a concomitant increase in surface tension. In the case of a pure surfactant at concentrations above the critical micelle concentration (CMC), this may not always happen. We find then that it is possible to generate a surface tension gradient in a foam film by stretching various elements of the film to different extents. The increase in surface tension due to stretching imparts an elasticity to the film. This property of foam films was first recognized by Gibbs [34] and is usually referred to as the Gibbs elasticity εG. It is defined as ε σ σ G AL AL dln dln = 2d 2d A h = − (1.3) where A is the film area and the factor 2 arises because of the two surfaces. A plot of εG against concentration for a submicellar aqueous solution of sodium dodecyl sulfate (SDS) is shown in Figure 1.5 by way of example. Here, we see that, except at very low concentrations, decreases in film thickness at constant concentra- tion produce increases in Gibbs elasticity so that (dεG/dh)c ≤ 0. Thus, as the film becomes thinner, stretching will cause a relatively greater depletion of surfactant in the intralamellar liquid and the surface tension will increase to a greater extent. The plot of Gibbs elasticity against concentration shown in Figure 1.5 clearly reveals a maximum at concentration cmax. At extremely low concentrations of surfac- tant, we find that upon stretching of the film, there is essentially no contribution from the intralamellar liquid, and the surfactant behaves as an insoluble monolayer. Here, with an increase in surfactant concentration, both the surface excess and the elastic- ity of the monolayer increase. However, further increases in the surfactant concen- tration will eventually mean that it significantly exceeds that required to compensate for stretching of the air–liquid surface, so εG → 0. These two opposing consequences
- 40. 8 The Science of Defoaming: Theory, Experiment and Applications of increasing concentration conspire to produce the maximum in a plot of Gibbs elasticity. Lucassen [15] considers the effect of Gibbs elasticity on the development of sta- bilizing surface tension gradients in a foam film in the gravity field. He points out that if εG decreases as the film is stretched, it will tend to be dynamically unstable. Under these circumstances, any stretching force will tend to increase the area of the thinnest part of the film. Such situations will tend to prevail for films formed at concentrations on the low side of cmax. Thus, we can write d d d d G G G ε ε ε h h c c h c h = ∂ ∂ + ∂ ∂ (1.4) where c is the concentration. We can therefore have dεG/dh 0 if c cmax because then (∂εG/∂c)h 0, dc/dh 0, and (∂εG/∂h)c→ 0 as c → 0. For vertical films prepared from concentrated solutions, the requirement that the surface tension gradient satisfy Equation 1.2 implies that rapid stretching will occur. In the case of micellar solutions of certain pure surfactants, this may require achievement of submicellar concentrations in order that εG 0 and therefore dσAL/dx 0. For verti- cal films prepared from extremely dilute surfactant solutions where c cmax, Lucassen [15] shows that the magnitude of εG for thin elements at the top of the film may be less than that of thinner elements lower down. Any force acting on the film, such as an increase in weight as it grows, could mean catastrophic extension of the thinnest ele- ments because of their lower Gibbs elasticities. In summary, then, we find that surface tension gradients are necessary if freshly formed foam films are to survive. These gradients may occur if surface tensions h = 1.0 microns h = 2.0 microns 5 10 Concentration of SDS (mM) cmc 10 20 30 40 50 εG (mN m−1) FIGURE 1.5 Gibbs elasticities of submicellar SDS solutions at two different film thick- nesses. (After Lucassen, J. Dynamic properties of free liquid films and foams, in Anionic Surfactants, Physical Chemistry of Surfactant Action (Lucassen-Reynders, E.H., ed.), Marcel Dekker, New York, Surfactant Sci. Series, Vol. 11, Chapter 6, p. 217, 1981.)
- 41. 9 Some General Properties of Foams depart from equilibrium values. This will happen when foam film air–liquid sur- faces are expanded at rates that are fast so that equilibrium with the bulk surfactant concentration cannot be maintained. They may also occur when films are thin so that stretching may deplete intralamellar bulk phase to give rise to a Gibbs elastic- ity. Unfortunately, there are few experimental observations that clearly reveal the importance of surface tension gradients in determining foam behavior. Perhaps the best examples are reported by Malysa et al. [35] and Prins [36]. 1.3.2 Drainage Processes in Foam Films Any freshly generated foam film that survives will now be subject to a capillary pressure exerted by the curved surfaces of the adjacent Plateau borders. That pres- sure will tend to suck liquid out of the foam film. The resultant process of film drain- age is surprisingly complex. The simplest description of foam film drainage is obtained if the film is supposed cylindrical with immobile plane-parallel surfaces. Such behavior is represented by the Reynolds equation [37] − = d d L h t h P r 2 3 3 2 Δ f η (1.5) where h is the film thickness, rf is the film radius, and ΔP is equal to the capil- lary pressure jump, pc PB , at the air–liquid surface of the Plateau border. pc PB is in turn equal to σAL∣κPB∣, where ∣κPB∣ is the modulus of the air–liquid curvature of the Plateau border. If the film is sufficiently thin (~100 nm), then the applied capillary pressure will be modified by the air–liquid–air (ALA) disjoining pressure, ΠALA(h), so that the total applied pressure becomes p h c PB ALA − Π ( ). Drainage will therefore cease if p h c PB ALA − Π ( ), where a metastable equilibrium can prevail. We consider the effects of disjoining pressure on film stability in Section 1.3.3. The behavior of real foam films is rarely represented by the Reynolds equation. Films of radius ~100 microns are, for example, generally not plane parallel [6]. The surface of a draining foam film is characterized by a balance of the surface tension gradient and the relevant viscous stress as indicated by Equation 1.1. In turn, this sur- face tension gradient implies a gradient in surfactant adsorption that will be subject to relaxation by diffusion from the bulk phase and along the surface. Maintenance of the balance of the surface tension gradient and viscous stress will therefore always require a countervailing flow in the surface in the direction of the bulk phase flow. Complete immobility of the air–liquid surface (so that the velocity uy(x = 0) = 0) is therefore never achieved [6]. In the case of relatively large foam films ~100 microns, where the air–liquid surface is close to a condition of immobility, drainage occurs in an axisymmetric manner but with a non-uniform thickness. A thick region develops in the center of the film, the so-called dimple, while a thinner region surrounds this dimple—the so- called barrier ring [38]. A schematic cross-section of a film with a dimple is depicted in Figure 1.6a. The behavior of such a film, formed inside a cylindrical glass cell (the
- 42. 10 The Science of Defoaming: Theory, Experiment and Applications Scheludko cell [13, 39]; see Chapter 2, Section 2.3.1), has been analyzed numerically by Joye et al. [38]. They deduce that the formation of a dimple requires that the capil- lary pressure inside the maximum putative dimple that can be formed be less than the capillary pressure in the Plateau border meniscus. This means that r R d PB − − ( ) 1 1 1 / , where rd is the radius of curvature of the maximum dimple and RPB the radius of cur- vature of the Plateau border as depicted in Figure 1.6b. Suction from the latter will not then be matched by sufficient pressure from the dimple so that liquid is selec- tively removed from the region adjacent to the meniscus, forming a thin barrier ring. If, on the other hand, the film is small and the capillary pressure in the maximum dimple is greater than the capillary pressure in the Plateau border, then liquid flow from any putative dimple readily matches that sucked into the border and neither a barrier ring nor a dimple are formed. If a dimple is formed, then the concomitant restricted liquid flow in the barrier ring contributes to a diminished rate of drainage Dimple in center of film Plateau border capillary pressure Thin barrier ring forms that inhibits drainage from dimple (a) (b) Rc rd RPB ≈ Rc Dimple Plateau border Cell wall FIGURE 1.6 Axisymmetric drainage of foam film with immobile air–water surfaces. (a) Dimple and barrier ring that form. (b) Defining maximum dimple size where barrier ring is absent and where film is formed in a cylindrical Scheludko cell (references [13, 39]; see Section 2.3.1). Radius of curvature, RPB, of Plateau border air–water surface is approximated by radius, Rc, of cell—rd is radius of curvature of air–water surface of the dimple. (Reprinted with per- mission from Joye, J. et al., Langmuir, 8, 3083. Copyright 1992 American Chemical Society.)
- 43. 11 Some General Properties of Foams relative to that predicted by the Reynolds equation. As the barrier ring becomes thin- ner, then eventually repulsive disjoining forces are predicted to become important in determining the final rates of film thinning. At high electrolyte concentrations, the barrier will become extremely thin so that the dimple drains extremely slowly [38]. In practice, axisymmetric foam film drainage is not usually observed in the case of solutions of simple surfactants. Although dimples are usually formed with such systems, the resulting films exhibit a hydrodynamic instability where dimples dis- gorge directly into one side of the Plateau borders [40]. This asymmetric drain- age is always significantly more rapid than the axisymmetric drainage. The latter appears to be observed only when asymmetric drainage is suppressed by extreme surface rheological behavior of the adsorbed surfactant. We illustrate this in Figure 1.7 where the drainage rates of horizontal foam films formed in a Scheludko cell [13, 39], and prepared from solutions of SDS and SDS–dodecanol mixtures, are com- pared with the surface shear viscosities of those solutions. Rapid asymmetric drain- age is associated with the low values of the surface shear viscosity typical of aqueous solutions of simple soluble surfactants. Joye et al. [40] offer an explanation, based on the earlier work of Stein [42, 43], for the hydrodynamic instability that produces asymmetric drainage. We illustrate this in Figure 1.8. Here it is supposed that thickness fluctuations occur in the barrier ring (linearized in the figure). In the thicker regions the flow of liquid from the dimple to the Plateau border meniscus will increase relative to the unperturbed flow rate. This will in turn produce an increased surface tension gradient in the direction of the flow. Conversely in the case of the thinner regions, the flow rate in the direction of the meniscus will be slow relative to the unperturbed flow rate. In turn this will produce a decreased surface tension gradient. As a consequence of these two oppos- ing trends in surface tension gradient in the direction of flow from the dimple to the meniscus, another surface tension gradient orthogonal to the unperturbed gradient 0 50 100 150 200 250 300 350 400 −5 −4 −3 −2 −1 0 log10 (Surface shear viscosity/g s−1) Film drainage time (s) Asymmetric drainage Axisymmetric drainage FIGURE 1.7 Film drainage time as function of surface shear viscosity for SDS and SDS + dodecanol (100:1) solutions using a Scheludko cell and film radius of 100 microns. (Reprinted with permission from Joye, J. et al., Langmuir, 8, 3083. Copyright 1994 American Chemical Society, using data from Djabbarah, N.F. et al., Colloid Polym. Sci., 256, 1002, 1978 [41].)
- 44. 12 The Science of Defoaming: Theory, Experiment and Applications will exist between the thin and thick regions where the surface tension of the former region will be slightly lower than the latter. That orthogonal surface tension gradient will therefore drive liquid from the thin regions to the thick regions, reinforcing the perturbation. The latter will therefore grow until discharge of the whole dimple is facilitated. Joye et al. [40] have subjected this putative process to a linear stability analysis and have shown that the hydrodynamic instability is suppressed, so that axisymmetric drainage occurs, if either the dimension of the film is small enough or if the surface shear viscosity is high enough. The latter is of course consistent with the experimental findings shown in Figure 1.7. However, we must emphasize that the instability is ubiquitous in draining foam films formed from aqueous solu- tions of common surfactants. Such films therefore exhibit rapid asymmetric drain- age by means of this hydrodynamic instability for which no analytical theoretical expression is available for the prediction of the rate of drainage. A related phenomenon to asymmetric drainage in horizontal cylindrical foam films is that of the so-called marginal regeneration. This process occurs in vertical foam films subject to both capillary suction from the Plateau borders and gravity. It is manifest as an apparent turbulent motion on the margin of foam films where thin elements of film are drawn out of the Plateau border and thick elements are sucked Barrier ring Dimple Plateau border a b c d Sinusoidal thickness fluctuation in the barrier ring Thick film elements ab permit faster bulk flow of fluid than thin film elements cd FIGURE 1.8 Origin of hydrodynamic instability driving asymmetric drainage in horizontal foam films. Thickness fluctuation in barrier ring means that region ab is thicker than con- tiguous region cd. Fluid flow rate from dimple to Plateau border therefore increases relative to cd in ab. Both relative shear stress and corresponding surface tension gradient must also increase in ab according to Equation 1.1. This means that surface tension in ab will be gener- ally greater than cd, which will mean additional surface tension gradient, orthogonal to that between dimple and Plateau border, driving fluid from the thin to the thick parts of barrier ring. This will reinforce instability. However, establishment of differences in surface tension gradients ab and cd between dimple and Plateau border will be resisted by high values of surface shear viscosity leading to suppression of instability. (Adapted with permission from Joye, J. et al., Langmuir, 8, 3083. Copyright 1994 American Chemical Society.)
- 45. 13 Some General Properties of Foams into the border causing continuous regeneration of the margin of the film. The thin elements of film then rise in the gravity field until they reach a height where the film has the same thickness. The phenomenon is readily observable in reflected white light where interference colors distinguish film elements of different thickness. Marginal regeneration can easily be seen in the large vertical foam films (of size ~1 cm) held in metal or glass frames, which are often used for study of aspects of antifoam action (see, e.g., references [44–46]). It is also often visible in the large bubbles formed at the top of foam columns generated by air entrainment (e.g., during hand dishwashing). A detailed description of the phenomenon is to be found in the monograph of Mysels et al. [3]. Images of films exhibiting marginal regeneration are given in Figure 1.9. Joye et al. [40] argue that marginal regeneration is driven by the same hydrody- namic instability as that which causes asymmetric drainage in horizontal cylindrical foam films. The main difference concerns the role of gravity in marginal regenera- tion, which causes thin film elements resulting from the growing perturbations to rise up the film. The net effect of the process is relatively rapid drainage of the film. As with asymmetric cylindrical film drainage, no analytical expression is available for the prediction of the rate of film drainage. The hydrodynamic instability is sup- pressed if the surface dilatational modulus is sufficiently high [15, 47]. It is probable that high surface shear viscosities may also lead to suppression of the instability as is found with small horizontal films. We note, however, that high surface shear viscosities often apparently accompany high surface dilatational moduli, which renders selection of the key property difficult. The resultant “rigid” films drain by a Poiseuille-like flow for which Mysels et al. [3] present an analytical expression. Vertical films, which exhibit marginal regeneration, typically drain at rates two orders of magnitude faster than rigid films in which the instability is suppressed [3]. It should be clear then that asymmetric drainage of horizontal cylindrical films and marginal regeneration in vertical foam films, although often apparent, are not always exhibited. Liquid flow by capillary suction from the Plateau border is, (a) (b) (c) FIGURE 1.9 Marginal regeneration. (a) Film in inverted triangular glass frame. (b) Film in rectangular glass frame illustrating marginal regeneration at bottom of film (it is absent from top of film). (c) Large film (~5.5 cm radius) in cylindrical vessel illustrating marginal regen- eration at angle of only ~10° to horizontal. In all cases, rising film elements move up to film regions of the same thickness with which they coalesce. (Image (a) after Mysels, K.J. et al., Soap Films, Studies of Their Thinning, Pergamon, London, 1959. Image (b) after Nierstrasz, V.A., Marginal regeneration, PhD Thesis, Delft University of Technology, 1996 [48].)
- 46. 14 The Science of Defoaming: Theory, Experiment and Applications however, always present. It is therefore misleading for Weaire and Hutzler [5] to state that “the traditional but somewhat obscure phrase for this (liquid flow by capillary suction) process is marginal regeneration.” For the relevant hydrodynamic instabili- ties to occur and cause marginal regeneration, the air–liquid surfaces should addi- tionally have the required rheological characteristics (i.e., low surface viscosity, etc.). 1.3.3 Disjoining Forces and Foam Film Stability By means of these processes, foam films drain until they either form metastable films or rupture. Which fate awaits a given film is decided largely by the nature of the forces that exist across a foam film as it thins. These forces are usually termed “dis- joining” forces and are positive if they resist the thinning of the film and negative if they assist thinning of the film. They are, however, only significant in comparatively thin films of ≤100 nm. The metastability of the film requires that resulting disjoining pressure balances the capillary pressure exerted by the Plateau borders. The disjoin- ing pressure is a function of the film thickness—plots of disjoining pressure against film thickness (disjoining pressure isotherms) largely determine the properties of those metastable films. The disjoining pressure, ΠALA(h) in air–liquid–air foam films, stabilized by adsorbed surfactant, is made up of contributions from several components. These include electrostatic forces (from overlapping electrostatic double layers), van der Waals forces, structural forces (arising from the presence of close-packed layers of micelles or nanoparticles in films [49]), steric forces (derived from the overlapping head group—head group interactions of surfactants or chain–chain interactions of polymers adsorbed on opposite sides of the film), and oscillatory forces, etc. These var- ious contributions conspire to produce disjoining pressure isotherms of various forms. Arguably, the simplest isotherm is one dominated entirely by the van der Waals forces, which are invariably present. Such an isotherm is shown schematically in Figure 1.10, where the disjoining pressure is always negative and where it becomes more negative as the film thins so that d d ALA Π ( ) . h h 0 (1.6) In consequence, any perturbation of the film thickness that produces thin and thick regions will tend to grow spontaneously because molecules in the thin regions will transfer to the thick regions. However, any perturbation of a film to produce thick and thin regions must also inevitably increase the surface area of the film. This will increase the number of molecules in the relatively weak attractive force field close to the air–liquid surface. Such an increase in surface area will therefore be resisted by an opposing force—the surface tension [51, 52]. Thickness perturbations, of a thermal or mechanical nature, may be consid- ered to be of a wavelike nature. A symmetrical sinusoidal perturbation is shown in Figure 1.11. Here the thin part of the film is subject to two opposing forces. Thus, a capillary pressure due to the surface tension tends to suck liquid back into the thin part of the film, and a disjoining force tends to push liquid away.
- 47. 15 Some General Properties of Foams The magnitude of the capillary pressure is determined by the curvature of the film surface. For a given amplitude of the perturbation, the curvature is determined by the wavelength. Thus, the shorter the wavelength, the more marked the curvature and the stronger the capillary pressure. Vrij and Overbeek [51, 52] deduce a criti- cal wavelength, λcrit, above which disjoining forces will dominate over the capillary pressure and the perturbation will spontaneously grow. The critical wavelength is ( )/ λ π σ crit AL ALA d d = 2 2 1 2 Π h h (1.7) The rate of growth of the perturbation will increase with increasing wavelength λ for λ λcrit because the damping effect of the capillary pressure will decrease. However, −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 10 20 30 40 50 60 70 h (nm) ΠALA(h) (104 N m−2) FIGURE 1.10 Plot of disjoining pressure isotherm for a plane-parallel air–water–air film with only van der Waals interactions. Here disjoining pressure, ΠALA, is given by ΠALA(h) = AH/6πh3 where AH is Hamaker constant of 3.7 × 10–20 J. (From Isrealachvilli, J.N. Intermolecular and Surface Forces with Applications to Colloidal and Biological Systems, Academic Press, London, 1985 [50].) Liquid movement due to disjoining pressure Liquid movement due to capillary pressure Unperturbed film surfaces FIGURE 1.11 Sinusoidal thickness perturbations in thin liquid foam film.
- 48. 16 The Science of Defoaming: Theory, Experiment and Applications for sufficiently long wavelengths, the rate of growth will eventually begin to decline because of the increased distances over which the film liquid has to be moved against viscous resistance. An optimum wavelength of 2crit for the maximum rate of growth of a perturbation therefore exists [51, 52]. As we show in Figure 1.10, dΠALA(h)/dh is strongly dependent on film thickness, which leads to λcrit decreasing in proportion to the square of the film thickness in the case where only van der Waals forces prevail. Film collapse is supposed to occur when the amplitude of the fastest-growing per- turbation equals the thickness of the film. Vrij and Overbeek [51, 52] have produced the simplest description of this process. They calculate the minimum total time for film drainage (unfortunately estimated using the Reynolds equation [37]) and subse- quent growth of the fastest-growing perturbation. The average thickness of the film at the moment of rupture is the critical thickness hcrit. Experimental measurements of hcrit for microscopic foam films are in the region of a few tens of nanometers (see, e.g., reference [53]). Films that are completely dominated by van der Waals forces and exhibit the type of instability described by Vrij and Overbeek [51, 52] are likely to be either free of surfactant or have low levels of surfactant adsorption. The presence of high levels of adsorbed surfactant means that positive contributions to the disjoining pressure become significant. In particular, long-range electrostatic and van der Waals forces, together with extremely short range steric forces, can produce isotherms with two minima. Such isotherms are often exhibited by aqueous surfactant solutions [6, 54]. This type of isotherm is illustrated schematically in Figure 1.12. It is characterized by two regions where dΠALA(h)/dh 0 and two regions where dΠALA(h)/dh 0. ΠALA(h) h E C D A B h1 h2 h3 Π max ALA(h3) pPB2 c = ΠALA(h2) pPB1 c = ΠALA(h1) Stable common black films where dΠALA(h)/dh 0 20–50 nm Stable Newton black films where dΠALA(h)/dh 0 ~5 nm FIGURE 1.12 Schematic diagram of disjoining pressure isotherm exhibiting two stable regions where common black foam films and Newton black films can be formed, respectively (see text for full explanation). ΠALA is air–liquid–air (ALA) foam film disjoining pressure and h is film thickness.
- 49. 17 Some General Properties of Foams Bergeron [55] has argued that adsorbed surfactant present at the surfaces of the film can also produce other complications in the analysis of the effect of thickness pertur- bations. Dilatational effects, for example, due to adsorbed surfactant, tend to dampen the fluctuations rendering rupture less probable. Moreover, fluctuations due to lateral adsorption density fluctuations could lead to lateral fluctuations in disjoining pressure to further complicate the picture. These issues are far from fully resolved. Despite these complications, disjoining pressure isotherms such as that shown in Figure 1.12 offer an explanation for much of the behavior of the thin liquid films, formed by aqueous solutions of simple surfactants, subject to an applied capillary pres- sure. Consider, for example, the regions in those isotherms where dΠALA(h)/dh 0. This means that the disjoining pressure increases as the film thins so that any perturbation in film thickness will be resisted. Such regions are therefore stable. Conversely in regions where dΠALA(h)/dh 0, any perturbation in film thickness will be enhanced leading to instability. For the isotherm shown in Figure 1.12, there are therefore thickness regions where stable films can be formed (regions BC and DE) and thickness regions where unstable films are formed (regions AB and CD). In consequence, regions AB and CD are not accessible to experimental measurement. In this type of isotherm, the region AD is largely determined by electrostatic and van der Waals contributions and can be readily accounted for by the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory (see, e.g., reference [6]). However, the region DE, if it is present, is usually largely due to short range steric interactions between molecules adsorbed on opposite sides of the film and cannot be accounted for by that theory. Consider now the likely evolution of a film with a disjoining pressure isotherm similar to that shown in Figure 1.12, which is subject to a capillary suction, pc PB1 , from the adjacent Plateau border. Such a film will drain in the unstable region AB until the critical thickness is reached whereupon perturbations in thickness will not grow to produce holes and film rupture but will rather produce stable regions, or “spots,” which continue to expand and cover the whole film until it reaches an equilibrium thickness h1 where the capillary pressure equals the disjoining pressure (i.e., p h c PB1 ALA = Π ( ) 1 ). The film thicknesses involved at this stage are in the region of 20–50 nm. Such films are so thin that destructive interference of light occurs so that they (and indeed the “spots”) appear black in reflected light. They are usually termed “common black films.” If now the capillary pressure increases to pc PB2 so that it exceeds the maximum at C in the isotherm then the film will drain to another unstable region and will jump from C to form another stable film at D. Equilibrium will again be established when the film thins to thickness h2 where p h c PB1 ALA = Π ( ) 2 . Such films are extremely thin, being little more than bilayer leaflets of thickness in the region of 5 nm. They are usually termed “Newton black films.” Further increases in capillary pressure to p h c PB3 ALA max Π ( ) 3 should simply lead to film rupture. However, thickness fluctuations cannot exist in such thin films. It has therefore been argued by Kashchiev and Exerowa [56] that rupture occurs by nucleation of holes caused by thermal fluctuations in these films. The transition from a disjoining pressure isotherm dominated by van der Waals forces, such as that shown in Figure 1.10, to one with at least a region where dΠALA(h)/ dh 0 is present can be observed at a certain surfactant concentration. That is of course represented by a transition from rupture to black spot formation. The
- 50. 18 The Science of Defoaming: Theory, Experiment and Applications concentration at which this transition occurs is designated cblack. It occurs because of the effects of changes in the adsorption layer of surfactant on the thickness depen- dence of ΠALA(h) and dΠALA(h)/dh. Foam film stability tends to increase markedly at concentrations above cblack. Values for cblack in microscopic films have been measured for a number of surfac- tants [57]. In the case of aqueous solutions of surfactants, cblack is generally signifi- cantly lower than the CMC. Thus, for example, for SDS, cblack = 1.6 × 10–6 M [57] and the CMC = 8.4 mM [58], and for dodecyl hexaethyleneglycol (C12EO6), cblack = 4.9 × 10–6 M [57] and the CMC = 8.7 × 10–6 M [59] at 25°C. We have seen then that film rupture may occur because surface tension gradients are not sufficiently high to enable the film to withstand stress, because dΠALA/dh is always positive so that rupture is inevitable at a certain critical thickness, or because the Plateau border capillary pressure exceeds any maximum in the relevant disjoining pressure isotherm. However these phenomena are associated with low concentrations of surfactant (at least if we consider films formed slowly so that equilibrium between the air–liquid surface and the intralamellar liquid is maintained). Thus for example, we have cmax ≪ CMC for dεG/dh 0 and cblack ≪ CMC. Elimination of both causes of rupture should therefore be readily achieved at sufficiently high concentrations of surfactant. The poor discrimination in foamability often found with relatively con- centrated aqueous micellar solutions of surfactants may well be attributable to that cause. Interesting differences in foamability are, however, often revealed when films are either formed rapidly so that equilibrium adsorption is not obtained and conditions for stability are thereby violated or if antifoam is added to the solution. 1.4 PROCESSES ACCOMPANYING AGING OF FOAM 1.4.1 Capillary Pressure Gradients Foam film stability is, as we have seen, determined in part by the lack of balance between the disjoining pressure and the capillary pressure applied to the films by the Plateau borders. The capillary pressure also drives the process of film thinning, which precedes film rupture. This in turn influences the frequency of foam film rupture. The relative magnitudes of the capillary pressure and the hydrostatic head in the foam also determine the bulk drainage behavior of the foam. If the capillary pressure at the top of the foam balances the hydrostatic head, then bulk drainage will not occur. As we show in later chapters, the stability of the films between antifoam entities and the gas liquid surface—the so-called pseudoemulsion films [60]—may also be determined by the lack of balance between the disjoining pressure in the pseudoemulsion film and the Plateau border capillary pressure. It is therefore important to clearly define the nature of the pressure distribution in the continuous phase of a foam as represented by the system of Plateau border channels. In this, we follow closely the arguments of Princen [61]. We first consider a polydisperse foam under mechanical equilibrium where drain- age is absent. The upper region of such a foam is shown schematically in Figure 1.13. The continuous gas–liquid surface at y = 0 covers the dome-shaped tops of bubbles and the upper Plateau borders. The Laplace pressure jump at the gas–liquid surface of the Plateau borders is the difference between the atmospheric pressure, Patm, and
- 51. 19 Some General Properties of Foams the pressure PPB(y = 0) in the Plateau borders at y = 0. We therefore define the Plateau border capillary pressure, p y c PB ( ) = 0 , by p y P y P c PB PB atm AL t ( ) ( ) = = = − = − 0 0 σ κ (1.8) where κt is the curvature of the air–liquid Plateau border surface at the top of the foam column where y = 0 and where we note that κt is negative with respect to the fluid in the border. As stated by Princen [61], κt is a constant everywhere at the top surface of the foam and is independent of the detailed shape of that surface. For the total Plateau border pressure PPB(y) at some distance y into the bulk of the foam, we must allow for the hydrostatic head and must therefore write P y P gy p y P PB atm L c PB atm ( ) ( ) = + + = = + 0 0 ρ ρL L AL t gy − σ κ (1.9) All this is illustrated in Figure 1.13. At equilibrium the pressure in the liquid phase at the bottom of a foam column must equal the atmospheric pressure plus that due to the weight of the foam. That pressure must also equal the Plateau border pressure at the bottom of the foam. Therefore, from Equation 1.9, we can write P y H P gH P g PB 0 e atm L 0 e AL t atm L ( ) = = + − = + ρ σ κ ρ 0 0 1 H y 0 e G foam d ∫ − ( ) Φ (1.10) where H0 e is the height of the equilibrium foam. The integral accounts for the weight of the foam column where ΦG foam is the gas phase volume fraction (and where the den- sity of the air is neglected). In the case of an equilibrium foam in the dry limit, the average gas volume fraction ΦG foam → 1. We can therefore often neglect the integral Plateau border pressure PPB(y = 0) = Patm – σAL |κt| κt κf Plateau border pressure PPB(y) = PPB(y = 0) + ρLgy Patm y = 0 y κfi κi Top of foam column FIGURE 1.13 Schematic illustration of factors contributing to Plateau border pressure in foam. (Adapted with permission from Princen, H.M. Langmuir, 4, 164. Copyright 1988 American Chemical Society.)
- 52. 20 The Science of Defoaming: Theory, Experiment and Applications in Equation 1.10 and deduce that the capillary pressure jump at the top of the foam σAL ∣κt∣ approximately matches the hydrostatic head ρLgH0 at the bottom of the foam. However in the case of a draining foam, the capillary pressure jump at the top of the foam, p y gH c PB AL t L ( ) = = 0 σ κ ρ so that the actual height H of the foam column exceeds the corresponding equilibrium height H0. If this foam is stable (in that foam films at the top of the foam do not rupture), it will continue to drain until H = H0 e as a result of increases in (the modulus of the) curvature, ∣κt∣, following increases in the gas phase volume fraction ΦG foam . An equilibrium condition is eventually achieved where σ κ ρ ρ AL t L L 0 e = = gH gH and where we must have κ κ t t where κt is the modulus of the curvature at the top of the foam when drainage has ceased. At equilibrium, the total capillary pressure applied to a foam film should be bal- anced by the disjoining pressure ΠALA(h) regardless of the position of the film in the foam. At the top of the foam column, the film will be subject to two capillary pres- sures—that due to the curvature of the film and that due to the Plateau border, both of which will conspire to force liquid out of the film, which will be resisted by a posi- tive disjoining pressure. Therefore, at equilibrium, the disjoining pressure is given by ΠALA c PB c film AL t f ( ) ( ) h p y p = − = + ( ) = + ( ) 0 σ κ κ (1.11) where κf is the curvature of the dome-shaped surface of the bubble films at the top of the foam column and pc film is the capillary pressure jump in those films. In a dry foam where ΦG foam → 1, then we have ∣κt∣ ≫ ∣κf∣. The capillary pressure in the interior of the foam where y 0 is given by p y c PB AL i ( ) = − 0 σ κ (1.12) where κi is the curvature of the relevant Plateauborders. At equilibrium, the disjoin- ing pressure in the films in the interior of the foam must therefore be given ΠALA (h) = σAL (∣κi∣ ± ∣κfi∣) (1.13) where κfi is the curvature of the film (the sign of which may be either positive or negative), and where, according to Princen [61], κfi is zero in the case of a monodis- perse Kelvin foam. We can use these arguments to describe the evolution of a foam in a gravity field. Immediately after generation, the foam is wet and Plateau borders are thick with relatively low curvatures. Drainage of the foam can therefore occur if Equation 1.10 is not satisfied. That process continues until the capillary pressure at the top of the foam column equals the hydrostatic head whereupon drainage ceases. During this process, foam films also drain until the disjoining pressure equals the capillary pres- sure everywhere in the foam and a condition of mechanical equilibrium is attained. However, this condition represents an unstable equilibrium because gas diffusion between bubbles will occur in response to differences in capillary pressure. This will in turn reduce the number of bubbles and therefore cause the Plateau borders to begin to swell so that drainage again commences. Moreover, the attainment of the
- 53. 21 Some General Properties of Foams equilibrium requires that the disjoining pressure isotherm exhibits a maximum suf- ficiently high to match the total capillary pressure at the top of the foam column so that Equation 1.10 is satisfied. If that equation cannot be satisfied, foam collapse will commence as the capillary pressure exceeds that maximum in the disjoining pres- sure isotherm. This process of foam collapse will start at the top of the foam column where, as indicated by Equations 1.11 through 1.13, the capillary pressure is highest. 1.4.2 Foam Drainage The drainage of the continuous phase liquid out of a foam is a surprisingly complex phenomenon. It is driven by a combination of gravitational, capillary, and viscous forces. The rate at which it occurs in the case of so-called “free drainage” is easily observed by monitoring the rate of accumulation of liquid at the bottom of a foam column. If the foam is generated from a gas of low liquid solubility, then, as we shall discuss in Section 1.4.3, changes in the structure of the foam due to gas diffusion are likely to be slow so that they may be neglected. General treatments must, however, take account of the coupling of gas diffusion with foam drainage (see, e.g., the treat- ment of Hilgenfeldt et al. [62]). The observed drainage behavior of foams has often been successfully described [63] by the classic foam drainage equation of Goldfarb et al. [64]. Derivation of this equation therefore provides a useful introduction to the nature of that process. This approach neglects any contribution to the process from structural changes due to gas diffusion or film rupture and makes the assumption that the liquid content of the foam is entirely contained within the Plateau borders. Drainage is therefore assumed to occur by flow down the interconnected system of Plateau borders (illustrated by a sim- ulation for the case of a foam made up of Kelvin cells in Figure 1.2b). Any contribution to drainage from foam films is neglected. The gas–liquid surfaces are assumed to be immobile. Essentially, the approach assumes that Poiseuille flow occurs in the Plateau borders of a monodisperse foam and that the liquid volume fraction of that foam, (y,t) (where ΦL foam (y,t) =1− ΦG foam (y,t)), at any position y and time t, is simply ΦL(y,t) = APB(y,t)LPB (1.14) where y is measured from the top of the foam column and APB(y,t) is the cross- sectional area of a Plateau border. LPB is the total length of Plateau borders per unit foam volume. The absence of gas diffusion and film rupture means that LPB is not a function of y or time t. The task of the theory then is to determine APB(y,t), and therefore ΦL(y,t), as a function of y and t. Following Weaire and coworkers [5, 23, 63], we initially suppose the border to be vertically orientated. The cross-sectional area of the border, APB(y,t), will change with time as it drains subject to gravitational, capillary, and viscous forces. Consider then a segment of Plateau border of length Δy, which is shown schematically in Figure 1.14. The volume of the segment is approximated by [APB(y,t) + (∂APB (y,t)/∂y)t Δy/2] Δy, which is the limit as Δy → 0 becomes simply APB(y,t)Δy. The total gravi- tational force driving drainage in the segment is ρg APB(y,t)Δy or simply ρg per unit volume. The gravitational force is resisted in part by a capillary pressure gradient,
- 54. 22 The Science of Defoaming: Theory, Experiment and Applications ( / ) ∂ ∂ p y c PB t which varies as a result of narrowing of the Plateau border with increasing y. The segment is therefore subject to a net capillary force of ( / ) ( , ) ∂ ∂ p y A y t y c PB t PB . Δ or simply ( / ) ∂ ∂ p y c PB t per unit volume. The gravitational force is also resisted in part by viscous friction. By analogy with Poiseuille flow in a cylinder (or even dimen- sional analysis), the relevant shear force per unit volume is k u y t A y t geom PB m L PB ( , ) / ( , ) η , where kgeom is a dimensionless geometrical constant dependent on the shape of the Plateau border (where kgeom ~ 50), uPB m (y,t) is the mean liquid velocity over the cross- section of the Plateau border [5, 23, 63]. We can therefore write for the net force/unit volume of the segment of Plateau border that ρ η L c PB t geom PB m L PB g p y k u y t A y t ( / ) ( , ) / ( , ) − ∂ ∂ − = 0 (1.15) where the capillary pressure is given by p A y t k c PB AL AL geom PB i = − = − ( , ) / σ κ σ (1.16) RPB = ĸi −1 pPB c = −σAL|ĸi| pc PB+ Δpc PB = pc PB + ∂pc PB ∂y t Δy Drainage driven by gravity resisted by capillary and viscous forces Cross-sectional area = APB(y,t) ∝ ĸi −2 (y,t) Cross-sectional area = ∆y RPB ( ) + ∂RPB ∂y t Δy APB ( ( ) ) (y,t) + ∂APB ∂y t Δy FIGURE 1.14 Plateau border segment subject to increase in cross-sectional area and decrease in capillary pressure along length Δy and where RPB and APB are radius of curvature and cross-sectional area of Plateau border, respectively.
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. did he come from? Did he drop from the moon? Where has he previously lived? What are his family? Where does his property lie?— in the funds, or in land, or in securities, or what? Most men, even though they do come as strangers into a neighbourhood, supply indications of some of these things, either accidentally or purposely.” “They have lived in London,” said Cecil. “London is a wide term,” answered Thomas Godolphin. “And I’m sure they have plenty of money.” “There’s where the chief puzzle is. When people possess so much money as Verrall appears to do, they generally make no secret of whence it is derived. Understand, my dear, I cast no suspicion on him in any way: I only say that we know nothing of him: or of the ladies either——” “They are very charming ladies,” interrupted Cecil again. “Especially Mrs. Verrall.” “Beyond the fact that they are very charming ladies,” acquiesced Thomas in a tone that made Cecil think he was laughing at her: “you should let me finish, my dear. But I would prefer that they were rather more open, as to themselves, before they became the too- intimate friends of Miss Cecilia Godolphin.” Cecil dropped the subject. She did not always agree with what she called Thomas’s prejudices. “How quaint that old doctor of ours is!” she exclaimed. “When he had looked at Mrs. Verrall’s arm, he made a great parade of getting out his spectacles, and putting them on, and looking again. ‘What d’ye call it—a burn?’ he asked her. ‘It is a burn, is it not?’ she answered, looking at him. ‘No,’ said he, ‘it’s nothing but a scorch.’ It made her laugh so. I think she was pleased to have escaped with so little damage.” “That is just like Snow,” said Thomas Godolphin. Arrived at home, Miss Godolphin was in the same place, knitting still. It was turned half-past nine. Too late for Thomas to pay his visit to Lady Sarah’s. “Janet, I fear you have waited tea for us!” said Cecil.
- 57. “To be sure, child. I expected you home to tea.” Cecil explained why they did not come, relating the accident to Mrs. Verrall. “Eh! but it’s like the young!” said Janet, lifting her hands. “Careless! careless! She might have been burned to death.” “What a loud ring!” exclaimed Cecil, as the hall-bell, pealed with no gentle hand, echoed and re-echoed through the house. “If it is Bessy come home, she thinks she will let us know who’s there.” It was not Bessy. A servant entered the room with a telegraphic despatch. “The man is waiting, sir,” he said, holding out the paper for signature to his master. Thomas Godolphin affixed his signature, and took up the despatch. It came from Scotland. Janet laid her hand upon it ere it was open: her face looked ghastly pale. “A moment of preparation!” she said. “Thomas, it may have brought us tidings that we have no longer a father.” “Nay, Janet, do not anticipate evil,” he answered, though his memory flew unaccountably to that ugly Shadow, and to what he had deemed would be Janet’s conclusions respecting it. “It may not be ill news at all.” He glanced his eye rapidly and privately over it, while Cecil came and stood near him with a stifled sob. Then he held it out to Janet, reading it aloud at the same time. “‘Lady Godolphin to Thomas Godolphin, Esquire. “‘Come at once to Broomhead. Sir George wishes it. Take the first train.’” “He is not dead, at any rate, Janet,” said Thomas quietly. “Thank Heaven.” Janet, her extreme fears relieved, took refuge in displeasure. “What does Lady Godolphin mean, by sending so vague a message as that?” she uttered. “Is Sir George worse? Is he ill? Is he in danger? Or has the summons no reference at all to his state of health?”
- 58. Thomas had taken it into his hand again, and was studying the words: as we are all apt to do in uncertainty. He could make no more out of them. “Lady Godolphin should have been more explicit,” he resumed. “Lady Godolphin has no right thus to play upon our fears, our suspense,” said Janet. “Thomas, I have a great mind to start this very night for Scotland.” “As you please, of course, Janet. It is a long and fatiguing journey for a winter’s night.” “And I object to being a guest at Broomhead, unless driven to it, you might add,” rejoined Janet. “But our father may be dying.” “I should think not, Janet. Lady Godolphin would certainly have said so. Margery, too, would have taken care that those tidings should be sent to us.” The suggestion reassured Miss Godolphin. She had not thought of it. Margery, devoted to the interests of Sir George and his children (somewhat in contravention to the interests of my lady), would undoubtedly have apprised them were Sir George in danger. “What shall you do?” inquired Janet of her brother. “I shall do as the despatch desires me—take the first train. That will be at midnight,” he added, as he prepared to pay a visit to Lady Sarah’s. Grame House, as you may remember, was situated at the opposite end of the town to Ashlydyat, past All Souls’ Church. As Thomas Godolphin walked briskly along, he saw Mr. Hastings leaning over the Rectory gate, the dark trees shading him from the light of the moon. “You are going this way late,” said the Rector. “It is late for a visit to Lady Sarah’s. But I wish particularly to see them.” “I have now come from thence,” returned Mr. Hastings. “Sarah Anne grows weaker, I hear.” “Ay. I have been praying over her.”
- 59. Thomas Godolphin felt shocked. “Is she so near death as that?” he asked, in a hushed tone. “So near death as that!” repeated the clergyman in an accent of reproof. “I did not expect to hear a like remark from Mr. Godolphin. My good friend, is it only when death is near that we are to pray?” “It is chiefly when death is near that prayers are said over us,” replied Thomas Godolphin. “True—for those who have not known when and how to pray for themselves. Look at that girl: passing away from amongst us, with all her worldly thoughts, her selfish habits, her evil, peevish temper! But that God’s ways are not as our ways, we might be tempted to question why such as these are removed; such as Ethel left. The one child as near akin to an angel as it is well possible to be, here; the other—— In our blind judgment, we may wonder that she, most ripe for heaven, should not be taken to it, and that other one left, to be pruned and dug around; to have, in short, a chance given her of making herself better.” “Is she so very ill?” “I think her so; as does Snow. It was what he said that sent me up there. Her frame of mind is not a desirable one: and I have been trying to do my part. I shall be with her again to-morrow.” “Have you any message for your daughter?” asked Thomas Godolphin. “I start in two hours’ time for Scotland.” And then, he explained why: telling of their uncertainty. “When shall you be coming back again?” inquired Mr. Hastings. “Within a week. Unless my father’s state should forbid it. I may be wishing to take a holiday at Christmas time, or thereabouts, so shall not stay away now. George is absent, too.” “Staying at Broomhead?” “No; he is not at Broomhead now.” “Will you take charge of Maria? We want her home.” “If you wish it, I will. But I should think they would all be returning very shortly. Christmas is intended to be spent here.”
- 60. “You may depend upon it, Christmas will not see Lady Godolphin at Prior’s Ash, unless the fever shall have departed to spend its Christmas in some other place,” cried the Rector. “Well, I shall hear their plans when I get there.” “Bring back Maria with you, Mr. Godolphin. Tell her it is my wish. Unless you find that there’s a prospect of her speedy return with Lady Godolphin. In that case, you may leave her.” “Very well,” replied Thomas Godolphin. He continued his way, and Mr. Hastings looked after him in the bright moonlight, till his form disappeared in the shadows cast by the roadside trees. It was striking ten as Thomas Godolphin opened the iron gates at Lady Sarah Grame’s: the heavy clock-bell of All Souls’ came sounding upon his ear in the stillness of the night. The house, all except from one window, looked dark: even the hall-lamp was out, and he feared they might all have retired. From that window a dull light shone behind the blind: a stationary light it had been of late, to be seen by any nocturnal wayfarer all night long; for it came from the sick- chamber. Elizabeth opened the door. “Oh, sir!” she exclaimed in the surprise of seeing him so late, “I think Miss Ethel has gone up to bed.” Lady Sarah came hastening down the stairs as he stepped into the hall: she also was surprised at the late visit. “I would not have disturbed you, but that I am about to leave for Broomhead,” he explained. “A telegraphic despatch has arrived from Lady Godolphin, calling me thither. I should like to see Ethel, if not inconvenient to her. I know not how long I may be away.” “I sent Ethel to bed: her head ached,” said Lady Sarah. “It is not many minutes since she went up. Oh, Mr. Godolphin, this has been such a day of grief! heads and hearts alike aching.” Thomas Godolphin entered the drawing-room, and Lady Sarah Grame called Ethel down, and then returned to her sick daughter’s room. Ethel came instantly. The fire in the drawing-room was still
- 61. alight, and Elizabeth had been in to stir it up. Thomas Godolphin stood over it with Ethel, telling her of his coming journey and its cause. The red embers threw a glow upon her face: her brow looked heavy, her eyes swollen. He saw the signs, and laid his hand fondly upon her head. “What has given you this headache, Ethel?” The ready tears came into her eyes. “It does ache very much,” she answered. “Has crying caused it?” “Yes,” she replied. “It is of no use to deny it, for you would see it by my swollen eyelids. I have wept to-day until it seems that I can weep no longer, and it has made my eyes ache and my head dull and heavy.” “But, my darling, you should not give way to this grief. It may render you seriously ill.” “Oh, Thomas! how can I help it?” she returned, with emotion, as the tears dropped swiftly over her cheeks. “We begin to see that there is no chance of Sarah Anne’s recovery. Mr. Snow told mamma so to-day: and he sent up Mr. Hastings.” “Ethel, will your grieving alter it?” Ethel wept silently. There was full and entire confidence between her and Thomas Godolphin: she could speak out all her thoughts, her troubles to him, as she could have told them to a mother—if she had had a mother who loved her. “If she were only a little more prepared to go, the pain would seem less,” breathed Ethel. “That is, we might feel more reconciled to losing her. But you know what she is, Thomas. When I have tried to talk a little bit about heaven, or to read a psalm to her, she would not listen: she said it made her dull, it gave her the horrors. How can she, who has never thought of God, be fit to meet Him?” Ethel’s tears were deepening into sobs. Thomas Godolphin involuntarily thought of what Mr. Hastings had just said to him. His hand still rested on Ethel’s head.
- 62. “You are fit to meet Him?” he exclaimed involuntarily. “Ethel, whence can have arisen the difference between you? You are sisters; reared in the same home.” “I do not know,” said Ethel simply. “I have always thought a great deal about heaven; I suppose it is that. A lady, whom we knew as children, used to buy us a good many story-books, and mine were always stories of heaven. It was that which first got me into the habit of thinking of it.” “And why not Sarah Anne?” “Sarah Anne would not read them. She liked stories of gaiety and excitement; balls, and things like that.” Thomas smiled; the words were so simple and natural. “Had the fiat gone forth for you, instead of for her, Ethel, it would have brought you no dismay?” “Only that I must leave all my dear ones behind me,” she answered, looking up at him, a bright smile shining through her tears. “I should know that God would not take me, unless it were for the best. Oh, Thomas! if we could only save her!” “Child, you contradict yourself. If what God does must be for the best—and it is—that thought should reconcile you to parting with Sarah Anne.” “Y—es,” hesitated Ethel. “Only I fear she has never thought of it herself, or in any way prepared for it.” “Do you know that I have to find fault with you?” resumed Thomas Godolphin, after a pause. “You have not been true to me, Ethel.” She turned her eyes upon him in surprise. “Did you not promise me—did you not promise Mr. Snow, not to enter your sister’s chamber while the fever was upon her? I hear that you were in it often: her head nurse.” A hot colour flushed into Ethel’s face. “Forgive me, Thomas,” she whispered; “I could not help myself. Sarah Anne—it was on the third morning of her illness, when I was getting up—suddenly began to
- 63. cry out for me very much, and mamma came to my bedroom and desired me to go to her. I said that Mr. Snow had forbidden me, and that I had promised you. It made mamma angry. She asked if I could be so selfish as to regard a promise before Sarah Anne’s life; that she might die if I thwarted her: and she took me by the arm and pulled me in. I would have told you, Thomas, that I had broken my word; I wished to tell you; but mamma forbade me to do so.” Thomas Godolphin stood looking at her. There was nothing to answer: he had known, in his deep and trusting love, that the fault had not lain with Ethel. She mistook his silence, thinking he was vexed. “You know, Thomas, so long as I am here in mamma’s home, her child, it is to her that I owe obedience,” she gently pleaded. “As soon as I shall be your wife, I shall owe it and give it implicitly to you.” “You are right, my darling.” “And it has produced no ill consequences,” she resumed. “I did not catch the fever. Had I found myself growing in the least ill, I should have sent for you and told you the truth.” “Ethel?” he impulsively cried—very impulsively for calm Thomas Godolphin; “had you caught the fever, I should never have forgiven those who led you into danger. I could not lose you.” “Hark!” said Ethel. “Mamma is calling.” Lady Sarah had been calling to Mr. Godolphin. Thinking she was not heard, she now came downstairs and entered the room, wringing her hands; her eyes were overflowing, her sharp thin nose was redder than usual. “Oh dear! I don’t know what we shall do with her!” she uttered. “She is so ill, and it makes her so fretful. Mr. Godolphin, nothing will satisfy her now but she must see you.” “See me!” repeated he. “She will, she says. I told her you were departing for Scotland, and she burst out crying, and said if she were to die she should never see you again. Do you mind going in? You are not afraid?”
- 64. “No, I am not afraid,” said Thomas Godolphin. “Infection cannot have remained all this time. And if it had, I should not fear it.” Lady Sarah Grame led the way upstairs. Thomas followed her. Ethel stole in afterwards. Sarah Anne lay in bed, her thin face, drawn and white, raised upon the pillow; her hollow eyes were strained forward with a fixed look. Ill as he had been led to suppose her, he was scarcely prepared to see her like this; and it shocked him. A cadaverous face, looking ripe for the tomb. “Why have you never come to see me?” she asked in her hollow voice, as he approached and leaned over her. “You’d never have come till I died. You only care for Ethel.” “I would have come to see you had I known you wished it,” he answered. “But you do not look strong enough to receive visitors.” “They might cure me, if they would,” she continued, panting for breath. “I want to go away somewhere, and that Snow won’t let me. If it were Ethel, he would take care to cure her.” “He will let you go as soon as you are equal to it, I am sure,” said Thomas Godolphin. “Why should the fever have come to me at all?—Why couldn’t it have gone to Ethel instead? She’s strong. She would have got well in no time. It’s not fair——” “My dear child, my dear, dear child, you must not excite yourself,” implored Lady Sarah, abruptly interrupting her. “I shall speak,” cried Sarah Anne, with a touch, feeble though it was, of her old peevish vehemence. “Nobody’s thought of but Ethel. If you had had your way,” looking hard at Mr. Godolphin, “she wouldn’t have been allowed to come near me; no, not if I had died.” Her mood changed to tears. Lady Sarah whispered to him to leave the room: it would not do, this excitement. Thomas wondered why he had been brought to it. “I will come and see you again when you are better,” he soothingly whispered. “No you won’t,” sobbed Sarah Anne. “You are going to Scotland, and I shall be dead when you come back. I don’t want to die. Why
- 65. do they frighten me with their prayers? Good-bye, Thomas Godolphin.” The last words were called after him; when he had taken his leave of her and was quitting the room. Lady Sarah attended him to the threshold: her eyes full, her hands lifted. “You may see that there’s no hope of her!” she wailed. Thomas did not think there was the slightest hope. To his eye— though it was not so practised an eye in sickness as Mr. Snow’s, or even as that of the Rector of All Souls’—it appeared that in a very few days, perhaps hours, hope for Sarah Anne Grame would be over for ever. Ethel waited for him in the hall, and was leading the way back to the drawing-room; but he told her he could not stay longer, and opened the front door. She ran past him into the garden, putting her hand into his as he came out. “I wish you were not going away,” she sadly said, her spirits, that night very unequal, causing her to see things with a gloomy eye. “I wish you were going with me!” replied Thomas Godolphin. “Do not weep, Ethel. I shall soon be back again.” “Everything seems to make me weep to-night. You may not be back until—until the worst is over. Oh! if she might but be saved!” He held her face close to him, gazing down at it in the moonlight. And then he took from it his farewell kiss. “God bless you, my darling, for ever and for ever!” “May He bless you, Thomas!” she answered, with streaming eyes: and, for the first time in her life, his kiss was returned. Then they parted. He watched Ethel indoors, and went back to Prior’s Ash.
- 66. CHAPTER XII. DEAD. “Thomas, my son, I must go home. I don’t want to die away from Ashlydyat!” A dull pain shot across Thomas Godolphin’s heart at the words. Did he think of the old superstitious tradition—that evil was to fall upon the Godolphins when their chief should die, and not at Ashlydyat? At Ashlydyat his father could not die; he had put that out of his power when he let it to strangers: in its neighbourhood, he might. “The better plan, sir, will be for you to return to the Folly, as you seem to wish it,” said Thomas. “You will soon be strong enough to undertake the journey.” The decaying knight was sitting on a sofa in his bedroom. His second fainting-fit had lasted some hours—if that, indeed, was the right name to give to it—and he had recovered, only to be more and more weak. He had grown pretty well after the first attack—when Margery had found him in his chamber on the floor, the day Lady Godolphin had gone to pay her visit to Selina. The next time, he was on the lawn before the house, talking to Charlotte Pain, when he suddenly fell to the ground. He did not recover his consciousness until evening; and nearly the first wish he expressed was a desire to see his son Thomas. “Telegraph for him,” he said to Lady Godolphin. “But you are not seriously ill, Sir George,” she had answered. “No; but I should like him here. Telegraph to him to start by first train.”
- 67. And Lady Godolphin did so, accordingly, sending the message that angered Miss Godolphin. But, in this case, Lady Godolphin did not deserve so much blame as Janet cast on her: for she did debate the point with herself whether she should say Sir George was ill, or not. Believing that these two fainting-fits had proceeded from want of strength only, that they were but the effect of his long previous illness, and would lead to no bad result, she determined not to speak of it. Hence the imperfect message. Neither did Thomas Godolphin see much cause for fear when he arrived at Broomhead. Sir George did not look better than when he had left Prior’s Ash, but neither did he look much worse. On this, the second day, he had been well enough to converse with Thomas upon business affairs: and, that over, he suddenly broke out with the above wish. Thomas mentioned it when he joined Lady Godolphin afterwards. It did not meet with her approbation. “You should have opposed it,” said she to him in a firm, hard tone. “But why so, madam?” asked Thomas. “If my father’s wish is to return to Prior’s Ash, he should return.” “Not while the fever lingers there. Were he to take it—and die— you would never forgive yourself.” Thomas had no fear of the fever on his own score, and did not fear it for his father. He intimated as much. “It is not the fever that will hurt him, Lady Godolphin.” “You have no right to say that. Lady Sarah Grame, a month ago, might have said she did not fear it for Sarah Anne. And now Sarah Anne is dying!” “Or dead,” put in Charlotte Pain, who was leaning listlessly against the window frame devoured with ennui. “Shall you be afraid to go back to Prior’s Ash?” he asked of Maria Hastings. “Not at all,” replied Maria. “I should not mind if I were going to- day, as far as the fever is concerned.”
- 68. “That is well,” he said. “Because I have orders to convey you back with me.” Charlotte Pain lifted her head with a start. The news aroused her. Maria, on the contrary, thought he was speaking in jest. “No, indeed I am not,” said Thomas Godolphin. “Mr. Hastings made a request to me, madam, that I should take charge of his daughter when I returned,” continued he to Lady Godolphin. “He wants her at home, he says.” “Mr. Hastings is very polite!” ironically replied my lady. “Maria will go back when I choose to spare her.” “I hope you will allow her to return with me—unless you shall soon be returning yourself,” said Thomas Godolphin. “It is not I that shall be returning to Prior’s Ash yet,” said my lady. “The sickly old place must give proof of renewed health first. You will not see either me or Sir George there on this side Christmas.” “Then I think, Lady Godolphin, you must offer no objection to my taking charge of Maria,” said Thomas courteously, but firmly, leaving the discussion of Sir George’s return to another opportunity. “I passed my word to Mr. Hastings.” Charlotte Pain, all animation now, approached Lady Godolphin. She was thoroughly sick and tired of Broomhead: since George Godol phin’s departure, she had been projecting how she could get away from it. Here was a solution to her difficulty. “Dear Lady Godolphin, you must allow me to depart with Mr. Godolphin—whatever you may do with Maria Hastings,” she exclaimed. “I said nothing to you—for I really did not see how I was to get back, knowing you would not permit me to travel so far alone —but Mrs. Verrall is very urgent for my return. And now that she is suffering from this burn, as Mr. Godolphin has brought us news, it is the more incumbent upon me to be at home.” Which was a nice little fib of Miss Charlotte’s. Her sister had never once hinted that she wished her home again; but a fib or two more or less was nothing to Charlotte.
- 69. “You are tired of Broomhead,” said Lady Godolphin. Charlotte’s colour never varied, her eye never drooped, as she protested that she should not tire of Broomhead were she its inmate for a twelvemonth; that it was quite a paradise upon earth. Maria kept her head bent while Charlotte said it, half afraid lest unscrupulous Charlotte should call upon her to bear testimony to her truth. Only that very morning she had protested to Maria that the ennui of the place was killing her. “I don’t know,” said Lady Godolphin shrewdly. “Unless I am wrong, Charlotte, you have been anxious to leave. What was it that Mr. George hinted at—about escorting you young ladies home—and I stopped him ere it was half spoken? Prior’s Ash would talk if I sent you home under his convoy.” “Mr. Godolphin is not George,” rejoined Charlotte. “No, he is not,” replied my lady significantly. The subject of departure was settled amicably; both the young ladies were to return to Prior’s Ash under the charge of Mr. Godolphin. There are some men, single men though they be, and not men in years, whom society is content to recognize as entirely fit escorts. Thomas Godolphin was one of them. Had my lady despatched the young ladies home under Mr. George’s wing, she might never have heard the last of it from Prior’s Ash: but the most inveterate scandalmonger in it would not have questioned the trustworthiness of his elder brother. My lady was also brought to give her consent to her own departure for it by Christmas, provided Mr. Snow would assure her that the place was “safe.” In a day or two Thomas Godolphin spoke to his father of his marriage arrangements. He had received a letter from Janet, written the morning after his departure, in which she agreed to the proposal that Ethel should be her temporary guest. This removed all barrier to the immediate union. “Then you marry directly, if Sarah Anne lives?” “Directly. In January, at the latest.”
- 70. “God bless you both!” cried the old knight. “She’ll be a wife in a thousand, Thomas.” Thomas thought she would. He did not say it. “It’s the best plan; it’s the best plan,” continued Sir George in a dreamy tone, gazing into the fire. “No use to turn the girls out of their home. It will not be for long; not for long. Thomas”—turning his haggard, but still fine blue eye upon his son—“I wish I had never left Ashlydyat!” Thomas was silent. None had more bitterly regretted the departure from it than he. “I wish I could go back to it to die!” “My dear father, I hope that you will yet live many years to bless us. If you can get through this winter—and I see no reason whatever why you should not, with care—you may regain your strength and be as well again as any of us.” Sir George shook his head. “It will not be, Thomas; I shall not long keep you out of Ashlydyat. Mind!” he added, turning upon Thomas with surprising energy, “I will go back before Christmas to Prior’s Ash. The last Christmas that I see shall be spent with my children.” “Yes, indeed, I think you should come back to us,” warmly acquiesced Thomas. “Therefore, if you find, when Christmas is close upon us, that I am not amongst you, that you hear no tidings of my coming amongst you, you come off at once and fetch me. Do you hear, Thomas? I enjoin it upon you now with a father’s authority; do not forget it, or disobey it. My lady fears the fever, and would keep me here: but I must be at Prior’s Ash.” “I will certainly obey you, my father,” replied Thomas Godolphin. Telegraphic despatches seemed to be the order of the day with Thomas Godolphin. They were all sitting together that evening, Sir George having come downstairs, when a servant called Thomas out of the room. A telegraphic message had arrived for him at the
- 71. station, and a man had brought it over. A conviction of what it contained flashed over Thomas Godolphin’s heart as he opened it— the death of Sarah Anne Grame. From Lady Sarah it proved to be. Not a much more satisfactory message than had been Lady Godolphin’s; for if hers had not been explanatory, this was incoherent. “The breath has just gone out of my dear child’s body. I will write by next post. She died at four o’clock. How shall we all bear it?” Thomas returned to the room; his mind full. In the midst of his sorrow and regret for Sarah Anne, his compassion for Lady Sarah— and he did feel all that with true sympathy—intruded the thought of his own marriage. It must be postponed now. “What did Andrew want with you?” asked Sir George, when he entered. “A telegraphic message had come for me from Prior’s Ash.” “A business message?” “No, sir. It is from Lady Sarah.” By the tone of his voice, by the falling of his countenance, they could read instinctively what had occurred. But they kept silence, all, —waiting for him to speak further. “Poor Sarah Anne is gone. She died at four o’clock.” “This will delay your plans, Thomas,” observed Sir George, after some minutes had been given to expressions of regret. “It will, sir.” The knight leaned over to his son, and spoke in a whisper, meant for his ear alone: “I shall not be very long after her. I feel that I shall not. You may yet take Ethel home at once to Ashlydyat.” Very early indeed did they start in the morning, long before daybreak. Prior’s Ash they would reach, all things being well, at nine at night. Margery was sent to attend them, a very dragon of a guardian, as particular as Miss Godolphin herself—had a guardian been necessary.
- 72. A somewhat weary day; a long one, at any rate; but at last their train steamed into the station at Prior’s Ash. It was striking nine. Mr. Hastings was waiting for Maria, and Mrs. Verrall’s carriage for Charlotte Pain. A few minutes were spent in collecting the luggage. “Shall I give you a seat as far as the bank, Mr. Godolphin?” inquired Charlotte, who must pass it on her way to Ashlydyat. “Thank you, no. I shall just go up for a minute’s call upon Lady Sarah Grame.” Mr. Hastings, who had been placing Maria in a fly, heard the words. He turned hastily, caught Thomas Godolphin’s hand, and drew him aside. “Are you aware of what has occurred?” “Alas, yes!” replied Thomas. “Lady Sarah telegraphed to me last night.” The Rector pressed his hand, and returned to his daughter. Thomas Godolphin struck into a by-path, a short cut from the station, which would take him to Grame House. Six days ago, exactly, since he had been there before. The house looked precisely as it had looked then, all in darkness, excepting the faint light that burned from Sarah Anne’s chamber. It burnt there still. Then it was lighting the living; now—— Thomas Godolphin rang the bell gently.—Does any one like to do otherwise at a house in which death is an inmate? Elizabeth, as usual, opened the door, and burst into tears when she saw who it was. “I said it would bring you back, sir!” she exclaimed. “Does Lady Sarah bear it pretty well?” he asked, as she showed him into the drawing-room. “No, sir, not over well,” sobbed the girl. “I’ll tell my lady that you are here.” He stood over the fire, as he had done the other night: it was low now, as it had been then. Strangely still seemed the house: he could almost have told that one was lying dead in it. He listened, waiting for Ethel’s step, hoping she would be the first to come to him.
- 73. Elizabeth returned. “My lady says would you be so good as to walk up to her, sir?” Thomas Godolphin followed her upstairs. She made for the room to which he had been taken the former night—Sarah Anne’s chamber. In point of fact, the chamber of Lady Sarah, until it was given up to Sarah Anne for her illness. Elizabeth, with soft and stealthy tread, crossed the corridor to the door, and opened it. Was she going to show him into the presence of the dead? He thought she must have mistaken Lady Sarah’s orders, and he hesitated on the threshold. “Where is Miss Ethel?” he whispered. “Who, sir?” “Miss Ethel. Is she well?” The girl stared, flung the door full open, and with a great cry flew down the staircase. He looked after her in amazement. Had she gone crazy? Then he turned and walked into the room with a hesitating step. Lady Sarah was coming forward to meet him. She was convulsed with grief. He took both her hands in his with a soothing gesture, essaying a word of comfort: not of inquiry, as to why she should have brought him to this room. He glanced to the bed, expecting to see the dead upon it. But the bed was empty. And at that moment, his eyes caught something else. Seated by the fire in an invalid chair, surrounded with pillows, covered with shawls, with a wan, attenuated face, and eyes that seemed to have a glaze over them, was—who? Sarah Anne? It certainly was Sarah Anne, and in life still. For she feebly held out her hand in welcome, and the tears suddenly gushed from her eyes. “I am getting better, Mr. Godolphin.” Thomas Godolphin—Thomas Godolphin—how shall I write it? For one happy minute he was utterly blind to what it could all mean: his whole mind was a chaos of wild perplexity. And then, as the dreadful
- 74. truth burst upon him, he staggered against the wall, with a wailing cry of agony. It was Ethel who had died.
- 75. CHAPTER XIII. UNAVAILING REGRETS. Yes. It was Ethel who had died. Thomas Godolphin leaned against the wall in his agony. It was one of those moments that can fall only once in a lifetime; in many lives never; when the greatest limit of earthly misery bursts upon the startled spirit, shattering it for all time. Were Thomas Godolphin to live for a hundred years, he never could know another moment like this: the power so to feel would have left him. It had not left him yet. Nay, it had scarcely come to him in its full realization. At present he was half stunned. Strange as it may seem, the first impression upon his mind, was—that he was so much nearer to the next world. How am I to define this “nearer?” It was not that he was nearer to it by time; or in goodness: nothing of that sort. She had passed within its portals; and the great gulf, which divides time from eternity, seems to be only a span now to Thomas Godolphin: it was as if he, in spirit, had followed her in. From being a place far, far off, vague, indefinite, indistinct, it had been suddenly brought to him, close and palpable: or he to it: Had Thomas Godolphin been an atheist, denying a hereafter,—Heaven in its compassion have mercy upon all such!—that one moment of suffering would have recalled him to a sense of his mistake. It was as if he looked above with the eye of inspiration and saw the truth; it was as a brief, passing moment of revelation from God. She, with her loving spirit, her gentle heart, her simple trust in God, had been taken from this world to enter upon a better. She was as surely living in it, had entered upon its mysteries, its joys, its rest, as that he was
- 76. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com